0), which we obtain from the complete active-space configuration interaction computation (CASCI) using the ground-state CASSCF orbitals. We refer to this procedure of constructing the model space as CASCI/CASSCF.\nTo guide our development of MR-ADC further, we introduce two requirements:\n* *Requirement 1*. At each order of perturbation theory *n*, the *n*-th-order MR-ADC approximation must reduce to the *n*-th-order SR-ADC approximation [ADC(n)] in the limit of the single-determinant $\\ket{\\Psi\\_0}$ and zero active orbitals.\n* *Requirement 2*. The zeroth-order approximation must yield exactly the CASCI/CASSCF excitation energies and transition amplitudes between the ground- ($\\ket{\\Psi\\_0}$) and excited-state ($\\ket{\\Psi\\_I}$, *I* > 0) model wavefunctions in the active space.\nThese two requirements make sure that MR-ADC produces the SR-ADC and the exact (i.e., full configuration interaction) excitation energies and transition amplitudes in the two limits, respectively: (i) zero active orbitals (requirement 1), and (ii) all orbitals are active (requirement 2).\nTo satisfy the requirement 1, we derive MR-ADC using the effective Liouvillean approach described in that we generalize for multi-determinant reference wavefunctions. Thus, we consider approximations to the ground-state correlated wavefunction in the form:\n \n$$\\begin{aligned}\n \\label{eq:mr\\_adc\\_wfn}\n \\ket{\\Psi} &= e^{A} \\ket{\\Psi\\_0} = e^{T - T^\\dag} \\ket{\\Psi\\_0}, \\quad T = \\sum\\_{k=1}^N T\\_k \\\\\n \\label{eq:mr\\_adc\\_t\\_amplitudes}\n T\\_k &= \\frac{1}{(k!)^2} {\\sum\\_{i'j'a'b'\\ldots}} t\\_{i'j'\\ldots}^{a'b'\\ldots} {a^\\dagger\\_{a'}}{a^\\dagger\\_{b'}}\\ldots{a^{\\ }\\_{j'}}{a^{\\ }\\_{i'}}, \\quad t\\_{xy\\ldots}^{wz\\ldots} = 0\\end{aligned}$$ \nwhere the operator *T* generates all internally-contracted excitations between core, active, and external orbitals. is equivalent to the wavefunction used in internally-contracted multi-reference unitary coupled cluster theory and the related approaches. As we will discuss in, excitations generated by the operator *T* are linearly-dependent and the redundant amplitudes need to be eliminated. A procedure for removing the redundant amplitudes is described in the Appendix.\nTo define the MR-ADC perturbative series, we must choose the zeroth-order Hamiltonian *H*(0). While the choice of *H*(0) is flexible, it must guarantee that the requirement 2 is satisfied. This suggests that *H*(0) must be an interacting Hamiltonian in the active space, rather than a Fock-like one-electron operator. In our MR-ADC development we choose *H*(0) to be the Dyall Hamiltonian, defined as:\n \n$$\\begin{aligned}\n \\label{eq:h\\_dyall\\_general}\n H^{(0)} \\equiv C + \\sum\\_{ij} {f\\_{{i}}^{{j}}} {a^\\dagger\\_{i}}{a^{\\ }\\_{j}} + \\sum\\_{ab} {f\\_{{a}}^{{b}}} {a^\\dagger\\_{a}}{a^{\\ }\\_{b}} + H\\_{act}\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n \\label{eq:f\\_gen}\n {f\\_{{p}}^{{q}}} &= {h\\_{{p}}^{{q}}} + \\sum\\_{rs} {{v}\\_{{pr}}^{{qs}}} {\\gamma\\_{{s}}^{{r}}} \\, \\quad {\\gamma\\_{{q}}^{{p}}} = \\braket{\\Psi\\_0|{a^\\dagger\\_{p}}{a^{\\ }\\_{q}}|\\Psi\\_0} \\\\\n \\label{eq:h\\_act}\n H\\_{act} &= \\sum\\_{xy}({h\\_{{x}}^{{y}}} + \\sum\\_{i} {{v}\\_{{xi}}^{{yi}}}) {a^\\dagger\\_{x}} {a^{\\ }\\_{y}} + \\frac{1}{4} \\sum\\_{xywz} {{v}\\_{{xy}}^{{zw}}} {a^\\dagger\\_{x}} {a^\\dagger\\_{y}} {a^{\\ }\\_{w}} {a^{\\ }\\_{z}} \\\\\n C &= \\sum\\_i {h\\_{{i}}^{{i}}} + \\frac{1}{2}\\sum\\_{ij}{{v}\\_{{ij}}^{{ij}}} - \\sum\\_i {f\\_{{i}}^{{i}}} = E\\_{fc} - \\sum\\_i {f\\_{{i}}^{{i}}} \\end{aligned}$$ \nThe Dyall Hamiltonian *H*(0) includes all active-space terms of the full electronic Hamiltonian *H* (), satisfying the eigenvalue problem\n \n$$\\begin{aligned}\n \\label{eq:h\\_zero\\_eval\\_problem}\n \\braket{\\Psi\\_J | H^{(0)} |\\Psi\\_I} = \\braket{\\Psi\\_J | H | \\Psi\\_I} = E\\_I \\delta\\_{IJ}\\end{aligned}$$ \nfor all CASCI/CASSCF model wavefunctions $\\ket{\\Psi\\_I}$ with eigenvalues *E**I*, which is a necessary condition for fulfilling the requirement 2. For convenience, we work in the basis of the diagonal core and external generalized Fock operators (*f**i**j* → *ɛ**i**δ**i**j*, *f**a**b* → *ɛ**a**δ**a**b*), where *H*(0) takes the form:\n \n$$\\begin{aligned}\n \\label{eq:h\\_dyall\\_diagonal}\n H^{(0)} &= C + \\sum\\_{i} {\\ensuremath{\\varepsilon\\_{i}}} {a^\\dagger\\_{i}}{a^{\\ }\\_{i}} + \\sum\\_{a} {\\ensuremath{\\varepsilon\\_{a}}} {a^\\dagger\\_{a}} {a^{\\ }\\_{a}} + H\\_{act}\\end{aligned}$$ \nWe now consider the expansion of the propagator with respect to the perturbation *V* = *H* − *H*(0) that defines the MR-ADC approximations. Conveniently, the general form of the MR-ADC equations is equivalent to that of SR-ADC presented in. The *n*-th order MR-ADC approximation, which will be termed as MR-ADC(n) henceforth, is defined by truncating the expansion for **G**+(*ω*) in after **G**+(*n*)(*ω*). The *n*-th-order contribution **G**+(*n*)(*ω*) is given by and the MR-ADC(n) eigenvalue problem can be written as in. Despite general similarities, in MR-ADC, the matrices **T**+(*n*), **M**+(*n*), and **S**+(*n*) that compose **G**+(*n*)(*ω*) are evaluated using the *multi-reference* model wavefunctions $\\ket{\\Psi\\_I}$, which incorporate the information about the active-space correlation and electronic states into the description of the propagator.\nPerturbative analysis of the MR-ADC equations\n---------------------------------------------\nIn this section, we perform a perturbative analysis of the MR-ADC equations to illustrate the most important features of this theory. To compute the MR-ADC(n) excitation energies and transition amplitudes, we need to evaluate the **M**+, **S**+, and **T**+ matrices up to the *n*-th order in perturbation theory. This requires: (i) deriving expressions for the effective Hamiltonian *H̃*(*k*), (ii) constructing the operator manifolds **h**+(*k*) †, (iii) solving equations for the *k*-th-order contributions to the *t**i*ʹ*j*ʹ…*a*ʹ*b*ʹ… amplitudes (), and (iv) evaluating the transformed operators $\\mathbf{\\tilde{q}}^{(k)\\dag}$. Since for the polarization propagator **G**+(*ω*) and **G**−(*ω*) contain the same information, we will only consider **G**+(*ω*) and drop the subscript + everywhere in the equations.\n### Zeroth-order contributions\nThe zeroth-order operators *H̃*(0) and $\\mathbf{\\tilde{q}}^{(0)\\dag}$ have a simple form:\n \n$$\\begin{aligned}\n \\label{eq:H\\_bch\\_0}\n \\tilde{H}^{(0)} &= H^{(0)} \\\\\n \\label{eq:q\\_bch\\_0}\n \\tilde{q}^{(0)\\dag}\\_\\mu &= q^\\dag\\_\\mu = a\\_{q}^{p} - \\braket{\\Psi\\_0|a\\_{q}^{p}|\\Psi\\_0}\\end{aligned}$$ \nwhere we define *a**q**p* ≡ *a**p*†*a**q*. Following procedure outlined in, from we determine that the zeroth-order operators **h**(0) † have the form of the single-particle operators *a**q**p*. Importantly, **h**(0) † must satisfy VAC with respect to the ground-state model wavefunction $\\ket{\\Psi\\_0}$, i.e.. Since $\\ket{\\Psi\\_0}$ is a CAS-type wavefunction, there is a total of nine different classes of the operators *a**q**p*, where indices *p* and *q* belong to different orbitals subspaces (i.e., core, active, or external). Out of nine classes, two classes with only core (*a**j**i*) or only external (*a**b**a*) indices are redundant as they do not produce excited configurations when acting on $\\ket{\\Psi\\_0}$. We do not include these operators in the operator manifold **h**(0) †. Among the remaining types, the operators *a**i**x*, *a**i**a*, and *a**x**a* can be added to **h**(0) †, while their adjoints contribute to **h**(0). The last operator class with all active indices (*a**y**x*) generates excitations in the active space, but does not satisfy VAC with respect to $\\ket{\\Psi\\_0}$, and thus cannot be included in **h**(0) †. These operators require a special treatment in our development.\nExpanding the active-space operators in the form *a**y**x* = ∑*I**Z**I*†*c**I*, *x**y*, where *Z**I*† is a complete set of the active-space eigenoperators with a property $Z^\\dag\\_I \\ket{\\Psi\\_0} = \\ket{\\Psi\\_I}$, we express the configurations generated by *a**y**x* as:\n \n$$\\begin{aligned}\n \\label{eq:xy\\_as\\_z\\_act\\_hilbert}\n a^{x}\\_{y}\\ket{\\Psi\\_0} = \\sum\\_{I} Z^\\dag\\_I \\ket{\\Psi\\_0} c\\_{I,xy} = \\sum\\_{I} \\ket{\\Psi\\_I}\\braket{\\Psi\\_I|a^{x}\\_{y}|\\Psi\\_0}\\end{aligned}$$ \nwhere the r.h.s. of is obtained by inserting the resolution of identity over a complete set of the active-space model states $\\ket{\\Psi\\_I}$ in the l.h.s. of that equation. suggests the form for the coefficients $c\\_{I,xy} = \\braket{\\Psi\\_I|a^{x}\\_{y}|\\Psi\\_0}$ and the eigenoperators *Z**I*†:\n \n$$\\begin{aligned}\n \\label{eq:z\\_ketbra}\n Z^\\dag\\_I &= \\ket{\\Psi\\_I}\\bra{\\Psi\\_0} \\end{aligned}$$ \nSince the CASCI/CASSCF model states $\\ket{\\Psi\\_I}$ are orthogonal, the excited-state eigenoperators *Z**I*† (*I* > 0) automatically fulfill VAC\n \n$$\\begin{aligned}\n \\label{eq:z\\_act}\n Z\\_I \\ket{\\Psi\\_0} &= 0 \\qquad (I > 0)\\end{aligned}$$ \nand thus can be included in the operator manifold **h**(0) †. Importantly, choosing *Z**I*† in the form of together with the choice for the zeroth-order Hamiltonian () ensures that the requirement 2 discussed in is satisfied. The completeness of the operator set *Z**I*† is equivalent to the availability of the complete model space $\\ket{\\Psi\\_I}$. For small active spaces, it is possible to operate with a complete set of $\\ket{\\Psi\\_I}$ such that the set of *Z**I*† is complete. However, in most computations, it will be necessary to truncate the set $\\ket{\\Psi\\_I}$ to include only the low-energy states of interest, introducing an approximation (see for details).\nWe summarize that the **h**(0) † operator manifold consists of the four classes of operators:\n \n$$\\begin{aligned}\n\\mathbf{h}^{(0)\\dag} = \\left\\{Z^\\dag\\_I; a^{a}\\_{i}; a^{x}\\_{i}; a^{a}\\_{x} \\right\\} \\end{aligned}$$ \nThe zeroth-order MR-ADC matrices have the general form:\n \n$$\\begin{aligned}\n \\mathbf{M}^{(0)} & = \\{\\mathbf{h}^{(0)}|\\tilde{\\mathcal{H}}^{(0)}|\\mathbf{h}^{(0)}\\} \\\\\n \\mathbf{S}^{(0)} & = \\{\\mathbf{h}^{(0)}|\\mathbf{h}^{(0)}\\} \\\\\n \\mathbf{T}^{(0)} & = \\{\\mathbf{\\tilde{q}}^{(0)}|\\mathbf{h}^{(0)}\\} \\end{aligned}$$ \nExplicit equations for **M**(0) and **S**(0) are shown in the Supporting Information. In contrast to SR-ADC, these matrices are non-diagonal, due to the non-orthogonal nature of the internally-contracted configurations (e.g., $a^{x}\\_{i}\\ket{\\Psi\\_0}$) and the non-diagonal form of the Dyall Hamiltonian. However, the off-diagonal blocks of **M**(0) and **S**(0) corresponding to different types of **h**(0) † vanish.\n[t!] [fig:adcM]\n### First-order contributions\nExpanding the BCH expansions in to the first order, we obtain:\n \n$$\\begin{aligned}\n \\label{eq:H\\_bch\\_1}\n \\tilde{H}^{(1)} &= V + [H^{(0)}, T^{(1)} - T^{\\dag(1)}] \\\\\n \\label{eq:q\\_bch\\_1}\n \\tilde{q}^{(1)\\dag}\\_\\mu &= [a\\_{q}^{p}, T^{(1)} - T^{\\dag(1)}]\\end{aligned}$$ \nSince *V* is a two-electron operator, the first-order excitation operator *T*(1) only includes up to the two-body terms: *T*(1) = *T*1(1) + *T*2(1). From we determine the first-order operator manifold **h**(1) †, which consists of eight different types of double excitations:\n \n$$\\begin{aligned}\n\\mathbf{h}^{(1)\\dag} = &\\left\\{ a\\_{ij}^{ab}; a\\_{ij}^{ax}; a\\_{ix}^{ab}; a\\_{ij}^{xy}; a\\_{xy}^{ab}; a\\_{ix}^{yz}; a\\_{xy}^{az}; a\\_{ix}^{ay} \\right\\} \\end{aligned}$$ \nwhere *a**r**s**p**q* ≡ *a**p*†*a**q*†*a**s**a**r*. The MR-ADC matrices have the following structure:\n \n$$\\begin{aligned}\n \\mathbf{M}^{(1)} & = \\{\\mathbf{h}^{(0)}|\\tilde{\\mathcal{H}}^{(1)}|\\mathbf{h}^{(0)}\\} + \\{\\mathbf{h}^{(1)}|\\tilde{\\mathcal{H}}^{(0)}|\\mathbf{h}^{(0)}\\} \\notag \\\\\n &+ \\{\\mathbf{h}^{(0)}|\\tilde{\\mathcal{H}}^{(0)}|\\mathbf{h}^{(1)}\\} \\\\\n \\mathbf{S}^{(1)} & = \\{\\mathbf{h}^{(1)}|\\mathbf{h}^{(0)}\\} + \\{\\mathbf{h}^{(0)}|\\mathbf{h}^{(1)}\\} \\\\\n \\mathbf{T}^{(1)} & = \\{\\mathbf{\\tilde{q}}^{(1)}|\\mathbf{h}^{(0)}\\} + \\{\\mathbf{\\tilde{q}}^{(0)}|\\mathbf{h}^{(1)}\\} \\end{aligned}$$ \nIn the MR-ADC(1) approximation, where **M** ≈ **M**(0) + **M**(1), the terms that depend on the double-excitation manifold **h**(1) do not contribute, since the $\\{\\mathbf{h}^{(1)}|\\tilde{\\mathcal{H}}^{(0)}|\\mathbf{h}^{(1)}\\}$ block of the effective Liouvillean matrix is a contribution to **M**(2). However, these **h**(1) contributions need to be included in MR-ADC(2) and the higher-order approximations. Evaluation of **M**(1) requires the first-order single-excitation (*t**i**a*(1), *t**x**a*(1), *t**i**x*(1)) and (semi-internal) double-excitation (*t**i**x**a**y*(1), *t**x**y**a**w*(1), *t**i**x**y**w*(1)) amplitudes. These amplitudes can be obtained by solving a system of the projected amplitude equations\n \n$$\\begin{aligned}\n \\label{eq:proj\\_amplitude\\_equations\\_1}\n \\braket{\\Psi\\_0|a\\_{q}^{p}\\tilde{H}^{(1)}|\\Psi\\_0} = 0 \\\\\n \\label{eq:proj\\_amplitude\\_equations\\_2}\n \\braket{\\Psi\\_0|a\\_{rs}^{pq}\\tilde{H}^{(1)}|\\Psi\\_0} = 0 \\end{aligned}$$ \nwhere *a**q**p* ∈ {*a**a**i*; *a**a**x*; *a**x**i*} and *a**r**s**p**q* ∈ {*a**a**y**i**x*; *a**a**w**x**y*; *a**y**w**i**x*}. We discuss the solution of these equations in more detail in the Appendix.\n### Second-order contributions\nThe second-order operators *H̃*(2) and $\\mathbf{\\tilde{q}}^{(2)\\dag}$ have the form:\n \n$$\\begin{aligned}\n \\label{eq:H\\_bch\\_2}\n \\tilde{H}^{(2)} &= [H^{(0)}, T^{(2)} - T^{\\dag(2)}] \\notag \\\\\n &+ \\frac{1}{2}[V + \\tilde{H}^{(1)}, T^{(1)} - T^{\\dag(1)}] \\\\\n \\label{eq:q\\_bch\\_2}\n \\tilde{q}^{(2)\\dag}\\_\\mu &= [a\\_{q}^{p}, T^{(2)} - T^{\\dag(2)}] \\notag \\\\\n &+ \\frac{1}{2} [[a\\_{q}^{p}, T^{(1)} - T^{\\dag(1)}], T^{(1)} - T^{\\dag(1)}]\\end{aligned}$$ \nThe second term in generates the three-body operators that compose the second-order operator manifold **h**(2) †. The MR-ADC matrices contain the following second-order terms:\n \n$$\\begin{aligned}\n \\mathbf{M}^{(2)} & = \\{\\mathbf{h}^{(0)}|\\tilde{\\mathcal{H}}^{(2)}|\\mathbf{h}^{(0)}\\} + \\{\\mathbf{h}^{(1)}|\\tilde{\\mathcal{H}}^{(1)}|\\mathbf{h}^{(0)}\\} \\notag \\\\\n & + \\{\\mathbf{h}^{(0)}|\\tilde{\\mathcal{H}}^{(1)}|\\mathbf{h}^{(1)}\\} + \\{\\mathbf{h}^{(1)}|\\tilde{\\mathcal{H}}^{(0)}|\\mathbf{h}^{(1)}\\} \\notag \\\\\n & + \\{\\mathbf{h}^{(2)}|\\tilde{\\mathcal{H}}^{(0)}|\\mathbf{h}^{(0)}\\} + \\{\\mathbf{h}^{(0)}|\\tilde{\\mathcal{H}}^{(0)}|\\mathbf{h}^{(2)}\\} \\\\\n \\mathbf{S}^{(2)} & = \\{\\mathbf{h}^{(1)}|\\mathbf{h}^{(1)}\\} + \\{\\mathbf{h}^{(2)}|\\mathbf{h}^{(0)}\\} + \\{\\mathbf{h}^{(0)}|\\mathbf{h}^{(2)}\\} \\\\\n \\mathbf{T}^{(2)} & = \\{\\mathbf{\\tilde{q}}^{(1)}|\\mathbf{h}^{(1)}\\} + \\{\\mathbf{\\tilde{q}}^{(2)}|\\mathbf{h}^{(0)}\\} + \\{\\mathbf{\\tilde{q}}^{(0)}|\\mathbf{h}^{(2)}\\} \\end{aligned}$$ \nTerms containing **h**(2) need to be included only in MR-ADC(4) and higher-order approximations. However, all terms involving the double-excitation manifold **h**(1) must be included already in MR-ADC(2). Computation of **M**(2) requires solving for all amplitudes of the *T*1(1) and *T*2(1) operators, as well as for the single- and semi-internal double-excitation amplitudes of *T*1(2) and *T*2(2).\ncompares the effective Liouvillean matrix **M** for the single-reference ADC(n) and multi-reference MR-ADC(n) approximations (n = 0, 1, 2). Although in MR-ADC there are more excitation classes than in SR-ADC, the perturbative structure of the MR-ADC(n) and ADC(n) matrices is very similar. These matrices become equivalent in the limit of the single-determinant $\\ket{\\Psi\\_{0}}$ and zero active orbitals.\nFinally, we comment about size-extensivity of the MR-ADC(n) approximations. The commutator structure of the MR-ADC equations ensures that all of the terms that appear in these equations are fully linked, which formally guarantees size-consistency of the MR-ADC excitation energies and transition amplitudes at any level of approximation, similar to that of the single-reference ADC. In practice, however, size-consistency can be violated during the numerical solution of the MR-ADC equations as a result of removing redundant excitations of the operator *T*. These size-consistency errors originate from the disconnected terms involving the single and semi-internal amplitudes (e.g., *t**i**a*(1) and *t**i**x**a**y*(1)) that become non-zero when linear dependencies are eliminated. We will investigate the size-consistency errors of the MR-ADC(1) approximation in.\nImplementation: MR-ADC(1)\n=========================\nWe have derived and implemented equations for the MR-ADC(1) approximation as a first step in constructing the hierarchy of the MR-ADC methods. The general structure of the MR-ADC(1) approximation was described in and the explicit equations are shown in the Supporting Information. Although the single-reference ADC(1) energies are equivalent to those obtained from the Tamm-Dancoff approximation (TDA), the MR-ADC(1) and the multi-configurational TDA (MC-TDA) energies are in general different. This is because in MR-ADC(1) the effective Liouvillean matrix contains additional non-zero terms that depend on the single-excitation (*t**x**a*(1), *t**i**x*(1)) and semi-internal (*t**i**x**a**y*(1), *t**x**y**a**w*(1), *t**i**x**y**w*(1)) amplitudes that are not present in MC-TDA. Neglecting these terms reduces the MR-ADC(1) equations to those of MC-TDA.\nThe main steps of the MR-ADC(1) implementation for a CASSCF reference wavefunction are summarized below:\n1. Choose active space, compute the ground-state CASSCF wavefunction $\\ket{\\Psi\\_0}$.\n2. Using the optimized CASSCF orbitals, compute the energies *E**I* and the wavefunctions $\\ket{\\Psi\\_I}$ for *N*CAS lowest-energy CASCI states.\n3. Compute active-space reduced density matrices (RDMs) for the ground state $\\ket{\\Psi\\_0}$ and transition RDMs between $\\ket{\\Psi\\_0}$ and the excited CASCI states $\\ket{\\Psi\\_I}$ (*I* > 0).\n4. Solve linear equations for the single-excitation (*t**x**a*(1), *t**i**x*(1)) and semi-internal (*t**i**x**a**y*(1), *t**x**y**a**w*(1), *t**i**x**y**w*(1)) amplitudes (see the Appendix for details).\n5. Compute the overlap (**S** = **S**(0)) and the effective Liouvillean matrices (**M** = **M**(0) + **M**(1)).\n6. Solve the generalized eigenvalue problem to compute excitation energies.\nIn the algorithm outlined above, the model space should contain all CASCI states $\\ket{\\Psi\\_I}$ that are important for the problem and spectral region of interest. Evaluation of the **M** matrix elements requires computation of up to the three-particle ground-state RDM (3-RDM) and three-particle transition RDM, which have O(*N*CAS × *N*det × *N*act6) computational scaling, where *N*det is the dimension of the active-space Hilbert space and *N*act is the number of active orbitals. In addition, solving the amplitude equations for the semi-internal excitations with three active-space indices (*t**x**y**a**w*(1) and *t**i**x**y**w*(1)) requires computing the ground-state 4-RDM, which has O(*N*det × *N*act8) computational cost. In our implementation, we avoid computation and storage of 4-RDM using the imaginary-time propagation algorithm outlined in the Appendix.\nComputational details\n=====================\nWe implemented MR-ADC(1) in a standalone Python program. To obtain one- and two-electron integrals and the CASSCF reference wavefunctions, our program was interfaced with Pyscf. The main steps of our implementation are described in. In all MR-ADC(1) computations, the CASSCF reference molecular orbitals were optimized for the ground electronic state with tight convergence parameters for the energy (10− 8 ). These orbitals were used in the following CASCI computation, which produced wavefunctions for the excited states in the active space. We refer to this procedure as CASCI/CASSCF. We denote active spaces used in CASCI/CASSCF as (*n*e, *m*o), where *n* is the number of active electrons and *m* is the number of orbitals. The MR-ADC(1) results were benchmarked against accurate reference data from full configuration interaction (FCI) and density matrix renormalization group (DMRG) and were compared to results from strongly-contracted *N*-electron valence second-order perturbation theory (sc-NEVPT2) and its quasidegenerate variant (sc-QD-NEVPT2). The NEVPT2 computations were performed using the Orca program and employed the state-averaged CASSCF orbitals (SA-CASSCF).\nIn addition to choosing the active space, the MR-ADC(1) results depend on three parameters: (i) the parameter Δ*c**o**n**v* for the imaginary-time propagation used to compute the semi-internal amplitudes of the effective Hamiltonian, (ii) the thresholds *η*[0ʹ] and *η*[ ± 1ʹ] for removing linear dependencies in the overlap matrices, and (iii) *N*CAS, the number of the CASCI states in the model space.\nWe refer the interested readers to the Appendix for details about the imaginary-time propagation and the overlap matrices. In short, our imaginary-time algorithm follows the procedure described in Ref.. The time propagation is performed using the embedded Runge-Kutta algorithm, which automatically determines the time step based on the accuracy parameter Δ*c**o**n**v*. Since the imaginary-time propagation is a relatively inexpensive step of our algorithm, we use a small value Δ*c**o**n**v* = 10− 6 in all computations, which allows to compute numerically accurate semi-internal amplitudes with ∼ 10 to 40 imaginary-time steps.\nTo remove linear dependencies, we first diagonalize the overlap matrices $\\mathbf{S^{[0']}}$, $\\mathbf{S^{[+1']}}$, and $\\mathbf{S^{[-1']}}$ defined in, respectively. We then arrange the resulting eigenvalues *s**p*[*i*] (*i* = 0ʹ, + 1ʹ, − 1ʹ) in the ascending order and project out the eigenvectors with the smallest *s**p*[*i*] that satisfy the following condition:\n \n$$\\begin{aligned}\n\\frac{\\sum\\_{p}^{trunc}s^{[i]}\\_{p}}{\\sum\\_{p}s^{[i]}\\_{p}} \\le \\eta^{[i]}\\end{aligned}$$ \nwhere the sum in the numerator runs over the truncated *s**p*[*i*] and the denominator contains the total sum of *s**p*[*i*]. In practice, the diagonalization needs to be performed only for the active-space blocks of $\\mathbf{S^{[0']}}$, $\\mathbf{S^{[+1']}}$, and $\\mathbf{S^{[-1']}}$ with the same core (*i* = *j*) and external (*a* = *b*) indices. Our numerical tests indicate that numerical instabilities due to the linear dependencies can be completely eliminated when *η*[0ʹ] and *η*[ ± 1ʹ] are chosen to be 10− 8 and 10− 3, respectively, which is consistent with the values used in implementations of other internally-contracted multi-reference theories. We use these values in all of our computations. We note that the procedure for removing linear dependencies used in this work treats redundancies in single and double (semi-internal) excitations on equal footing, but is not unique. Other strategies for eliminating linear dependencies can be used, where either one of the excitation classes is omitted or excitations of one class are removed from excitations of another class. We refer to the work by Hanauer and Köhn in Ref. for a detailed numerical analysis of these alternative methods within the framework of internally-contracted multi-reference coupled cluster theory.\n[t!] [tab:sizeconsistency]\nL1cmC2.75cmC2.75cmC2.75cmC2.75cmC2.75cm & \nState & 10− 1 & 10− 2 & 10− 3 & 10− 4 & 10− 5 \n \nS1 & 4.7 × 10− 3 & 8.0 × 10− 3 & − 9.7 × 10− 4 & 2.8 × 10− 2 & 1.1 × 10− 2 \nS2 & 8.7 × 10− 3 & 1.5 × 10− 3 & − 6.2 × 10− 3 & 4.2 × 10− 2 & − 1.3 × 10− 2 \nS3 & 4.3 × 10− 3 & 1.1 × 10− 2 & 2.0 × 10− 3 & 8.7 × 10− 3 & 1.8 × 10− 2 \nS4 & 5.0 × 10− 3 & 5.3 × 10− 2 & − 1.3 × 10− 2 & − 2.8 × 10− 3 & − 4.0 × 10− 3 \n \nS1 & 4.0 × 10− 3 & 4.1 × 10− 3 & 2.1 × 10− 3& 2.2 × 10− 3 & 2.2 × 10− 3 \nS2 & 5.3 × 10− 3 & 1.5 × 10− 3 & − 6.9 × 10− 5 & − 4.2 × 10− 5 & − 5.8 × 10− 5 \nS3 & 2.2 × 10− 2 & 5.5 × 10− 3 & 3.2 × 10− 3& 3.3 × 10− 3 & 3.2 × 10− 3 \nS4 & 1.0 × 10− 2 & 4.0 × 10− 3 & 2.9 × 10− 3& 2.9 × 10− 3 & 2.8 × 10− 3 \nFinally, the MR-ADC(1) results depend on the number of the CASCI states included in the model space (*N*CAS, see ). The optimal value of *N*CAS depends on the system and should include all active-space states in the energy range of interest. In our computations, we usually start with *N*CAS = 10 and increase it until the excitation energies for the relevant states are converged. For the systems and active spaces considered in this work, the optimal value of *N*CAS ranged from 20 to 50 states, with the exception of the Be atom with the (2*s*3*s*2*p*3*p*3*d*) active space, where we had to use *N*CAS = 80 to obtain converged energies for all excited states. An important feature of MR-ADC(1) is that the excited-state CASCI wavefunctions are only used to compute the transition reduced density matrices and thus can be discarded after their computation is complete.\nResults\n=======\nSize-consistency of excitation energies\n---------------------------------------\nWe first examine size-consistency of the MR-ADC(1) energies for a system of two non-interacting water molecules. As we discussed in, the MR-ADC approximations are formally size-consistent, but removing redundancies from the cluster operator *T* can give rise to size-consistency errors. shows size-consistency errors (Δ*E* in eV) of the MR-ADC(1) excitation energies for the four lowest-energy singlet excited states of the non-interacting water molecules with two different O–H bond lengths (1.0 and 2.0 ). Our numerical tests indicate that the size-consistency errors originate from the terms that depend on the core-active and active-external amplitudes (denoted as [ + 1ʹ] and [ − 1ʹ] in the Appendix) and vanish when these amplitudes are set to zero. To study the effect of the overlap truncation on the magnitude of the size-consistency errors, we computed Δ*E* for different values of the overlap truncation parameter *η*[ ± 1ʹ] (). For all values of *η*[ ± 1ʹ], the Δ*E* errors are small ( ∼ 10− 3 eV) with the largest error of only 0.053 eV. Changing *η*[ ± 1ʹ] from 10− 1 to 10− 5 does not significantly affect the Δ*E* values, most of them change by less than an order of magnitude.\nWe note that the observed size-consistency errors are intrinsic to the procedure used to project out linear dependencies and are not unique to MR-ADC. For example, Hanauer and Köhn demonstrated that eliminating redundancies in the equations of internally-contracted multi-reference coupled cluster theory also gives rise to size-extensivity and size-consistency errors. They developed alternative truncation schemes that allow to reduce or eliminate size-extensivity errors. These schemes can be readily adopted in MR-ADC to remove size-consistency errors, which will be the subject of our future work.\nExcitation energies of the atom\n-------------------------------\n[t!] [tab:becascinevpt]\nL2cmC2cmC3.25cmC3.25cmC3.25cmC2cm & CASCI/ & CASCI/ & CASCI/ & sc-NEVPT2/ & FCI \nState & CASSCF & SA(9)-CASSCF & SA(20)-CASSCF & SA(20)-CASSCF & \n2*s*12*p*1 1*P*0 & 6.08 & 5.33\\* & 5.38\\* & 5.37\\* &5.32 \n2*s*13*s*1 1*S* & 10.76 & 6.75\\* & 6.69\\* & 6.77\\* &6.77 \n2*p*2 1*D* & 8.11 & 7.29 & 7.04\\* & 7.14\\* &7.09 \n2*s*13*p*1 1*P*0 & 17.21 & 8.81 & 7.75 & 7.53 &7.46 \n2*s*13*d*1 1*D* & 18.90 & 12.76 & 11.16 & 9.20 &8.03 \n2*s*12*p*1 3*P*0 & 2.92 & 2.69\\* & 2.63\\* & 2.73\\* &2.73 \n2*s*13*s*1 3*S* & 11.94 & 6.41\\* & 6.33\\* & 6.46\\* &6.44 \n2*s*13*p*1 3*P*0 & 7.93 & 7.42 & 7.24\\* & 7.33\\* &7.30 \n2*p*2 3*P* & 16.73 & 8.38 & 7.39\\* & 7.45\\* &7.42 \n2*s*13*d*1 3*D* & 16.47 & 11.54 & 9.44 & 8.30 &7.74 \n& 5.07 & 1.13 & 0.56 & 0.20 & \n& 4.30 & 1.74 & 1.08 & 0.38 & \n[t!] [tab:be]\nL2cmC2.75cmC2.75cmC2.75cmC2cmC2.5cm & MR-ADC(1) & MR-ADC(1) & MR-ADC(1) & FCI & Experiment \nState & (2*s*2*p*) & (2*s*2*p*3*s*3*p*) & (2*s*3*s*2*p*3*p*3*d*) & & \n2*s*12*p*1 1*P*0 & 5.43 & 5.31 & 5.37 & 5.32 & 5.28 \n2*s*13*s*1 1*S* & 6.96 & 6.77 & 6.78 & 6.77 & 6.78 \n2*p*2 1*D* & 7.18 & 7.15 & 7.15 & 7.09 & 7.05 \n2*s*13*p*1 1*P*0 & 7.62 & 7.46 & 7.48 & 7.46 & 7.46 \n2*s*13*d*1 1*D* & 8.24 &8.07 &8.07 &8.03 &7.99 \n2*s*14*s*1 1*S* & 8.26 & 8.08 & 8.09 & 8.08 & 8.09 \n2*s*14*p*1 1*P*0 & 8.48 & 8.30 & 8.32 & 8.30 & 8.31 \n2*s*14*d*1 1*D* & 8.74 &8.55 &8.55 &8.54 &8.53 \n2*s*15*s*1 1*S* & 8.79 & 8.60 & 8.61 & 8.60 & 8.60 \n2*s*15*p*1 1*P*0 & 8.88 & 8.69 & 8.70 & 8.69 & 8.69 \n2*s*16*s*1 1*S* & 9.17 & 8.98 & 8.99 & 8.98 & 8.84 \n2*s*12*p*1 3*P*0 &2.72 & 2.72 & 2.73 & 2.73 & 2.73 \n2*s*13*s*1 3*S* &6.52 & 6.43 & 6.44 & 6.44 & 6.46 \n2*s*13*p*1 3*P*0 &7.45 & 7.29 & 7.30 & 7.30 & 7.30 \n2*p*2 3*P* &7.65 & 7.65 & 7.55 & 7.42 & 7.40 \n2*s*13*d*1 3*D* &7.91 & 7.74 & 7.75 & 7.74 & 7.69 \n2*s*14*s*1 3*S* &8.15 & 7.98 & 7.99 & 7.99 & 8.00 \n2*s*14*p*1 3*P*0 &8.45 & 8.27 & 8.28 & 8.27 & 8.28 \n2*s*14*d*1 3*D* &8.63 & 8.45 & 8.46 & 8.45 & 8.42 \n2*s*15*s*1 3*S* &8.74 & 8.56 & 8.57 & 8.56 & 8.56 \n2*s*15*p*1 3*P*0 &8.87 & 8.68 & 8.69 & 8.69 & 8.69 \n2*s*16*s*1 3*S* &9.06 & 8.88 & 8.90 & 8.89 & 8.82 \n2*s*16*p*1 3*P*0 &9.14 & 8.96 & 8.97 & 8.96 & 8.89 \n&0.16 & 0.02 & 0.02 & & \n&0.05 & 0.05 & 0.03 & & \nSee Ref. for references to experimental results.\nIn this section, we study the dependence of the MR-ADC(1) results on the size of the active space. In our benchmark, we consider the beryllium atom (Be), for which the accurate results from full configuration interaction (FCI) are available in the literature. We use the same basis set as in Ref. and employ three active spaces in our reference CASSCF computations, including all orbitals in parenthesis: (2*s*2*p*), (2*s*2*p*3*s*3*p*), and (2*s*3*s*2*p*3*p*3*d*). The largest active space corresponds to (2e, 13o).\nWe first investigate accuracy of the Be excitation energies computed using conventional methods, such as CASCI and strongly-contracted NEVPT2 (sc-NEVPT2). shows results of CASCI and sc-NEVPT2 obtained using the largest (2*s*3*s*2*p*3*p*3*d*) active space. The CASCI energies were computed using the ground-state CASSCF orbitals (CASCI/CASSCF), as well as the CASSCF orbitals averaged over *n* lowest-energy states [CASCI/SA(*n*)-CASSCF, *n* = 9 and 20]. For sc-NEVPT2, only the SA(20)-CASSCF orbitals were used. For all electronic states, the CASCI excitation energies strongly depend on the choice of molecular orbitals. In particular, using the ground-state CASSCF orbitals leads to very large mean absolute errors () and standard deviations () of 5.07 and 4.30 eV, respectively, relative to FCI. These errors are substantially reduced when using the state-averaged CASSCF orbitals. In this case, the best agreement with the FCI benchmark results is observed for states that were included in state-averaging (errors of ≤ 0.11 eV), while much larger errors (up to ∼ 3 eV) are obtained for the remaining states. Incorporating the description of dynamic correlation using the sc-NEVPT2/SA(20)-CASSCF method further reduces the errors, yielding excitation energies in excellent agreement with FCI for states included in state-averaging. Although sc-NEVPT2 improves the description of the remaining states, their computed transition energies are significantly overestimated (up to 1.17 eV).\nWe now turn our attention to the MR-ADC(1) results presented in. Importantly, while CASCI and sc-NEVPT2 can only be used to compute energies of transitions between orbitals in the active space, MR-ADC(1) provides information about excitations between all orbitals. Although MR-ADC(1) is the simplest approximation in the MR-ADC hierarchy, its accuracy can be improved by increasing the size of the active space. demonstrates that the MR-ADC(1) excitation energies converge towards the FCI limit as the active space is expanded from (2*s*2*p*) to (2*s*3*s*2*p*3*p*3*d*). Including more orbitals in the active space also improves the description of excitations between active and external orbitals. For example, the error in excitation energy for the 2*s*14*p*1 3*P*0 state reduces from 0.17 eV to 0.01 eV by increasing the active space from (2*s*2*p*) to (2*s*3*s*2*p*3*p*3*d*). Overall, for the largest (2*s*3*s*2*p*3*p*3*d*) active space, the MR-ADC(1) results are in a close agreement with FCI, with mean absolute errors () and standard deviations () of 0.02 and 0.03 eV, respectively. Although the MR-ADC(1) method used the ground-state CASSCF orbitals, its and computed using the smallest (2*s*2*p*) active space (0.16 and 0.05 eV, respectively) are already smaller than those of sc-NEVPT2/SA(20)-CASSCF employing the largest (2*s*3*s*2*p*3*p*3*d*) active space (0.20 and 0.38 eV).\nInterestingly, the MR-ADC(1) excitation energies agree very closely to the energies from multi-configurational linear response theory (MC-LR) reported in Ref. for almost all states but the 2*s*12*p*1 1*P*0 state, where the difference of ∼ 0.05 eV is observed. Although both theories are based on the first-order approximation to the polarization propagator, MC-LR is formulated as a non-Hermitian eigenvalue problem, whereas the MR-ADC(1) energies are computed by diagonalizing a Hermitian matrix of a smaller dimension.\nAvoided crossing in\n-------------------\nNext, we test MR-ADC(1) for the description of an avoided crossing between the ground *X* 1Σ*g*+ and the excited 2 1Σ*g*+ states of LiF. At short bond distances, the wavefunctions of these two states are dominated by the ionic and covalent configurations, respectively. As the bond distance increases, the potential energy curves of the two states closely approach each other and their wavefunctions strongly interact, exchanging their character. In this section, we use the 6-31+G\\* basis set and compare the MR-ADC(1) results to those obtained from FCI, state-averaged CASSCF (SA-CASSCF), strongly-contracted quasidegenerate NEVPT2 (sc-QD-NEVPT2), and multi-configurational Tamm-Dancoff approximation (MC-TDA). As described in, MC-TDA can be considered as an approximation to MR-ADC(1) where all terms that depend on the single- and semi-internal double-excitation amplitudes are neglected. The FCI results were computed using the semistochastic heat-bath configuration interaction algorithm (SHCI) implemented in the Dice program. The 1*s* orbital of fluorine was not correlated in the SHCI computations. All active-space methods used the active space. In SA-CASSCF and sc-QD-NEVPT2, both *X* 1Σ*g*+ and 2 1Σ*g*+ states were included in state-averaging to compute the reference CASSCF orbitals. In MR-ADC(1) and MC-TDA, the CASSCF orbitals were optimized for the ground *X* 1Σ*g*+ state.\n [fig:LiF]\n[t!] [tab:c2]\nL1.5cmC2.5cmC3.5cmC2.5cmC2.5cmC2.5cm State & TDA & CASCI/CASSCF & MR-ADC(1) & sc-NEVPT2 & DMRG \n*A* 1Π*u* & − 1.30 & 2.31 & 1.97 & 1.42 & 1.29 \n*B* 1Δ*g* & & 4.08 & 4.08 & 2.45 & 2.17 \n*B*ʹ 1Σ*g*+ & & 4.07 & 4.04 & 2.73 & 2.45 \n*C* 1Π*g* & 6.42 & 6.27 & 5.98 & 4.83 & 4.61 \n*a* 3Π*u* & − 2.41 & 0.97 & 0.64 & 0.34 & 0.21 \n*b* 3Σ*g*− & & 3.17 & 3.17 & 1.56 & 1.29 \n*c* 3Σ*u*+ & − 1.08 & 1.39 & 1.28 & 1.35 & 1.29 \n*d* 3Π*g* & & 3.74 & 3.57 & 2.85 & 2.65 \n1 3Δ*u* & 7.35 & 8.03 & 7.17 & 7.02 & 6.66 \n& 2.01 & 1.27 & 1.03 & 0.21 & \n& 2.11 & 0.59 & 0.69 & 0.10 & \nThe DMRG results employing the active space are from Ref..\nExcited state with double excitation character, absent in TDA.\nplots the *X* 1Σ*g*+ → 2 1Σ*g*+ excitation energy (Δ*E*) for various bond distances computed using MR-ADC(1), MC-TDA, sc-QD-NEVPT2, SA-CASSCF, and FCI. The FCI curve exhibits a minimum corresponding to an avoided crossing at 12.25 *a*0 with Δ*E* = 0.07 eV. SA-CASSCF shows the worst agreement with FCI predicting an avoided crossing at 8.75 *a*0 with Δ*E* = 0.55 eV. MR-ADC(1) qualitatively reproduces the shape of the FCI curve locating an avoided crossing at 15.25 *a*0 with Δ*E* = 0.12 eV. Although at each geometry the MR-ADC(1) error in Δ*E* is large (0.3 to 0.5 eV), relative to FCI, the MR-ADC(1) curve is quite parallel to FCI for the short ( < 10 *a*0) and long ( > 16 *a*0) bond distances. In addition, MR-ADC(1) demonstrates a significant improvement over the reference CASCI/CASSCF results, which do not exhibit an avoided crossing at any geometry. The MC-TDA curve is quite close to MR-ADC(1) for the short bond distances ( < 14 *a*0), but significantly deviates from MR-ADC(1) at the longer distances ( > 16 *a*0), showing larger non-parallelity errors relative to FCI. The best results are shown by the sc-QD-NEVPT2 method, which locates an avoided crossing at 11.5 *a*0 with Δ*E* = 0.09 eV, indicating the importance of the two-electron dynamic correlation effects that are not included in MR-ADC(1).\nDoubly excited states in\n------------------------\nFinally, we consider a challenging example of the molecule, whose excited states require very accurate description of static and dynamic correlation. reports the MR-ADC(1) vertical excitation energies for at near equilibrium bond distance () computed using the cc-pVDZ basis set. In addition, we report results from the single-reference Tamm-Dancoff approximation (TDA), CASCI computed using the ground-state CASSCF orbitals (CASCI/CASSCF), and strongly-contracted NEVPT2 (sc-NEVPT2). For all active-space methods, the CASSCF reference wavefunction was computed using the active space. As a benchmark, we employ the accurate density matrix renormalization group (DMRG) results obtained by Wouters *et al*.\nAll low-lying electronic states of considered in have a significant multi-reference character. This is evidenced by the TDA method, which incorrectly predicts the ground state and completely misses four out of nine excited states, due to their doubly excited nature. On the contrary, the MR-ADC(1) method correctly assigns the ground state to be *X* 1Σ*g*+ and provides excitation energies for the doubly excited states. However, for most of the electronic states, the MR-ADC(1) energies significantly overestimate the DMRG results and are close to CASCI/CASSCF. Although for some states (e.g., *B* 1Δ*g* or *b* 3Σ*g*−) the MR-ADC(1) and CASCI/CASSCF results are virtually identical, for other states MR-ADC(1) predicts somewhat lower excitation energies, reducing the error relative to DMRG by up to 0.35 eV, with an exception of the 1 3Δ*u* state where a substantial improvement of 0.86 eV is observed. The large errors of MR-ADC(1) are due to the missing description of the two-electron dynamic correlation between core, active, and external orbitals, which is very important in the excited states of. This is evidenced by the results of the sc-NEVPT2 method, which are in a much closer agreement with DMRG (= 0.21, = 0.10 eV) than MR-ADC(1) (= 1.03, = 0.69 eV). The two-electron dynamic correlation effects will be incorporated in the second-order MR-ADC approximation (MR-ADC(2)) and are expected to significantly lower the excitation energies, reducing the errors relative to reference values.\nConclusions\n===========\nIn this work, we considered a multi-reference formulation of the algebraic diagrammatic construction theory (MR-ADC) for excited states of strongly correlated systems. The MR-ADC approach is an alternative to multi-reference perturbation theories and multi-reference propagator methods and has several attractive properties: (i) it allows for including and systematically improving the description of the dynamic correlation effects outside of the active space, (ii) it describes electronic transitions involving all orbitals (i.e., core, active, and external), (iii) it is based on the Hermitian eigenvalue problem and thus ensures that the excitation energies have real values, (iv) it enables efficient computation of spectroscopic properties (such as transition amplitudes and spectral densities), and (v) allows for simulations of various spectroscopic processes (e.g., valence or core excitations, photoionization). In contrast to the original (single-reference) ADC theory, MR-ADC is more reliable in situations where the multi-reference effects are important in the ground or excited electronic states. Our formulation of MR-ADC is based one the effective Liouvillean formalism of Mukherjee and Kutzelnigg, originally developed for the single-determinant reference wavefunctions. By generalizing this formalism to multi-determinant states, we arrived at the MR-ADC formulation, which naturally reduces to the conventional ADC theory in the single-reference limit. We performed a perturbative analysis of MR-ADC and outlined a procedure for constructing MR-ADC approximations at each order in perturbation theory.\nAs a first step in defining the hierarchy of the MR-ADC methods, we presented an implementation of the first-order MR-ADC approximation for the polarization propagator (MR-ADC(1)) and benchmarked its results for two non-interacting water molecules, excitation energies of the Be atom, an avoided crossing in, and doubly excited states in. For the water molecules, we showed that the MR-ADC(1) excitation energies exhibit small size-consistency errors, which originate due to removing linear dependencies in the overlap metric. For the Be atom, we demonstrated that the MR-ADC(1) results converge to the full configuration interaction limit with increasing active space size. In a study of, we showed that MR-ADC(1) qualitatively correctly describes an avoided crossing due to the mixing of the ionic and covalent configurations. For, MR-ADC(1) predicts excitation energies of the doubly excited states, but shows large errors relative to reference values, missing the description of the two-electron dynamic correlation outside of the active space. These correlation effects are incorporated in the second-order MR-ADC approximation and are expected to significantly improve the results.\nWe envision many possible extensions of the current work. An immediate extension is the development of the second-order MR-ADC(2) approximation for the polarization propagator, which will incorporate the description of the two-electron dynamic correlation effects that are essential for accurate predictions of the excitation energies and excited-state properties. A further direction is to develop the MR-ADC methods for simulations of other spectroscopic properties, such as photoelectron, X-ray absorption or two-photon absorption spectra. Although the conventional ADC theory has been applied to these problems, MR-ADC is expected to be more reliable for systems with challenging electronic structure, such as open-shell molecules and transition metal complexes. Additionally, for systems containing heavy elements, it will be important to combine MR-ADC with the description of relativistic effects. Finally, MR-ADC can be combined with density matrix renormalization group or selected configuration interaction approaches, which will enable simulations of spectroscopic properties of multi-reference systems with a large number of strongly correlated electrons. Work along these directions is ongoing in our group.\nAcknowledgements\n================\nThis work was supported by the start-up funds provided by the Ohio State University. The author would like to thank Francesco Evangelista for comments on this manuscript and insightful discussions.\nAppendix: MR-ADC(1) amplitude equations\n=======================================\nHere, we describe how to solve the MR-ADC(1) amplitude equations. The general form of the equations is shown in. The action of the first-order effective Hamiltonian on the reference state can be written as:\n \n$$\\begin{aligned}\n \\label{eq:H1eff\\_psi0}\n \\tilde{H}^{(1)}\\ket{\\Psi\\_0} = V\\ket{\\Psi\\_0} + (H^{(0)} - E\\_0) T^{(1)} \\ket{\\Psi\\_0} \\end{aligned}$$ \nwhere *T*(1) = *T*1(1) + *T*2(1) and *E*0 is the ground-state CASCI/CASSCF energy.\nCore-external amplitudes\n------------------------\nFirst, we consider equations for the core-external *t**i**a*(1) and *t**i**x**a**y*(1) amplitudes. For brevity, we omit the perturbation order in our notation, e.g. *t**i**a*(1) ≡ *t**i**a*. Projecting by $a^{a}\\_{i}\\ket{\\Psi\\_0}$ and $a^{ay}\\_{ix}\\ket{\\Psi\\_0}$ on the left, the amplitude equations can be written in a tensor form:\n \n$$\\begin{aligned}\n \\label{eq:t\\_amp\\_0p}\n \\mathbf{K^{[0']}} \\mathbf{T^{[0']}} = \\mathbf{V^{[0']}}\\end{aligned}$$ \nwhere [0ʹ] is a standard notation for the core-external single excitations used in *N*-electron valence perturbation theory (NEVPT) and the tensors are defined as:\n \n$$\\begin{aligned}\n \\mathbf{K^{[0']}} =\n \\begin{pmatrix} \n \\bra{\\Psi\\_0}a^{i}\\_{a} [H^{(0)}, a^{b}\\_{j}] \\ket{\\Psi\\_0} & \\bra{\\Psi\\_0}a^{i}\\_{a} [H^{(0)}, a^{bw}\\_{jz}] \\ket{\\Psi\\_0} \\\\\n \\bra{\\Psi\\_0} a^{ix}\\_{ay} [H^{(0)}, a^{b}\\_{j}] \\ket{\\Psi\\_0} & \\bra{\\Psi\\_0} a^{ix}\\_{ay} [H^{(0)}, a^{bw}\\_{jz}] \\ket{\\Psi\\_0} \n \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\mathbf{T^{[0']}} = \n \\begin{pmatrix} \n {{t}\\_{{j}}^{{b}}} \\\\\n {{t}\\_{{jz}}^{{bw}}}\n \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\mathbf{V^{[0']}} = -\n \\begin{pmatrix} \n \\bra{\\Psi\\_0}a^{i}\\_{a} V\\ket{\\Psi\\_0} \\\\\n \\bra{\\Psi\\_0} a^{ix}\\_{ay} V\\ket{\\Psi\\_0} \n \\end{pmatrix}\\end{aligned}$$ \nThe elements of $\\mathbf{K^{[0']}}$ reduce to\n \n$$\\begin{aligned}\n \\label{eq:K\\_0p\\_1}\n \\bra{\\Psi\\_0}a^{i}\\_{a} [H^{(0)}, a^{b}\\_{j}] \\ket{\\Psi\\_0} &= {\\delta\\_{{i}}^{{j}}} {\\delta\\_{{a}}^{{b}}} ({\\ensuremath{\\varepsilon\\_{a}}} - {\\ensuremath{\\varepsilon\\_{i}}}) \\\\\n \\label{eq:K\\_0p\\_2}\n \\bra{\\Psi\\_0}a^{i}\\_{a} [H^{(0)}, a^{bw}\\_{jz}] \\ket{\\Psi\\_0} &= {\\delta\\_{{i}}^{{j}}} {\\delta\\_{{a}}^{{b}}} ({\\ensuremath{\\varepsilon\\_{a}}} - {\\ensuremath{\\varepsilon\\_{i}}}) \\braket{\\Psi\\_0|a^{w}\\_{z}|\\Psi\\_0} \\\\\n \\label{eq:K\\_0p\\_3}\n \\bra{\\Psi\\_0} a^{ix}\\_{ay} [H^{(0)}, a^{b}\\_{j}] \\ket{\\Psi\\_0} &= {\\delta\\_{{i}}^{{j}}} {\\delta\\_{{a}}^{{b}}} ({\\ensuremath{\\varepsilon\\_{a}}} - {\\ensuremath{\\varepsilon\\_{i}}}) \\braket{\\Psi\\_0|a^{x}\\_{y}|\\Psi\\_0} \\\\\n \\label{eq:K\\_0p\\_4}\n \\bra{\\Psi\\_0} a^{ix}\\_{ay} [H^{(0)}, a^{bw}\\_{jz}] \\ket{\\Psi\\_0} &= {\\delta\\_{{i}}^{{j}}} {\\delta\\_{{a}}^{{b}}} \\left[({\\ensuremath{\\varepsilon\\_{a}}} - {\\ensuremath{\\varepsilon\\_{i}}}) \\braket{\\Psi\\_0|a^{x}\\_{y}a^{w}\\_{z}|\\Psi\\_0} \\right. \\notag \\\\\n &\\left. + \\braket{\\Psi\\_0|a^{x}\\_{y}[H\\_{act},a^{w}\\_{z}]|\\Psi\\_0}\\right]\\end{aligned}$$ \nFor the elements of $\\mathbf{V^{[0']}}$, we obtain:\n \n$$\\begin{aligned}\n \\bra{\\Psi\\_0}a^{i}\\_{a} V\\ket{\\Psi\\_0} &= {h\\_{{a}}^{{i}}} + \\sum\\_{j} {{v}\\_{{aj}}^{{ij}}} + \\sum\\_{zw} {{v}\\_{{aw}}^{{iz}}} \\braket{\\Psi\\_0|a^{w}\\_{z}|\\Psi\\_0} \\\\\n \\bra{\\Psi\\_0} a^{ix}\\_{ay} V\\ket{\\Psi\\_0} &= {h\\_{{a}}^{{i}}} \\braket{\\Psi\\_0|a^{x}\\_{y}|\\Psi\\_0} + \\sum\\_{j} {{v}\\_{{aj}}^{{ij}}} \\braket{\\Psi\\_0|a^{x}\\_{y}|\\Psi\\_0} \\notag \\\\\n &+ \\sum\\_{zw} {{v}\\_{{aw}}^{{iz}}} \\braket{\\Psi\\_0|a^{x}\\_{y}a^{w}\\_{z}|\\Psi\\_0} \\end{aligned}$$ \nTo solve, we first consider the generalized eigenvalue problem for the matrix $\\mathbf{K^{[0']}}$:\n \n$$\\begin{aligned}\n \\label{eq:eig\\_problem\\_0p}\n \\mathbf{K^{[0']}} \\mathbf{X^{[0']}} = \\mathbf{S^{[0']}} \\mathbf{X^{[0']}} \\boldsymbol{\\epsilon^{[0']}} \\end{aligned}$$ \nwhere the overlap metric has a general form\n \n$$\\begin{aligned}\n \\label{eq:S\\_0p}\n \\mathbf{S^{[0']}} =\n \\begin{pmatrix} \n \\bra{\\Psi\\_0}a^{i}\\_{a} a^{b}\\_{j} \\ket{\\Psi\\_0} & \\bra{\\Psi\\_0}a^{i}\\_{a} a^{bw}\\_{jz} \\ket{\\Psi\\_0} \\\\\n \\bra{\\Psi\\_0} a^{ix}\\_{ay} a^{b}\\_{j} \\ket{\\Psi\\_0} & \\bra{\\Psi\\_0} a^{ix}\\_{ay} a^{bw}\\_{jz} \\ket{\\Psi\\_0} \n \\end{pmatrix}\\end{aligned}$$ \nDefining $\\mathbf{\\tilde{X}^{[0']}} = (\\mathbf{S^{[0']}})^{1/2} \\mathbf{X^{[0']}}$, the $\\mathbf{K^{[0']}}$ matrix in can be expressed as:\n \n$$\\begin{aligned}\n \\mathbf{K^{[0']}} &= (\\mathbf{S^{[0']}})^{1/2} \\mathbf{\\tilde{X}^{[0']}} \\boldsymbol{\\epsilon^{[0']}} \\mathbf{\\tilde{X}^{[0']\\dag}} (\\mathbf{S^{[0']}})^{1/2}\\end{aligned}$$ \nThis allows us to obtain expression for the amplitudes $\\mathbf{T^{[0']}}$ in terms of the eigenvalues $\\boldsymbol{\\epsilon^{[0']}}$, eigenvectors $\\mathbf{\\tilde{X}^{[0']}}$ and the overlap matrices:\n \n$$\\begin{aligned}\n \\mathbf{T^{[0']}} = (\\mathbf{S^{[0']}})^{-1/2} \\mathbf{\\tilde{X}^{[0']}} (\\boldsymbol{\\epsilon^{[0']}})^{-1} \\mathbf{\\tilde{X}^{[0']\\dag}} (\\mathbf{S^{[0']}})^{-1/2} \\mathbf{V^{[0']}}\\end{aligned}$$ \nSolving the eigenvalue problem requires diagonalizing the overlap metric $\\mathbf{S^{[0']}}$ and projecting out the eigenvectors corresponding to small eigenvalues (see for details). Since the $\\mathbf{K^{[0']}}$ matrix elements in are zero for *i* ≠ *j* or *a* ≠ *b*, can be solved for each block with *i* = *j* and *a* = *b* independently, which greatly reduces the cost of computing the amplitudes. Although in this work we only deal with the first-order amplitudes, we note that in MR-ADC the extraction of linearly independent amplitudes has to be done separately at each order of perturbation theory.\nAvoiding computation of 4-RDM for the core-active and active-external amplitudes\n--------------------------------------------------------------------------------\nNext, we consider the core-active amplitudes *t**i**x* and *t**i**x**y**w* belonging to the [ + 1ʹ] excitation class of NEVPT. The amplitude equations can be obtained by projecting by the $a^{x}\\_{i}\\ket{\\Psi\\_0}$ and $a^{xw}\\_{iz}\\ket{\\Psi\\_0}$ configurations and can be written in the form similar to : $\\mathbf{K^{[-1']}} \\mathbf{T^{[-1']}} = \\mathbf{V^{[-1']}}$. As previously, this system of linear equations can be solved by diagonalizing and inverting the matrix $\\mathbf{K^{[-1']}}$. However, evaluation of $\\mathbf{K^{[-1']}}$ requires the four-particle reduced density matrix (4-RDM), which has a steep O(*N*det × *N*act8) computational scaling with the number of active orbitals *N*act and the dimension of the active-space Hilbert space *N*det.\nHere, we present a different algorithm, which does not require computation of 4-RDM. We start by multiplying both sides of by and and set the corresponding projections to zero. For example, for the $a^{x}\\_{i}\\ket{\\Psi\\_0}$ projection we obtain:\n \n$$\\begin{aligned}\n \\label{eq:H1eff\\_psi0\\_2}\n \\bra{\\Psi\\_0} a^{i}\\_{x} T^{(1)} \\ket{\\Psi\\_0} &= - \\bra{\\Psi\\_0} a^{i}\\_{x} (H^{(0)} - E\\_0)^{-1} V\\ket{\\Psi\\_0} \\notag \\\\\n &= - \\int\\_0^\\infty \\bra{\\Psi\\_0} a^{i}\\_{x} e^{-(H^{(0)} - E\\_0)\\tau} V\\ket{\\Psi\\_0} \\mathrm{d} \\tau\\end{aligned}$$ \nwhere we expressed the operator resolvent (*H*(0) − *E*0)− 1 using a Laplace transform as an integral over imaginary time *τ*. The resulting amplitude equations can be written in the matrix form:\n \n$$\\begin{aligned}\n \\label{eq:t\\_amp\\_+1p}\n \\mathbf{T^{[+1']}} = (\\mathbf{S^{[+1']}})^{-1} \\mathbf{W^{[+1']}}\\end{aligned}$$ \nwhere the tensor product is carried out over the unique sets of amplitudes only (i.e., *t**j**v**y**u*, *y* > *u*) and the tensors are defined as:\n \n$$\\begin{aligned}\n \\mathbf{T^{[+1']}} = \n \\begin{pmatrix} \n {{t}\\_{{j}}^{{y}}} \\\\\n {{t}\\_{{jv}}^{{yu}}}\n \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\label{eq:S\\_+1p}\n \\mathbf{S^{[+1']}} =\n \\begin{pmatrix} \n \\bra{\\Psi\\_0} a^{i}\\_{x} a^{y}\\_{j} \\ket{\\Psi\\_0} & \\bra{\\Psi\\_0} a^{i}\\_{x} a^{yu}\\_{jv} \\ket{\\Psi\\_0} \\\\\n \\bra{\\Psi\\_0} a^{iz}\\_{xw} a^{y}\\_{j} \\ket{\\Psi\\_0} & \\bra{\\Psi\\_0} a^{iz}\\_{xw} a^{yu}\\_{jv} \\ket{\\Psi\\_0} \n \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\mathbf{W^{[+1']}} \n &= -\n \\begin{pmatrix} \n \\int\\_0^\\infty \\bra{\\Psi\\_0} a^{i}\\_{x} e^{-(H^{(0)} - E\\_0)\\tau} V\\ket{\\Psi\\_0} \\mathrm{d} \\tau \\\\\n \\int\\_0^\\infty \\bra{\\Psi\\_0} a^{iz}\\_{xw} e^{-(H^{(0)} - E\\_0)\\tau} V\\ket{\\Psi\\_0} \\mathrm{d} \\tau\n \\end{pmatrix} \\notag \\\\\n \\label{eq:W\\_p1}\n &= -\n \\begin{pmatrix} \n \\int\\_0^\\infty e^{{\\ensuremath{\\varepsilon\\_{i}}} \\tau} \\bra{i(\\tau)} {a^\\dagger\\_{x}} \\ket{\\Psi\\_0} \\mathrm{d} \\tau \\\\\n \\int\\_0^\\infty e^{{\\ensuremath{\\varepsilon\\_{i}}} \\tau} \\bra{i(\\tau)} {a^\\dagger\\_{x}} {a^\\dagger\\_{w}} {a^{\\ }\\_{z}}\\ket{\\Psi\\_0} \\mathrm{d} \\tau\n \\end{pmatrix}\\end{aligned}$$ \nThe matrix elements of $\\mathbf{W^{[+1']}}$ can be evaluated by the imaginary-time propagation of the active-space state $\\ket{i(\\tau)} = e^{-(H^{(0)} - E\\_0)\\tau}\\ket{i}$, where\n \n$$\\begin{aligned}\n \\ket{i} &\\equiv \\sum\\_{y} {h\\_{{y}}^{{i}}} {a^\\dagger\\_{y}} \\ket{\\Psi\\_0} + \\sum\\_{yk} {{v}\\_{{yk}}^{{ik}}} {a^\\dagger\\_{y}} \\ket{\\Psi\\_0} \\notag \\\\\n &+ \\frac{1}{2} \\sum\\_{wzy} {{v}\\_{{yw}}^{{iz}}} {a^\\dagger\\_{y}} {a^\\dagger\\_{w}} {a^{\\ }\\_{z}} \\ket{\\Psi\\_0}\\end{aligned}$$ \nand integrating numerically over a set of time steps *τ**k*. The details of the imaginary-time propagation algorithm can be found in and in Ref.. Once $\\mathbf{W^{[+1']}}$ is determined, we diagonalize the overlap metric $\\mathbf{S^{[+1']}}$, project out the linearly dependent eigenvectors with small eigenvalues, and compute the amplitudes $\\mathbf{T^{[+1']}}$ according to.\nFor the active-external excitations *t**x**a* and *t**x**y**a**w* corresponding to the [ − 1ʹ] excitation class of NEVPT, the amplitude equations have a similar form:\n \n$$\\begin{aligned}\n \\mathbf{T^{[-1']}} = (\\mathbf{S^{[-1']}})^{-1} \\mathbf{W^{[-1']}}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\mathbf{T^{[-1']}} = \n \\begin{pmatrix} \n {{t}\\_{{y}}^{{b}}} \\\\\n {{t}\\_{{yv}}^{{bu}}}\n \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\label{eq:S\\_-1p}\n \\mathbf{S^{[-1']}} =\n \\begin{pmatrix}\n \\bra{\\Psi\\_0} a^{x}\\_{a} a^{b}\\_{y} \\ket{\\Psi\\_0} & \\bra{\\Psi\\_0} a^{x}\\_{a} a^{bu}\\_{yv} \\ket{\\Psi\\_0} \\\\\n \\bra{\\Psi\\_0} a^{xz}\\_{aw} a^{b}\\_{y} \\ket{\\Psi\\_0} & \\bra{\\Psi\\_0} a^{xz}\\_{aw} a^{bu}\\_{yv} \\ket{\\Psi\\_0} \n \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\mathbf{W^{[-1']}} \n &= -\n \\begin{pmatrix} \n \\int\\_0^\\infty e^{-{\\ensuremath{\\varepsilon\\_{a}}} \\tau} \\bra{a(\\tau)} {a^{\\ }\\_{x}} \\ket{\\Psi\\_0} \\mathrm{d} \\tau \\\\\n \\int\\_0^\\infty e^{-{\\ensuremath{\\varepsilon\\_{a}}} \\tau} \\bra{a(\\tau)} {a^\\dagger\\_{w}} {a^{\\ }\\_{z}} {a^{\\ }\\_{x}}\\ket{\\Psi\\_0} \\mathrm{d} \\tau\n \\end{pmatrix}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\ket{a} &\\equiv \\sum\\_{y} {h\\_{{a}}^{{y}}} {a^{\\ }\\_{y}} \\ket{\\Psi\\_0} + \\sum\\_{yk} {{v}\\_{{ak}}^{{yk}}} {a^{\\ }\\_{y}} \\ket{\\Psi\\_0} \\notag \\\\\n &+ \\frac{1}{2} \\sum\\_{wzy} {{v}\\_{{aw}}^{{yz}}} {a^\\dagger\\_{w}} {a^{\\ }\\_{z}} {a^{\\ }\\_{y}} \\ket{\\Psi\\_0}\\end{aligned}$$ \nAvoiding the evaluation of 4-RDM lowers the computational cost of solving the core-active and active-external amplitude equations from O(*N*det × *N*act8) to and, respectively, where *N**τ* is the number of time steps in imaginary time ( ∼ 10 to 40), *N*core is the number of core orbitals, and *N*ext is the number of external orbitals.\n100\nA. 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Malrieu, and P. Rancurel, J. Chem. Phys. **58**, 5745 (1973).\nG. K.-L. Chan and S. Sharma, Annu. Rev. Phys. Chem. **62**, 465 (2011).\n"},"data_id":{"kind":"string","value":"arxiv_0183109"}}},{"rowIdx":45792,"cells":{"content":{"kind":"string","value":"+ 3 &*P**k* + 1 = *H*(*N**k* + 1) & for *b**n* < *b* < *b**n* + 3:& \n & &1em find **first** LV such that & \n & &2em *H*(*P**k* + 1∣∣*L* ⊕ *P**k* + 1) = *V* & \n & &**if found in block** *b̂*:& \n & &1em**unless** ∃*b* ∈ (*b**n*, *b̂*), *L*′ ∈ *B**b* : & \n & &2em*H*(*L*′ ⊕ *P**k* + 1) = *P**k*& Post \n & &**else**& Reject \n1em To unlock the data object *M**k* compute $M={\\bf D}^\\*\\_{L\\oplus P\\_{k+1}}(P\\_{k+1},S)$ for every S-message in the interval [*b**n*, *b̂*) and accept the first valid *M*. Here *E**k*\\*(*a*, *b*) is the encryption of *b* under key *k* in PCBC mode with *a* as IV; *D**k*\\*(*a*, *b*) is the matching decryption.\n#### Operations.\nTransactions can be posted by both and human users. Each has one or more *masters*, which are typically users (but could be other ). Not only does each check the posting of each of its SLVP messages on the blockchain, it also monitors the postings of all its masters and any relevant counterparties, and validates their data objects by applying the SLVP protocol. Alternatively, the can participate in a smart contract which would only require it to follow and validate messages from the contract engine. In this paper we limit ourselves to the mechanics of transaction processing, while leaving higher-level protocol to further work. We will assume in the sequel that each is interacting with a very small number of other actors and needs to follow a few SLVP threads (perhaps 2 or 3). Our focus will be on how to make these interactions as computationally and communicationally efficient as possible.\n#### Addresses.\nEach PLS user has an address, which is a small number. Since we concern ourselves with a localised enterprise solution (e.g. a smart hospital) covered by a direct link radio network (e.g. smart sensors equipped with a LoRa transceiver), we do not expect the number of greater than circa 1000. The total number of actors should be a small factor of that to account for human users and smart contracts, so 2–4K addresses is our target. Transactions have an originating address and a destination.\n#### Frequency.\nIn IoT networks of interest, communication is limited by the duty cycle to save the limited bandwidth that all have to share. This is in addition to the constraints imposed by the energy budget of an individual IoT device. Consequently a small fraction (typically a few percent) of the registered users will be posting a transaction in any given block.\nBlock structure and optimisation challenge\n==========================================\n#### Immutable dictionary.\nIn the previous section blockchain blocks were shown in the diagrams that consisted of records attributed to various users as key-value pairs, where the key is the user ID. In a given block only some of the registered users would be represented by records. Since the frequency of posting on the blockchain in our case is severely limited by the ’ communication duty cycle (if using LoRa) or energy budget (if using public networks or LoRa), the proportion of users posting to any given block is expected to be very small. Still, the user can only authenticate the block by the S-message of the PLS protocol, which, when unlocked, contains the block hash. On the other hand, as we mentioned in the previous section the user is typically interested in two or three other users’ contributions, which, given a typical enterprise IoT network of a thousand, is still much less than the expected volume of activity in a block. Indeed, given a block production rate of 4 blocks per hour and a data production rate of 2–4 samples per day, and bearing in mind that each data post requires 3 blocks according to the SLVP protocol, we arrive at 6 to 12 blocks per device per day and circa 100 blocks per day in total. This means that in the absence of correlated activities we should expect about 5 to 10 percent of the swarm to post in every given block. For a 1000-strong swarm, the block may contain an estimated 50 to 100 user records authenticated by a single hash. If a wishes to access just a few of these, it would have to first read the whole block and check the hash to validate it, and then dispose of most of these records as they would be irrelevant.\n#### Merkle tree.\nThe standard solution to the above problem is called the *Merkle Tree*(MT), see figure [fig:MT]. It is a labelled binary[5](#fn5) tree each node *ν* of which has two children, with some labels *ν**L* and *ν**R*, and its own label is *ν* = *H*(*ν**L*∣∣*ν**R*). A child can either be a leaf or a full node in its own right; in both cases it has a label but in the latter case it also has two children of its own. It is quite clear that a change in any leaf will change the root label (also known as the *root hash*), so the authenticated root makes the whole tree authentic. For each node except the root there exists one other node with the same parent, which we call *adjunct*. What makes the MT useful is that it can also authenticate a single leaf by providing a root-path list of adjunct nodes’ labels, or root-adjunct path for short. For example, to authenticate the leaf *ν*101 shown in red in the figure, given the root label *r*, we only need to know the labels of the blue (adjunct) nodes: *ν*100, *ν*11 and *ν*0, since \n*r* = *H*(*ν*0∣∣*H*(*H*(*ν*100∣∣*ν*101)∣∣*ν*11)) . \n Generally speaking, for a tree with *K* leaves one needs to communicate *h* = ⌈log2*K*⌉ hashes, which is much less than *K* for the number of leaves in the hundreds that we are considering. The tree thus represents an *array* of leaves indexed by the path: a left edge represents 0 and a right edge 1; the edges traversed en route to the leaf form a bit-string that represents the key. The leaf itself represents the value of the key-value pair.\n#### Blocks represented as Merkle trees.\nCommon practice in Blockchain construction is to represent a block as an MT, each leaf of which carries the hash of a user’s record included in the block, with the user ID being the key gleaned from the leaf’s root path. A user requesting another user’s record (or the one of its own) from an intelligent CAS could just receive the root-adjunct path corresponding to the requested ID and hash it through to match with the root hash value. If the PBC signs the root hash of every block it creates, no further security is required to authenticate any user records. Our investigation is of a special case when the maximum number of users is small and is known in advance, and where good communication efficiency is important. We could use an MT with the tree height *h* close to 10 (to accommodate our expected 210 ∼ 1000 users). Since, as we have mentioned earlier, we expect only around 50 (maybe up to 100) users to contribute to any given block, a great majority of the leaves will not be used.\n#### Mask-controllable sparse MT.\nThe number of leaves in an MT does not have to be a power of 2. Also leaves can have no value associated the root-path key. We can think of such leaves as unoccupied. An MT with no-value leaves is called a *sparse* MT, or SMT. There are several ways of organising an SMT, but proposals usually focus on *mutable* trees that are used for secure updatable key-value storage. Our interest is in *immutable* SMTs, where efficiency is understood in narrow terms as efficiency of retrieval *only*. Below we define our own version of the SMT, geared towards our objectives.\nWe can assume that a leaf without value has a special label NULL and the parent of two NULL nodes has the label NULL as well. This assumption does not diminish security due to the fact that a NULL child of any node is implicitly associated with the node height. Consequently, the shape of the NULL subtree associated with the child is completely defined by its root position. All such NULL trees are identical anyway, so a single label value fully represents them.\nFor the verifier to be able to verify a path with NULL nodes, it requires a bit mask of length *h*, where bit-value 1 indicates that the corresponding adjunct node is non-NULL; and the bit-value 0, that it is NULL. The NULL labels can then be omitted from the path. Finally we extend the domain of *H*(*x*) to include NULL-concatenated strings by defining that for any bit-string *x*\n \n$$\\begin{gathered}\nH(x\\parallel{\\rm NULL}) = H(x\\parallel x^\\prime)\\,,\\label{eq:none-case}\\\\\nH({\\rm NULL}\\parallel x) = H(x^\\prime\\parallel x)\\,,\\nonumber\\end{gathered}$$ \nwhere *x*′ is the bit string obtained form *x* by flipping all bits. Interestingly, a simpler extension \n$$H(x\\parallel{\\rm NULL}) = H({\\rm NULL}\\parallel x) = H(x)$$ \n would not be secure, as it allows one to construct a second preimage by rotating the subtree or swapping nodes along a NULL path. It is easy to see that the hashing process introduced by Eq [eq:none-case] is not invariant to any such transformation. It is impossible to create a new valid SMT with the same root hash and a different leaf sequence without solving the second preimage problem.\nIf the path is mask-controlled, CAS only needs to communicate **up to *h*** hashes in addition to the bit mask for the verifier to successfully compute the root hash. Extending the above example, if *ν*11 were unoccupied, CAS could supply the bit mask 101 (the second adjunct is missing in path order), and the values of *ν*0 and *ν*100. Notice that the bit mask does not need to be secured: if it is incorrect, the verifier will compute an incorrect path expression and the result will not match the root hash. Also notice that the mask is very small compared to the hash length: for a tree of 1024 leaves (counting both NULL and non-NULL) the root-adjunct path contains from 1 to 10 hashes 256 bits each, i.e. 256 to 2560 bis, but the mask length is only 10 bits regardless.\n#### Merkle-Patricia Trie.\nThe idea of mask-control path is similar to that of the so called *Merkle-Patricia Trie*(MPT) where not only the nodes but also the edges can be labelled. If a node has a single active edge (i.e the other edge leads to a NULL subtree), the node is eliminated and its parent uses the prefix of the other edge as its label, see figure [fig:MPT]. The example in the figure is of a block where, out of the maximum 8, only users 2, 4, 5, and 7 (010, 100, 101, 111 in binary) are present. Notice that we still have a binary tree, but the root-adjunct path augmented with edge labels requires from one (for 010) to three (for 100 and 101) adjunct hashes for validation, depending on the leaf. The edge labels are typically much shorter bit-strings than a single cryptographic hash (log2*K* ≪ 256) and so can be neglected in determining the communication efficiency of the access scheme. The same is true of masks with our version of the SMT.\nHow is the edge label secured? It is simply hashed together with its child content in determining the node label: \n*ν* = *H*(*λ*0∣∣*λ*1∣∣*ν*0∣∣*ν*1) , \nwhere *λ*0, 1 are the edge labels of the left and right child, respectively.\nIt is easy to see that there is a direct correspondence between the MPT and the SMT with mask-controlled paths. Our construction requires more work when validating a path: each node, irrespective of its path quality involves *h* hash calculations for verification, where *h* is the height of the tree, but in the MPT case the number of times a hash is calculated is the same as the number of adjunct hashes supplied with the MPT path, although each hash calculation also involves edge labels, which may increase the cost. The total *length* of edge labels along the root path in the MPT case is equal to the length of the mask in the SMT case. However, an MPT path requires markers to partition the path string into individual edge labels. Our construction is slightly more frugal in this respect, and it is simpler, which is why we prefer it.\nMotivation and optimisation idea[sec:motivation]\n================================================\nIt is obvious from the SLVP protocol that an actor engaging in transactions with another on the PLSB must check each block to determine the presence of a transaction message from the counterparty. Due to the low duty-cycle of -to-FS communication, the counterparty will not be present in a great majority of blocks. However, to securely establish the absence the actor must traverse the block and verify that the counterparty’s record is not there. In the MT case we can use the mask-controlled path to the unoccupied leaf which can contain up to *h* adjunct hashes. In the MPT case CAS will supply the longest path *in the direction of*, rather than to, the unoccupied leaf. By examining the last node on that path the user will be able to verify that the necessary edge is missing. For example, looking at the MPT in figure [fig:MPT], if an actor requested the unoccupied leaf 110, the one-step path *r* → *ν*1 with adjunct material will be sent back for validation:\n \n$$\\begin{gathered}\n010,1,\\nu\\_{010}\\nonumber\\\\\n0,11,\\nu\\_{10},\\nu\\_{111}\\nonumber\\end{gathered}$$ \nThe number of hashes to be communicated is the same as that for the mask-controlled MT: in the current example the latter would require hashes *ν*111, *ν*10, and *ν*0. An advantage of the MPT is that it saves the verifier extra hash computations by providing segments of the path as (hashed-controlled) edge labels. While saving *some* compute time, the effect of it is negligible, since the maximum root-adjunct path length is logarithmic in the number of leaves ($\\lesssim12$), and a modern microcontroller can compute tens of thousands of hashes per second. A disadvantage of the MPT is that it requires communication of edge labels in addition to the hash for each node on the path, but again, compared to the hash length this is negligible, too.\nWhat is considerably more important here is that neither mask-controlled MT, nor MPT reduce the maximum root-adjunct path length. In our example the number of leaves present is 4, but the hight of the MPT as a binary tree is 3, not log24 = 2; it is as if all leaves were present. As a result, the system would require to accommodate longer communication packets, which may affect the guaranteed duty cycle limit of an IoT device.\nThis brings us to the **central idea** of the paper: to index a block, one might prefer to locally renumber the users to achieve a contiguous range of IDs rather than a scattering over a regular structure with subsequent remedies such as the MPT.\nHowever, before proceeding to our solution, we would like to evaluate the base case, the mask-controlled MT. We would like to establish some quantitative characteristics of MT paths under a random distribution of leaf occupancy.\nSparse MT Statistics[sec:stats]\n-------------------------------\nLet us number the levels of the MT from the leaves up, starting with 0. We will call the number of adjunct hashes associated with a path its *weight*. Let function $\\PDF\\_k(i)$ of integer *i* be the probability for a path from a given leaf to a hight-*k* node on the tree to be of weight *i*. Recall that we only count nodes on the path with both children being non-NULL as those require an adjunct hash for a Merkle proof. Let us take a closer look at an example path, see figure [fig:path]. From level 0 we move to level 1 as a left child, then to level 2 as the left child, and finally to the level 3 as the right child. Clearly different leaves’ paths differ by the choice of left- and right- ascension at each level, but the significance of the node does not depend on it: the node is only counted when *both* its children are non-NULL. The shaded triangles signify the subtrees that represent the other, *non-path* child of the corresponding path node, which, if non–NULL, produces what we termed above the adjunct hash. If the non-path node is NULL, this fact is noted in the path bit mask, but no adjunct hash is produced.\nAt level 0, the subtree is of height 0 (it is a leaf), at level 1 it is of height 1 (connects two leaves), etc.\n#### Statistical model.[pg:stats]\nWe adopt a model relevant to the IoT case that the PLS blockchain was developed for. A submits a message for inclusion in a block very infrequently. It does so at random with some small probability *p*, whose value depends on the duty cycle restriction, urgency of the sensor data and the available energy budget. Without loss of generality let us assume that *p* ∼ 0.1 in our examples, which will give us some intuition of what kind of figures may arise in practical work. This level of activity means that a participates in roughly one block out of 10 or that about 1/10 of all blockchain users are active in any given PLS round. We also assume that the activities of different are uncorrelated, so any given leaf is either present or absent (NULL) irrespective of the presence/absence of other leaves.\n#### Path weight.\nThe subtrees in figure [fig:path] will consequently be NULL-valued with the probability\n \n*α**k* = (1 − *p*)2*k* , \nwhere *k* is the level at which the subtree is rooted. Let us introduce the Probability Distribution Function $\\PDF\\_k(L)$ as the probability for the weight of a path from level *k* to a leaf to be equal to *L*. Clearly $\\PDF\\_k(L>k)=0$, and we also assume for convenience that for all *k*, $\\PDF\\_k(L<0)=0$. It is easy to calculate $\\PDF\\_1$ directly:\n \n$$\\PDF\\_1(0) = 1-p,\\; \\PDF\\_1(1) = p\\,.\\label{eq:rec0}$$ \nIndeed, the other child of a given leaf of a height-1 tree is NULL with the probability 1 − *p*, producing no adjunct hash, so *L* = 0 with that probability; otherwise (with the probability *p*) the other leaf is non-NULL, supplying a single adjunct hash.\nFor a hight-*k* path we have a combinatorial problem of calculating the probabilities of 2*k* combinations of absence/presence of each adjunct hash (remember that these probabilities are completely independent as per our chosen statistical model). Instead of doing this, we observe the following recurrence relation between the paths to neighbouring levels:\n \n$$\\PDF\\_{k+1}(i)=\\alpha\\_k\\PDF\\_k(i)+(1-\\alpha\\_k)\\PDF\\_k(i-1)\\label{eq:rec}$$ \nIndeed, if the non-path child of the height-(*k* + 1) path node is NULL (this happens with the probability *α**k*) the number of adjunct hashes that the path to height *k* + 1 produces is the same as that to height *k*. Alternatively, if the non-path child is non-NULL, it produces one adjunct hash, and so the probability to produce L hashes for the whole path is the same as the probability to produce *L* − 1 for the path to height *k*. The above equation is the weighted sum of those two outcomes, a mixture distribution.\nThe significance of Eqs. [eq:alpha]–[eq:rec] is in the fact that they permit direct calculation of the PDF at any level above 1 very cheaply given the value of *p*. The PDF obtained can deliver various practical parameters: the average path weight: \n$$\\bar{L}\\_k = \\sum\\_i i\\times\\PDF\\_k(i)\\,,$$ \n the standard deviation, the probability that a certain limit $L\\_{\\rm max}$ is exceeded, etc., which are useful in designing bandwidth-limited communication protocols.\nc\n```\n k mean 0 1 2 3 4 5 6 7 8 9 10 \n==============================================================================\n 2 0.200 81.0 18.0 1.0\n 3 0.390 65.6 30.0 4.2 0.2\n 4 0.734 43.0 42.2 13.1 1.6 0.1\n 5 1.303 18.5 42.7 29.7 8.1 0.9 0.0\n 6 2.118 3.4 23.0 40.3 25.7 6.8 0.8 0.0\n 7 3.084 0.1 4.1 23.6 39.8 25.0 6.6 0.7 0.0\n 8 4.083 0.0 0.1 4.1 23.6 39.8 25.0 6.6 0.7 0.0\n 9 5.083 0.0 0.0 0.1 4.1 23.6 39.8 25.0 6.6 0.7 0.0\n10 6.083 0.0 0.0 0.0 0.1 4.1 23.6 39.8 25.0 6.6 0.7 0.0\n```\nTable [tab:MT-stat] presents the outcome of a direct calculation of Eqs. [eq:alpha]–[eq:rec] for *p* = 0.1 and also includes the value of *L̄**k* in the second column (heading “mean”). The table shows the value (%) of $\\PDF\\_k(i)$, where *i* runs horizontally. For obvious reasons nontrivial evolution only happens until *α**k* drops to small values, whereupon Eq. [eq:rec] degenerates to \n$$\\PDF\\_{k+1}(i)\\approx\\PDF\\_k(i-1)\\,$$ \n making the $\\PDF(i)$ shift to the right by 1 without change of shape as *k* increases. For *p* = 0.1 sparsity is present in the first 7 levels of the tree; from level 8 up the tree becomes dense.\nAnother noteworthy feature of the distribution is its breadth: 95% of the paths require from 4 to 8 hashes, with the mean being around 6, which would necessitate variable length communication, since a factor of 2 difference cannot be ignored. This variability comes despite the compression we have already applied by introducing the bitmap-controlled MT.\nThe sparse MT is indexed by the user ID, and a set of active users for an individual block is random as defined by our statistical model. To get a feel of how efficient the sparse MT is in terms of path weights, we compare its mean path weight with that of a truncated dense tree carrying the same number of non-NULL leaves. We use the least sufficient hight of the dense tree to accommodate all non-NULL leaves, and place all NULLs on the right hand side of level 0, so that the non-NULL leaves may be contiguous, and use Eqs: [eq:none-case] to deal with none values (that is what we mean by truncation). Figure [fig:path-stats] compares the path weights of the two trees. The dependence of the path-weight averaged across the truncated tree on the number of non-NULL leaves is not smooth, as the tree hight leaps up when the number of non-NULLs crosses power-of-two boundaries. Nevertheless, one can clearly see that the dense tree has a lighter path weight at the majority of probability values, becoming slightly worse just before the probability reaches a value that the equivalent dense tree must grow at.\n#### Challenge.\nNow we are prepared to argue our case. A user engaging in the SLVP protocol as verifier must examine every consecutive block to see if the prover has placed a message in it (i.e. an S-, LV- or P-message). According to our statistical model, in a great majority of the blocs, in fact, in a factor of 1 − *p* of them, the prover’s message is likely to be absent. Nevertheless, the verifier needs to satisfy itself that indeed, no message from the prover is present. With the classic MT as well as MPT and our own version of sparse MT, the absence of a leaf is almost as expensive to prove as its presence with a particular value. The difference is that for an absent leaf the label is NULL and it is not communicated, but that is a difference of 1 against, as our calculation shows, circa 6 adjunct hashes to be communicated when 1024 users participate at probability *p* = 0.1. This means that on average 60 (!) hashes would be required to certify the start of an SLVP round. Worse still, each active user, even when all it does is wait for a possible signed message from a counterparty, will be actively requesting the counterparty’s root-adjunct path from CAS every time a block is released, which pretty much destroys the advantages of a low-bandwidth Guy Fawkes protocol.\nHowever, a simple remedy exists, which we consider next.\nTunstall-Merkel Tree[sec:MTT]\n=============================\n#### Basic idea.\nWe kill two birds with one stone by providing a one-time renumbering of users in each block while broadcasting the renumbering information together with the root of the tree. The purpose of the renumbering is to achieve a contiguous range of ID numbers. This way absent users will be recognised as such immediately by any counterparty involved. As a result the cost of absence proof will be zero (plus the cost of the one-for-all broadcast message, which need not be requested). The indexing structure in terms of new IDs will be the kind of tree we have already studied and shown the superior access cost of: a dense, truncated one.\nHow do we enumerate users that are present? Imagine a bitmap sized 2*h*, where *h* is the height of the original (sparse) MT. In the bitmap 1s mark the presence of the corresponding user/leaf and 0s its absence. Under our statistical model (see page ) on average 2*k**p* bits (as per binomial distribution) of the bitmap will be 1s. Users are renumbered according to the bitmap: the user’s new ID is the number of 1s in the bitmap preceding the bit that corresponds to the user’s actual ID.\nOur statistical model assumes that all users are engaged the whole time. A user that decides not to use the blockchain for a while will not be able to maintain a factor of *p* messages per block on average until the user becomes active again; the user’s bit position in the bitmap will be 0 during that period. If there are many such users, the bitmap may have significantly fewer 1s than the aforementioned expectation 2*k**p*. In this sense the expectation is pessimistic.\nThe maximum number of users is within a near-unity factor from the number of in the swarm, since non-IoT users have typically a one-to-many relation with : a human or a server would be in control of several IoT devices. The majority of the users tend to be always-on, active, which work according to a near-periodic schedule. Another useful circumstance here is that the bitmap can be effectively and efficiently compressed to a fraction of its length, provided that the distribution of 1s is close to random and that the number of 1s is known to both the sender and the recipient. The former can be made true by pseudorandomisation, and the later is easy to achieve by including a small integer (typically 10-12 bits in length) in the message that broadcasts the bitmap. In this section we describe the compression technique, and in the next one we will propose a simple and efficient pseudorandomisation.\n#### Tunstall code.\nGiven a bit string of length *n* which is expected to contain *m* = *p**n*, *p* < 1 ones in random positions (which makes *p* a true probability), or alternatively a bit string which is known to contain *m* ones, *m* = *p**n* in random positions (which makes *p* an *empirical* probability), with the positions of ones pairwise uncorrelated (this is called a *zero-order* environment), we set ourselves the task of finding a bijective function *C* : *B**n* → *B**r**w*, *r**w* < *n* that maps the string to a sequence of *r* codewords of length *w*. We wish to minimise *r**w*, or, for a given *w*, to minimise *r*. The theoretical limit of compression is well known from information theory: *r**w* ≥ *H*0*n*, where the zero-order per-bit entropy *H*0 is defined thus: \n*H*0 = − *p*log2*p* − (1 − *p*)log2(1 − *p*) . \n The mapping *C* is realised by partitioning the source bit string into (generally unequal length) chunks and assigning a codeword to each. A chunk *b*0, *b*1, …, *b**k* is found at any given position in our random string with the likelihood\n \n*p**c* = ∏*i* = 0*k**p**b**i*(1 − *p*)1 − *b**i* . \nThe best code with the word length *w* should assign its 2*w* codewords to the 2*w* chunks with the highest likelihood. It must also make sure that the code is complete, i.e. any bit sequence can be represented as a sequence of codewords. It is intuitive that such a code would be optimal, and it can be proven that it is also asymptotically effective, i.e. that its compression ratio tends to the entropy limit as *w* tends to infinity.\nIt is not easy, however, to turn Eq. [eq:lh] into a practical encoder/decoder. The main reason for it is that the value of *k* is not bounded, and neither is the search for suitable chunks to find the top 2*w* ones in terms of their likelihood. The problem is not so much the amount of work required for the search, since we could take the logarithm of Eq [eq:lh] and maximise the linear form\n \nlog*p**c* = *l*log*p* + (*k* − *l*)log(1 − *p*) , \nwhere \n*l* = ∑*i* = 0*k**b**i* , \n in the (0 ≤ *l* ≤ *k*, *k* > 0) area of the (*l*, *k*)-plane starting at the maximum (0, 1). The problem is that each (*l*, *k*) point corresponds to ${k \\choose l}$ chunks of length *k* with *l* 1s in each. Their enumeration and mapping at different *k* would be rather awkward.\nTunstall in his PhD thesis proposed a greedy search which at the same time builds a compact dictionary structure (the Tunstall tree) that can be used for encoding/decoding efficiently, *without* sharing the dictionary (as long as *p* is known to both the encoder and the decoder). The greedy search turns out to be of excellent quality, too, delivering the entropy limit asymptotically, and, as a recent study shows, with a rapidly decreasing redundancy as *w* increases. The redundancy formula from being useful for our analysis, we present it below (without derivation and rewritten in our notation):\n \n$$rw \\le nH\\_0 + O(nH\\_0{\\log(1/p)\\over w})\\,.\\label{eq:redundancy}$$ \nWe will return to Eq [eq:redundancy] later and present our own measurements for the relevant range of parameters, but let us first introduce the dictionary idea, see figure [fig:t-tree]. The dictionary is a (generally imbalanced) labelled binary tree. Both the nodes and the edges are labelled. The edge labels are 0 and 1 as usual, and a node’s label is the likelihood value of the chunk composed by reading the edge labels along the path from the root to the node. The algorithm builds the tree node-by-node, as follows:\n1. Create a root node with two edges labelled 1 and 0 to two child nodes labelled with the value of *p* and 1 − *p*, respectively.\n2. Find and mark the maximum likelihood leaf node. Denote its label as *R*.\n3. Create two leaf children of the marked node and connect them with edges labelled 1 and 0. Make the node labels the value of *R**p* and *R*(1 − *p*), respectively.\n4. Repeat steps 2 and 3 until the tree has 2*w* nodes besides the root.\n5. Now relabel each leaf by its consecutive leaf number while visiting the leaves in some order agreed between the encoder and decoder[6](#fn6).\nTunstall encoding is achieved by running the source bit-string down the Tunstall tree bit by bit until a leaf is reached, at which point the codeword is read off from the leaf label and the process returns to the root. Tunstall decoding requires a 2*w*-entry table where variable-length path sequences are set against codewords, with the former read off from the path to the leaf labelled by the latter. We notice with satisfaction that Tunstall decoding has the cost *O*(1).\n#### Implementation.\nTunstall encoding (and especially decoding) is very undemanding, well within reach of a small, system-on-chip smart sensor. To avoid accuracy/underflow problems with repeated multiplication in generating the dictionary at the receiver (Step 3 of the algorithm above), one could use log-likelihoods as node labels. Then instead of multiplication, log*p* and log(1 − *p*) are *added* to the parent label to produce labels for the 1- and 0-child, respectively[7](#fn7). This way for any reasonable table size, computational accuracy will not be a problem. We implemented the algorithm to see what kind of residual redundancy we could be getting from a specific Tunstall code. The results of our running a Tunstall compressor through 1 million random bits are presented in table [tab:tun-sim]. Comparing this with Eq [eq:redundancy], we conclude that at *p* = 0.05 the compressor already reaches asymptotic mode when doubling the codeword length roughly halves the redundancy *ρ*. At the same time, the dependency *ρ*(*p*) in the interesting range of *p*, i.e. in the area around *p* = 0.1 is not quite asymptotic: the contrast in redundancy between *p* = 0.05 and *p* = 0.15 is nowhere near a factor of 2 that the formula suggests.\nFrom the practical point of view, if we target a blockchain with ∼ 1*K* users, with 10% of them posting messages in any given block, *p* = 0.1 suggests a compressed bitmap of at least 1024 × 0.47 = 482 bits or 61 bytes. An 8% residual redundancy would increase this by only 5 bytes. However, the bitmap is broadcast together with the root hash, 32 bytes long, and a few extra bytes of forced redundancy for the purposes of S-message verification (as per PLS protocol). This increases the length of the S-message up to nearly 100 bytes, and at this level a redundancy of the compressor to the tune of 5 to 10 bytes makes little difference.\n| *w* | *p* | *κ* | *H*0 | *ρ*(%) |\n| --- | --- | --- | --- | --- |\n| 4 | 0.05 | 0.37 | 0.29 | 30.4 |\n| 4 | 0.10 | 0.50 | 0.47 | 7.5 |\n| 4 | 0.15 | 0.65 | 0.61 | 6.9 |\n| 8 | 0.05 | 0.32 | 0.29 | 13.4 |\n| 8 | 0.10 | 0.49 | 0.47 | 4.6 |\n| 8 | 0.15 | 0.63 | 0.61 | 3.8 |\nIf the number of users drops to 0.05, even the poor compression quality for *w* = 4 results in only 379 bits (though 82 bits, or 11 bytes, more than the entropy limit), which is still less than the already acceptable 482 bits we observed for *p* = 0.1. An alternative is to use a list of raw ID numbers, circa 52 in total, each requiring 10 bits. This is 520 bits, far worse than the compressor’s output, but not significantly worse than 482, and the list length would decrease in proportion to *p*. If *p* were to drop further below 0.05, and if the Tunstall compressor further deteriorated, the uncompressed ‘list’ option could at some point be preferred, with the switch controlled by a single additional bit in the message.\n| codeword | − log2*p**c* | chunk |\n| --- | --- | --- |\n| 0000 | 3.06 | 0000000000000 |\n| 0001 | 5.55 | 0000000000001 |\n| 0010 | 5.32 | 000000000001 |\n| 0011 | 5.08 | 00000000001 |\n| 0100 | 4.85 | 0000000001 |\n| 0101 | 4.61 | 000000001 |\n| 0110 | 4.38 | 00000001 |\n| 0111 | 4.14 | 0000001 |\n| 1000 | 3.91 | 000001 |\n| 1001 | 3.67 | 00001 |\n| 1010 | 3.44 | 0001 |\n| 1011 | 3.20 | 001 |\n| 1100 | 3.20 | 010 |\n| 1101 | 5.70 | 011 |\n| 1110 | 2.97 | 10 |\n| 1111 | 5.47 | 11 |\nWe conclude that a four-bit Tunstall code is all that is required to implement the PLS S-message within half of the maximum LoRa message length (250 bytes). To aid the reader’s intuition, we present an example of a 4-bit Tunstall code for *p* = 0.15 in table [tab:tun-code]. For each codeword we additionally show its log-likelihood. Notice that unless the log-likelihoods are exactly identical, as is the case for codewords 1010 and 1100 which correspond to chunks with the same number of 1s and 0s, the differences between log-likelihoods manifest themselves in the first (decimal) fractional digit already, so computational accuracy should not be a concern[8](#fn8).\n#### New structure of the root of trust.\nIn the original PLS protocol the S-record was a message that contains the block’s root of trust *J**i*, which was the root hash of the Merkle tree representing the new block *B**i*. In the light of our analysis of indexing costs presented in Section [sec:stats] and the properties of Tunstall encoding described in the current section, we propose to modify the root of trust *J**i* as shown in Table [tab:root].\n| | | | |\n| --- | --- | --- | --- |\n| offset | field | size | description |\n| (bits) | | (bits) | |\n| 0 | *T**i* | 256 | root hash of the Merkle Tree for the new block *B**i* |\n| | | | built using new user-IDs |\n| 256 | *n*,*m* | 24 | *n*: total number of users, *m*: how many present |\n| 280 | flags *ϕ* | 8 | bits 0,1: bitmap type (plain, compressed, list, empty) |\n| | | | bit 2: (0: *w* = 4, 1: *w* = 8) |\n| | | | bits 3–7: pre-randomisation parameter (see next section) |\n| 288 | bitmap *M* | *L* ≤ 1024 | processed bitmap content |\n| *L* + 288 | redundancy | 32 | all zeros, for PLS validation |\nThe total message length is *L* + 320 bits or *L* + 40 bytes. We expect *L* to be close to 60 bytes in most cases (which is the entropy limit for 1024 users at 10% occupancy per block on average), which makes the S-message circa 100 bytes long, but if necessary *L* can be increased to 128 bytes resulting in the packet length 168 bytes, still well within the length limit (255 bytes) for LoRa communications. A 128-byte bitmap would support the number of users up to 1024 without Tunstall compression, or about twice as many if Tunstall compression is used at 10% occupancy.\nThe hash *T* requires the server to renumber the users, building a new Merkle tree and computing its root hash. The client will recompute *T* from any leaf hash, adjunct hashes and the path mask – all sent to it by CAS (unauthenticated, unsigned) at request, and then check it against the *T* in the S-record.\nNotice that the redundancy field is only 32 bits, since it is impossible to crack the S-message directly: *both* the plaintext and the key are unknown, the former due to the XORing of the next P-message, yet to be received, to the plain text, see Figure [fig:pls-prot]. As mentioned earlier, the purpose of the redundancy field is to thwart a *random message* attack for the DoS purposes, and so a 32-bit redundancy translates into a less than 1-in-a-billion chance to cause the recipient to accept a false message, which is more than sufficient in the IoT world.\nFinally, let us dwell a little on the block’s MT whose root *T**i* is included in the block’s root-of-trust *J**i*. The leaves of that tree are hashes of the user records with the user ID corresponding to the path label sequence as usual, except the IDs are now new IDs calculated from the block bitmap and occupying a range from 0 to *m* − 1 without gaps. Since *m* is not necessarily a power of 2, the MT generally consists of a complete half with leaf labels in the interval [0, 2⌊log2*m*⌋) without gaps and a truncated half with labels in the interval [2⌊log2*m*⌋, *m*), also without gaps, with the rest of the leaves labelled with NULL. The shape of the MT depends solely on one parameter, *m*, which is part of the root-of-trust. Consequently, no further information, such as path masks, etc, is required for access and validation of the root hash, *T**i*, except, of course, we must use Eqs. [eq:none-case] to calculate the root-adjunct paths in the truncated half. For our running example of 10% occupancy and the total number of users 1024, the value of *m* will have an expectancy of around 102, which means this path will be between 1 and 7, and never longer. Table [tab:MT-stat] indicates that the standard (MPT or mask-controlled MP) would require from 3 to 9 adjunct hashes. The difference between 7 and 9 is not big, but notice that 8 hashes would already require more than one LoRa packet to transmit.\nWe would like to emphasise here that the main effect of using the Tunstall-Merkle Tree (TMT, which is how we wish to call our construction) rather than, say, MPT is **not** that fewer hashes have to be communicated with the former than the latter, but the fact that the latter requires a full path irrespective of the presence or absence of the leaf for secure retrieval. By contrast, a TMT provides an absence proof directly from the root-of-trust bypassing the Merkle Tree entirely. Since the SLVP protocol requires every to check the presence of its own S- and LV- messages before advancing the protocol, and since a would typically monitor another user’s infrequent activity, the cost of absence proof dominates over the cost of secure retrieval. Nevertheless, it is reassuring to see that the latter is also improved, in terms of limits if not necessarily average, by our approach. The price we are paying is some additional calculations well within the capabilities of resource-limited systems such as most tend to be.\nReturning to the compression issue, there is one factor yet to be accounted for. We remarked earlier that our statistical calculations are based on the zero-order assumption, i.e. that different users’ behaviours are uncorrelated. Obviously it is not the case when users engage in a higher-level protocol with one another, e.g. producer/consumer. This may skew the chunk statistics resulting in a longer codeword sequence for the block bitmap. In the next section we will propose a simple remedy.\nPre-randomisation[sec:rand]\n===========================\nThe idea is to apply a bijective function to the source user ID which depends on an extra parameter, block number *i*. A different block number should result in a very different permutation. This way a position in the block bitmap will have the value 1 in a proportion of bitmaps that does not depend on the value of other positions. A user ID will be associated with a pseudorandom sequence of positions as new blocks are produced. Invertibility (bijection) is very important, as it prevents different users from being mapped on the same bit-position in the bitmap, thus ensuring that the mapping a pseudorandom permutation.\nA simple and effective pseudorandom permutation based solely on the block number *i* can be achieved by analogy with randomising the order in a deck of playing cards. One player performs the deck shuffle: for the card in position *i* in the deck *i* ∈ [0, *n* = 2*d*) represented in binary as *i* = *i**d**i**d* − 1…*i*0, the new position of the card *i*′, represented in binary as *i**d*′*i**d* − 1′…*i*0′ is obtained from the current position by applying the following operator \n*i**d*′*i**d* − 1′…*i*0′ = *i**d* − 1*i**d* − 2…*i*0*i**d* = *σ*(*i*) , \nwhich is a cyclic shift left. This corresponds to dividing the deck into two halves and interleaving them, exactly as an experienced dealer would.\nWhen the shuffle is finished, the other player *shifts* the deck, i.e. divides it into two unequal parts and transposes them. In terms of card numbers, this corresponds to adding a pseudorandom value *v* modulo 2*d*: \n$$i^\\prime = i + v \\mod n = \\tau\\_v(i)$$ \n Applying the shuffle-shift operator *Q**v* = *σ* ∘ *τ**v* to a range *ρ* = [0, *n*) *t* times with a pseudorandom choice of *v*: \n*Q**v**t*…*Q**v*1*Q**v*0 *ρ* \n has the same effect as repeatedly shuffling/shifting a deck of cards, which, the intuition suggests, delivers a rather arbitrary permutation. We call *t* the number of rounds.\nNote that the operators *σ* and *τ**v* have a negligible cost even when executed by the least powerful platform, as they take literally a few machine instructions. The cost of pseudorandom generator that produces a series of *v*-values is similarly small if we use a standard Linear Congruential Generator (LCG): \n$$v\\_{k+1} = F\\times v\\_k+1 \\mod n,\\;0\\le k<t\\,,$$ \n where *n* is a power of 2, *v*0 is set to the block number, and the factor *F* is any positive integer that satisfies the well-known Hull-Dobell constraint: $F= 5 \\mod 8 $. We chose for *F* the hex value 5EED which satisfies the constraint and which has more than enough significant digits for any reasonable *n*.\nOur solution appears quite attractive from the point of view of its cost; however, while bijectiveness is guaranteed by construction, we need to be reassured that the solution can deliver sufficient *randomness* of mapping at a reasonably small number of rounds *t*.\n#### Avalanche test.\nHow do we judge the quality of a pseudorandom mapping? A common test is based on the so-called avalanche criterion, used in evaluation of symmetric ciphers and hash functions. We consider it next in relation to our mapping *Q*.\nSelect the block-number *v*0 for the test. Next, select a number *x* from the range *ρ* and some integer 0 ≤ *k* < *d* and prepare two numbers *x*1 = *x* and *x*2 same as *x*1, except bit *k* of it is flipped. Apply the mapping *Q* to both and take the bitwise XOR of the results: *A**i*(*x*, *k*) = *Q*(*x*1) ⊕ *Q*(*x*1). Let *A**i*(*x*, *k*, *l*) be the *l*th bit of *A**i*(*x*, *k*). Define the correlation matrix ${\\bf K}\\_{kl}$ thus:\n \n$${\\bf K}\\_{kl} = \\langle A\\_i(x,k,l) \\rangle\\_{x,i}\\,,\\label{eq:corr}$$ \nHere the averaging is done over values of *x* ∈ *ρ* and block numbers *v*0. Good randomness of the mapping manifests itself in the closeness of all matrix elements of ${\\bf K}\\_{kl}$ to 1/2:\n \n$$\\max\\_{kl} ({\\bf K}\\_{kl}-1/2) \\ll 1/2\\,.\\label{eq:avalanche}$$ \nThis means that if we flip a random bit in a random value *x* the probability that any bit in the image of *x* under *Q* flips in response is close to 1/2. In other words, if we flip one bit in *x*, on average close to one half of the bits in the result will flip. The name “avalanche effect” is to do with the fact that small changes cascade through the rounds of the computation causing further changes until all bits of the result are affected in a complex and unpredictable, though of course deterministic, way.\nWe have applied the avalanche criterion to our proposed randomiser to estimate the acceptable minimum value of *t*, the number of rounds. As it is impossible to average over all potential values of the block number *v*0, we limited ourselves to 50 random samples taken from the interval [0, 10000] The results have proven quite insensitive to the averaging over *v*0, which in not surprising given that we established that the required *t*-numbers are in the hundreds. The averaging over *x* was done by sweeping the whole range *ρ*.\n| *d* | *t* | *δ* | *d* | *t* | *δ* | *d* | *t* | *δ* |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| 10 | 100 | 0.061 | 11 | 100 | 0.113 | 12 | 100 | 0.164 |\n| 10 | 150 | 0.014 | 11 | 150 | 0.026 | 12 | 150 | 0.042 |\n| 10 | 200 | 0.008 | 11 | 200 | 0.009 | 12 | 200 | 0.013 |\nThe results of the avalanche test are presented in Table [tab:avalanche]. For practical purposes we limited ourselves to *n* = 1024, 2048 and 4096, since more users are unlikely to be supported by the communication infrastructure of a single site. The results show that a surprisingly large number, around 200, of rounds is required to achieve good randomisation. It is large compared to the number of rounds one expects to be necessary to randomise a deck of cards (of the order of 10), but it is not large technically: a microprocessor would have to execute only a few thousand instructions to compute the image *i*′ given *i*. This has to be done as many times per block as the number of counterparties that the user has to monitor on the blockchain. For a this would be a number of the order unity, hence the cost would be negligible even on a tight energy budget. On the other hand, setting *t* to 200 would ensure that possible correlations between bits in the image do not exceed 3% (0.013 normalised by 1/2), which should be good enough for practical purposes.\nFinally, let us recall that the IoT platform is low-power, but the server running the PLS protocol via the Sequencer is not. It has ample capacity to analyse the quality of the permutation in terms of its influence on Tunstall compression. We have reserved 5 bits in *ϕ* to pass to the shuffle-shifter an integer value in the interval [0, 32). It is convenient to use the value 0 to indicate that a random permutation is not required[9](#fn9), whereas a nonzero value is added to the round counter *t*. The server can try up to 31 additional rounds and choose the one that gives the best compression. The users receiving the root of trust will be aware of how many additional rounds should be performed and will maintain consistency.\nPutting it all together[sec:together]\n=====================================\nNext we consider a complete example of a user attempting to retrieve a contribution to a block that has been made either by itself or a counterparty.\nFigure [fig:overall-1] presents the flow of data when a block-269 S-message is received and successfully unlocked by a user. The user is about to request the contribution to block 269 from user ID 45. To accomplish this, it needs to decode the received block bitmap by feeding it to the Tunstall decoder together with the parameters *n* and *m* (total number of users and the number of users contributing to block 269, respectively). The decoder produces the uncompressed bitmap. At the same time the user ID (45) and the block number (269) along with the total number of users and configuration parameters are fed to the shuffle-shifter, which will extract the number of additional rounds from *ϕ* and produce its output value, 20. The bit in position 20 of the uncompressed bitmap happens to be 1, which means that the contribution from ID 45 is present in block 269. The number of 1s in the bitmap to the left of position 20 is 4, so the index in the truncated Merkle tree for the contribution in question will be 4. The path to leaf 4 is highlighted in red in the figure.\nThe user’s CAS request will include the block number, 269, and the ID index, 4. CAS will respond with the leaf hash *h*4 and the adjunct sequence \n*V*0 = *h*5, *V*2 = *H*(*H*(*h*0 ∥ *h*1) ∥ *H*(*h*2 ∥ *h*3)) , \n which consists of the labels of the two nodes of the tree marked in blue. Because the user acquired *m* from the unlocked S-message, i.e. the root of trust, it knows the shape of the tree. Consequently, no mask is communicated, but the user is able to reconstruct the mask anyway.\nTo validate the requested *h*4, the user checks that the following equation holds: \n*H*(*V*2 ∥ *H*(*h*4 ∥ *V*0) ∥ *H*(*h*4 ∥ *V*0)′)) = *T* . \nwhere *T* is the root hash received with the unlocked S-message.\nAn alternative scenario is shown in figure [fig:overall-0]. When attempting to retrieve the contribution of user ID=17 to block 305, it turns out that the output of the shuffle-shifter points to a 0 in the uncompressed block bitmap. Since the unlocked S-message is the root of trust, this constitutes a proof that block 305 has no contribution from user 17. Notice that CAS is not involved in the process at all.\nRelated work\n============\nThe PLS blockchain and the protocols in basic form were proposed in. The idea of sparse Merkle Tree has an unclear origin. To the best of our knowledge it was first put forward by Bauer and was recently improved on in. Both studies are concerned with mutable trees, with objectives very different from ours, although, like ourselves, the authors remark on the importance of proofs of absence (non-membership). Tree statistics is tackled theoretically in in the context of optimising mutable MTs for the Bitcoin blockchain in the context of Bitcoin transactions. The objectives of this study are similar to ours as the authors attempt to group the leaves together to minimise the proof length, but they do it using tree transformations (taking the data structure red-black tree as a starting point), while we achieve a similar objective by renumbering the keys (user IDs in our case).\nThe compression technique we use is due to Tunstall and this seems to be uniquely suitable for our case since it is based on empyrical probability of leaf occupancy, which is available to the Fog Server running PLS and which takes next to no resources to communicate to the client. The efficiency of our technique depends on this method.\nWe used our own pseudorandom permutation as a combination of a perfect shuffle and a random shift, using a classical LCG source. There exist various methods of pseudorandom permutation, an oft-cited one being Fisher-Yates shuffle, first published in the 1930s (citation unavailable). The idea there is to choose a (pseudo)random element of a sequence of source items and exchange it with the first element on the sequence. Clearly, if this is repeated enough times then any possible permutation could be achieved[10](#fn10). A recent paper presents a fast, parallel algorithm that mimics the technique of merge-sort except the merge makes a pseudorandom choice when ordering two elements for the output.\nHowever, our situation is quite different. Not because we are dealing with a contiguous range of numbers rather than an abstract sequence of objects: one could enumerate the objects and the problem would boil down to the one we are faced with. Our situation is different because the sender and the recipient must choose the *same* permutation. To encode an arbitrary permutation of *n* numbers would take close to (*m* − 1)log2*m* bits, which is the same order of magnitude as the block bitmap we are trying to make more compact. Of course the ability to perform an arbitrary permutation is not required: all we want is break correlations between user IDs in a series of block bitmaps, and for this any sufficiently rich subgroup would do.\nConclusions\n===========\nStatistical analysis of a sparse Merkel Tree under the assumption of uniform, uncorrelated leaf occupancy has been presented. The model obtained allows direct computation of the Probability Distribution Function for path lengths given the leaf-value probabilities. The path weight was quantified in terms of the number of adjunct hashes required for its leaf proof. We determined that the mean path weight of a sparse MT tree is close to that of a dense, truncated MT tree, with the latter being slightly better at most leaf-probability values *p* in the practically interesting interval. We proposed an alternative structure, a Tunstall-Merkle tree, which combines a dense, truncated MT and a Tunstall-compressed bitmap indicating leaf ocupancy. We tested the compressor at several practical values of code size and quantified its residual redundancy. We found that a very small code table (16 or 256 codewords) proves sufficient for achieving near-limit compression, which means that Tunstall decoding presents no storage problem whatsoever to an IoT platform. To improve the effect of compression we further proposed a decorellation facility in the form of a shuffle-shifting algorithm and tested its properties using the standard avalanche criterion to determine the number of rounds. Both the Tunstall decoder and the shuffle-shifter with the codes size and the number of rounds, respectively, sufficient for our purposes are quite processor-efficient as well, since they involve inexpensive operations (table indexing, cyclic shift and binary addition) and short instruction sequences in implementation.\nThe main effect of the proposed technology is a drastic improvement in the cost of the SLVP protocol. Indeed an SLVP verifier has to check every block for the presence of counterparty (prover) contributions, and no such contribution would be present in a great majority of blocks. Our proposed Tunstall-Merkle tree has zero proof-of-absence cost, and when a leaf is present the communication cost of retrieval is in most cases better than that for the standard MT and MPT. Obviously our technique offers no advantage to a system with an unlimited and dynamic number of users, but it is beneficial for at least the PLS blockchain situation. The statistical analysis of a sparse MT/MPT has significance beyond the area of our study; it could be useful for planning and designing any secure storage structure that involves Merkle trees.\nFuture work will concentrate on higher-level protocols which control the interaction of with a smart contract within the same limited-resource set of assumptions.\n---\n1. Correspondence address: University of Hertfordshire, U.K.; email: <a.shafarenko@herts.ac.uk>[↩](#fnref1)\n2. In the sequel, when it is not important what kind of actor is meant we will call all actors users for short and apply the pronoun ’it’.[↩](#fnref2)\n3. Since the attacker does not know the preimage of *P* at the time when an attack is possible, it can only send an arbitrary message; after unlocking, it would produce a near-random bitstring as a would-be block hash. The requirement for it to have *r* trailing zeros will only be satisfied with the probability 2− *r*[↩](#fnref3)\n4. The original paper has a slightly different arrangement for S-messages since in the original design CAS was trusted for progress, but in the present paper we eliminate this requirement.[↩](#fnref4)\n5. The tree does not have to be binary, but higher-based trees, and higher-based MPTs, discussed later, are inefficient for a small number of leaves.[↩](#fnref5)\n6. For example, left to right, or in prefix order. The properties of the code remain the same under any permutation of the codeword assignments but the practicalities of encoding/decoding require a shared order[↩](#fnref6)\n7. We use the fact that the greatest number has the greatest logarithm[↩](#fnref7)\n8. the main danger would be that the sender and the receiver use different floating-point arithmetic, incur different rounding errors and end up using different dictionaries; this example shows that for a small dictionary there is no such danger[↩](#fnref8)\n9. This could be advantageous when, for example, no compression is used.[↩](#fnref9)\n10. unless the pseudorandom generator producing the selections is caught in a cycle first[↩](#fnref10)\n\nIndexing structures for the PLS blockchain\n==========================================\nDeclarations\n============\nThis paper studies known indexing structures from a new point of view: minimisation of data exchange between an IoT device acting as a blockchain client and the blockchain server running a protocol suite that includes two Guy Fawkes protocols, PLS and SLVP. The PLS blockchain is not a cryptocurrency instrument; it is an immutable ledger offering guaranteed non-repudiation to low-power clients *without* use of public key crypto. The novelty of the situation is in the fact that every PLS client has to obtain a proof of absence in all blocks of the chain to which its counterparty does not contribute, and we show that it is possible without traversing the block’s Merkle tree.\nWe obtain weight statistics of a leaf path on a sparse Merkle tree theoretically, as our ground case. Using the theory we quantify the communication cost of a client interacting with the blockchain. We show that large savings can be achieved by providing a bitmap index of the tree compressed using Tunstall’s method. We further show that even in the case of correlated access, as in two IoT devices posting messages for each other in consecutive blocks, it is possible to prevent compression degradation by re-randomising the IDs using a pseudorandom bijective function. We propose a low-cost function of this kind and evaluate its quality by simulation, using the avalanche criterion.\n**keywords:** PLS blockchain, Guy Fawkes protocol, content-addressable storage, data–structure statistics, Tunstall coding, pseudorandom bijections\nIntroduction[intro]\n===================\nThis paper gives statistical analysis of some known data structures required for the implementation of the PLS (permissioned) blockhain or PLSB for short, whose purpose is to support a *swarm* of IoT devices, or operating on the premises of a single administrative authority, for example a smart hospital. The use of a blockchain is for the purposes of audit trail, authentication and non-repudiation of **all** actors, both human and unmanned, including small, bare-metal microcontrollers that supply critical sensor data and those which drive actuators.\nThe utility of permissioned distributed ledger systems (permissioned blockchains, or PBCs for short) is based on two fundamentals: (i) distributed validity check of messages and (ii) an immutable, linearly-ordered ledger. In IoT applications, especially in sensor-networks, (ii) tends to be more important than (i). Indeed, typically messages are not transactions in the financial sense, so checks such as double spending are not relevant; value checks are domain-specific and are best performed by smart contracts, which leaves the authenticity and provenance of each message posted on the ledger as the only general validity concerns. The PLS blockchain assures (ii) by employing Guy-Fawkes Protocols (GFPs).\nA GFP is a post-quantum signature protocol based on an unlimited series of interlocking cryptographic hashes. GFP computations are fast, messages short and secrets neither moved nor kept for a long time; the GFPs are resistant to quantum computing as they do not use operations such as prime-number factorisation or discrete logarithm. Finally, by their recursive nature, GFPs define a single sequence of signatures that is very hard to split; this makes them quite suitable as a basis of a blockchain.\nIn the next section we will briefly outline the architecture and protocols of the PLS blockchain published in an earlier paper. Operational differences between the PLS and other blockchains, such as Etherium, call for re-evaluation of major data structures required for its implementation. In Section [sec:motivation] we argue that the limited number of users (IoT devices on the premises and human actors[2](#fn2)) and the limit on their communication duty cycle and disposable energy need efficient secure data structures to avoid communicating irrelevant data. We propose a Merkle-Tunstall Tree for that purpose (Section [sec:MTT]) and provide a statistical evaluation of its efficiency. The efficiency of the Tunstall compressor depends on the lack of correlations between different user’s contributing to the same block. To decorrelate block access we propose to use a random permutation function to map users’ true IDs onto local IDs for a given block, see Section [sec:rand]. To illustrate how proposed technologies work together we give one illustrated example in section [sec:together]. Finally, there is a section on related work and some conclusions.\n#### The main contributions of the paper are as follows:\n1. *Statistical analysis of a sparse Merkle tree with uniform, uncorrelated probabilistic leaf occupancy.* We have obtained the path-weight probability distribution function (as a recurrence relation in the tree height) analytically, without Monte-Carlo simulation. It is easy to quantify the function numerically for any given height.\n2. *The proposal and evaluation of a compressed bitmap and local enumeration of block users.* This makes it possible for a user to obtain the proof of absence in the block directly from the broadcast root of trust without accessing the Merkle tree. We have also shown that the local enumeration results in a path weight similar to that on the original sparse Merkle Tree on average, but the variation is tightly bounded from above, which makes it possible to limit the packet length when communicating a secure leaf path using our structure. By contrast, the path across the original tree varies more widely depending on the leaf statistics and may result in paths exceeding the maximum packet size.\n3. *The proposal and evaluation of shift-shuffle* as a low-cost pseudorandom permutation technique sufficient to break a possible correlation between occurrences of different users’ records in block contents. We quantified the number of rounds in the permutation algorithm to be used taking the avalanche criterion as a basis.\nPLS blockchain: architecture and protocols.\n===========================================\nThe details of the protocols and their security analysis are available from. We present them here for completeness. However, for the contributions of the present paper we only need to discuss the logistics of the PLSB, whose architecture is shown in figure [fig:arch].\n#### Blocks\nare formed from transactions communicated by via *proxies* that make it possible for all to rely on low-power radio communication. To authorise a transaction, run another GFP protocol, called SLVP. That protocol’s messages are forwarded by one or more proxies to the Fog Server (FS) to be included in the next block. The FS forms blocks regularly, on a fixed wall-clock schedule, by validating incoming SLVP messages from, and adding them to the current block.\n#### Chain.\nBy regular deadlines the current blocks are stored in CAS and their hash is signed by producing messages of the main protocol, PLS. All PLS messages are generated and transmitted by radio via a sealed unit, Sequencer, which receives the current block’s hash from the FS on a private radio channel. The Sequencer does not contain a changeable program and is not connected to the Internet, so it is not hackable. The PLS sequence, i.e. the sequence of PLS messages, requires a short-term secret, which is produced inside the Sequencer using a physical source of randomness in one time interval and is revealed in the next interval at the same time as selecting a new random secret. All must receive each PLS message, validate it, and unlock the corresponding block’s hash, which is a file name of the block in CAS, see fig [fig:pls-prot]. P- and L- messages cross-validate as shown in the figure, and S-messages contain some redundancy, which, after deciphering, indicates whether the message is valid or not. For example, *J* can include a run of zeros at the end; this would be sufficient to thwart a “random message” attack, which is a possible DoS action of the attacker jamming the radio channel[3](#fn3). Also notice that blocks of the blockchain are, as usual, key-value collections, where the key is the originator’s ID.\nAny invalid messages, possibly sent by an attacker will fail the validity check with a very high probability. Progress is assured by limiting the number of invalid messages using various techniques discussed in, but those are exclusively DoS countermeasures which do not influence the semantics of the blockchain. The initial message *P*0 is authenticated by all blockchain users via external credentials. Users joining the system later would require external authentication of the latest P-message instead of *P*0. The unlocked hashes of all the subsequent blocks are as secure as the weaker of the credential and the computational hardness of the full hash preimage problem (i.e. finding *all* bit-strings of a given length whose hash is a given value). The latter is not feasible for a SHA-2 hash even Post Quantum. Also notice that the verification and unlocking computations are fast (single microseconds) even for a small bare-metal microcontroller equipped with a crypto-accelerator, e.g. ESP32.\n1em -0.25in\n|c|l|l|l| Interval& Transmit/Receive & Verify & Unlock \n[*T*0, *T*1]&*L*0 = *H*(*N*1) ⊕ *N*0 && \n&$S\\_0={\\bf E}\\_{N\\_0} (J\\_0\\oplus H(N\\_{1}))$ & *P*0 out of band & \n&*P*0 = *H*(*N*0) & & \n[*T*1, *T*2]&*L*1 = *H*(*N*2) ⊕ *N*1 & & \n&$S\\_1={\\bf E}\\_{N\\_1} (J\\_1\\oplus H(N\\_{2}))$ & *H*(*L*0 ⊕ *P*1) = *P*0 & $H(B\\_0) = P\\_1 \\oplus {\\bf D}\\_{L\\_0\\oplus P\\_1}(S\\_0)$ \n&*P*1 = *H*(*N*1) & & \n...&...&...&... \n[*T**k*, *T**k* + 1]&*L**k* = *H*(*N**k* + 1) ⊕ *N**k* & & \n&$S\\_k={\\bf E}\\_{N\\_k} (J\\_k\\oplus H(N\\_{k+1}))$ & *H*(*L**k* − 1 ⊕ *P**k*) = *P**k* − 1 &$H(B\\_{k-1}) = P\\_k \\oplus {\\bf D}\\_{L\\_{k-1}\\oplus P\\_k}(S\\_{k-1})$ \n&*P**k* = *H*(*N**k*) & & \n1em Notation: \n=1em *H*( ⋅ ): a cryptographic hash function \n*J**i*: a digest of blockchain block *B**i*, e.g. *H*(*B**i*) \n${\\bf E}\\_\\alpha$: encipherment under symmetric key *α*; use PCBC mode if *J* exceeds cipher block size \n${\\bf D}\\_\\alpha$: decipherment under symmetric key *α*, matching ${\\bf E}\\_\\alpha$ \n#### Transactions.\nAs mentioned earlier, a publishes a transaction on the blockchain by running the SLVP protocol with the FS. A transaction requires one round of the SLVP protocol, which takes three blockchain blocks. For the security of the protocol it is required that the originating check that the latest sent SLVP message has appeared on a block. As soon as it has, the next protocol message can be sent. The first message to send is an S-message, which contains the data object to be signed. Then an LV-message is posted on the blockchain, which provides interlocking hashes and verification data (the latter is needed to thwart jam-spoof attacks, see ). Finally, the posts a (proof) P-message. The FS validates the P-message using the data contained in the previous round’s P-message and the content of the LV-message sent in between. The FS will only include a P-message in a block if the P-message is valid, while LV- and S- are posted right away, the reason being that invalid LV- or S-messages will be recognised as such by the protocol itself only after the next P-message is posted. In practice The FS and a user may share a secret to help the FS to authenticate incoming messages early to make it difficult for an attacker to post a large number of invalid S- and LV-messages. However, this does not help a counterparty that must be mistrustful of the FS. So additional authentication, if present, is purely a DoS countermeasure; we needn’t focus on it as we concern ourselves only with the machinery of the blockchain.\nThe protocol is summarised in figure [fig:slvp-prot]. The diagram shows two users, blue and brown, posting their SLVP messages on the blockchain using the transmissions shown in the table below, which is presented on behalf of a single blockchain user. Users are independent in transmitting protocol messages for themselves and in verifying messages sent by others. The verification formulae in the penultimate column enable users (as well as the FS that does it first) to prove to themselves that the other party has genuinely signed its data object *M*. If the next P-message, *P**k* + 1 checks out, they use its value to unlock the data object *M**k* as defined in the bottom of the figure[4](#fn4). Just like blockchain blocks, the data objects have some redundancy. For example, a certain number of leading zeroes, not necessarily very few as this does not require computation, will adequately defend against random S messages, sent by an attacker. Notice that the SLVP protocol defines *variable length* encryption for S-messages using a block cipher in PCBC mode. Encryption is bijective, i.e. information-preserving, and the redundancy required for validation in the presence of a random-message attack is just a few bytes (e.g. 4 bytes gives the attacker 1 chance in a few billion to post valid random data, but even then it only subverts a single S-message).\nAlso note that the FS has authority to introduce a new user by posting their very first P-message. The first P-message is always marked as such on the blockchain for other users exchanging transactions to recognise it as the originator’s identity.\n-1in\n1em\n|c|l|l|l| **Block& **Transmit & **Verify & **BC Action \n*b*0 &*P*1 = *H*(*N*1) & Out of Band (Enrolment) & Post \n*b*1& $S\\_1={\\bf E}^\\*\\_{N\\_1} (H(N\\_{2}), M\\_1)$ & — & Post \n*b*2& *L**V*1 = *H*(*N*2) ⊕ *N*1 ∣∣ *H*(*H*(*N*2)∣∣*N*1) & — & Post********\n...&...&...&... \n*b**n* &*P**k* = *H*(*N**k*) &*as for *b**n* + 3, assume success*& Post \n*b**n* + 1& $S\\_{k}={\\bf E}^\\*\\_{N\\_{k}}(H(N\\_{k+1}), M\\_{k})$ & — & Post \n*b**n* + 2& *L**V**k* = *H*(*N**k* + 1) ⊕ *N**k* ∣∣ *H*(*H*(*N**k* + 1)∣∣*N**k*) & — & Post \n*b**n* + 3 &*P**k* + 1 = *H*(*N**k* + 1) & for *b**n* < *b* < *b**n* + 3:& \n & &1em find **first** LV such that & \n & &2em *H*(*P**k* + 1∣∣*L* ⊕ *P**k* + 1) = *V* & \n & &**if found in block** *b̂*:& \n & &1em**unless** ∃*b* ∈ (*b**n*, *b̂*), *L*′ ∈ *B**b* : & \n & &2em*H*(*L*′ ⊕ *P**k* + 1) = *P**k*& Post \n & &**else**& Reject \n1em To unlock the data object *M**k* compute $M={\\bf D}^\\*\\_{L\\oplus P\\_{k+1}}(P\\_{k+1},S)$ for every S-message in the interval [*b**n*, *b̂*) and accept the first valid *M*. Here *E**k*\\*(*a*, *b*) is the encryption of *b* under key *k* in PCBC mode with *a* as IV; *D**k*\\*(*a*, *b*) is the matching decryption.\n#### Operations.\nTransactions can be posted by both and human users. Each has one or more *masters*, which are typically users (but could be other ). Not only does each check the posting of each of its SLVP messages on the blockchain, it also monitors the postings of all its masters and any relevant counterparties, and validates their data objects by applying the SLVP protocol. Alternatively, the can participate in a smart contract which would only require it to follow and validate messages from the contract engine. In this paper we limit ourselves to the mechanics of transaction processing, while leaving higher-level protocol to further work. We will assume in the sequel that each is interacting with a very small number of other actors and needs to follow a few SLVP threads (perhaps 2 or 3). Our focus will be on how to make these interactions as computationally and communicationally efficient as possible.\n#### Addresses.\nEach PLS user has an address, which is a small number. Since we concern ourselves with a localised enterprise solution (e.g. a smart hospital) covered by a direct link radio network (e.g. smart sensors equipped with a LoRa transceiver), we do not expect the number of greater than circa 1000. The total number of actors should be a small factor of that to account for human users and smart contracts, so 2–4K addresses is our target. Transactions have an originating address and a destination.\n#### Frequency.\nIn IoT networks of interest, communication is limited by the duty cycle to save the limited bandwidth that all have to share. This is in addition to the constraints imposed by the energy budget of an individual IoT device. Consequently a small fraction (typically a few percent) of the registered users will be posting a transaction in any given block.\nBlock structure and optimisation challenge\n==========================================\n#### Immutable dictionary.\nIn the previous section blockchain blocks were shown in the diagrams that consisted of records attributed to various users as key-value pairs, where the key is the user ID. In a given block only some of the registered users would be represented by records. Since the frequency of posting on the blockchain in our case is severely limited by the ’ communication duty cycle (if using LoRa) or energy budget (if using public networks or LoRa), the proportion of users posting to any given block is expected to be very small. Still, the user can only authenticate the block by the S-message of the PLS protocol, which, when unlocked, contains the block hash. On the other hand, as we mentioned in the previous section the user is typically interested in two or three other users’ contributions, which, given a typical enterprise IoT network of a thousand, is still much less than the expected volume of activity in a block. Indeed, given a block production rate of 4 blocks per hour and a data production rate of 2–4 samples per day, and bearing in mind that each data post requires 3 blocks according to the SLVP protocol, we arrive at 6 to 12 blocks per device per day and circa 100 blocks per day in total. This means that in the absence of correlated activities we should expect about 5 to 10 percent of the swarm to post in every given block. For a 1000-strong swarm, the block may contain an estimated 50 to 100 user records authenticated by a single hash. If a wishes to access just a few of these, it would have to first read the whole block and check the hash to validate it, and then dispose of most of these records as they would be irrelevant.\n#### Merkle tree.\nThe standard solution to the above problem is called the *Merkle Tree*(MT), see figure [fig:MT]. It is a labelled binary[5](#fn5) tree each node *ν* of which has two children, with some labels *ν**L* and *ν**R*, and its own label is *ν* = *H*(*ν**L*∣∣*ν**R*). A child can either be a leaf or a full node in its own right; in both cases it has a label but in the latter case it also has two children of its own. It is quite clear that a change in any leaf will change the root label (also known as the *root hash*), so the authenticated root makes the whole tree authentic. For each node except the root there exists one other node with the same parent, which we call *adjunct*. What makes the MT useful is that it can also authenticate a single leaf by providing a root-path list of adjunct nodes’ labels, or root-adjunct path for short. For example, to authenticate the leaf *ν*101 shown in red in the figure, given the root label *r*, we only need to know the labels of the blue (adjunct) nodes: *ν*100, *ν*11 and *ν*0, since \n*r* = *H*(*ν*0∣∣*H*(*H*(*ν*100∣∣*ν*101)∣∣*ν*11)) . \n Generally speaking, for a tree with *K* leaves one needs to communicate *h* = ⌈log2*K*⌉ hashes, which is much less than *K* for the number of leaves in the hundreds that we are considering. The tree thus represents an *array* of leaves indexed by the path: a left edge represents 0 and a right edge 1; the edges traversed en route to the leaf form a bit-string that represents the key. The leaf itself represents the value of the key-value pair.\n#### Blocks represented as Merkle trees.\nCommon practice in Blockchain construction is to represent a block as an MT, each leaf of which carries the hash of a user’s record included in the block, with the user ID being the key gleaned from the leaf’s root path. A user requesting another user’s record (or the one of its own) from an intelligent CAS could just receive the root-adjunct path corresponding to the requested ID and hash it through to match with the root hash value. If the PBC signs the root hash of every block it creates, no further security is required to authenticate any user records. Our investigation is of a special case when the maximum number of users is small and is known in advance, and where good communication efficiency is important. We could use an MT with the tree height *h* close to 10 (to accommodate our expected 210 ∼ 1000 users). Since, as we have mentioned earlier, we expect only around 50 (maybe up to 100) users to contribute to any given block, a great majority of the leaves will not be used.\n#### Mask-controllable sparse MT.\nThe number of leaves in an MT does not have to be a power of 2. Also leaves can have no value associated the root-path key. We can think of such leaves as unoccupied. An MT with no-value leaves is called a *sparse* MT, or SMT. There are several ways of organising an SMT, but proposals usually focus on *mutable* trees that are used for secure updatable key-value storage. Our interest is in *immutable* SMTs, where efficiency is understood in narrow terms as efficiency of retrieval *only*. Below we define our own version of the SMT, geared towards our objectives.\nWe can assume that a leaf without value has a special label NULL and the parent of two NULL nodes has the label NULL as well. This assumption does not diminish security due to the fact that a NULL child of any node is implicitly associated with the node height. Consequently, the shape of the NULL subtree associated with the child is completely defined by its root position. All such NULL trees are identical anyway, so a single label value fully represents them.\nFor the verifier to be able to verify a path with NULL nodes, it requires a bit mask of length *h*, where bit-value 1 indicates that the corresponding adjunct node is non-NULL; and the bit-value 0, that it is NULL. The NULL labels can then be omitted from the path. Finally we extend the domain of *H*(*x*) to include NULL-concatenated strings by defining that for any bit-string *x*\n \n$$\\begin{gathered}\nH(x\\parallel{\\rm NULL}) = H(x\\parallel x^\\prime)\\,,\\label{eq:none-case}\\\\\nH({\\rm NULL}\\parallel x) = H(x^\\prime\\parallel x)\\,,\\nonumber\\end{gathered}$$ \nwhere *x*′ is the bit string obtained form *x* by flipping all bits. Interestingly, a simpler extension \n$$H(x\\parallel{\\rm NULL}) = H({\\rm NULL}\\parallel x) = H(x)$$ \n would not be secure, as it allows one to construct a second preimage by rotating the subtree or swapping nodes along a NULL path. It is easy to see that the hashing process introduced by Eq [eq:none-case] is not invariant to any such transformation. It is impossible to create a new valid SMT with the same root hash and a different leaf sequence without solving the second preimage problem.\nIf the path is mask-controlled, CAS only needs to communicate **up to *h*** hashes in addition to the bit mask for the verifier to successfully compute the root hash. Extending the above example, if *ν*11 were unoccupied, CAS could supply the bit mask 101 (the second adjunct is missing in path order), and the values of *ν*0 and *ν*100. Notice that the bit mask does not need to be secured: if it is incorrect, the verifier will compute an incorrect path expression and the result will not match the root hash. Also notice that the mask is very small compared to the hash length: for a tree of 1024 leaves (counting both NULL and non-NULL) the root-adjunct path contains from 1 to 10 hashes 256 bits each, i.e. 256 to 2560 bis, but the mask length is only 10 bits regardless.\n#### Merkle-Patricia Trie.\nThe idea of mask-control path is similar to that of the so called *Merkle-Patricia Trie*(MPT) where not only the nodes but also the edges can be labelled. If a node has a single active edge (i.e the other edge leads to a NULL subtree), the node is eliminated and its parent uses the prefix of the other edge as its label, see figure [fig:MPT]. The example in the figure is of a block where, out of the maximum 8, only users 2, 4, 5, and 7 (010, 100, 101, 111 in binary) are present. Notice that we still have a binary tree, but the root-adjunct path augmented with edge labels requires from one (for 010) to three (for 100 and 101) adjunct hashes for validation, depending on the leaf. The edge labels are typically much shorter bit-strings than a single cryptographic hash (log2*K* ≪ 256) and so can be neglected in determining the communication efficiency of the access scheme. The same is true of masks with our version of the SMT.\nHow is the edge label secured? It is simply hashed together with its child content in determining the node label: \n*ν* = *H*(*λ*0∣∣*λ*1∣∣*ν*0∣∣*ν*1) , \nwhere *λ*0, 1 are the edge labels of the left and right child, respectively.\nIt is easy to see that there is a direct correspondence between the MPT and the SMT with mask-controlled paths. Our construction requires more work when validating a path: each node, irrespective of its path quality involves *h* hash calculations for verification, where *h* is the height of the tree, but in the MPT case the number of times a hash is calculated is the same as the number of adjunct hashes supplied with the MPT path, although each hash calculation also involves edge labels, which may increase the cost. The total *length* of edge labels along the root path in the MPT case is equal to the length of the mask in the SMT case. However, an MPT path requires markers to partition the path string into individual edge labels. Our construction is slightly more frugal in this respect, and it is simpler, which is why we prefer it.\nMotivation and optimisation idea[sec:motivation]\n================================================\nIt is obvious from the SLVP protocol that an actor engaging in transactions with another on the PLSB must check each block to determine the presence of a transaction message from the counterparty. Due to the low duty-cycle of -to-FS communication, the counterparty will not be present in a great majority of blocks. However, to securely establish the absence the actor must traverse the block and verify that the counterparty’s record is not there. In the MT case we can use the mask-controlled path to the unoccupied leaf which can contain up to *h* adjunct hashes. In the MPT case CAS will supply the longest path *in the direction of*, rather than to, the unoccupied leaf. By examining the last node on that path the user will be able to verify that the necessary edge is missing. For example, looking at the MPT in figure [fig:MPT], if an actor requested the unoccupied leaf 110, the one-step path *r* → *ν*1 with adjunct material will be sent back for validation:\n \n$$\\begin{gathered}\n010,1,\\nu\\_{010}\\nonumber\\\\\n0,11,\\nu\\_{10},\\nu\\_{111}\\nonumber\\end{gathered}$$ \nThe number of hashes to be communicated is the same as that for the mask-controlled MT: in the current example the latter would require hashes *ν*111, *ν*10, and *ν*0. An advantage of the MPT is that it saves the verifier extra hash computations by providing segments of the path as (hashed-controlled) edge labels. While saving *some* compute time, the effect of it is negligible, since the maximum root-adjunct path length is logarithmic in the number of leaves ($\\lesssim12$), and a modern microcontroller can compute tens of thousands of hashes per second. A disadvantage of the MPT is that it requires communication of edge labels in addition to the hash for each node on the path, but again, compared to the hash length this is negligible, too.\nWhat is considerably more important here is that neither mask-controlled MT, nor MPT reduce the maximum root-adjunct path length. In our example the number of leaves present is 4, but the hight of the MPT as a binary tree is 3, not log24 = 2; it is as if all leaves were present. As a result, the system would require to accommodate longer communication packets, which may affect the guaranteed duty cycle limit of an IoT device.\nThis brings us to the **central idea** of the paper: to index a block, one might prefer to locally renumber the users to achieve a contiguous range of IDs rather than a scattering over a regular structure with subsequent remedies such as the MPT.\nHowever, before proceeding to our solution, we would like to evaluate the base case, the mask-controlled MT. We would like to establish some quantitative characteristics of MT paths under a random distribution of leaf occupancy.\nSparse MT Statistics[sec:stats]\n-------------------------------\nLet us number the levels of the MT from the leaves up, starting with 0. We will call the number of adjunct hashes associated with a path its *weight*. Let function $\\PDF\\_k(i)$ of integer *i* be the probability for a path from a given leaf to a hight-*k* node on the tree to be of weight *i*. Recall that we only count nodes on the path with both children being non-NULL as those require an adjunct hash for a Merkle proof. Let us take a closer look at an example path, see figure [fig:path]. From level 0 we move to level 1 as a left child, then to level 2 as the left child, and finally to the level 3 as the right child. Clearly different leaves’ paths differ by the choice of left- and right- ascension at each level, but the significance of the node does not depend on it: the node is only counted when *both* its children are non-NULL. The shaded triangles signify the subtrees that represent the other, *non-path* child of the corresponding path node, which, if non–NULL, produces what we termed above the adjunct hash. If the non-path node is NULL, this fact is noted in the path bit mask, but no adjunct hash is produced.\nAt level 0, the subtree is of height 0 (it is a leaf), at level 1 it is of height 1 (connects two leaves), etc.\n#### Statistical model.[pg:stats]\nWe adopt a model relevant to the IoT case that the PLS blockchain was developed for. A submits a message for inclusion in a block very infrequently. It does so at random with some small probability *p*, whose value depends on the duty cycle restriction, urgency of the sensor data and the available energy budget. Without loss of generality let us assume that *p* ∼ 0.1 in our examples, which will give us some intuition of what kind of figures may arise in practical work. This level of activity means that a participates in roughly one block out of 10 or that about 1/10 of all blockchain users are active in any given PLS round. We also assume that the activities of different are uncorrelated, so any given leaf is either present or absent (NULL) irrespective of the presence/absence of other leaves.\n#### Path weight.\nThe subtrees in figure [fig:path] will consequently be NULL-valued with the probability\n \n*α**k* = (1 − *p*)2*k* , \nwhere *k* is the level at which the subtree is rooted. Let us introduce the Probability Distribution Function $\\PDF\\_k(L)$ as the probability for the weight of a path from level *k* to a leaf to be equal to *L*. Clearly $\\PDF\\_k(L>k)=0$, and we also assume for convenience that for all *k*, $\\PDF\\_k(L<0)=0$. It is easy to calculate $\\PDF\\_1$ directly:\n \n$$\\PDF\\_1(0) = 1-p,\\; \\PDF\\_1(1) = p\\,.\\label{eq:rec0}$$ \nIndeed, the other child of a given leaf of a height-1 tree is NULL with the probability 1 − *p*, producing no adjunct hash, so *L* = 0 with that probability; otherwise (with the probability *p*) the other leaf is non-NULL, supplying a single adjunct hash.\nFor a hight-*k* path we have a combinatorial problem of calculating the probabilities of 2*k* combinations of absence/presence of each adjunct hash (remember that these probabilities are completely independent as per our chosen statistical model). Instead of doing this, we observe the following recurrence relation between the paths to neighbouring levels:\n \n$$\\PDF\\_{k+1}(i)=\\alpha\\_k\\PDF\\_k(i)+(1-\\alpha\\_k)\\PDF\\_k(i-1)\\label{eq:rec}$$ \nIndeed, if the non-path child of the height-(*k* + 1) path node is NULL (this happens with the probability *α**k*) the number of adjunct hashes that the path to height *k* + 1 produces is the same as that to height *k*. Alternatively, if the non-path child is non-NULL, it produces one adjunct hash, and so the probability to produce L hashes for the whole path is the same as the probability to produce *L* − 1 for the path to height *k*. The above equation is the weighted sum of those two outcomes, a mixture distribution.\nThe significance of Eqs. [eq:alpha]–[eq:rec] is in the fact that they permit direct calculation of the PDF at any level above 1 very cheaply given the value of *p*. The PDF obtained can deliver various practical parameters: the average path weight: \n$$\\bar{L}\\_k = \\sum\\_i i\\times\\PDF\\_k(i)\\,,$$ \n the standard deviation, the probability that a certain limit $L\\_{\\rm max}$ is exceeded, etc., which are useful in designing bandwidth-limited communication protocols.\nc\n```\n k mean 0 1 2 3 4 5 6 7 8 9 10 \n==============================================================================\n 2 0.200 81.0 18.0 1.0\n 3 0.390 65.6 30.0 4.2 0.2\n 4 0.734 43.0 42.2 13.1 1.6 0.1\n 5 1.303 18.5 42.7 29.7 8.1 0.9 0.0\n 6 2.118 3.4 23.0 40.3 25.7 6.8 0.8 0.0\n 7 3.084 0.1 4.1 23.6 39.8 25.0 6.6 0.7 0.0\n 8 4.083 0.0 0.1 4.1 23.6 39.8 25.0 6.6 0.7 0.0\n 9 5.083 0.0 0.0 0.1 4.1 23.6 39.8 25.0 6.6 0.7 0.0\n10 6.083 0.0 0.0 0.0 0.1 4.1 23.6 39.8 25.0 6.6 0.7 0.0\n```\nTable [tab:MT-stat] presents the outcome of a direct calculation of Eqs. [eq:alpha]–[eq:rec] for *p* = 0.1 and also includes the value of *L̄**k* in the second column (heading “mean”). The table shows the value (%) of $\\PDF\\_k(i)$, where *i* runs horizontally. For obvious reasons nontrivial evolution only happens until *α**k* drops to small values, whereupon Eq. [eq:rec] degenerates to \n$$\\PDF\\_{k+1}(i)\\approx\\PDF\\_k(i-1)\\,$$ \n making the $\\PDF(i)$ shift to the right by 1 without change of shape as *k* increases. For *p* = 0.1 sparsity is present in the first 7 levels of the tree; from level 8 up the tree becomes dense.\nAnother noteworthy feature of the distribution is its breadth: 95% of the paths require from 4 to 8 hashes, with the mean being around 6, which would necessitate variable length communication, since a factor of 2 difference cannot be ignored. This variability comes despite the compression we have already applied by introducing the bitmap-controlled MT.\nThe sparse MT is indexed by the user ID, and a set of active users for an individual block is random as defined by our statistical model. To get a feel of how efficient the sparse MT is in terms of path weights, we compare its mean path weight with that of a truncated dense tree carrying the same number of non-NULL leaves. We use the least sufficient hight of the dense tree to accommodate all non-NULL leaves, and place all NULLs on the right hand side of level 0, so that the non-NULL leaves may be contiguous, and use Eqs: [eq:none-case] to deal with none values (that is what we mean by truncation). Figure [fig:path-stats] compares the path weights of the two trees. The dependence of the path-weight averaged across the truncated tree on the number of non-NULL leaves is not smooth, as the tree hight leaps up when the number of non-NULLs crosses power-of-two boundaries. Nevertheless, one can clearly see that the dense tree has a lighter path weight at the majority of probability values, becoming slightly worse just before the probability reaches a value that the equivalent dense tree must grow at.\n#### Challenge.\nNow we are prepared to argue our case. A user engaging in the SLVP protocol as verifier must examine every consecutive block to see if the prover has placed a message in it (i.e. an S-, LV- or P-message). According to our statistical model, in a great majority of the blocs, in fact, in a factor of 1 − *p* of them, the prover’s message is likely to be absent. Nevertheless, the verifier needs to satisfy itself that indeed, no message from the prover is present. With the classic MT as well as MPT and our own version of sparse MT, the absence of a leaf is almost as expensive to prove as its presence with a particular value. The difference is that for an absent leaf the label is NULL and it is not communicated, but that is a difference of 1 against, as our calculation shows, circa 6 adjunct hashes to be communicated when 1024 users participate at probability *p* = 0.1. This means that on average 60 (!) hashes would be required to certify the start of an SLVP round. Worse still, each active user, even when all it does is wait for a possible signed message from a counterparty, will be actively requesting the counterparty’s root-adjunct path from CAS every time a block is released, which pretty much destroys the advantages of a low-bandwidth Guy Fawkes protocol.\nHowever, a simple remedy exists, which we consider next.\nTunstall-Merkel Tree[sec:MTT]\n=============================\n#### Basic idea.\nWe kill two birds with one stone by providing a one-time renumbering of users in each block while broadcasting the renumbering information together with the root of the tree. The purpose of the renumbering is to achieve a contiguous range of ID numbers. This way absent users will be recognised as such immediately by any counterparty involved. As a result the cost of absence proof will be zero (plus the cost of the one-for-all broadcast message, which need not be requested). The indexing structure in terms of new IDs will be the kind of tree we have already studied and shown the superior access cost of: a dense, truncated one.\nHow do we enumerate users that are present? Imagine a bitmap sized 2*h*, where *h* is the height of the original (sparse) MT. In the bitmap 1s mark the presence of the corresponding user/leaf and 0s its absence. Under our statistical model (see page ) on average 2*k**p* bits (as per binomial distribution) of the bitmap will be 1s. Users are renumbered according to the bitmap: the user’s new ID is the number of 1s in the bitmap preceding the bit that corresponds to the user’s actual ID.\nOur statistical model assumes that all users are engaged the whole time. A user that decides not to use the blockchain for a while will not be able to maintain a factor of *p* messages per block on average until the user becomes active again; the user’s bit position in the bitmap will be 0 during that period. If there are many such users, the bitmap may have significantly fewer 1s than the aforementioned expectation 2*k**p*. In this sense the expectation is pessimistic.\nThe maximum number of users is within a near-unity factor from the number of in the swarm, since non-IoT users have typically a one-to-many relation with : a human or a server would be in control of several IoT devices. The majority of the users tend to be always-on, active, which work according to a near-periodic schedule. Another useful circumstance here is that the bitmap can be effectively and efficiently compressed to a fraction of its length, provided that the distribution of 1s is close to random and that the number of 1s is known to both the sender and the recipient. The former can be made true by pseudorandomisation, and the later is easy to achieve by including a small integer (typically 10-12 bits in length) in the message that broadcasts the bitmap. In this section we describe the compression technique, and in the next one we will propose a simple and efficient pseudorandomisation.\n#### Tunstall code.\nGiven a bit string of length *n* which is expected to contain *m* = *p**n*, *p* < 1 ones in random positions (which makes *p* a true probability), or alternatively a bit string which is known to contain *m* ones, *m* = *p**n* in random positions (which makes *p* an *empirical* probability), with the positions of ones pairwise uncorrelated (this is called a *zero-order* environment), we set ourselves the task of finding a bijective function *C* : *B**n* → *B**r**w*, *r**w* < *n* that maps the string to a sequence of *r* codewords of length *w*. We wish to minimise *r**w*, or, for a given *w*, to minimise *r*. The theoretical limit of compression is well known from information theory: *r**w* ≥ *H*0*n*, where the zero-order per-bit entropy *H*0 is defined thus: \n*H*0 = − *p*log2*p* − (1 − *p*)log2(1 − *p*) . \n The mapping *C* is realised by partitioning the source bit string into (generally unequal length) chunks and assigning a codeword to each. A chunk *b*0, *b*1, …, *b**k* is found at any given position in our random string with the likelihood\n \n*p**c* = ∏*i* = 0*k**p**b**i*(1 − *p*)1 − *b**i* . \nThe best code with the word length *w* should assign its 2*w* codewords to the 2*w* chunks with the highest likelihood. It must also make sure that the code is complete, i.e. any bit sequence can be represented as a sequence of codewords. It is intuitive that such a code would be optimal, and it can be proven that it is also asymptotically effective, i.e. that its compression ratio tends to the entropy limit as *w* tends to infinity.\nIt is not easy, however, to turn Eq. [eq:lh] into a practical encoder/decoder. The main reason for it is that the value of *k* is not bounded, and neither is the search for suitable chunks to find the top 2*w* ones in terms of their likelihood. The problem is not so much the amount of work required for the search, since we could take the logarithm of Eq [eq:lh] and maximise the linear form\n \nlog*p**c* = *l*log*p* + (*k* − *l*)log(1 − *p*) , \nwhere \n*l* = ∑*i* = 0*k**b**i* , \n in the (0 ≤ *l* ≤ *k*, *k* > 0) area of the (*l*, *k*)-plane starting at the maximum (0, 1). The problem is that each (*l*, *k*) point corresponds to ${k \\choose l}$ chunks of length *k* with *l* 1s in each. Their enumeration and mapping at different *k* would be rather awkward.\nTunstall in his PhD thesis proposed a greedy search which at the same time builds a compact dictionary structure (the Tunstall tree) that can be used for encoding/decoding efficiently, *without* sharing the dictionary (as long as *p* is known to both the encoder and the decoder). The greedy search turns out to be of excellent quality, too, delivering the entropy limit asymptotically, and, as a recent study shows, with a rapidly decreasing redundancy as *w* increases. The redundancy formula from being useful for our analysis, we present it below (without derivation and rewritten in our notation):\n \n$$rw \\le nH\\_0 + O(nH\\_0{\\log(1/p)\\over w})\\,.\\label{eq:redundancy}$$ \nWe will return to Eq [eq:redundancy] later and present our own measurements for the relevant range of parameters, but let us first introduce the dictionary idea, see figure [fig:t-tree]. The dictionary is a (generally imbalanced) labelled binary tree. Both the nodes and the edges are labelled. The edge labels are 0 and 1 as usual, and a node’s label is the likelihood value of the chunk composed by reading the edge labels along the path from the root to the node. The algorithm builds the tree node-by-node, as follows:\n1. Create a root node with two edges labelled 1 and 0 to two child nodes labelled with the value of *p* and 1 − *p*, respectively.\n2. Find and mark the maximum likelihood leaf node. Denote its label as *R*.\n3. Create two leaf children of the marked node and connect them with edges labelled 1 and 0. Make the node labels the value of *R**p* and *R*(1 − *p*), respectively.\n4. Repeat steps 2 and 3 until the tree has 2*w* nodes besides the root.\n5. Now relabel each leaf by its consecutive leaf number while visiting the leaves in some order agreed between the encoder and decoder[6](#fn6).\nTunstall encoding is achieved by running the source bit-string down the Tunstall tree bit by bit until a leaf is reached, at which point the codeword is read off from the leaf label and the process returns to the root. Tunstall decoding requires a 2*w*-entry table where variable-length path sequences are set against codewords, with the former read off from the path to the leaf labelled by the latter. We notice with satisfaction that Tunstall decoding has the cost *O*(1).\n#### Implementation.\nTunstall encoding (and especially decoding) is very undemanding, well within reach of a small, system-on-chip smart sensor. To avoid accuracy/underflow problems with repeated multiplication in generating the dictionary at the receiver (Step 3 of the algorithm above), one could use log-likelihoods as node labels. Then instead of multiplication, log*p* and log(1 − *p*) are *added* to the parent label to produce labels for the 1- and 0-child, respectively[7](#fn7). This way for any reasonable table size, computational accuracy will not be a problem. We implemented the algorithm to see what kind of residual redundancy we could be getting from a specific Tunstall code. The results of our running a Tunstall compressor through 1 million random bits are presented in table [tab:tun-sim]. Comparing this with Eq [eq:redundancy], we conclude that at *p* = 0.05 the compressor already reaches asymptotic mode when doubling the codeword length roughly halves the redundancy *ρ*. At the same time, the dependency *ρ*(*p*) in the interesting range of *p*, i.e. in the area around *p* = 0.1 is not quite asymptotic: the contrast in redundancy between *p* = 0.05 and *p* = 0.15 is nowhere near a factor of 2 that the formula suggests.\nFrom the practical point of view, if we target a blockchain with ∼ 1*K* users, with 10% of them posting messages in any given block, *p* = 0.1 suggests a compressed bitmap of at least 1024 × 0.47 = 482 bits or 61 bytes. An 8% residual redundancy would increase this by only 5 bytes. However, the bitmap is broadcast together with the root hash, 32 bytes long, and a few extra bytes of forced redundancy for the purposes of S-message verification (as per PLS protocol). This increases the length of the S-message up to nearly 100 bytes, and at this level a redundancy of the compressor to the tune of 5 to 10 bytes makes little difference.\n| *w* | *p* | *κ* | *H*0 | *ρ*(%) |\n| --- | --- | --- | --- | --- |\n| 4 | 0.05 | 0.37 | 0.29 | 30.4 |\n| 4 | 0.10 | 0.50 | 0.47 | 7.5 |\n| 4 | 0.15 | 0.65 | 0.61 | 6.9 |\n| 8 | 0.05 | 0.32 | 0.29 | 13.4 |\n| 8 | 0.10 | 0.49 | 0.47 | 4.6 |\n| 8 | 0.15 | 0.63 | 0.61 | 3.8 |\nIf the number of users drops to 0.05, even the poor compression quality for *w* = 4 results in only 379 bits (though 82 bits, or 11 bytes, more than the entropy limit), which is still less than the already acceptable 482 bits we observed for *p* = 0.1. An alternative is to use a list of raw ID numbers, circa 52 in total, each requiring 10 bits. This is 520 bits, far worse than the compressor’s output, but not significantly worse than 482, and the list length would decrease in proportion to *p*. If *p* were to drop further below 0.05, and if the Tunstall compressor further deteriorated, the uncompressed ‘list’ option could at some point be preferred, with the switch controlled by a single additional bit in the message.\n| codeword | − log2*p**c* | chunk |\n| --- | --- | --- |\n| 0000 | 3.06 | 0000000000000 |\n| 0001 | 5.55 | 0000000000001 |\n| 0010 | 5.32 | 000000000001 |\n| 0011 | 5.08 | 00000000001 |\n| 0100 | 4.85 | 0000000001 |\n| 0101 | 4.61 | 000000001 |\n| 0110 | 4.38 | 00000001 |\n| 0111 | 4.14 | 0000001 |\n| 1000 | 3.91 | 000001 |\n| 1001 | 3.67 | 00001 |\n| 1010 | 3.44 | 0001 |\n| 1011 | 3.20 | 001 |\n| 1100 | 3.20 | 010 |\n| 1101 | 5.70 | 011 |\n| 1110 | 2.97 | 10 |\n| 1111 | 5.47 | 11 |\nWe conclude that a four-bit Tunstall code is all that is required to implement the PLS S-message within half of the maximum LoRa message length (250 bytes). To aid the reader’s intuition, we present an example of a 4-bit Tunstall code for *p* = 0.15 in table [tab:tun-code]. For each codeword we additionally show its log-likelihood. Notice that unless the log-likelihoods are exactly identical, as is the case for codewords 1010 and 1100 which correspond to chunks with the same number of 1s and 0s, the differences between log-likelihoods manifest themselves in the first (decimal) fractional digit already, so computational accuracy should not be a concern[8](#fn8).\n#### New structure of the root of trust.\nIn the original PLS protocol the S-record was a message that contains the block’s root of trust *J**i*, which was the root hash of the Merkle tree representing the new block *B**i*. In the light of our analysis of indexing costs presented in Section [sec:stats] and the properties of Tunstall encoding described in the current section, we propose to modify the root of trust *J**i* as shown in Table [tab:root].\n| | | | |\n| --- | --- | --- | --- |\n| offset | field | size | description |\n| (bits) | | (bits) | |\n| 0 | *T**i* | 256 | root hash of the Merkle Tree for the new block *B**i* |\n| | | | built using new user-IDs |\n| 256 | *n*,*m* | 24 | *n*: total number of users, *m*: how many present |\n| 280 | flags *ϕ* | 8 | bits 0,1: bitmap type (plain, compressed, list, empty) |\n| | | | bit 2: (0: *w* = 4, 1: *w* = 8) |\n| | | | bits 3–7: pre-randomisation parameter (see next section) |\n| 288 | bitmap *M* | *L* ≤ 1024 | processed bitmap content |\n| *L* + 288 | redundancy | 32 | all zeros, for PLS validation |\nThe total message length is *L* + 320 bits or *L* + 40 bytes. We expect *L* to be close to 60 bytes in most cases (which is the entropy limit for 1024 users at 10% occupancy per block on average), which makes the S-message circa 100 bytes long, but if necessary *L* can be increased to 128 bytes resulting in the packet length 168 bytes, still well within the length limit (255 bytes) for LoRa communications. A 128-byte bitmap would support the number of users up to 1024 without Tunstall compression, or about twice as many if Tunstall compression is used at 10% occupancy.\nThe hash *T* requires the server to renumber the users, building a new Merkle tree and computing its root hash. The client will recompute *T* from any leaf hash, adjunct hashes and the path mask – all sent to it by CAS (unauthenticated, unsigned) at request, and then check it against the *T* in the S-record.\nNotice that the redundancy field is only 32 bits, since it is impossible to crack the S-message directly: *both* the plaintext and the key are unknown, the former due to the XORing of the next P-message, yet to be received, to the plain text, see Figure [fig:pls-prot]. As mentioned earlier, the purpose of the redundancy field is to thwart a *random message* attack for the DoS purposes, and so a 32-bit redundancy translates into a less than 1-in-a-billion chance to cause the recipient to accept a false message, which is more than sufficient in the IoT world.\nFinally, let us dwell a little on the block’s MT whose root *T**i* is included in the block’s root-of-trust *J**i*. The leaves of that tree are hashes of the user records with the user ID corresponding to the path label sequence as usual, except the IDs are now new IDs calculated from the block bitmap and occupying a range from 0 to *m* − 1 without gaps. Since *m* is not necessarily a power of 2, the MT generally consists of a complete half with leaf labels in the interval [0, 2⌊log2*m*⌋) without gaps and a truncated half with labels in the interval [2⌊log2*m*⌋, *m*), also without gaps, with the rest of the leaves labelled with NULL. The shape of the MT depends solely on one parameter, *m*, which is part of the root-of-trust. Consequently, no further information, such as path masks, etc, is required for access and validation of the root hash, *T**i*, except, of course, we must use Eqs. [eq:none-case] to calculate the root-adjunct paths in the truncated half. For our running example of 10% occupancy and the total number of users 1024, the value of *m* will have an expectancy of around 102, which means this path will be between 1 and 7, and never longer. Table [tab:MT-stat] indicates that the standard (MPT or mask-controlled MP) would require from 3 to 9 adjunct hashes. The difference between 7 and 9 is not big, but notice that 8 hashes would already require more than one LoRa packet to transmit.\nWe would like to emphasise here that the main effect of using the Tunstall-Merkle Tree (TMT, which is how we wish to call our construction) rather than, say, MPT is **not** that fewer hashes have to be communicated with the former than the latter, but the fact that the latter requires a full path irrespective of the presence or absence of the leaf for secure retrieval. By contrast, a TMT provides an absence proof directly from the root-of-trust bypassing the Merkle Tree entirely. Since the SLVP protocol requires every to check the presence of its own S- and LV- messages before advancing the protocol, and since a would typically monitor another user’s infrequent activity, the cost of absence proof dominates over the cost of secure retrieval. Nevertheless, it is reassuring to see that the latter is also improved, in terms of limits if not necessarily average, by our approach. The price we are paying is some additional calculations well within the capabilities of resource-limited systems such as most tend to be.\nReturning to the compression issue, there is one factor yet to be accounted for. We remarked earlier that our statistical calculations are based on the zero-order assumption, i.e. that different users’ behaviours are uncorrelated. Obviously it is not the case when users engage in a higher-level protocol with one another, e.g. producer/consumer. This may skew the chunk statistics resulting in a longer codeword sequence for the block bitmap. In the next section we will propose a simple remedy.\nPre-randomisation[sec:rand]\n===========================\nThe idea is to apply a bijective function to the source user ID which depends on an extra parameter, block number *i*. A different block number should result in a very different permutation. This way a position in the block bitmap will have the value 1 in a proportion of bitmaps that does not depend on the value of other positions. A user ID will be associated with a pseudorandom sequence of positions as new blocks are produced. Invertibility (bijection) is very important, as it prevents different users from being mapped on the same bit-position in the bitmap, thus ensuring that the mapping a pseudorandom permutation.\nA simple and effective pseudorandom permutation based solely on the block number *i* can be achieved by analogy with randomising the order in a deck of playing cards. One player performs the deck shuffle: for the card in position *i* in the deck *i* ∈ [0, *n* = 2*d*) represented in binary as *i* = *i**d**i**d* − 1…*i*0, the new position of the card *i*′, represented in binary as *i**d*′*i**d* − 1′…*i*0′ is obtained from the current position by applying the following operator \n*i**d*′*i**d* − 1′…*i*0′ = *i**d* − 1*i**d* − 2…*i*0*i**d* = *σ*(*i*) , \nwhich is a cyclic shift left. This corresponds to dividing the deck into two halves and interleaving them, exactly as an experienced dealer would.\nWhen the shuffle is finished, the other player *shifts* the deck, i.e. divides it into two unequal parts and transposes them. In terms of card numbers, this corresponds to adding a pseudorandom value *v* modulo 2*d*: \n$$i^\\prime = i + v \\mod n = \\tau\\_v(i)$$ \n Applying the shuffle-shift operator *Q**v* = *σ* ∘ *τ**v* to a range *ρ* = [0, *n*) *t* times with a pseudorandom choice of *v*: \n*Q**v**t*…*Q**v*1*Q**v*0 *ρ* \n has the same effect as repeatedly shuffling/shifting a deck of cards, which, the intuition suggests, delivers a rather arbitrary permutation. We call *t* the number of rounds.\nNote that the operators *σ* and *τ**v* have a negligible cost even when executed by the least powerful platform, as they take literally a few machine instructions. The cost of pseudorandom generator that produces a series of *v*-values is similarly small if we use a standard Linear Congruential Generator (LCG): \n$$v\\_{k+1} = F\\times v\\_k+1 \\mod n,\\;0\\le k<t\\,,$$ \n where *n* is a power of 2, *v*0 is set to the block number, and the factor *F* is any positive integer that satisfies the well-known Hull-Dobell constraint: $F= 5 \\mod 8 $. We chose for *F* the hex value 5EED which satisfies the constraint and which has more than enough significant digits for any reasonable *n*.\nOur solution appears quite attractive from the point of view of its cost; however, while bijectiveness is guaranteed by construction, we need to be reassured that the solution can deliver sufficient *randomness* of mapping at a reasonably small number of rounds *t*.\n#### Avalanche test.\nHow do we judge the quality of a pseudorandom mapping? A common test is based on the so-called avalanche criterion, used in evaluation of symmetric ciphers and hash functions. We consider it next in relation to our mapping *Q*.\nSelect the block-number *v*0 for the test. Next, select a number *x* from the range *ρ* and some integer 0 ≤ *k* < *d* and prepare two numbers *x*1 = *x* and *x*2 same as *x*1, except bit *k* of it is flipped. Apply the mapping *Q* to both and take the bitwise XOR of the results: *A**i*(*x*, *k*) = *Q*(*x*1) ⊕ *Q*(*x*1). Let *A**i*(*x*, *k*, *l*) be the *l*th bit of *A**i*(*x*, *k*). Define the correlation matrix ${\\bf K}\\_{kl}$ thus:\n \n$${\\bf K}\\_{kl} = \\langle A\\_i(x,k,l) \\rangle\\_{x,i}\\,,\\label{eq:corr}$$ \nHere the averaging is done over values of *x* ∈ *ρ* and block numbers *v*0. Good randomness of the mapping manifests itself in the closeness of all matrix elements of ${\\bf K}\\_{kl}$ to 1/2:\n \n$$\\max\\_{kl} ({\\bf K}\\_{kl}-1/2) \\ll 1/2\\,.\\label{eq:avalanche}$$ \nThis means that if we flip a random bit in a random value *x* the probability that any bit in the image of *x* under *Q* flips in response is close to 1/2. In other words, if we flip one bit in *x*, on average close to one half of the bits in the result will flip. The name “avalanche effect” is to do with the fact that small changes cascade through the rounds of the computation causing further changes until all bits of the result are affected in a complex and unpredictable, though of course deterministic, way.\nWe have applied the avalanche criterion to our proposed randomiser to estimate the acceptable minimum value of *t*, the number of rounds. As it is impossible to average over all potential values of the block number *v*0, we limited ourselves to 50 random samples taken from the interval [0, 10000] The results have proven quite insensitive to the averaging over *v*0, which in not surprising given that we established that the required *t*-numbers are in the hundreds. The averaging over *x* was done by sweeping the whole range *ρ*.\n| *d* | *t* | *δ* | *d* | *t* | *δ* | *d* | *t* | *δ* |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| 10 | 100 | 0.061 | 11 | 100 | 0.113 | 12 | 100 | 0.164 |\n| 10 | 150 | 0.014 | 11 | 150 | 0.026 | 12 | 150 | 0.042 |\n| 10 | 200 | 0.008 | 11 | 200 | 0.009 | 12 | 200 | 0.013 |\nThe results of the avalanche test are presented in Table [tab:avalanche]. For practical purposes we limited ourselves to *n* = 1024, 2048 and 4096, since more users are unlikely to be supported by the communication infrastructure of a single site. The results show that a surprisingly large number, around 200, of rounds is required to achieve good randomisation. It is large compared to the number of rounds one expects to be necessary to randomise a deck of cards (of the order of 10), but it is not large technically: a microprocessor would have to execute only a few thousand instructions to compute the image *i*′ given *i*. This has to be done as many times per block as the number of counterparties that the user has to monitor on the blockchain. For a this would be a number of the order unity, hence the cost would be negligible even on a tight energy budget. On the other hand, setting *t* to 200 would ensure that possible correlations between bits in the image do not exceed 3% (0.013 normalised by 1/2), which should be good enough for practical purposes.\nFinally, let us recall that the IoT platform is low-power, but the server running the PLS protocol via the Sequencer is not. It has ample capacity to analyse the quality of the permutation in terms of its influence on Tunstall compression. We have reserved 5 bits in *ϕ* to pass to the shuffle-shifter an integer value in the interval [0, 32). It is convenient to use the value 0 to indicate that a random permutation is not required[9](#fn9), whereas a nonzero value is added to the round counter *t*. The server can try up to 31 additional rounds and choose the one that gives the best compression. The users receiving the root of trust will be aware of how many additional rounds should be performed and will maintain consistency.\nPutting it all together[sec:together]\n=====================================\nNext we consider a complete example of a user attempting to retrieve a contribution to a block that has been made either by itself or a counterparty.\nFigure [fig:overall-1] presents the flow of data when a block-269 S-message is received and successfully unlocked by a user. The user is about to request the contribution to block 269 from user ID 45. To accomplish this, it needs to decode the received block bitmap by feeding it to the Tunstall decoder together with the parameters *n* and *m* (total number of users and the number of users contributing to block 269, respectively). The decoder produces the uncompressed bitmap. At the same time the user ID (45) and the block number (269) along with the total number of users and configuration parameters are fed to the shuffle-shifter, which will extract the number of additional rounds from *ϕ* and produce its output value, 20. The bit in position 20 of the uncompressed bitmap happens to be 1, which means that the contribution from ID 45 is present in block 269. The number of 1s in the bitmap to the left of position 20 is 4, so the index in the truncated Merkle tree for the contribution in question will be 4. The path to leaf 4 is highlighted in red in the figure.\nThe user’s CAS request will include the block number, 269, and the ID index, 4. CAS will respond with the leaf hash *h*4 and the adjunct sequence \n*V*0 = *h*5, *V*2 = *H*(*H*(*h*0 ∥ *h*1) ∥ *H*(*h*2 ∥ *h*3)) , \n which consists of the labels of the two nodes of the tree marked in blue. Because the user acquired *m* from the unlocked S-message, i.e. the root of trust, it knows the shape of the tree. Consequently, no mask is communicated, but the user is able to reconstruct the mask anyway.\nTo validate the requested *h*4, the user checks that the following equation holds: \n*H*(*V*2 ∥ *H*(*h*4 ∥ *V*0) ∥ *H*(*h*4 ∥ *V*0)′)) = *T* . \nwhere *T* is the root hash received with the unlocked S-message.\nAn alternative scenario is shown in figure [fig:overall-0]. When attempting to retrieve the contribution of user ID=17 to block 305, it turns out that the output of the shuffle-shifter points to a 0 in the uncompressed block bitmap. Since the unlocked S-message is the root of trust, this constitutes a proof that block 305 has no contribution from user 17. Notice that CAS is not involved in the process at all.\nRelated work\n============\nThe PLS blockchain and the protocols in basic form were proposed in. The idea of sparse Merkle Tree has an unclear origin. To the best of our knowledge it was first put forward by Bauer and was recently improved on in. Both studies are concerned with mutable trees, with objectives very different from ours, although, like ourselves, the authors remark on the importance of proofs of absence (non-membership). Tree statistics is tackled theoretically in in the context of optimising mutable MTs for the Bitcoin blockchain in the context of Bitcoin transactions. The objectives of this study are similar to ours as the authors attempt to group the leaves together to minimise the proof length, but they do it using tree transformations (taking the data structure red-black tree as a starting point), while we achieve a similar objective by renumbering the keys (user IDs in our case).\nThe compression technique we use is due to Tunstall and this seems to be uniquely suitable for our case since it is based on empyrical probability of leaf occupancy, which is available to the Fog Server running PLS and which takes next to no resources to communicate to the client. The efficiency of our technique depends on this method.\nWe used our own pseudorandom permutation as a combination of a perfect shuffle and a random shift, using a classical LCG source. There exist various methods of pseudorandom permutation, an oft-cited one being Fisher-Yates shuffle, first published in the 1930s (citation unavailable). The idea there is to choose a (pseudo)random element of a sequence of source items and exchange it with the first element on the sequence. Clearly, if this is repeated enough times then any possible permutation could be achieved[10](#fn10). A recent paper presents a fast, parallel algorithm that mimics the technique of merge-sort except the merge makes a pseudorandom choice when ordering two elements for the output.\nHowever, our situation is quite different. Not because we are dealing with a contiguous range of numbers rather than an abstract sequence of objects: one could enumerate the objects and the problem would boil down to the one we are faced with. Our situation is different because the sender and the recipient must choose the *same* permutation. To encode an arbitrary permutation of *n* numbers would take close to (*m* − 1)log2*m* bits, which is the same order of magnitude as the block bitmap we are trying to make more compact. Of course the ability to perform an arbitrary permutation is not required: all we want is break correlations between user IDs in a series of block bitmaps, and for this any sufficiently rich subgroup would do.\nConclusions\n===========\nStatistical analysis of a sparse Merkel Tree under the assumption of uniform, uncorrelated leaf occupancy has been presented. The model obtained allows direct computation of the Probability Distribution Function for path lengths given the leaf-value probabilities. The path weight was quantified in terms of the number of adjunct hashes required for its leaf proof. We determined that the mean path weight of a sparse MT tree is close to that of a dense, truncated MT tree, with the latter being slightly better at most leaf-probability values *p* in the practically interesting interval. We proposed an alternative structure, a Tunstall-Merkle tree, which combines a dense, truncated MT and a Tunstall-compressed bitmap indicating leaf ocupancy. We tested the compressor at several practical values of code size and quantified its residual redundancy. We found that a very small code table (16 or 256 codewords) proves sufficient for achieving near-limit compression, which means that Tunstall decoding presents no storage problem whatsoever to an IoT platform. To improve the effect of compression we further proposed a decorellation facility in the form of a shuffle-shifting algorithm and tested its properties using the standard avalanche criterion to determine the number of rounds. Both the Tunstall decoder and the shuffle-shifter with the codes size and the number of rounds, respectively, sufficient for our purposes are quite processor-efficient as well, since they involve inexpensive operations (table indexing, cyclic shift and binary addition) and short instruction sequences in implementation.\nThe main effect of the proposed technology is a drastic improvement in the cost of the SLVP protocol. Indeed an SLVP verifier has to check every block for the presence of counterparty (prover) contributions, and no such contribution would be present in a great majority of blocks. Our proposed Tunstall-Merkle tree has zero proof-of-absence cost, and when a leaf is present the communication cost of retrieval is in most cases better than that for the standard MT and MPT. Obviously our technique offers no advantage to a system with an unlimited and dynamic number of users, but it is beneficial for at least the PLS blockchain situation. The statistical analysis of a sparse MT/MPT has significance beyond the area of our study; it could be useful for planning and designing any secure storage structure that involves Merkle trees.\nFuture work will concentrate on higher-level protocols which control the interaction of with a smart contract within the same limited-resource set of assumptions.\n---\n1. Correspondence address: University of Hertfordshire, U.K.; email: <a.shafarenko@herts.ac.uk>[↩](#fnref1)\n2. In the sequel, when it is not important what kind of actor is meant we will call all actors users for short and apply the pronoun ’it’.[↩](#fnref2)\n3. Since the attacker does not know the preimage of *P* at the time when an attack is possible, it can only send an arbitrary message; after unlocking, it would produce a near-random bitstring as a would-be block hash. The requirement for it to have *r* trailing zeros will only be satisfied with the probability 2− *r*[↩](#fnref3)\n4. The original paper has a slightly different arrangement for S-messages since in the original design CAS was trusted for progress, but in the present paper we eliminate this requirement.[↩](#fnref4)\n5. The tree does not have to be binary, but higher-based trees, and higher-based MPTs, discussed later, are inefficient for a small number of leaves.[↩](#fnref5)\n6. For example, left to right, or in prefix order. The properties of the code remain the same under any permutation of the codeword assignments but the practicalities of encoding/decoding require a shared order[↩](#fnref6)\n7. We use the fact that the greatest number has the greatest logarithm[↩](#fnref7)\n8. the main danger would be that the sender and the receiver use different floating-point arithmetic, incur different rounding errors and end up using different dictionaries; this example shows that for a small dictionary there is no such danger[↩](#fnref8)\n9. This could be advantageous when, for example, no compression is used.[↩](#fnref9)\n10. unless the pseudorandom generator producing the selections is caught in a cycle first[↩](#fnref10)\n"},"data_id":{"kind":"string","value":"arxiv_0183110"}}},{"rowIdx":45793,"cells":{"content":{"kind":"string","value":"A Population of Radio-Loud Narrow Line Seyfert1 Galaxies with Blazar-like Properties?\n=====================================================================================\nBlazars with strong emission lines were found to be associated mostly with broad-line type 1 Active Galactic Nuclei (AGN). Hitherto, evidence for blazars identified with Narrow Line Seyfert1 (NLS1) AGN was limited to very few individual cases. Here we present a comprehensive study of a sample of 23 genuine radio-loud NLS1 galaxies which have the radio-loudness parameters, the ratio of radio (21cm) to optical (4400Å) luminosity, greater than 100. The sample, drawn from the SDSS and FIRST, is homogeneous and the largest of this kind. A significant fraction of the sample objects show interesting radio to X-ray properties that are unusual to most of the previously known radio-loud NLS1 AGN, but are reminiscent of blazars. These include flat radio spectra, large amplitude flux and spectral variability, compact VLBI cores, very high brightness temperatures (1011 − 14K) derived from variability, enhanced optical emission in excess of the normal ionising continuum, flat X-ray spectra, and blazar-like SEDs. We interpret them as evidence for the postulated blazar nature of these very radio-loud NLS1 AGN, which might possess at least moderately relativistic jets. We suggest that those steep spectrum radio-loud NLS1 AGN in the sample are of the same population but with their radio jets aligned at large angles to the lines-of-sight. Intrinsically, some of the objects have relatively low radio power and would have been classified as radio-intermediate AGN.\nThe black hole masses, estimated from the broad Balmer line width and luminosity, are within 106 − 8, and the inferred Eddington ratios are around unity. Unless the black hole masses are largely under-estimated, our result stretches the low mass end of the black holes of luminous, fast accreting radio-loud AGN to a smaller mass regime (the order of 106) in the black hole mass–radio-loudness space where other normal AGN are seldom found. The results imply that radio-loud AGN may be powered by black holes with moderate masses ( ∼ 106 − 7) accreting at high rates (Eddington ratios up to unity or higher). The host galaxies of a few nearby objects appear to be disk-like or merger; and some of the objects show imprints of young stellar populations in their SDSS spectra. We find that some of the objects, despite having strong emission lines, resemble high-energy peaked BL Lacs in their SED with the synchrotron component peaked at around the UV; such objects constitute an intriguingly high fraction of the sample. The radio sources of the sample are ubiquitously compact. They are smaller than at most several tens of kilo-parsecs, suggesting a possible link with compact steep-spectrum radio sources. Given the peculiarities of blazar-like NLS1 galaxies, questions arise as to whether they are plain downsizing extensions of normal radio-loud AGN, or whether they form a previously unrecognised population.\nIntroduction\n============\nAs a minority of AGN, radio-loud (RL) AGN differ from their radio-quiet (RQ) counterparts mainly in possessing prominent radio jets and/or lobes that produce strong radio radiation. In observations, a widely used division between RL and RQ AGN is the radio-loudness parameter *R* ≈ 10, defined as the ratio between the radio 5GHz to optical *B*-band luminosity, though it is still a controversy whether *R* has a bimodal distribution below and above this value. RL AGN are an important laboratory to study the formation of relativistic jets, which is not yet understood so far. Nevertheless, it is widely thought that radio-loudness (formation of radio jets) is possibly depending on the accretion rate/state, the spin of the black hole, black hole mass, and host galaxy morphology, or a combination of some of these factors.\nAs an important sub-class of RL AGN, blazar is a collective term for BL Lac objects and flat-spectrum radio quasars (FSRQs). Blazars are characterised by flat radio spectra at above ∼ 1GHz, fast variability, high and variable polarization, superluminal motion, and high brightness temperatures. They are now believed to be RL AGN with the orientation of relativistic jets close to the line-of-sight, and hence their non-thermal jet emission is highly Doppler boosted. Blazars have distinctive spectral energy distributions (SED) that are characterised by two broad humps in the log*ν* − log*ν**f**ν* representation. The low-energy hump is commonly interpreted as synchrotron emission, while the high-energy one as inverse Compton emission scattering off the same electron population which produces the synchrotron emission. Depending on the peak frequency of the synchrotron hump, BL Lac objects are divided into Low-energy-peaked BL Lacs (LBL) with the peak around IR–optical wavelengths and High-energy-peaked BL Lacs (HBL) with the peak around UV/soft X-ray energies. Classical FSRQs have generally the peak frequencies of the synchrotron hump similar to or even lower than LBL. Interestingly, it was suggested that the whole blazar family can be arranged in a sequence from HBL to LBL and to FSRQ in the order of decreasing synchrotron peak frequencies and increasing source power—the so called blazar sequence[1](#fn1). However, exceptions were also reported in recent years as the presence of powerful FSRQs with HBL-like SEDs (though this remains a matter of controversy), whose synchrotron hump is peaked around the UV/soft X-ray band. Following, we term FSRQs with HBL-like SED HFSRQs (High-energy-peaked FSRQs), and those with LBL-like SED—the classical FSRQs—LFSRQs (Low-energy-peaked FSRQs), as parallels to HBLs and LBLs.\nUntil recently, RL AGN (blazars) with strong broad emission lines were found to ubiquitously have the Balmer line widths greater than 2000in full width at half maximum (FWHM). It was found that the line width is correlated with the orientation of the radio axis, with narrower lines as well as stronger optical emission in objects with smaller viewing angles to the radio axis. This was interpreted as arising from a disk-like emission line region. A marked absence of radio-loud quasars (RLQs) was noted below the 2000line-width cutoff, that is the characteristic line-width range for narrow line Seyfert1 galaxies—a sub-class of type1 AGN in contrast to the classical Broad Line AGN (BLAGN). The situation has changed since the detection of radio emission from NLS1 galaxies.\nApart from narrow line-widths of the broad Balmer lines, NLS1 galaxies also show other extreme properties compared to normal broad line AGN, such as strong permitted optical/UV emission lines, steep soft X-ray spectra, and rapid X-ray variability. Observational evidence suggests that NLS1 galaxies tend to have small black hole masses and high Eddington ratios (defined as the bolometric to the Eddington luminosity ratio), $\\equiv L\\_{\\rm bol}/L\\_{\\rm Edd}\\approx 1$. In fact, they were found to locate at one extreme end, opposite to BLAGN, of eigenvector1 of the correlation matrix which is believed to be driven primarily by. NLS1 galaxies were once thought to be radio-quiet. This was simply a consequence of the fact that NLS1 galaxies have a low radio-loud fraction and the small number of NLS1 galaxies known till then.\nPrevious studies of the radio properties of NLS1 galaxies are all based on small samples. These showed that the radio sources, if detected, are compact (less than a few hundred parsec), at modest radio power, of steep radio spectra, mostly in the RQ regime, and of low RL fraction. Until recently, there were only several RL NLS1 galaxies identified and studied individually. Studies using (small) samples of RL NLS1 galaxies were carried out by and for non-radio selected and radio selected objects, respectively. These studies confirmed that, as in ‘normal’ NLS1 galaxies (their RQ counterparts), RL NLS1 AGN are accreting at a high rate close to the Eddington limit. More importantly, as pointed out by, with relatively low and high *R*, they occupy a previously sparsely populated region in the –*R* diagram. Most of the sources of that sample are compact steep spectrum sources. It was confirmed that RL NLS1 galaxies are rare ( ∼ 7%), and very radio-loud objects with *R* > 100 are even more sparse, compared to about 10–15% for normal BLAGN and quasars. Several of these studies discussed the starburst versus AGN contribution to the radio emission, and concluded that it is AGN dominated.\nOf particular interest, a few outstanding RL NLS1 galaxies came to be known to exhibit blazar properties. RXJ16290+4007 was known as a blazar and its X-ray emission is dominated by synchrotron emission, or else by soft X-ray emission that is typical for NLS1 galaxies, perhaps from the accretion disk. The SED of PKS2004-447[2](#fn2) was well modeled with the blazar type. Using information on the radio flux variations, very high radio brightness temperatures ($\\gtrsim10^{13}$K) were inferred for J0948+0022 and 0846+513[3](#fn3) —both included in our sample, arguing for relativistic beaming. They also show some other blazar-like behavior. Recently, performed high-resolution VLBI observations for several radio-loud NLS1 AGN (three are included in our sample[4](#fn4)) and found that they are unresolved with milli-arcsec resolutions, setting direct lower limits on the brightness temperatures in the range of 107 − 109K. They also found that inverted radio spectra are common in the radio-loudest objects. The authors suggested that Doppler beaming, presumably resulting from highly relativistic jets, can explain naturally the observations. In this respect, the most remarkable object of this kind is perhaps 2MASXJ0324+3410, found in our recent work. It showed rapid variability in the radio, optical, and X-ray bands, and was even claimed to be marginally detected in TeV *γ*-rays. Its non-thermal SED clearly resembles that of HBLs. Moreover, 2MASXJ0324+3410 is hosted by a relatively small, apparent disk galaxy with one-armed spiral or ring galaxy morphology. These enigmatic objects present a challenge to current models of both NLS1 galaxies and blazars, and render a unique opportunity to study jet formation in black hole systems accreting at high rates. Apparently, they do not seem to belong to any types of AGN currently known. An immediate question is whether they are some exceptional individuals or there exists a population of such objects. What are their general properties in continuum and line emission, central engine and host galaxies? What are their relations with other types of AGN?\nMotivated by these issues, we compiled a RL NLS1 galaxy sample from the Sloan Digital Sky Survey (SDSS) database, aimed at a systematic study with a large and homogeneous sample. Since, as the first step, we focus on genuine RL objects, we consider in this paper objects with the radio-loudness parameter *R* > 100 (see §[sect:samplesel] for the definition of *R*) for the following reasons. Firstly, the classification of AGN into RL and RQ is somewhat ambiguous in the range *R* = 10 − 50 due to the presence of (a small number of) so called radio-intermediate AGN spanning in between the bulks of the two classes, which may have a different origin from genuine RL objects. The same may also be true for NLS1 galaxies. Secondly, in *R* > 100 objects contamination of radio emission from the host galaxy is negligible (see §[sect:discsed]). Thirdly, their jet component, if present, can be detected and studied relatively easily in other wavebands than the radio, such as optical and X-rays. Last, but not the least, they are the least studied objects *as a sample*, due to their extreme rarity, in contrast to less RL NLS1 galaxies as the bulk in the samples of and. We defer to a later paper a comprehensive and complete treatment of the SDSS–FIRST detected NLS1 galaxies sample, including radio-intermediate objects, to address questions such as the RL–RQ dichotomy for NLS1 galaxies. Our results show that there exists a population of NLS1 galaxies that are blazar-like, despite their rarity.\nThe compilation of the sample and the analyses of multi-waveband data are described in §[sect:sample]. The broad band continuum and emission line properties are presented in §[sect:cont] and §[sect:emline], respectively. In §[sect:hostgal] the properties of their host galaxies are presented. We discuss other properties such as black hole mass and accretion rate, and isotropic radio emission, as well as the implications of the results in §[sect:discus]. Throughout the paper, we assume a cosmology with *H*0= 70 km s− 1Mpc− 1, Ω*M* = 0.3, and ΩΛ = 0.7. Errors are quoted at the 68% confidence level unless mentioned otherwise. We use the following conventions to denote power-law spectral indices: in the frequency domain [*S*(*ν*) ∝ *ν**α**ν*, specifically ≡ in the radio band], and in the wavelength domain [*S*(*λ*) ∝ *λ**α**λ*]; and the X-ray photon index Γ, defined as $f\\_{\\rm pho}(E) \\propto E^{-\\Gamma}$, where $f\\_{\\rm pho}(E)$ is X-ray photon number flux density.\nSample compilation and multiwavelength data analysis\n====================================================\nSDSS NLS1 galaxy sample\n-----------------------\nWe have carried out a systematic search for NLS1 galaxies from the SDSS spectroscopic samples and the first results have been published in. The procedures of data analysis have been documented in that paper in detail and only an outline is summarised here. We carefully fit emission line spectra, AGN continua, and host galaxy starlight in a self-consistent manner. Firstly, galaxy starlight and AGN continuum, as well as the optical emission complex are modeled (see Appendix[sect:sdssspec] for a brief account) and then subtracted. The optical multiplets are modeled with the spectral data given by for both the broad and narrow components. Emission line spectra are fitted with the following models, using a code similar to that described in. The Balmer emission lines are de-blended into a narrow and a broad component, which are modeled by a Gaussian and a Lorentz profile, respectively. All narrow emission lines are fitted with a single Gaussian except the *λ**λ*4959/5007 doublet. Each line of the doublet is fitted with two Gaussians, one of which is used to account for a possible blue wing as seen in a few objects. As shown in, our analysis procedures yield reliable and accurate measurements of emission line parameters.\nThe SDSS spectroscopic samples of both galaxies and QSOs in the redshift range of *z* < 0.8 were analyzed based on the SDSS data release 5. Following, we classify NLS1 galaxies as those having the ``broad\" component of H*β* or H*α* which is detected at the 10 *σ* or higher confidence level and is narrower than 2200in FWHM. As discussed in, objects selected as such fulfill naturally the second of the conventional criteria of NLS1 classification, i.e. < 3, and the sample is reliable and uniform. As a result, about 3300 NLS1 galaxies are found, superseding the previously published 2011 drawn from the SDSS DR3.\nSearch for radio counterparts\n-----------------------------\nHaving compiled a large NLS1 galaxies sample from the SDSS, we search for their radio emission using data of the FIRST[5](#fn5) survey. The survey has a spatial resolution of 5and typical 90% source positional uncertainty ellipses less than 1. The identification is performed by matching the SDSS positions of NLS1 AGN against those of radio sources following the procedure used by. This procedure was designed to search for SDSS quasars detected in the FIRST survey, and is summarised here. Since extended radio sources often have multiple (diffuse) components in morphology and sometimes their positions are vague to define, we treat extended sources differently from compact sources in search of radio counterparts. By compact FIRST source we mean that only one source is found within 3of the optical position and is unresolved; for them, we use a 2matching radius. We search for possible extended radio sources associated with the NLS1 AGN in two steps. First, candidates are selected if they match one of the following criteria. (1) Only one resolved radio source is found within 3of the optical position, and the optical position is located within the size of the radio source. (2) Two radio sources are located nearly symmetrically around the optical position within 3. (3) More than two radio sources are scattered around the optical position within 3. Then we visually inspect $6\\arcmin \\times 6\\arcmin$ cutouts of the FIRST images centered at an object to reject false matches. As discussed in, this approach has proved to be effective for finding extended radio sources associated with AGN; about 24% of the radio quasars in the sample of are resolved by the FIRST survey, among which half show FRII type morphology.\nThis procedure yields a sample of SDSS NLS1 galaxies detected in the FIRST survey. Interestingly, we note that *all* the radio sources associated with these NLS1 AGN are compact in morphology. For each object we calculate the radio-loudness parameter defined as $\\equiv f\\_{\\nu}({\\rm 1.4\\,GHz})/f\\_{\\nu}(4400\\,\\mbox{\\AA})$, where the fluxes are in the rest frame of the objects[6](#fn6). The rest frame 4400Å flux densities are calculated from the SDSS g-magnitudes assuming a spectral slope = − 0.5. An index of − 0.5 is also assumed for the K-correction for the radio fluxes. We focus on very radio-loud objects only with > 100 in this paper, and will present the whole sample elsewhere.\nWe note that three NLS1 galaxies with apparent > 100 stand out for their extremely red optical continua ($\\ga 2.7$), namely, J095919.14+090659.4, J111354.66+124439.0, and J233903.82-091221.3. Our original fit to the continuum spectrum with a fixed power law slope = − 1.7 yields substantial reddening of *E*(*B* − *V*)=0.7–1. Among them, J233903.82-091221.3 (*z*=0.66) has been found to show convincing evidence for substantial dust extinction [7](#fn7). Given the fact that the other two objects resemble closely J2339-0912 in their SDSS spectra, their optical light is most likely subject to heavy extinction as well. After correction for extinction, their radio-loudness parameters are largely reduced to ≃ 10 − 20, below or merely close to the RL/RQ dividing line. We therefore exclude these three objects from our current sample.\nVery radio-loud NLS1 galaxy Sample\n----------------------------------\nThe above selection procedures result in 23 NLS1 galaxies having > 100, which compose our sample. By excluding less RL objects with < 100 our sample can be regarded as very radio-loud to some extent. We believe that the radio-loudness estimates of our sample are not significantly affected by optical extinction, as is discussed in §[sect:disopt]. The objects are listed in Table[tbl:sample], along with some relevant parameters of the continua and emission lines. As demonstration, we show in Figure[fig:sdssspec] examples of the SDSS spectra and their best fit models for two representative objects; they are chosen to represent the ranges of some characterising parameters of the sample. Measurable starlight contribution is present in only one object, J1633+4718, accounting for at most ∼ 20% of its observed spectral flux density at 5100Å. Most of our objects show strong emission complexes (see Section[sect:emline]). The lines are weak in general, all having < 1.2. All of the objects fulfill the conventional definition of NLS1, and are therefore bona-fide NLS1 galaxies.\nThe redshift and radio-loudness distributions are plotted in Figure[fig:distrl]. Our sample has a relatively high redshift distribution peaked at *z* ∼ 0.5 (the median). We calculate the *B*-band absolute magnitudes *M**B* assuming = − 1.5 for the K-correction, which are listed in Table[tbl:sample]. The *M**B* values range from − 20.8 to − 25.6, with a median of − 22.8. Among the sample, 10 objects are as bright as *M**B* ≤ − 23 and hence can be classified as narrow-line typeI quasars. Three objects in the sample have been previously identified with RL NLS1 galaxies and studied in detail, namely, J094857.32+002225.5, J084957.98+510829.0, and J172206.03+565451.6. In addition, several objects were known as general RL AGN and have been studied previously, whose properties are summarised individually in Appendix[sect:indv].\nX-ray and UV data analysis\n--------------------------\nWe have searched for X-ray emission from the sample objects using ROSAT source catalogues of the RASS and pointed observations of both the RXP and WGA catalogues. The matching radii are 5 times the given positional errors of ROSAT sources, which are typically 10–20. For each match, we visually inspect their SDSS optical images to avoid spurious matches in which the X-ray source is actually associated with another nearby object within the matching radius. Of the 23 objects, 12 were detected in the RASS and 4 in pointed observations, among which 2 detected in both. Thus a total of 14 objects—more than half of the sample—were detected in X-rays with the ROSAT PSPC[8](#fn8).\nAll of the objects except J1633+4718 do not have sufficient X-ray counts to allow proper spectral modeling. Following,, and, we estimate the X-ray photon index Γ from the two hardness ratios[9](#fn9), wherever available, assuming an absorbed power-law spectral model. Both Γ and absorption column density can be estimated as free parameters, or the index alone assuming to be the Galactic column density. This method had been verified to be reliable and robust under the assumption that intrinsic spectral shapes are indeed a simple absorbed power-law. The X-ray fluxes in the ROSAT PSPC band (0.1–2.4keV) are calculated from the count rates using the energy to counts conversion factor (ECF) for a power law spectrum and Galactic absorption (ROSAT AO-2 technical appendix, 1991). The ECF for each X-ray source is explicitly calculated from the ROSAT PSPC effective area[10](#fn10), by using information on individual Γ if obtainable, or using the mean otherwise (Γ = 2.49, see below). We list some of the information about the X-ray observations in Table[tbl:xray].\nFor objects not detected in X-rays, we determine upper limits on their X-ray fluxes from count rate limits set by the RASS. Since the X-ray backgrounds are very low in most of the RASS observations, detection of a source is mostly determined by the source photon counts that follow Poisson statistics. We set an upper limit of source counts as 12 for an object not detected in the RASS, as often adopted. The corresponding count rate limit is then calculated by using the effective exposure time at the source position as available from the RASS exposure map. Then the flux limit for each object is calculated using the same method above and assuming Γ as the mean.\nAmong the ROSAT detected objects, J1644+2619 was observed with Chandra with a large number of photon counts accumulated. Moreover, for J0849+5108 the hardness ratios (*H**R*1 = 0.97 ± 0.11, *H**R*2 = 0.28 ± 0.15) yield an abnormally flat spectrum Γ = 0.6− 0.52+ 0.36. For this object we also investigate its ROSAT data in spite of the relatively low number of counts ( ∼ 60). We perform X-ray spectral and timing analysis for J1633+4718, J1644+2619, and J0849+5108. We briefly describe here the data reduction procedures and present results in §[sect:res-xray]. We use version 12.2 of XSPEC for spectral modeling.\n**J1633+4718** (RXJ16333+4719): The target was observed with the ROSAT PSPC-b with an exposure time of 3732 seconds on July 24th, 1993 (Obs-ID: 701549; PI: N. Bade). We retrieved the ROSAT data from the archive and use the XSELECT (version 2.3) tool of FTOOLS for data analysis. The X-ray spectrum is extracted from a circle of 150radius, and a background spectrum is extracted from an annulus with inner/outer radii of 195/300. There are ∼ 976 net source counts, and the count rate is 0.26 ± 0.01. The spectrum in the 0.1–2.4keV band is binned to have at least 30 counts in each bin. J1633+4718 was also detected in the RASS with a weighted exposure of 909s, yielding ∼ 185 net source counts and a count rate 0.20 ± 0.02. We extract the RASS spectrum following the procedure described in.\n**J1644+2619**: The object was targeted with the Chandra ACIS-S (PI: S. Laurent-Muehleisen) with an exposure time of 2946s on June 02, 2003. The data were retrieved from the Chandra data archive and were reduced from the level-2 data set following the standard procedure using CIAO (version 3.4). There are 553 ± 44 net source counts detected, with a count rate of 0.188. The 0.2–5keV spectrum (very few counts accumulated above 5keV) is binned to have at least 20 counts in each energy bin.\n**J0849+5108**: The object was observed with the ROSAT PSPC-b in a pointed observation as a target in 1993, April with an exposure time of 4496s (Obs-ID: 700821). There are only 63 net source counts detected in the 0.1–2.4keV band. We fit the unbinned spectrum by minimizing the C-statistic, which is valid for Poisson statistics appropriate to the low counts regime.\nWe also searched for UV data from the GALEX GR2/GR3 data release[11](#fn11). Twelves objects in our sample have available photometric measurements in the far and/or near UV imaging bands with an effective wavelength of 1516Å and 2267Å, respectively. Their UV images are mostly point source like. The GALEX magnitudes (in the AB magnitude system) are corrected for Galactic extinction using $A\\_{\\rm FUV}/E(B-V)=8.376$ and $A\\_{\\rm NUV}/E(B-V)=8.741$, following.\nBroad band continuum radiation\n==============================\nGHz radio emission\n------------------\n### Radio morphology and spectrum\nThe FIRST radio images of our RL NLS1 sources are ubiquitously unresolved at a resolution of 5.4, which is remarkable. This sets upper limits on the projected size of 10–40kpc, depending on the redshift. Among our sample there are several objects observed with VLBI, including J0948+0022, J1633+4718 and J1644+2619, and J1505+0326, as reported in the literature; they remain unresolved at resolutions ranging from a few milli-arcsec to a few tens of milli-arcsec, which correspond to several to several-tens parsec in physical scale. These observations set constraints on the brightness temperatures of $T\\_{\\rm B}>10^{8-9}$K. Since all the radio sources in our sample are compact, we use hereafter the average of the FIRST and NVSS[12](#fn12) measurements as their 1.4GHz flux densities in the following analysis; this is to minimize the effects of fluctuations on the results caused by large flux variations. Similarly, at other frequencies, in cases where multiple epoch measurements are available at the same frequency, the average flux is used.\nMulti-frequency and multi-epoch radio data were collected from the NASA/IPAC Extragalactic Database (NED). Of the 23 objects, 11 have flux measurements at 5GHz, for which the 1.4GHz–5GHz spectral indices are estimated, though the observations were not simultaneous. The 5GHz fluxes and are listed in Table[tbl:sample], and the histogram of is shown in Figure[fig:radslp][13](#fn13). It can be seen that the radio spectra are systematically flat, with a median of − 0.30 and a mean of − 0.13. Adopting the conventional dividing line between flat- and steep-spectrum sources, = − 0.5, 8 out of the 11 objects with measurement turn out to be flat-spectrum radio sources ( > − 0.5). In particular, several objects show an inverted radio slope ( > 0) near 5GHz and toward higher frequencies based on simultaneous multi-frequency measurements, namely, J0948+0022, J1633+4718, J1644+2619, and J1505+0326. Only 3 objects have steeper than, yet very close to, − 0.5.\n [fig:radslp]\nAmong the remaining 12 objects without available slopes near 5GHz, 9 have lower frequency 327MHz–1.4GHz indices measurable (see Section[sect:lowfemi]), out of which 6 have flat spectra ( > − 0.5). Thus, it is likely that most of these 12 objects may have the same flat (or even flatter) indices at higher frequencies as well, unless there is a break near 1GHz, as in Gigahertz-peaked spectrum sources (GPS). This fact supports further the flat spectral index distribution found above for our sample. Though in part of the sample the radio slopes are estimated from non-simultaneous data, the current distribution should be correct in the statistical sense; therefore, we expect that the bulk of our sample objects should be flat-spectrum sources. As is known, radio sources with flat spectrum, which can be produced by relativistic jet models, have a fairly good chance to be blazars.\nNow we can elaborate the radio-loudness parameter calculation for our sample by using the averaged 1.4GHz flux densities and the estimated for individual sources for the K-correction (if not available, = − 0.5 is assumed as above). We also make use of the measured optical slopes as presented in §[sect:optcont]. The improved radio-loudness values are listed in Table[tbl:sample]. The differences between the previous estimates and the improved values are small, typically within 10% in log; such differences are comparable with changes in caused by source flux variations typical of our objects (see below).\n### Radio variability and brightness temperature\nA number of objects showed significant flux variations based on multi-epoch observations taken at the same wavelengths [14](#fn14). In Table[tbl:radvar] we list those having the significance of variations higher than ∼ 3 *σ* between two observations, where *σ* is defined as $\\sigma=(S\\_1-S\\_2) /\\sqrt{\\sigma\\_{s1}^2+\\sigma\\_{s2}^2}$. We also give in the table the amplitudes of variations Δ *S*, the fractional amplitudes Δ*S*/⟨*S*⟩ where ⟨*S*⟩ is the average of the two fluxes, and the time spans Δ*t* between the two observational epochs. Their variability is consistent with the fact that the variables have ubiquitously flat or inverted radio spectra when available. The fractional amplitudes given in Table[tbl:radvar] have a median of 43%. In particular, several objects show large amplitude variations ( ∼ 40–75%) on relatively short timescales, J0948+0022, J0849+5108, and J1505+0326 within a few years, and J1548+3511 by 44% within 207 days (140 days in the source rest frame).\nLarge amplitude variability on short timescales has been commonly used to set lower limits on the *apparent* brightness temperature ($T\\_{\\rm B}$) of a radio source, since the size of the variable part of the source can be constrained from the light-crossing time. Assuming that the variable portion of the radio flux is emitted from a region with a size smaller than that corresponding to the light-crossing time, which is the time span between the two observations, the *apparent* brightness temperature is estimated to be [15](#fn15)\n \n$$T\\_{\\rm B} \\ga \\frac{\\Delta P\\_{\\nu e}}{2 \\pi^2 k \\nu^2 (\\Delta t)^2}\n= \\frac{2D^2\\_{L}\\Delta S\\_{\\nu}}{(1+z) \\pi k \\nu^2 (\\Delta t)^2}$$ \nwhere *k* is the Boltzmann constant, *D**L* the luminosity distance, Δ*S**ν* the variable portion of the observed flux density, Δ*P**ν**e* the corresponding radio power at the emission frequency in the source rest frame, *ν* the observing frequency and Δ*t* the time span in the observer’s frame. The estimated $T\\_{\\rm B}$ are listed in Table[tbl:radvar].\nOf the 8 objects, 6 have estimated $T\\_{\\rm B}$ exceeding the equipartition brightness temperature ∼ 3 × 1011K as proposed by, which is expected to conform by most radio sources. In particular, 4 of the objects, all of flat-spectrum, have $T\\_{\\rm B}$ exceeding the inverse Compton limit ≃ 1012K. The highest $T\\_{\\rm B}$ is found to be 1014K in J1505+0326, from two VLBI observations at 8.3GHz with the variability significance of 3*σ*. The extremely high $T\\_{\\rm B}$ values are commonly explained as emission originating from relativistic jets. For instance, taking the inverse Compton limit as a conservative limit, the minimum Doppler factor can be estimated as $\\delta\\_{\\rm min}=(T\\_{\\rm B}/10^{12}\\,K)^{1/3}$. This results in $\\delta\\_{\\rm min}=$1.5–4.7 for the 4 objects (Table[tbl:radvar]), consistent with the range of the Doppler factors of relativistic jets inferred for classical RL AGN.\n### Summary of the GHz radio properties\n(1) All the radio sources of our sample are compact, being unresolved at the FIRST resolution of several arc-seconds. (2) The majority of objects with available radio indices are of flat-spectrum (around 1.4–5GHz), some even with inverted spectra. (3) Most of the flat-spectrum sources show significant variability, from which the brightness temperatures are inferred to be as high as $T\\_{\\rm B}>10^{11}$K, and even exceeding the inverse Compton limit of 1012K in 4 objects. These high $T\\_{\\rm B}$ values may suggest the presence of at least mildly relativistic beaming of the radio emission.\nThe properties of the radio emission at low frequencies of our sample objects are summarised in Appendix[sect:lowfemi], which is important for understanding the intrinsic radio power and radio-loudness of these objects, as discussed in Section[sect:intrpower].\nOptical continuum emission\n--------------------------\nOur standard algorithm to select NLS1 galaxies as above does not give information about the optical continuum shape. However, for RL objects the optical continuum may deviate from the ‘canonical’ shape in consideration of a potential contribution of jet emission. To properly take this effect into account, we parameterise the optical continuum shape especially for our RL sample. We model the AGN continuum in the 2500Å–5500Å rest frame range with a power-law or a broken power-law, and repeat the above spectral fitting with the slope as a free parameter. The fits are performed in the same way as described in §[sect:nls1] assuming no intrinsic reddening. The improved fits give a more realistic description of the AGN continuum; while, on the other hand, we find that the new fits have almost no effects on the parameterisation of the emission line spectra. The optical continuum fluxes in the source rest frame are measured directly from the power-law fits of the AGN continuum.\nIt has been established that the optical luminosity $L\\_{\\rm opt}$ of NLS1 AGN is tightly correlated with the line luminosity based on large samples ; hence is a good indicator of $L\\_{\\rm opt}$. We test this scaling relation for RL NLS1 galaxies as shown in Figure[fig:lo-lb], where the measured 5100Å luminosities are plotted versus. As a comparison the ∼ 2 000 predominantly RQ NLS1 galaxies in the Zhou’06 sample are over-plotted, along with the fitted *λ**L**λ*5100– relation. As can be seen, the RL NLS1 galaxies reveal the same trend of correlation, but, of particular interest, lie systematically above the known $L\\_{\\rm opt}$–relation for normal NLS1 galaxies. We estimate the ratio of the observed luminosity to that expected from the luminosity (*L**λ*5100/*L**λ*5100*H**β*) using the *λ**L**λ*5100– relation. We find that the ratios range from 0.93 to 5.2, with a mean of 1.9. The ratios differ systematically between flat- and steep-spectrum sources as groups, with the mean being 2.4 for the former and 1.8 for the latter, and 1.6 for those without radio indices.\n [fig:lo-lb]\nThe optical continua in the rest frame 2500Å–5500Å range can be described with a power-law for most of the objects. The fits are relatively poor for two objects, namely, J1037+0036 and J1634+4809, which show convex-shaped spectra in the optical–UV band with a drop short-ward of *λ*2800Å in wavelength. For one object, J1138+3653, the spectrum cannot be fitted with a power-law due to a rapid drop in the blue part, which results possibly from mild extinction and is discussed further in §[sect:disopt]. The best-fit slopes show a large range from modestly red (= − 0.2) to blue (= − 2.4) spectra. Their distribution is shown in Figure[fig:optslp] (excluding J1138+3653), which has a median of − 1.54 and a mean of − 1.43. We compare this distribution with that of a RQ NLS1 sample, which comprises 55 SDSS-selected objects with reliably measured optical slopes. It should be noted that that RQ sample has the redshift and luminosity distributions indistinguishable from those of our sample [the two-sided Kolmogorov-Smirnov (K-S) test probability of 0.48 and 0.52, respectively]. It can be seen that, though there is a broad range of overlap, our very radio-loud NLS1 AGN appear to have systematically bluer continua than the RQ objects. The RQ sample have a median of − 1.24 and a mean of − 1.15. The chance probability that the two distributions are the same is only $\\la0.01$, as given by the K-S test. A comparison with the smaller NLS1 galaxies sample of, which have UV–optical slope measurements with the Hubble Space Telescope (HST), yields a similar conclusion; the closest value of their sample to ours is the continuum slope of the median composite spectra, ∼ − 0.79 ( ∼ − 1.21).\n [fig:optslp]\nX-ray radiation\n---------------\nAssuming an absorbed power-law spectral model and using the aforementioned hardness ratio method, the effective photon indices are estimated for most of the objects with valid ROSAT/PSPC hardness ratios extracted from the ROSAT source catalogues. The estimated absorption values are found to be well consistent with the Galactic values within errors, indicating that there is no significant X-ray absorption in these objects. Hence, we adopt the photon indices Γ (Table[tbl:xray]) obtained by assuming =for better constraining the parameters. Furthermore, for 5 of the objects no physically meaningful result can be obtained by this method, because either the spectrum is complex and highly deviating from the assumed absorbed power-law, or the measured hardness ratios are displaced outside the valid zone by Poisson fluctuations. For objects with multiple X-ray data, we use the weighted mean for Γ and fluxes in the following analyses. Results from detailed spectral analysis for the 3 objects, J1633+4718, J1644+2619, and J0849+5108, are summarised below.\n**J1633+4718** (RXJ16333+4719): The spectrum deviates apparently from a simple absorbed power-law model, with prominent soft X-ray emission dominating *E* < 0.5keV and a hard tail at *E* > 0.5keV. The best fit is achieved using a model of a power-law plus a redshifted blackbody component, and the absorption is found to be in excellent agreement with the Galactic =1.79 × 1020. Hence is fixed to. The data and the best fit model are shown in Figure[fig:xsp1633]. The rest frame blackbody temperature is found to be 32.5− 6.0+ 8.0eV and the photon index of the underlying power law Γ = 1.37 ± 0.49 (90% confidence for one parameter of interest). The latter is less well constrained due to the limited bandpass of the PSPC in the ‘hard’ X-rays. No temporal variability is found within the observation interval. The spectrum measured in the RASS has almost identical spectral shape as that of the pointed observation. The best-fit parameters are in excellent agreement with those obtained from the pointed observation, with Γ = 1.47− 0.92+ 0.77 and the blackbody temperature 30− 10+ 12eV (90% confidence level), though the errors are large given the small source counts. No variation in either the spectral shape or the flux is found between the RASS and the pointed observation.\n[]0.48 \n[]0.48 \n[fig:xsp1633]\n**J1644+2619**: A simple power-law model is found to be an excellent fit to the spectrum (see Figure[fig:xsp1644]), resulting in a reduced =13 (for 23 d.o.f). The fitted absorption is almost identical to the Galactic value =5.12 × 1020. The best-fit photon index is steep, Γ = 2.19 ± 0.27 (90% confidence for one parameter of interest). Adding an extra soft X-ray component (e.g. blackbody) does not improve the fit, though with two more free parameters; the power-law photon index becomes flatter, Γ ∼ 1.8− 0.3+ 0.6, but is subject to a large uncertainty range which encompasses the single power-law model. We therefore suggest that the simple power-law model with a relatively steep index is the most likely spectral shape, though models composed of a flatter power-law with a weak soft component cannot be ruled out from the current data. Compared to the observation in the RASS, the X-ray source was about 2.5 times fainter at the Chandra observation epoch, though the photon index seems not to vary.\n [fig:xsp1644]\n**J0849+5108**: The spectrum can be fitted with a power-law with absorption close to the Galactic value =3.0 × 1020. Fixing column density at yields a photon index Γ = 1.77− 0.60+ 0.44 (90% confidence for one parameter of interest). We note that the extremely hard hardness ratios given in the source catalogue are incorrect, which may resulted from inappropriate background subtraction around this object in the presence of nearby, seemingly extended X-ray emission.\nFigure[fig:gam] shows the distribution of the *effective* X-ray photon indices Γ, which span a wide range from 1.7 to 3.3. Since the uncertainties of most of the derived Γ are relatively large, we try to disentangle the intrinsic distribution of Γ from the measurement uncertainties, using the Maximum-Likelihood method introduced by. Assuming that the intrinsic distribution of Γ is Gaussian, we find that the distribution is intrinsically broad, with a standard deviation of 0.39− 0.19+ 0.41 and a mean of 2.60− 0.43+ 0.37 (at the 90% confidence level for two interesting parameters). We conclude that the soft X-ray continuum shape, as described by the power-law photon index Γ, exhibits a considerable variety among our RL NLS1 galaxies, with both flat and steep spectra.\n [fig:gam]\nIt should be noted that the thus estimated Γ from hardness ratios are only an indicator of the overall spectral shape in the 0.1–2.4keV range. A steep Γ value does not rule out the presence of a flat ‘hard’ X-ray component if both a steep soft X-ray and a flat ‘hard’ X-ray component co-exist. As an example, for J1633+4718, a slope of Γ ≃ 2.8 is derived from its hardness ratios if the spectrum is approximated as a power-law. This is particularly true for NLS1 galaxies, since a soft X-ray excess component is commonly seen in them. Better quality X-ray spectra in a wider energy band than that of the ROSAT/PSPC are needed to determine the true underlying power-law in these objects.\nBroad band SED\n--------------\nTo quantify the broad band SED, we calculate the commonly used effective spectral indices between the 5GHz radio, 2500Å optical/UV, and 2keV X-ray band, namely,,, and ; they are defined as $\\alpha\\_{\\rm 12}=-\\log (S\\_{\\nu1}/S\\_{\\nu2})/\\log (\\nu\\_1/\\nu\\_2)$, where *S**ν*1 and *S**ν*2 are the fluxes at the frequencies *ν*1 and *ν*2, respectively, in the object rest frame[16](#fn16). The rest frame 2500Å fluxes are computed from the *u*-band PSF magnitudes, which have the effective wavelength (3542Å) very close to the redshifted 2500Å light (within a range of 500Å); for the K-correction, the measured optical slopes for individual objects are adopted (for one object without the slope available, = − 1.5 is assumed). For the 3 objects for which X-ray spectral fits were enabled, the X-ray flux densities at 2keV ($f\\_{\\rm 2keV}$) are measured directly from the best-fit spectral model; while for the others $f\\_{\\rm 2keV}$ are computed from the 0.1–2.4keV integrated fluxes. In doing so, as well as the K-correction, the derived Γ values are used for individual sources whenever available, or the mean ⟨Γ⟩ = 2.6 is used otherwise. For those without X-ray detection, lower limits on and are calculated using the upper limits on the X-ray fluxes constrained by the RASS. The obtained,, and are listed in Table[tbl:arox].\nIn Figure[fig:alprox] we plot our RL NLS1 galaxies on the commonly used –blazar diagnostic diagram, separating them into three groups: flat (underlying) spectra (Γ < 2.0, filled dots), steep spectra (Γ > 2.0, open circles), and those without Γ estimates (asterisks). For comparisons, we over-plot RLQs from the ROSAT detected quasar sample of. That sample is composed of both FSRQ and steep-spectrum radio quasars (SSRQ), whose loci largely overlap with each other. It can be seen that some of the objects do follow the trend defined by FSRQ/SSRQs, indicating their SEDs are similar to those of FSRQ/SSRQs. Specifically, all the 3 objects in our sample known to have a steep radio spectrum are well consistent with normal SSRQs. It is known that FSRQs (and SSRQs) occupy almost the same region on the –diagram as LBLs, owing to their similar SEDs.\n [fig:alprox]\nHowever, a considerable fraction of the objects clearly offsets. We expect that these outliers must have different SEDs from those of classical FSRQ and LBLs. By invoking the classification scheme of BL Lac objects, we divide the –diagram into the loci of LBLs and HBLs. A simple yet commonly used criterion separating HBLs and LBLs is ; a nominal value is set to be around $\\alpha\\_{\\rm rx}'\\sim 0.75-0.78$ (defined between 5GHz and 1keV) suggested by various authors, with HBLs having smaller (flatter) values. We adopt 0.78 as the dividing line , which corresponds to =0.787 for the definition adopted in this paper and is represented by the dashed line in Figure[fig:alprox]. Following, a more elaborated locus for HBL is also indicated schematically, which is populated by X-ray selected BL Lac objects. Of particular interest, most of the objects offsetting from the classical FSRQ and LBL locus actually fall within the HBL locus[17](#fn17). This indicates that the broad band SEDs of these NLS1 AGN are similar to those of HBLs. Following our terminology introduced in §1, we refer to them as NLS1 HFSRQs, and those with SEDs typical of classical FSRQs (LBL-like) as NLS1 LFSRQs. These NLS1 HFSRQs have ubiquitously flat radio spectra whenever available, consistent with a potential blazar nature. Those without X-ray detection (arrows) have their and limits consistent with the and values for either LFSRQs/SSRQs or HFSRQs. Also, a few objects lie very close to the border line, on both the HBL and the LBL sides, suggesting a smooth transition between the two types. As a comparison, we also mark in the plot the position of 2MASXJ0324+3410, an extreme NLS1–blazar composite object, which has a HBL-type SED with the synchrotron peak in the UV/soft X-rays. Clearly, 2MASXJ0324+3410 fulfills the HBL classification in terms of its radio–optical–X-ray SED, as expected. We note there are a few objects lying in the vicinity of 2MASXJ0324+3410 in the –diagram, suggesting their similarities and thus the previously known NLS1–blazar hybrid 2MASXJ0324+3410 is not unique.\nFSRQs/LBLs and HBLs are known to differ in their X-ray spectral shapes as FSRQs/LBLs have relatively flat spectra while HBLs have steep spectra ; specifically, Γ ∼ 2.0 were found for FSRQs/LBLs and Γ ∼ 2.5 or steeper for HBLs in the ROSAT band. As such, the overall spectral shapes from the optical to the X-ray band are concave in FSRQs/LBLs ($>\\alpha\\_{\\rm x}$, where $\\alpha\\_{\\rm x}\\equiv \\Gamma-1$ is the energy index), while they are convex ($<\\alpha\\_{\\rm x}$) in HBLs. Parallelly, the X-ray emission of HFSRQs is hypothesized to be dominated by synchrotron radiation with a steep X-ray spectrum, as claimed to be seen in a few individuals. For the NLS1 HFSRQs in our sample with measured X-ray slopes, we find that all but one (J1633+4718) have steep X-ray spectra (Γ > 2.0). We also examine the optical–to–X-ray spectral shapes for the objects with available by plotting versus in Figure[fig:aox-ax]. As can be seen, the two objects (J1633+4718 and J0849+5108) with flat X-ray spectra < 1.0 show prominently concave optical-to-X-ray spectral shapes; whereas most of the steep X-ray spectrum objects are consistent with convex spectra, including 2MASXJ0324+3410 with Γ ≃ 2.2. We conclude that some of our radio-loud NLS1 AGN have SEDs similar to those of LBLs/RLQs, while the remaining similar to HBLs/HFSRQs. The X-ray spectral shapes are also broadly consistent with these two types of SED.\n [fig:aox-ax]\nFinally, in Figure[fig:sed] we plot the broad band SED in the log*ν* − log*ν**L**ν* representation for the 14 objects with X-ray detection. Though there is only one broad radiation hump detected (due to a lack of hard X-ray and *γ*-ray data), they do reveal some similarities with the low-energy (synchrotron) hump characteristic of blazars. In particular, some of the HBL-like candidates are found to have the radiation hump peaked around the UV, supporting the results from the above –distribution; these include J1146+3236, J1238+3942, J1644+2619, and J1722+5654, etc. They also show SEDs resembling that of 2MASXJ0324+3410.\n[fig:sed]\nEmission lines and AGN eigenvectors\n===================================\nThe Eigenvector-1 space involves primarily a set of well known correlations among the following parameters: the line width *F**W**H**M*(),,, and soft X-ray spectral index. is the relative strength of the multiplets in terms of the -to-flux ratio, *R*4570 ≡ /H*β*, where is the flux of the multiples in the range of 4434–4684Å, and is the total flux. is the strength of relative to that of. Here we examine these correlations for our RL NLS1 galaxies, and compare them with those for the large NLS1 sample of Zhou’06, which are predominantly radio-quiet (plotted as small dots in the figures in this section).\nFigure[fig:feii-fwhm] shows the correlation between the line-width and the relative strength of ; again, a significant anti-correlation is present for the RQ NLS1 sample (with a large scatter though), as shown in, whereas no significant correlation is found for the RL objects. A similar result is found for the correlation between and, as shown in Figure[fig:fe-o3]. While an anti-correlation between the and strength is evident for ‘normal’ NLS1 galaxies, there appears no such correlation for the RL objects. However, it should be noted that, given the weakness of most of these correlations found for the RQ sample, our RL NLS1 sample is too small to draw statistically significant inferences. Also, we note that the high values for the RL objects span a relatively narrow range of in the diagram, which may partly account for the lack of the –correlation.\nAnother well known Eigenvector-1 correlation is the anti-correlation between soft X-ray photon index Γ and line-width, that was found to extend down to *F**W**H**M* ∼ 1000. We examine this property for the 14 RL NLS1 galaxies with measured Γ, and find that they do not follow the Γ–*F**W**H**M* trend defined by their RQ counterparts. However, the sample size is small, and the scatter in the known relation is huge. Alternatively, the finding could be attributed to a different X-ray radiation mechanism dominating the ROSAT PSPC band in RL and RQ NLS1 galaxies (see below).\nWe also search for potential correlations between radio-loudness and as well as within our RL NLS1 sample; no significant correlations are found, however.\nIn the process of optical spectral analysis, we have noticed that the multiplets are generally strong in our RL NLS1 galaxies. We plot in Figure[fig:feii] the distribution of, and, as a comparison, also the distribution for the NLS1 galaxies in the Zhou’06 sample (normalized to the number of RL objects). The two-sided K-S test testing if the two distributions are drawn from the same population yields a probability level of 0.2% only; it is evident that the RL NLS1 galaxies have systematically higher values than their RQ counterparts. The average is 1.03, compared to 0.82 for the ‘normal’ NLS1 galaxies with reliable measurements in the Zhou’06 sample. To examine whether the large values are caused by systematically weak (total) emission, we compare the flux distribution of the RL and RQ samples; we find that the two flux distributions are indistinguishable between the two samples (the K-S test). Moreover, the RL NLS1 sample has *F**W**H**M*() and distributions indistinguishable from those of the Zhou’06 sample (the K-S test). This result, together with the absence of the anti-correlations of –FWHM() and –indicate that the high values for the RL sample are real, rather than spuriously induced from dependence on other parameters. We thus conclude that the optical emission in our very RL NLS1 galaxies is on average stronger than that in RQ NLS1 galaxies. In fact, this effect has been noted in the sample of, but the statistical significance is not as high due to a factor of 2 smaller of the sample size.\nWe have identified four blue outliers in our sample, with the lines significantly blueshifted by *V*([]) > 200relative to [] [18](#fn18). The phenomenon of blue outliers has been observed in a few radio galaxies, and is especially frequent among NLS1 galaxies. We find a fraction of blue outliers with *V*([]) > 200of 17% among our sample, about a factor of 2.4 higher than that in the RQ NLS1 sample of. These authors also discovered that, for RQ blue outliers, the velocity offsets are correlated with ionization potential and line width. Such correlations persist for RL blue outliers. In all four we detect blueshifted [] lines with velocity shifts up to ∼ 2300(J1443+4725). The high frequency of blue outliers in RL NLS1 galaxies may indicate that jet-cloud interactions contribute to, or dominate, the *λ*5007 blueshifts in the RL sources.\nMost of the objects show symmetric profiles of the *λ*5007 line. In three objects, particularly J1633+4718, the doublets show a blue wing that can be modeled by an extra Gaussian component (see Figure[fig:sdssspec]). This indicates the presence of possible outflows associated with the narrow line region. In addition, J103123.73+423439.3 shows double-peaked profiles in both *λ*5007 and.\n [fig:feii-fwhm]\n [fig:fe-o3]\n [fig:feii]\nHost galaxies\n=============\nThe host galaxies of radio-loud NLS1 AGN are poorly understood due to very sparse observations. The only object, as far as we are aware of, observed with the HST is 2MASXJ0324+3410. Its HST image reveals a ring or one-armed galaxy morphology, indicating a possible galaxy interacting/merging history. As for our current sample, given the relatively high redshifts, most of their host galaxies cannot be examined with the SDSS images. The exceptions are J1633+4718 and J1644+2619, which have redshifts of 0.116 and 0.144, respectively. Interestingly, the host galaxy of J1633+4718 has a size of more than 20kpc along the major axis and is in an interacting pair/merger system, in which the second galaxy hosts a reddened starburst nucleus. Both of the member galaxies appear to be gas rich with disturbed morphology, and thus of late morphological type. The host galaxy of J1644+2619, though being compact ( ∼ 15kpc), also seems to be disk-like. We note that the black hole masses in both galaxies are small, 2.0 × 106 and 8.4 × 106for J1633+4718 and J1644+2619, respectively (see §[sect:mbh]).\nThe SDSS spectrum of J1633+4718 includes a significant contribution of the host galaxy light within the fiber of 3 diameter, which appears to be dominated by a young stellar population, as can be seen in Figure[fig:sdssspec] (the contamination from the starburst nucleus of 4away should be negligible). Besides, marginal imprints of high-order Balmer absorption lines, though being week, are likely present in the SDSS spectra of other three objects at higher redshifts (J1146+3236, J1505+0326, and J1634+4809; *z* = 0.4 − 0.5), indicating possible young stellar populations in their host galaxies.\nWe thus suggest that some of the radio-loud NLS1 AGN (at least those with a small black hole mass; see §[sect:mbh]) may reside in gas-rich, disk galaxies, and some reside in galaxies with possible young stellar populations. This result contrasts clearly with classical RLQs and BL Lac objects, that preferentially reside in giant elliptical or massive bulge-dominated galaxies, or strongly passively evolving galaxies. While it is, to some extent, in line with the finding of and for the Seyfert galaxies in their samples, which are much less luminous and more sub-Eddington than our radio-loud NLS1 AGN studied here, however. As is shown in §[sect:mbh], most of the objects in our sample have relatively small black hole masses ( ∼ 106 − 8). Hence, considering the –spheroid mass relation known for nearby galaxies, our finding about the host galaxies of radio-loud NLS1 AGN is not unexpected. Deeper observations are needed to reveal the host galaxy properties for the remaining radio-loud NLS1 AGN with larger, which are not an easy task given their relatively high redshifts, however.\nInterpretation and discussion\n=============================\nThe sample\n----------\nOur objects fulfill the conventional definition of NLS1 galaxies, and show typical NLS1 characteristics such as strong optical emission. They also share similar distributions in black hole mass and Eddington ratio with ‘normal’ NLS1 galaxies (see §[sect:mbh]). Therefore, they should be bona-fide NLS1 galaxies. The 5GHz radio luminosities $L\\_{\\rm 5GHz}$ range from 1024.17 to 1026.96WHz− 1 with a median of 1025.23. Thus, most of our objects would remain qualified as radio-loud even if the alternative criterion of RL AGN is applied, which is based solely on radio luminosity, $L\\_{\\rm 5GHz} \\ga 10^{25}$WHz− 1 as proposed by some authors. Of our sample, 19 were spectroscopically targeted in the SDSS as quasar candidates based on their optical colors, and 3 were targeted serendipitously; only one object was selected solely as a ROSAT/FIRST counterpart. Hence our very RL NLS1 sample can be regarded as an essentially optically-selected one, similar to other SDSS radio-quiet NLS1 samples.\nThe sample spans a luminosity range from Seyfert galaxies to quasars, with roughly half of the sample in each category. Compared to the ‘normal’ NLS1 galaxies in the Zhou’06 sample, which have an average luminosity ⟨⟩ = 3.0 × 1042, our RL objects have relatively higher luminosities (⟨⟩ = 6.0 × 1042, see also Figure[fig:lo-lb]). This difference is likely a consequence of the trend that RL objects are more likely to be found in AGN with more massive black holes (and thus higher luminosities), as suggested recently. We find that such very RL objects represent only 0.7% of NLS1 galaxies drawn from the SDSS, though this fraction may be somewhat uncertain given all kinds of selection effects introduced in the surveys.\nCompared to the other two previously published RL NLS1 samples of and, which comprise 11 and 16 objects respectively, our current sample differs mainly in that it has a much higher radio-loudness cutoff. Consequently, the sample is biased for selecting flat-spectrum sources dominated by beamed radio emission. In addition, our sample has a statistically higher redshift distribution than those two samples. In comparison, only three and four objects from those two samples satisfy > 100, respectively, among which, two from the former and one from the latter sample are included in our current sample.\nBroad band continuum emission\n-----------------------------\n### Radio emission: relativistic beaming and jets\nThe radio luminosities of our sample objects are well above those expected from the most radio-luminous starburst galaxies ∼ 1022.3 − 23.4WHz− 1. In addition, no imprints of vigorous starbursts are found in the optical SDSS spectra of sources of our sample; these spectra sample the central region of the host galaxies within a radius of 3–11kpc. None of the objects are found to be associated with IRAS sources except for J1633+4718, for which the infrared emission is expected (at least predominantly) from the starburst nucleus 4 away, rather than from the active nucleus, however. These results suggest that for essentially *all* of the objects the radio emission is of AGN origin, rather than of starburst origin.\nLarge amplitude flux variations up to ∼ 70% have been observed in some of our objects with flat radio spectra. To make sure that this is not spurious due to contamination of nearby sources, we visually examined the FIRST and NVSS images of the variable objects. We find that for 6 of the 8 objects no other sources are detected within 3; for the remaining 2, faint sources are present but are all well separated from them. Therefore, the observed variations are not caused by flux contamination from nearby sources. In addition, we argue in this paper (see Appendix[sect:riss] for details) that those observed variations are largely intrinsic, rather than extrinsic that is caused by refractive interstellar scintillation (RISS).\nWe have shown that most of the sources known to be flat-spectrum in our sample have very high brightness temperatures 1011 − 13K; in 4 of the objects $T\\_{\\rm B}$ even exceeds the inverse Compton limit of 1012K. Such high brightness temperatures indicate that the radio emission is highly Doppler beamed and hence must originate from (at least mildly) relativistic jets, as in blazars. Among the 4 with $T\\_{\\rm B} > 10^{12}$K, J0849+5108 is a known OVV/HPQ blazar and the high brightness temperature is well expected; in turn, this gives us confidence in the results for the other objects. J0948+0022 has been previously suggested to be so by. For the remaining two, J1505+0326 and J1548+3511, the evidence for relativistic beaming is presented for the first time. For J1505+0326, this result is consistent with its VLBI images made at 2.3GHz, which are dominated by a very compact component, marginally resolved at a few tens milli-arcsec resolution. Besides, two objects, J1633+4718, and J1644+2619 have also been suggested to be blazar-like based on VLBI observations. Thus, in a total of 6 objects of our sample there is evidence for relativistic beaming. Assuming that the brightness temperatures of 1011 − 12K are probably also an indication of beaming, the number of such objects will increase to 9, i.e. all the 8 flat-spectrum sources and one with unknown radio index.\nOur suggestion for the presence of (at least mildly) relativistic jets is also supported by polarization detected in a few individual objects, e.g. J0849+5108 (see Appendix[sect:indv]) and 2MASXJ0324+3410. Furthermore, this is consistent with the observed large amplitude variability in the optical in a few objects monitored, e.g. J0849+5108, J0948+0022, since normal NLS1 galaxies are found to be much less variable in the optical band.\nThese results are most naturally explained in terms of beamed emission; and thus we postulate that relativistic jets are likely present in most, if not all, of the flat-spectrum, very radio-loud NLS1 AGN. However, it should be noted that part of the observed properties above may also be explained by some other non-blazar, compact sources. For instance, flat and inverted spectra are also typical of Gigahertz-Peaked Spectrum (GPS) sources, though they generally lack of large amplitude flux variations. Moreover, flat spectra are often seen also in the radio sources of some ‘normal’ Seyfert galaxies, though mostly with much less power compared to the objects studied here. Incorporating the properties of these flat-spectrum NLS1 AGN in other wave-bands, we regard the blazar-like scenario as the most likely origin of their multi-waveband (non-thermal) emission in this paper, though other possibilities cannot be ruled out completely at this stage. We note that the majority of the sources of sample ( > 10) are steep-spectrum sources, in which strong beaming is generally not expected. The difference is largely attributed to the much higher radio-loudness cutoff of our sample, as mentioned above.\n### Optical continuum: dust extinction and jet contribution?\nThe majority of our sample objects (16/23) have optical continuum slopes bluer than = − 1. If the continuum is intrinsically blue in all objects (assuming = − 1.7) but is reddened by extinction, a slope of = − 1 corresponds to an amount of extinction E(B-V) ≃ 0.2, i.e. a correction factor of only ∼ 2 in the *B*-band flux. So, even if this was the case, most of our objects would still have $\\ga$100, above the radio-loudness cutoff of our sample. There are 5 objects at low redshifts (*z* < 0.38) having both the and lines measurable, including J1138+3653, the only one showing a significant deviation from a power-law optical continuum. J1138+3653 has a broad line Balmer decrement of /=4.43, indicative of mild extinction. Adopting the intrinsic /ratio of 3.37 [with a dispersion of 0.06dex in log(/)] for blue radio-loud quasars (free from dust extinction) found in our recent work and the Galactic extinction curve, we find an internal extinction correction $A\\_{\\rm B}\\simeq 0.97$ [E(B-V)=0.236], and an extinction corrected a factor of 2.4 lower, leading to an intrinsic radio-loudness ≃ 86. For the remaining 4 objects, the broad line /ratios range from 2.8–3.8 with a mean of 3.4, that are well consistent with the intrinsic /distribution and hence with little or no extinction; therefore their radio-loudness measurements as above are almost un-affected. Particularly, J1031+4234 shows a red optical spectrum with =0.55, indicating that in at least some of the objects the red optical spectrum is intrinsic, rather than resulting from dust extinction. Another example is the blazar J0849+5108 with the reddest slope, = − 0.20, that is close to the previous measurement at its high light state. Its slope steepens with decreasing luminosity and changes rapidly on timescales on several months, and hence the red slope is preferentially explained as being intrinsic[19](#fn19), similar to red BL Lac objects.\nObservations have revealed that in a large fraction of FSRQs the optical light can be contributed significantly by the non-thermal emission from relativistic jets that is highly beamed. We have discovered that the observed optical fluxes are systematically in excess of those predicted from the broad luminosity, presumably from an accretion disk, assuming that radio-loud NLS1 AGN have the same conditions in the broad line region as normal NLS1 AGN (see §[sect:optcont]). In addition, the optical slopes of our radio-loud objects are systematically bluer than that of radio-quiet NLS1 AGN. These facts point to the presence of a second component superposed on the normal AGN-like (thermal) continuum. In light of the similar radio properties of our RL NLS1 galaxies to FSRQs, we attribute the excess optical emission to arising from the non-thermal radiation from jets, as in FSRQs, i.e. the extension to higher energies of the radio-to-infrared synchrotron radiation. This model explains naturally the observed difference in the observed-to-predicted luminosity ratios between flat- and steep-spectrum NLS1 AGN. In flat-spectrum sources, the relativistic beaming effect is strong—as suggested above—and hence the jet component over-shines the thermal optical component; while in steep-spectrum sources, the former becomes comparable to or weaker than the latter.\nThe observed optical continua have a variety of slopes, ranging from moderately red to blue. They are reminiscent of the optical emission of BL Lac objects, which have both red and blue continua, corresponding to the low- and high-peak frequency of the synchrotron hump of LBLs and HBLs, respectively. In fact, this comparison is particularly relevant in light of our finding that the RL NLS1 AGN have both LBL- and HBL-like SEDs: the blue optical continuum may be explained as contribution from the synchrotron emission peaked at high energies (UV or higher); while the red slope may be contributed by that peaked in the infrared. In turn, this is consistent with our above inference that the red slopes in our objects are mostly intrinsic, rather than resulting from extinction.\n### X-ray radiation and broad band SED\nSo far, we have shown that relativistically beamed jet emission, as found in blazars, provides a good explanation of the observed radio and optical properties in flat-spectrum RL NLS1 AGN. This scenario is also consistent with the fact that some of the objects show SEDs similar to FSRQs/SSRQs, while the others similar to HFSRQs in the,, and parameter space (Figure[fig:alprox]). In fact, a few of these objects were simply classified as FSRQs or SSRQs in previous studies. It is thus natural to speculate that the similar radiation mechanisms are operating in these objects as in normal FSRQs/SSRQs and HFSRQs for producing the broad-band non-thermal emission.\nFrom the soft X-ray spectral shapes alone it is not possible to distinguish the origin of the X-ray emission, i.e.that associated with accretion disk as in normal RQ NLS1 galaxies or non-thermal jet emission as in blazars, since RQ NLS1 galaxies also show a similar Γ distribution with a large scatter. The only exception is perhaps the flat hard X-ray component (Γ = 1.37 ± 0.49) in J1633+4718, whose best estimate value is hardly found in RQ NLS1 galaxies but is typical of FSRQs.\nOne way is to search for possible enhancement in the X-ray emission of RL NLS1 AGN compared to RQ objects in a statistical sense, which, if is the case, may be considered as an indirect evidence for the presence of a non-thermal jet X-ray component. Here, we simply use the X-ray detection rate in the RASS as an indicator of possible enhancement in the X-ray emission of our objects over normal NLS1 galaxies by comparing the RL and RQ samples. This is plausible since our RL NLS1 sample is essentially an optically selected one (see discussion in §[sect:dissample]), as the SDSS RQ NLS1 samples. The RASS detection rate for RQ NLS1 galaxies is ≈ 31% from the SDSS DR3 NLS1 sample ; while it is 50( ± 11)% (11/22) for the present RL sample (excluding one solely RASS-selected target). If we consider the objects (both detections and upper limits) falling within the HBL locus only (Figure[fig:alprox]; excluding 2MASXJ0324+3410), the detection rate is even higher 58( ± 14)% (7/12). Therefore, the higher RASS detection rate in our RL sample compared to the RQ NLS1 sample is suggestive of the presence of a significant excess X-ray component, presumably from the relativistic jets.\nIn Appendix[app:xray], we discuss in detail the possible mechanisms of the X-ray radiation of our RL NLS1 AGN, which are summarised here. In all of the objects, the X-rays produced in the accretion process as in normal RQ NLS1 galaxies are likely not negligible and make a considerable contribution. In addition, we suggest that the non-thermal X-ray emission from relativistic jets may also play a similarly important role. Such a component is most likely to originate from inverse Compton radiation and has a flat X-ray spectrum (in J1633+4718 and possibly in J0849+5108), as seen in FSRQs, for objects having the LFSRQ-like SEDs (NLS1 LFSRQs); whereas it is possibly the high-energy tail of synchrotron radiation with a steep spectrum for objects with the HFSRQ-like SEDs (NLS1 HFSRQs).\n### Dichotomy of LFSRQ- and HFSRQ-like NLS1 AGN\nIn Figure[fig:alprox], there are about 7–12 objects, or 1/3–1/2 of the sample, falling within the box-shaped locus and thus having HBL/HFSRQ-like SEDs (the uncertainty comes from those with lower limits). Considering that radio/optically-selected (‘broad’-line) FSRQs have predominately LBL/LFSRQ-like SED, such a high fraction of HFSRQ-like objects is surprising for our sample, which is essentially optically/radio-selected. The figure also shows that there are no objects resembling extreme HBL/HFSRQs (the lower-left corner of the HBL locus), which have the synchrotron component peaked in the X-ray regime. In fact, nearly all the candidate HFSRQ-type SEDs have the synchrotron humps peaked likely around the UV band.\nStrong emission lined–FSRQs with HBL-type SEDs have recently been reported as a result of the identification of radio–X-ray (ROSAT) selected sources. One major distinction with our objects is that those objects have much broader lines with *F**W**H**M* > 2000, typical of classical FSRQs. The fraction of HBLs/HFSRQs is sample-selection dependent; the authors proposed a fraction (HBLs + HFSRQs) of about 10–15% among the whole blazar population and about 30% in the RASS–Green Bank (RGB) sample. The RGB sample is comparable in the way of selection to our RASS-detected sub-sample here, which comprises 12 objects; among them, 8 objects are possibly of the HFSRQ-type (Figure[fig:alprox]), making the fraction as high as ∼ 2/3. It seems that the HFSRQ fraction of our radio-loud NLS1 sample is higher than that of HFSRQs in.\nWithin the unified blazar sequence, the energy of the synchrotron peak was found to be well anti-correlated with 5GHz radio luminosity[20](#fn20), and with, which was suggested to be a tracer of the peak energy. We plot in Figure[fig:arx-lr] versus 5GHz radio luminosity for the sample, with those objects without X-ray detection indicated by arrows. It clearly reveals a good correlation; the Spearman correlation test incorporating lower limits of using ASURV yields a significant correlation at the probability level PS. =0.4%. It can be seen that the transition between LFSRQs and HFSRQs (using ≈ 0.78) happens at around *ν**L**ν* ∼ 1042, consistent with what was previously found between HBLs and LBLs. If does trace the synchrotron peak energy, this result shows that our radio-loud NLS1 AGN do follow the blazar sequence well. In fact, the objects at the low luminosity end are among the least radio luminous blazars known so far, falling close to or even below the low-end of the FSRQ luminosity function according to unification schemes. However, there seems to be no or at most a weak correlation between and luminosity (PS. =0.1). Such a blazar sequence also explains naturally the high fraction of HFSRQs, as the majority of our sample have lower radio powers compared to classical FSRQs. In fact, the objects of our sample have intrinsic powers mostly in the range for FRI type radio sources (see §[sect:intrpower]).\n [fig:arx-lr]\nIntrinsic radio power and radio-loudness\n----------------------------------------\nOptically thin radio emission at low frequencies is thought to be isotropic, and thus can be used as a measure of the intrinsic power of the radio sources free from Doppler boosting. The low frequency power of radio galaxies can be separated into two distinct classes at the fiducial value $L\\_{\\rm 178MHz}\\simeq 2\\times 10^{25}$WHz− 1sr− 1: the FRIs with the radio emission decreasing outwards away from the nucleus, and the FRIIs which have lobes with prominent hot spots and bright edges. It has been suggested that RLQs are found in FRII sources, and the distinction of the FRI and FRII types is likely determined by the accretion process of the central engine, as well as by the density of the environment. Despite a lack of adequate observations on their radio morphology, it would be informative to compare the isotropic radio power of our RL NLS1 galaxies with the FR types.\nWe estimate the rest frame 151MHz luminosities,, for our objects, based on their radio data at low frequencies, which are summarised in Appendix[sect:lowfemi]. For 4 objects with 151MHz fluxes, are calculated directly by using the 151–327MHz indices for K-correction. Among the rest, 11 objects have fluxes at 327MHz, for which are estimated by extrapolating the spectra down to 151MHz assuming the average = − 0.34 found from the above 4. Using the steepest ( − 0.62) and the flattest ( + 0.09) found above would increase and decrease by a factor of up to 1.4 and 1.8, respectively; and thus the estimated are relatively insensitive to the choice of. For the 15 objects with either detected or estimated 151MHz fluxes, range from 4.5 × 1024 to 3.5 × 1027, with a median of 7.1 × 1025. Among these, three have luminosities in the FRII range (J1047+4725, J1305+5116, and J1443+4725, all detected in the 151MHz survey); while the remaining majority have similar luminosities as FRIs or are close to the transition line. With estimated =9 × 1024, 2MASXJ0324+3410 also has a luminosity typical of a FRI galaxy. However, it is not clear whether the clear separation in both radio power and morphology for radio galaxies is applicable to RL NLS1 galaxies, considering their apparently peculiar, compact morphology.\nIt has been suggested that, in classical RL AGN, isotropic radio emission at low frequencies can be used as a good indicator of the power of the jets. If this is applicable to RL NLS1 galaxies, the above results imply that our sample has much lower jet power compared to classical RLQs, which have radio power and morphology predominantly in the FRII category. We will address this issue in a detailed study in the future.\nThe radio-loudness, by its definition, is largely an apparent rather than intrinsic parameter for the sources in which the beaming effect is important above 1GHz in the radio but less so in the optical band. As a consequence, some radio-loud AGN would have been classified as radio-intermediate or even radio-quiet, which might also possess weak and mildly relativistic jets in at least some, were their jets not directed close to our line-of-sight. A more physically meaningful parameter is the ‘intrinsic’ radio-loudness free from the beaming effect. We calculate the intrinsic radio-loudness using the un-beamed radio component, that can be extrapolated from a low-frequency flux and dominates fluxes at several GHz after the jet emission is de-boosted. The intrinsic radio-loudness at 5GHz, $R\\_{\\rm 5GHz}^{\\rm intr}$, is estimated by using 327MHz fluxes for those having this flux measurement. We find $R\\_{\\rm 5GHz}^{\\rm intr}$ in the range 34–2600 (with a median of 110) assuming a spectral slope of − 0.7 for the un-beamed component, or 17–1380 (with a median of 56) assuming a slope of − 1. It should be noted that such estimates are subject to large uncertainties, partly due to the above finding that the optical light is also partially beamed in some of the objects. Nevertheless, they seem to indicate that a sizable fraction of the sample, including 2MASXJ0324+3410, are likely radio-intermediate. Among the sample, there are almost no genuine intrinsically RQ objects with *R* ≪ 10, however.\nBlack hole masses of radio-loud NLS1 galaxies\n---------------------------------------------\n### Estimation of black hole mass and Eddington ratio\nIt has become possible recently to estimate the black hole masses of AGN from single epoch spectroscopic observational data, though the uncertainties of both systematics and scatter remain large. Assuming that the motion of the line-emitting clouds are virialised, the black hole mass can be determined by $M\\_{\\rm BH} = f R\\_{\\rm BLR} \\Delta v^2 / G$, where *f* is a scale factor depending on the kinematics and geometry of the BLR, Δ*v* the emission line width, and $R\\_{\\rm BLR}$ the BLR radius. We assume that the commonly used $R\\_{\\rm BLR}$–luminosity relation, derived via reverberation mapping for nearby, normal AGN, is applicable to NLS1 AGN. We use the relation for the luminosity given by ; we intend not to use the continuum 5100Å luminosity for possible contamination from jet emission (see Section[sect:optcont]). A common practice, among a few others[21](#fn21), is to take *F**W**H**M* as Δ*v* and $f=(\\sqrt{3}/2)^2=0.75$, assuming an isotropic distribution with random orbital inclinations of the broad line clouds. We follow the practice in this work, the same as in.\nThe estimated black hole masses of our sample range from 1 × 106 to 1.5 × 108, peaked at near the median 1.3 × 107. The black hole mass distribution is well consistent with that of the sample. However, by studying the systematic effects in estimation, argued that the use of *F**W**H**M* rather than line-dispersion ($\\sigma\\_{\\rm line}$) would introduce systematic bias in the scaling factor *f*. We also give alternative estimation by correcting this effect using the formalism proposed by. For the objects in our sample (FWHM < 2200) specifically, this correction sets a value of the scale factor *f* ≃ 1.5, about twice the value in the former estimation. This results in estimates a factor of 2 larger than the values obtained above, ranging from 2 × 106 to 3 × 108with a median 2.6 × 107. The black hole masses thus estimated are listed in Table[tbl:arox], which are used in the following analysis.\nHaving estimated the masses of the central black holes, now we can derive the Eddington ratios,. For bolometric luminosity correction, we assume $L\\_{\\rm bol}$=9*λ**L**λ*5100, as suggested by, e.g. ,, and. The *λ**L**λ*5100 luminosity is estimated from the broad line luminosity using Eq.(5) of to eliminate the effect of the extra jet contribution. It should be noted that such an estimate is subject to large systematic uncertainties inherent in estimation of, as well as of bolometric luminosity. Using the values obtained with the scale factor *f* = 0.75 (see above), we find the Eddington ratios ranging from =0.52 to 2.6 and peaked around a mean of 1.0, similar to those found by with a mean of 1.2. However, when the correction of is applied, the values become about twice larger and, consequently, the obtained values become about twice smaller, i.e.ranging from 0.27 to 1.28 with an average of 0.52.\nNow we compare the distributions of and between our RL sample and the normal NLS1 sample of Zhou’06. Our RL sample has significantly higher and distributions compared to the RQ one. This agrees with the general trend that, as aforementioned in §[sect:dissample], RL objects are more likely to be found in AGN with massive black holes. Given their narrow ranges of the line-width, the black hole masses of NLS1 samples, as determined from the /line-width and luminosity, is strongly correlated with luminosity, and in turn, with redshift for flux limited samples. This may explain the higher redshift distribution of our RL sample compared to that of the normal NLS1 sample of Zhou’06 (see Figure[fig:distrl]).\nIt should be pointed out that the virial estimation is subject to assumption of an isotropic geometry of the BLR with random orbital inclination of the clouds. If the BLR is flat, a small line width would be a result of small inclination, and then would be largely underestimated. Indeed there are some suggestions in the literature for a flattened geometry of the BLR in some RL AGN ; but see. This caveat concerns particularly those of our objects, that have compact radio morphology and those that are blazar-like, indicating a small viewing angle. Given that the distribution of our sample is indistinguishable from that of the RQ NLS1 sample (within the same redshift range), this comes back to the old question whether NLS1 galaxies, as a population, have flattened BLRs and are viewed preferentially face-on, rather than having small ; but see e.g. .\nIf RL NLS1 galaxies do possess relativistic jets and hence anisotropic radio emission, as we have shown here, they can be used as a probe to this question since their inclination can be inferred from radio observations in a statistic manner. However, with the relatively few data available currently, we do not find any statistically significant trends of the dependence of line-width () on probable inclination indicators, such as radio-loudness or radio slope. Therefore, though there is no compelling evidence supporting the planar BLR nature of RL NLS1 from the present data, we cannot rule out this possibility, either. A much larger RL NLS1 sample covering a wide range of radio-loudness is desirable to shed light to this important question in the future.\n### Radio-loudness versus black hole mass\nAs one of the fundamental relations in understanding AGN jets, the dependence of radio-loudness *R* on has been studied extensively in recent years, with the advent of the easy estimation of in AGN. A *R*–correlation was proposed by that all RLQs in the PG quasar sample have > 3 × 108, while the radio-quiet quasars have a much lower distribution. This trend was confirmed in some studies, but was found to be weak or absent in others. Furthermore, by including Seyferts and low luminosity AGN, found that *R* is mostly correlated with rather than. In a recent comprehensive treatment of this question, demonstrated that *R* depends on both and, with two distinct inverse *R*–dependence sequences for AGN hosted by elliptical (systematically higher *R*) and disk galaxies (systematically lower *R*), respectively. Regardless whether there is indeed a trend of dependence or not, most aforementioned studies show that, on the –*R* plane, there is a scarcity of RL AGN with low. Even for those low-RL AGN suggested in the literature, there remains a controversy as to the estimation of and *R*.\nObviously, as first pointed out by and, radio-loud NLS1 AGN do *not* follow the above trend found for normal RL AGN (provided the virial as estimated above are largely correct). Moreover, demonstrated explicitly that RL NLS1 galaxies are located in a region of the –*R* diagram that is sparsely populated by other types of AGN (*R* ≃ 101 − 3 and < 108).\nSimilarly, we plot versus for our very RL NLS1 galaxies in Figure[fig:mhb-rdln]. Also plotted are the loci of the objects in the samples of and for comparison. Two pieces of results can be inferred immediately from Figure[fig:mhb-rdln]. Firstly, within our sample, there is no correlation found between and, nor between and (PS. =0.44 and 0.27, respectively); however, we note that these results are rather uncertain given the small sample size and the limited ranges in both and. Secondly, our new result confirms and strengthens that of, and improves upon it by extending the RL NLS1 locus to even lower and higher. In particular, the bulk of our RL NLS1 galaxies lies within a locus of ≃ 106 − 7.5and *R* ≃ 101 − 3, which appears to be preferentially avoided by other types of RL AGN [compared to, e.g., see their Figure4], except for low power Seyfert galaxies and LINERs; however, those exceptions have exclusively low Eddington ratios < 10− 2 and are drastically different from NLS1 with high. Compared to Figure4 in, our RL NLS1 galaxies appear to have the largest radio-loudness up to ∼ 103 − 4 among all types of AGN within this range. This means that the RL NLS1 galaxies show strong excess radio emission with respect to their, when the suggested radio power–relation for normal AGN is compared[22](#fn22). Finally, we note that the above two results remain valid even if the Doppler de-boosted radio-loudness values (see §[sect:intrpower]) are used instead, which are in the range of *R* ≃ 101 − 3.\n[fig:mhb-rdln]\nHence, the existence of RL NLS1 galaxies as a *population* with small, if estimated correctly, fills up the previously sparsely populated region in the –*R* parameter space for normal RL AGN. This result indicates that there indeed exist AGN with high radio-loudness (*R* ∼ 102 − 3) but intermediate (106 − 8). Though there is the systematic uncertainty in estimating ascribed to the possible BLR inclination effect, we argue that this result may not be affected qualitatively. This is because the Komossa et al. sample, though composed of mainly steep spectrum sources (thus relatively larger inclination), led to similar results. Furthermore, in a few objects where the host galaxies are resolved, their small are supported by a lack of massive galactic bulges, as predicted by the –bulge relation. Our result also implies that the inverse relation between *R* and as suggested previously may not work at the high end close to the Eddington rate. Furthermore, argued that RL AGN hosted by elliptical galaxies are much radio-louder than AGN hosted by disk galaxies. Though this statement may be true in general, it is perhaps more appropriate in terms of RL fraction, i.e. the fraction of RL objects is lower in AGN hosted by disk galaxies compared to that of AGN hosted by ellipticals; in addition, the difference in radio-loudness is now less than what was thought previously.\nOn the nature of radio-loud NLS1 galaxies\n-----------------------------------------\n### A previously unknown population of radio-loud AGN?\nOn the black hole mass–radio-loudness plane our RL NLS1 galaxies form a natural extension to low of classical RLQ/BLRGs (Broad Line Radio Galaxies) with roughly constant radio-loudness. This is not surprising, since the transition between BLAGN and NLS1 AGN is *smooth* in the line-width of the broad lines, and, as a consequence, in black hole mass and luminosity. If the are underestimated systematically (see §[sect:mbh]), the loci of the radio-loud NLS1 AGN will move even closer to those of normal RL AGN. In this regard, it is natural to treat RL NLS1 galaxies as the low–end of normal RL AGN.\nOn the other hand, however, RL NLS1 galaxies show some peculiarities in observed properties that are unusual in comparison with, or at an extreme of, normal RL AGN. The most distinctive features are in the radio properties. Even for steep-spectrum sources, the radio sources of RL NLS1 galaxies appear to be compact, with the size less than several tens of kpc or even smaller wherever higher resolution observations are available. In contrast, the radio sources of RLQ/BLRGs are about 100–400kpc in a similar redshift and luminosity range as our radio-loud NLS1 AGN. Moreover, most of the RL NLS1 galaxies have radio power in the range of FRI radio galaxies, whereas RLQ/BLRGs tend to be associated with FRII type sources[23](#fn23).\nIn the following aspects the two types are not mutually exclusive but overlap to some extent, yet they seem to differ in a systematic way. Firstly, There seems to be a higher fraction of HBL/HFSRQ-like blazars in optically selected RL NLS1 galaxies than in normal RLQ/BLRGs of radio/optically selected samples, though further confirmation using complete samples is needed. Secondly, as commonly suggested, NLS1 AGN are accreting at close to, or larger than the Eddington limit. At such high rates the accretion flow proceeds via a slim disk where energy is partially advected into the black hole before being radiated due to relatively long cooling time. Thus the accretion processes in radio-loud NLS1 AGN may differ drastically from those in normal RLQ/BLRGs, where a standard thin disk is believed to be operating, though they do overlap at ≃ 1 to some extent. The final point—on the host galaxies—is only suggestive because of a lack of sufficient evidence. RL NLS1 nuclei, at least some, seem to be hosted in disk or interacting galaxies, and/or galaxies with relatively young stellar populations. In contrast, classical RLQ/BLRGs [24](#fn24) are preferentially hosted in giant elliptical/massive bulge dominated galaxies or at least strongly passively evolving galaxies, although there was also evidence reported for star-formation in some hosts beyond $z \\ga 0.4$.\nIn consideration of their unusual properties, we are left with two options regarding the nature of RL NLS1 galaxies: either they are a downsized extension of normal RL AGN to the lower-(and perhaps higher-) range, or they form a previously unknown population distinct from normal RL AGN. The former would invoke some unknown processes that can explain the observed differences between RL NLS1 and normal RL AGN with the smooth change of. For the latter, the operating physical processes in likely black hole accretion and jet formation may be radically different for the two classes. A related, interesting question is whether the differences in the observed properties between RL NLS1 and normal RL AGN can be naturally explained by the same underlying physical processes that make NLS1 different from normal AGN, or whether some additional mechanisms are needed. To further understand these questions, a large sample of RL NLS1 galaxies with the full range of radio-loudness is needed, as well as extensive observations, especially in the radio band. As complementary discussion, we also compare them with two special types of RL AGN below, namely, BL Lac objects and compact steep-spectrum sources. As will be seen, RL NLS1 galaxies may be a heterogeneous population themselves.\n### Relationship with BL Lac objects\nOne object in our sample, J0849+5108 (0846+51W1), was formerly classified as a BL Lac object at its high light state based on previous observations (see Appendix[sect:indv] for details). As such, it presents an interesting case linking possibly RL NLS1 galaxies with BL Lac objects, as pointed out by, though its black hole mass (2.6 × 107) is lower compared to those typical of BL Lac objects. In fact, it has been speculated in the past decade that concealed Seyfert nuclei exist in a few BL Lac objects, including BL Lac itself. It is therefore tempting to postulate that some RL NLS1 galaxies may actually be BL Lac objects (with low ) at their low light states.\nThe bulk of our RL NLS1 galaxies have a BLR luminosity[25](#fn25) distribution lying within $\\log\\,(L\\_{\\rm BLR}$/)=43.5–44.5, the low end of which overlaps partially with those of BL Lac objects. Specifically, J0849+5108 has $L\\_{\\rm BLR}\\sim 10^{43.7}$. However, this is likely not the case for most of the RL NLS1 galaxies given their relatively higher $L\\_{\\rm BLR}$ compared to the upper limits for BL Lac objects. On the other hand, this possibility means that BL Lac objects might not be a homogeneous population, and some may just be Seyfert nuclei/NLS1 at a high state of the non-thermal radiation from the jet. Similarly, this concerns only BL Lac objects at the low-end, as is the case of J0849+5108, rather than the bulk of classical BL Lac objects that habour massive black holes and accrete likely via a low efficiency accretion flow.\n### Relationship with CSS sources\nCompact steep-spectrum sources are compact, powerful radio sources with spectral peaks near 100MHz. They are inherently compact with linear sizes ≤ 20kpc, having (often symmetric) double/triple morphology with complete jets and lobes on small scales. The most popular scenarios to explain their nature advocate that they are either young and evolving radio sources or confined by dense gas of their environment. Very recently, a few lines of observational evidence point to an interesting link between RL NLS1 galaxies and CSS, which share some degree of similarity as the following. (1) As mentioned in §1, the radio sources of all RL NLS1 galaxies are compact, including steep-spectrum sources. In particular, a few NLS1 AGN having higher resolution observations show small-size structure: e.g.most of the radio emission of 2MASXJ0324+3410 is from a region smaller than 6.8mas (8.2pc projected size), as seen from VLBA images at 8.4GHz (in preparation). (2) The low frequency spectrum of J1047+4725, one of the steep-spectrum objects in our sample, appears to flatten at around a few hundred MHz as indicated by its flat slope of − 0.28 between 327 and 151MHz; such a spectral break is characteristic of CSS. In fact, the RL NLS1–CSS connection was advocated by and for the case of PKS2004-447, although it is unclear as to whether PKS2004-447 is a genuine NSL1 galaxy. (3) There are observational indications that the host galaxies of a large fraction of CSS are interacting/merging and may contain significant amount of dense gas, similar to what was found in some RL NLS1 galaxies.\nThe radio power distribution of our RL NLS1 galaxies overlaps with the lower part of that of CSS. As such, if there is an overlap between CSS and RL NLS1 galaxies, they should be mostly the CSS having low radio power, and, relatively small. Since CSS sources also possess (small scale) jets, in cases where the jets are directed close to our line-of-sight, we expect to see blazar-like objects. Some of those could be the flat-spectrum RL NLS1 galaxies studied here, in which relativistic jets are inferred to be present. Conclusive remarks have to await future proper and detailed comparisons of the two classes, as well as more radio observations with higher resolution, however.\n### Implication for jet formation in AGN\nGiven the high Eddington ratios of NLS1 galaxies in general, as commonly suggested and also found here, our RL NLS1 AGN present a strong case where jets, possibly at least mildly relativistic, can be produced in high systems. This strengthens the similar situation in powerful RLQs with much larger. This contrasts clearly with black hole X-ray binaries where jets are found at the low/hard state, despite the suggestion that transient jets (or mass ejection) may occur at the so called very-high state where is higher. The formation of jets is thought to be related to the black hole and accretion process. The finding of relativistic jets in black hole accretion systems with high (where the accretion proceeds via probably a new form other than the standard disk) and, moreover, with relatively low is important for addressing basic questions as to how jets are formed. Jet formation in high systems (slim disks) is far from clear, though there are some suggestions that outflows as well as mildly beamed and collimated emission are likely present in such systems.\nAs has been shown above, apart from the continuum radiation, the RL NLS1 galaxies resemble closely their RQ counterparts in many ways, e.g. optical emission properties (except for stronger emission), the distributions of black hole masses and the Eddington ratios. These facts suggest that RL NLS1 galaxies are simply otherwise normal NLS1 galaxies but having relativistic jets, reminiscent of the RL/RQ dichotomy of more powerful, classical AGN. It would be interesting to find out whether there is a bimodal distribution in the radio-loudness, as reported for the broad-line quasar population (though this is still a matter of debate), in future work considering objects with the whole range of radio-loudness. The nearly indistinguishable properties other than the presence of radio jets possessed by the two subsets of NLS1 galaxies, RL and RQ, seemingly hint at a possibly important role of black hole spin in the making of relativistic jets in RL NLS1 galaxies.\nSummary of conclusions and implications\n=======================================\nA homogeneous sample of genuine RL NLS1 galaxies (radio-loudness $\\ga 100$) is presented for the first time, comprising 23 objects drawn from SDSS spectroscopic objects with FIRST detections. Their NLS1 nature is discovered by analysing systematically the spectroscopic data from the QSO and galaxy database in the SDSS DR5. The sample can be treated as an optically–radio selected one. We carry out systematic investigations into their multiwaveband properties from radio to X-rays using data available from the archives. The main results are the following.\nAll radio sources of the sample are compact, less than a few tens of kilo-parsec at least. Several with VLBI observations are unresolved or marginally resolved at scales of about tens of parsec. Among those with available radio spectral indices, the majority are flat-spectrum sources (*α* > − 0.5 around 1.4–5GHz), and a few even have inverted spectra (*α* > 0). Radio variability is found to be common in most of the flat-spectrum sources. We were able to estimate the brightness temperatures of the radio sources, that are as high as $T\\_{\\rm B}>10^{11}$K and even exceeding the inverse Compton limit (1012K) in four objects. The high brightness temperatures indicate the presence of relativistic beaming. The majority of RL NLS1 galaxies has a low-frequency (isotropic) radio power falling in the range of FRI type radio galaxies ($\\la 10^{25}$), and the remaining minority in the range of the FRII type. When isotropic, rather than Doppler boosted, emission is considered, a large fraction of the objects can be considered as intrinsically radio-intermediate.\nThe observed optical luminosities are systematically in excess of the thermal continua predicted from the line luminosities by a factor of up to several. The optical slopes of our RL objects, while spanning a wide range from moderately red to blue, are systematically bluer than that of radio-quiet NLS1 AGN. We interpret this effect as the contribution from a (non-thermal) component, presumably from relativistic jets, to the optical band superimposing the central ionising continuum seen in radio-quiet NLS1 AGN. The optical emission in the radio-loud NLS1 in our sample is on average stronger than ‘normal’ NLS1 galaxies. Except this, the two subsets of NLS1 galaxies seem to be indistinguishable in other emission line properties.\nOn the radio–optical–X-ray effective spectral indices plane (vs. ), our objects have broad-band SEDs well consistent with those of blazars, i.e. both the HFSRQ and LFSRQ types as well as SSRQs. Of particular interest, the fraction of the candidate HFSRQ-type objects seems to be high, though their synchrotron component peaks are only moderately high (at around the UV). This finding extends the recently discovered strongly-lined (FSRQ) HBL-type blazars to NLS1 galaxies.\nThe continuum shape in the soft X-ray band exhibits a large variety among our RL NLS1 galaxies. There is evidence in one, or possibly two objects for a flat X-ray spectral component ($\\Gamma\\la 1.8$), which we interpret as an inverse-Compton origin from jets as in FSRQs. Most of the objects with estimation of the X-ray spectral slopes show steep spectra. The steep X-ray spectrum is also consistent with what is expected for HFSRQs, that originates from the high energy end of the synchrotron component. Based on this circumstantial evidence, we suggest that such a component may be a significant contributor to the observed X-ray emission, though the X-ray component as seen in radio-quiet NLS1 galaxies may also play a role.\nIn view of the independent lines of evidence presented here, as well as those found in previous studies for several individuals (either inclusive or exclusive of our sample), we suggest that there likely exists a population of RL NLS1 galaxies showing the broad-band continuum emission characteristic of blazars, though alternative explanations cannot be ruled out completely. In this scenario, beamed non-thermal emission from jets, that are at least mildly relativistic, dominates the high-frequency radio band, and contributes significantly to the optical and possibly the X-ray bands. Our finding extends the radio-loud quasar population into a narrow line-width regime of the broad emission lines. We postulate that all genuine radio-loud NLS1 galaxies possess small scale, at least mildly relativistic jets, reminiscent of classical radio-loud AGN. In this scenario, steep-spectrum NLS1 sources (perhaps including some of those with smaller radio-loudness within *R* = 10–100) are simply objects of the same population whose radio jets are aligned at a relatively large angle to the line-of-sight.\nThe host galaxies of two nearby RL NLS1 nuclei, whose images are resolved, have disk morphology, one even in a merger, in line with the finding for the previously published blazar-NLS1 hybrid object 2MASXJ0324+3410. In several objects imprints of young stellar populations are likely to be present in form of high-order Balmer absorption lines in the SDSS spectra.\nBased on the commonly used relation of black hole mass with broad line width and luminosity, our RL NLS1 galaxies have estimated in the range of 106 − 108, provided that this scaling relation is applicable to NLS1 and the BLR is not largely flattened. The thus estimated Eddington ratios range from =0.5 to 3, and peak at ≈ 1 (assuming $L\\_{\\rm bol}=9 \\lambda L\\_{\\lambda5100}$). RL NLS1 galaxies deviate from the known trends of radio-loudness versus the Eddington ratio and black hole mass, and are populated in a region in the radio-loudness–diagram where normal AGN are seldom found, i.e. moderate and large *R* ( ≃ 106 − 7.5, *R* ≃ 101 − 3), confirming and improving upon the previous results suggested by. As such, RL NLS1 show strong excess radio emission with respect to their, when compared to the previously suggested radio power–relation for normal AGN. Our results imply that large radio-loudness (*R* = 101 − 3) can be produced in AGN with moderate down to ∼ 106 − 107and accreting at/close to the Eddington rate; and some of these AGN are hosted in late-type galaxies, some even in interacting systems. The presence of relativistic jets in such systems may have interesting implications for understanding jet formation in AGN in general.\nIn spite of a likeness to normal RL AGN in many aspects, RL NLS1 galaxies show some extreme or unusual properties, though some of the evidence are only circumstantial. These include compact radio morphology, relatively low radio power, a high incidence of HBL/HFSRQ-type SEDs, host galaxies of probably late-type and/or with young stellar populations, and the common properties characteristic of NLS1 galaxies, e.g. smaller black hole masses, higher Eddington ratios, and stronger emission. All of these lead us to postulate that either they are an extension of normal RL AGN to the lower-range, and all those differences can be explained as a consequence of decreasing ; or they form a previously unknown population distinct from normal radio-loud AGN. As one interesting possibility, we speculate that low-power CSS might be the parent population of the flat-spectrum RL NLS1 galaxies with beamed jet emission. For the future, a much larger sample and extensive multi-waveband observations are essential for understanding the nature of radio-loud NLS1 galaxies.\nFinally, if the broad-band radiation mechanisms in flat-spectrum radio-loud NLS1 AGN are indeed the same as in blazars, we expect some of these objects to be *γ*-ray emitters in the GeV band. This can be tested by observations of GLAST in the near future.\nWe thank the anonymous referee for helpful comments and suggestions. This work is supported by the Chinese Natural Science Foundation through projects No. NSF10533050 and NSF10473013, and the BaiRen Project of the Chinese Academy of Sciences. Funding for the creation and the distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Princeton University, the United States Naval Observatory, and the University of Washington. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.\nOptical spectral modeling\n=========================\nThe starlight and nuclear continua are modeled as\n \n$$\\label{eq1}\nS(\\lambda)=A\\_{host}(E\\_{B-V}^{host},\\lambda)~A(\\lambda)+A\\_{nucleus}(E\\_{B-V}^{nucleus},\\lambda)~[bB(\\lambda)+ c\\_{\\rm b} C\\_{\\rm b}(\\lambda) +\nc\\_{\\rm n} C\\_{\\rm n}(\\lambda) + dD(\\lambda)]$$ \nwhere *S*(*λ*) is the observed spectrum. *A*(*λ*) = ∑*i* = 16*a**i* *I**C**i*(*λ*, *σ*\\*) represents the starlight component modeled by our 6 synthesized galaxy templates, which had been built up from the spectral template library of Simple Stellar Populations (SSPs) of using our new method based on the Ensemble Learning Independent Component Analysis (EL-ICA) algorithm. The details of the galaxy templates and their applications are presented in. *A*(*λ*) was broadened by convolving with a Gaussian of width *σ*\\* to match the stellar velocity dispersion of the host galaxy; in this process the velocity difference between emission and absorption lines is taken into account. The un-reddened nuclear continuum is assumed to be *B*(*λ*) = *λ*− 1.7 as given in. We modeled the optical emission, both broad and narrow, using the spectral data of the multiplets for IZwI in the *λ**λ*3535–7530Årange provided by. We assume that the broad lines ($C\\_{\\rm b}$ in Eq.1) have the same profile as the broad line, and the narrow lines ($C\\_{\\rm n}$), both permitted and forbidden, have the same profile as the that of the narrow component, or of *λ*5007 if is weak. *D*(*λ*) represents the templates of higher-order Balmer emission lines (10 ≤ *n* ≤ 50) and Balmer continuum generated in the same way as in. *A**h**o**s**t*(*E**B* − *V**h**o**s**t*, *λ*) and *A**n**u**c**l**e**u**s*(*E**B* − *V**n**u**c**l**e**u**s*, *λ*) are the color excesses due to possible extinction of the host galaxy and the nuclear region, respectively, assuming the extinction curve for the Small Magellanic Cloud of. The fitting is performed by minimizing the *χ*2 with *E**B* − *V**h**o**s**t*, *E**B* − *V**n**u**c**l**e**u**s*, *a**i*, *σ*\\*, *b*, $c\\_{\\rm b}$, $c\\_{\\rm n}$, and *d* being free parameters.\nNotes on individual objects\n===========================\n**J0849+5108** (0846+51W1): This object has drawn substantial attention ever since its first identification with a BL Lac object (at a bright state) by, at a claimed high redshift of *z* = 1.86[26](#fn26). Its radio flux also showed large amplitude variations up to 50% at 1.4GHz. This object exhibited a remarkable optical brightening with an amplitude of Δ *V* ∼ 5 within less than a month and high and variable optical polarization, making it one of the optically violent variables (OVV) and highly polarized QSOs (HPQ). Night-to-night and even intra-night optical variability was also found. At maximum light outburst, the continuum is featureless, whereas its spectra taken at normal/minimum light states show strong emission lines. Thus it is a transition between a quasar and a BL Lac object, i.e. appeared to be a BL Lac object only at its flaring state. The large and variable polarization in the optical band suggests strongly that this emission is due to synchrotron radiation. The conspicuous characteristic emission lines identified in the SDSS spectrum of J0849+5108 give a redshift of *z* = 0.583. A detailed study of its SDSS spectrum and other properties was presented by. The optical spectral slope obtained from fitting the SDSS spectrum is red, with ≃ − 0.2.\n**J0948+0022**: The object was first discovered for its radio emission in various radio surveys ($S\\_{\\rm 5GHz}\\sim$0.2–0.3Jy). It was identified with a NLS1 galaxy, based on its SDSS spectrum, by and, and was studied in detail as one of the few previously known very RL NLS1 galaxies. The radio source has an inverted radio spectrum and was unresolved with VLBI at milli-arcsec resolution. These observations were explained in terms of relativistic beaming. A very high brightness temperature ( ∼ 1013K) inferred from radio flux variations suggest the presence of a jet with relativistic beaming. Long-term optical variability with an amplitude of about 1mag in the *B*-band was also noted in.\n**J1505+0326**: It has radio fluxes peaked at ∼ 7GHz. Its VLBI images are dominated by a very compact component, marginally resolved at 2.3GHz band. The radio emission at centimeter wavelengths shows variability.\n**J1633+4718** (RXJ16333+4719): The object was identified with a NLS1 galaxy in optical identification of ROSAT All-Sky Survey (RASS) sources. The AGN (see Figure[fig:sdssspec] for its SDSS spectrum) is actually one of the two nuclei, separated by 4, residing in an interacting/merger system. The other nucleus has a starburst spectrum. The radio source, which is associated with the AGN, is unresolved with VLBI at milli-arcsec resolution. Its spectral shape above 5GHz is inverted, as measured by simultaneous multiwavelength observations by. The system was noticed to be associated with the IRAS source IRAS16319+4725 (RA=16h33m23.03s, Dec=+47d19m03.5s), whose positional uncertainty—much larger than the separation of the two optical nuclei—encompasses both of the nuclei. The doublet shows a noticeable blue wing in its SDSS spectrum.\n**J1644+2619**: Its radio source has an inverted spectrum and was unresolved with VLBI at milli-arcsec resolution.\nRadio emission at low frequencies\n=================================\nIn the above sections, evidence has been presented for the blazar-like properties of the (flat-spectrum) RL NLS1 galaxies, arguing for the presence of relativistic jets in them. As is known in normal radio quasars/galaxies, radio jets are associated with large scale radio lobes. These show diffuse radio emission with steep spectrum which dominates the radio fluxes at low frequencies (several hundred MHz), via synchrotron radiation from optically thin plasma. Here we examine the radio properties of our sample at low frequencies, though the data are sparse in general, especially at around 100MHz. The radio emission is searched for using the WENSS[27](#fn27) survey at 327MHz and using NED for surveys at lower frequencies. We find 15 objects with detections at 327MHz, for which the 1.4GHz to 327MHz spectral indices can be computed. We found a median of − 0.47, and ∼ 47% being steeper than − 0.5 with the steepest as − 0.83, systematically steeper than the 1.4–5GHz indices (3/11 are steep-spectrum). This means that the spectral indices steepen progressively towards low frequencies.\nThis behavior can well be explained by onset of a steep-spectrum component dominating the low frequencies, as commonly seen in normal RL AGN. Therefore, we postulate that in these objects there exists diffuse, optically thin emission, presumably from putative radio lobes that are powered by (relativistic) jets. If this is the case, those blazar-like (flat-spectrum) NLS1 AGN must possess the general core–halo morphology as seen in some blazars. Given the compact morphology unresolved at the FIRST resolution (smaller than several tens kpc), there are two possibilities: either they have large scale extended emission—similar to the classical radio sources—but are viewed close to the radio axis, or are intrinsically compact, similar to CSS (compact steep-spectrum sources).\nAt the even lower frequency of 151MHz, 4 objects have detected radio emission, all from the 6C/7C survey, namely J1047+4725 (1.81Jy), J1305+5116 (0.32Jy), J1443+4725 (0.5Jy), and J1548+3511 (0.26Jy). The 151–327MHz indices are = − 0.28, − 0.54, − 0.62, + 0.09, respectively, and the average is − 0.34. These values are much flatter than the ‘canonical’ slope for diffuse optically thin emission from powerful sources, indicating that the spectra have already turned over, or started to turn over, in this frequency range due possibly to synchrotron self-absorption. In fact, the radio spectra of J1047+4725 and J1548+3511 show a clear trend of turnover at low frequencies. Considering the relatively large redshifts of the sample, the break frequencies in some sources could be as high as several hundred MHz in the source rest frame. Spectral turnovers at such frequencies are often seen in radio sources with small scale structure, such as in CSS. The previously reported extreme NLS1/blazar hybrid object 2MASXJ0324+3410 was also detected at 151MHz (1.02Jy); its radio slope also steepens significantly towards low frequencies, = − 0.44, suggesting the dominance of a steep-spectrum component at low frequencies.\nEffects of refractive interstellar scintillation on radio flux variations\n=========================================================================\nFor compact radio sources (milli-arcsec scale), it is known that refractive interstellar scintillation (RISS), caused by inhomogeneities in the interstellar medium, may give rise to variations of radio flux densities. RISS is wavelength-dependent and concerns predominantly the low frequency domain (below 1GHz). At centimeter wavelengths, RISS mostly causes “flickering” at a lever of a few percent, as was found in observations and is also consistent with the theoretical models. Although variations at a larger amplitude were sometimes ascribed to RISS, they were limited to a small number of individual objects which ubiquitously are at low Galactic latitude ( < 10∘) or even have a line-of-sight passing through the Galactic plane. Monitoring observations of radio sources at frequencies above 1GHz showed that both intrinsic and extrinsic (RISS) variability contribute to observed radio flux variations, with the former having timescales from months to years and the latter being faster and having rms amplitude typically < 10%.\nIn consideration of the large variation amplitudes in some of the objects of our sample and their high Galactic latitudes (b=31∘ − 51∘, Table[tbl:radvar]), we argue that the variations are predominantly intrinsic for the majority of the variables reported here, instead of arising from RISS. This is especially true for variations at 5GHz or even higher frequencies. Furthermore, the brightness temperatures deduced from variability at 1.4GHz are in the same range as those from 5GHz (Table[tbl:radvar]); this favours the intrinsic variability explanation, since RISS at 1.4GHz is expected to be more pronounced than at 5GHz due to its strong wavelength dependence. We therefore suggest that, although a non-negligible role played by RISS may not be ruled out in individual objects, especially those with relatively low amplitudes, our results for the sample as a whole should not be affected significantly.\nOn the X-ray radiation mechanisms of radio-loud NLS1 AGN\n========================================================\nAs one of the defining characteristics, the X-ray spectra of FSRQs are flat and extend from the soft to the hard X-ray band, which is interpreted as inverse Compton radiation from the relativistic jets and beamed. In most RLQs the contribution from a hot disk corona to the observed X-rays is negligible (in the hard X-ray band > 2keV), except the steep-spectrum soft X-ray excess below 1keV. From the data analysis above, we find a flat X-ray spectrum in J1633+4718 with Γ = 1.37 ± 0.49, and possibly in J0849+5108 with Γ = 1.77− 0.60+ 0.44 (90% errors). Their concave optical to X-ray spectral shapes (see Figure[fig:aox-ax]) also agree with the inverse Compton scenario. For J0849+5108, the LBL-like SED and flat X-ray spectrum are well expected, based on its OVV/HPQ properties (see Appendix[sect:indv]). The prominent soft X-ray excess in J1633+4718 is also typical of normal NLS1 galaxies. It is puzzling, however, that J1633+4718 shows a broad band SED similar to that of HFSRQs, for which a steep X-ray spectrum and a convex –X-ray shape are generally expected. Future multi-waveband observations, especially in the X-ray band, are needed to confirm this peculiarity.\nAs discussed above, in the presence of an excess soft X-ray component, the effective spectral slopes derived from the hardness ratios are generally steep. The same is expected to be true for radio-loud NLS1 galaxies, as found in J1633+4718 above. For reference, FSRQs at redshifts (*z* < 1) are found to have Γ ∼ 2.1 with a dispersion *σ* ≃ 0.2 in the ROSAT PSPC band. Therefore, we expect that a few other objects, e.g. those having FSRQ/LBL-type SEDs yet steep Γ values in the PSPC bandpass (Figure[fig:alprox]) would similarly have a flat underlying X-ray continuum.\nHBLs are commonly observed to show steep X-ray spectra, which are intrinsically curved, progressively steepening with increasing photon energy. Their X-ray emission is thought to be attributed to the high-energy tail of the synchrotron radiation, i.e. the synchrotron bump reaches the soft X-ray band. There are only a few flat-spectrum radio-loud NLS1 AGN studied previously with multi-waveband data and, interestingly, most of these show HBL/HFSRQ-like SEDs. For example, RXJ16290+4007, formerly classified as a FSRQ, was suggested by to be the first confirmed HFSRQ for its modeled synchrotron peak at 2 × 1016Hz and a steep soft X-ray spectrum Γ ≃ 3. 2MASXJ0324+3410 also shows a continuum SED and a luminosity closely resembling that of Mkn421, a well-known HBL; the object was claimed to be even marginally detected in TeV *γ*-rays, as seen in some nearby bright HBL. In the X-ray band, 2MASXJ0324+3410 also shows a steep spectrum.\nMost of the objects falling into the HBL locus in Figure[fig:alprox] show steep X-ray spectra wherever available (Γ = 2.0 − 3.3). Meanwhile, their –SED shapes are either convex or close to a straight line (i.e. power-law; see Figure[fig:aox-ax]). 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Amer. Astro. Society, 22, 917\nIvezi*ć*, *Ž*., et al. 2002,, 124, 2364\nJackson, N., Browne, I.W.A. 1991,, 250, 422\nJarvis, M.J., & McLure, R.J. 2002,, 336, L38\nJarvis, M.J. & McLure, R.J. 2006,, 369, 182\nJiang, D.R"},"data_id":{"kind":"string","value":"arxiv_0183115"}}},{"rowIdx":45794,"cells":{"content":{"kind":"string","value":"The Secretary Recommendation Problem\n====================================\nIn this paper we revisit the basic variant of the classical secretary problem. We propose a new approach in which we separate between an agent that evaluates the secretary performance and one that has to make the hiring decision. The evaluating agent (the sender) signals the quality of the candidate to the hiring agent (the receiver) who must make a decision. Whenever the two agents’ interests are not fully aligned, this induces an information transmission (signaling) challenge for the sender. We study the sender’s optimization problem subject to persuasiveness constraints of the receiver for several variants of the problem.\nOur results quantify the loss in performance for the sender due to online arrival. We provide optimal and near-optimal persuasive mechanisms that recover at least a constant fraction of a natural utility benchmark for the sender. The separation of evaluation and decision making can have a substantial impact on the approximation results. While in some scenarios, techniques and results closely mirror the conditions in the standard secretary problem, we also explore conditions that lead to very different characteristics.\n**Keywords:** Bayesian Persuasion, Secretary Problem, Online Algorithms, Approximation Algorithms\nIntroduction\n============\nIn the classical secretary problem a sequence of *n* candidates with unknown valuations arrives sequentially in uniform random order. Upon arrival, a decision maker observes the value of a candidate and must make an irreversible decision of whether to hire the candidate or observe the subsequent one. The lion’s share of this literature studies optimal or near-optimal algorithms for the decision maker and provides many elegant results on how well she can perform (see, e.g., and many more).\nIn this problem, the decision maker is tasked with two responsibilities: (1) evaluate the candidate, and (2) make the decision of whether or not to hire. In many settings, these two tasks are separated and the evaluation is done separately from the hiring decision. In this paper, we initiate the study of (near-)optimal persuasive mechanisms for the secretary problem when the evaluation is done by one agent while the decision is taken by another. Following tradition in the literature, we shall often refer to the latter as a ‘receiver’ and to the former as a ’sender’. The separation between evaluation and hiring decision introduces incentives that are immaterial to the original setting.\nThe aforementioned separation is wide-spread in applications. Consider, for example, the original motivating story of hiring an employee (e.g., a secretary). It is often the case that firms separate between the evaluation process, often led by a specialized HR department, and the hiring individual (the prospective boss). Another setting that has such a separation is in financial projects and investment opportunities. Many firms hire third-party consultants to evaluate a sequence of financial opportunities and use the consultant’s report in their decision process. An opportunity not seized is often an opportunity foregone.\nTo capture the incentive discrepancies between the sender and receiver, we endow each candidate with two valuations, one that is associated with the sender and one with the receiver. Candidates arrive according to a uniform random order. Upon arrival, the valuation pair is disclosed to the sender, who then makes some recommendation to the receiver. The receiver in turn takes one of two irrevocable actions, to hire or not to hire. Throughout the paper, we assume the sender has commitment power, and we design mechanisms for her recommendation such that the receiver always finds it in her interest to adopt the recommended action. Our goal is to design mechanisms that achieve (near-)optimal expected utility for the sender.\nInitially, we study an elementary scenario – primarily for didactic reasons – in which we assume that the valuation pairs of all candidates are known to both players in advance. Both players a-priori have full information about (1) the set of valuation pairs, (2) the fact that candidates arrive in uniform random order, and (3) the mechanism the sender uses for recommendation. In round *t*, the sender gets to see the arriving candidate and makes a recommendation using the mechanism. By applying Bayesian updates based on the a-priori information and the signals received in this and the previous rounds, the receiver can then decide whether it is in her interest to follow the recommendation or not. We provide optimal persuasive mechanisms for the sender, and we use this scenario as benchmark for the subsequent analysis. Hence, this is our *basic* or *benchmark scenario*.\nOur main interest lies in the *secretary scenario* as it is more reminiscent of the original secretary problem. Here the valuation pairs are unknown to the two players and are set by an adversary, but their arrival order is known to be uniform at random. The unknown adversarial design of valuations raises questions about modeling the incentives of the receiver. To determine her interests, the receiver should be able to evaluate what is “in store” for her if she adopts the sender’s recommendation or deviates to another hiring strategy. To enable this decision, we adopt a robust approach and require that the mechanism is persuasive even if the receiver knows all valuation pairs (and, thus, has the same a-priori information as in the basic scenario above). Note that such a mechanism stays persuasive even if the receiver does not know the valuation pairs, since persuasiveness is merely a guarantee that it is in the receiver’s interest to always follow the recommendation. Therefore, on a technical level, the secretary scenario differs from the basic scenario by the *sender* a-priori lacking information about valuation pairs. In the secretary scenario, we provide a persuasive mechanism that guarantees a constant-factor approximation in the expected utility for the sender when compared to the optimum from the basic scenario. Hence, for a sender in a signaling context, *secretary-style online arrival only results in a performance deterioration by a small constant factor*.\nMoreover, we study both scenarios in two different variants: In the first one, the receiver only receives the sender’s signals; in the second one, she is also informed about the valuations of rejected candidates. We refer to this variant as a scenario *with disclosure*. For motivation, one can think of candidates that are foregone as candidates that go on the market which, inter-alia, extracts and makes public all the relevant information about the candidate. In the example with financial investments, one can think of a sequence of investment opportunities given to some prominent investor (e.g., a top-ten private equity fund) before they go on the market. In the basic scenario with disclosure, we again obtain a mechanism with a constant approximation ratio. However, disclosure in the secretary scenario can lead to a deterioration in the expected utility by a factor of Θ(1/*n*).\nWe examine all four scenarios (basic and secretary, both with and without disclosure) with different utilities that are standard in the literature on the secretary problem. With *cardinal utility* a player strives to maximize her expected valuation of the hired candidate. In contrast, with *ordinal utility* she strives to maximize the probability that her best candidate is hired. Our goal in all variants is to design good persuasive mechanisms for the sender. In almost all variants, we show constant approximation ratios, thus, *even with online arrival and/or disclosure the expected utility for the sender deteriorates only by a small constant factor*. Also, we show that many ratios are optimal or asymptotically optimal with respect to any persuasive mechanism.\nRelated Literature\n------------------\nOur work can be seen as an extension of the celebrated *secretary problem* which first appeared in print in Martin Gardner’s 1960 *Scientific American* column (but apparently originated much earlier, see ). The problem gained considerable popularity, and subsequently various extensions have been studied (c.f. for an early survey and for recent extensions). In all the literature we are familiar with, a separation between candidate evaluation and hiring decision is non-existent.\nIn our model the sender has commitment power and hence our work contributes to the burgeoning literature on Bayesian persuasion, which originated in Aumann and Maschler and more recently enjoyed a renaissance through the work of Gentzkow and Kamenica. In particular, one can think of our model as an online Bayesian persuasion model, where the state of nature is revealed in a round-wise fashion to the sender who in turn sends a signal to the receiver in each round. Whereas dynamic Bayesian persuasion problems have been studied by various authors (e.g., ), we are not familiar with an analysis of an online variant similar to what we study. Ely studies the case that the sender observes the current state and sends a message to the receiver. Using this information, the receiver’s beliefs are updated, and she takes an action according to her current belief. The state of nature evolves according to a stochastic process, namely Poisson transitions over states. In principle, our approach might also be cast within a framework of a single, evolving state of nature, but the specific details of this approach would turn out to be very different from . Au studies an approach, where in each round the receiver either takes an action or goes on to the next round. Unlike our model, the action is the same throughout. Once the action is taken, the process ends and both parties get their respective utilities. The sender extracts positive utility from the fact that the receiver has taken the action, regardless of the state of nature. Hence, the sender’s objective is persuading the receiver to take the action. The receiver’s utility depends on the state. A somewhat different approach is taken by Ely et al. where the authors study the notions of suspense and surprise, i.e. the variance in the next update of belief and the distance of two consecutive beliefs, resp. The sender uses these objectives to design the disclosure policies.\nIn the algorithmic literature, Dughmi and Xu study the offline variant of Bayesian persuasion where candidates’ valuations are drawn independently from known distributions. In , the authors study algorithms and approximation for Bayesian persuasion with multiple receivers, each with a binary choice of actions. More recent additions to this literature include, e.g., duality considerations or the study of risk-conscious agents . In contrast, our approach extends the Bayesian persuasion model with a single sender and a single receiver in a different direction, by studying an online approach with adversarial valuations and random-order arrival.\nSomewhat related is previous work in the context of delegation . Here, a principal delegates a search problem to an agent. Both have individual interests which might be misaligned. The principal can either accept or reject the agent’s proposed solution and will do so according to a specified mechanism. A key difference between the two models lies in the commitment power. In our model, the party that observes the information designs the mechanism while in delegation, the mechanism is designed by the principal who only has a-priori information. Kleinberg and Kleinberg recently studied how techniques and results from stochastic online optimization can be used to approximately solve delegation problems. In contrast to our work, they study application of these techniques to approximately solve an *offline* delegation problem.\nModel\n-----\nWe consider a Bayesian persuasion problem with online arrival. There is a *sender* S and a *receiver* R. A set of candidates arrives sequentially in *n* rounds over time in *uniform random order*. Each candidate comes with a pair of valuations, one for S and one for R. In the beginning, R knows the set of all these pairs and that their arrival order is uniform at random. S always knows that the arrival order is uniform at random. In the *basic* scenario, she also knows the set of valuation pairs, in the *secretary* scenario she does not.\nIn round *t*, a candidate arrives and a state of nature *θ**t* ∈ Θ*t* is revealed to S. We use ${{\\bm{{\\theta}}}}= ({\\theta}\\_1,\\ldots,{\\theta}\\_n)$ to denote the vector of the states of nature revealed in all rounds *t* ∈ {1, …, *n*} = [*n*]. Associated with *θ**t* is the pair of non-negative values *ξ*(*θ**t*), *ρ*(*θ**t*) ≥ 0, where *ρ*(*θ**t*) is the utility of R and *ξ*(*θ**t*) is the utility of S when the candidate gets hired. We use ${\\bm{{\\rho}}}$ and ${\\bm{{\\xi}}}$ to denote the vectors of all possible utility values of all candidates in all states of nature, and *ρ*max = max*θ**t**ρ*(*θ**t*) and *ξ*max = max*θ**t**ξ*(*θ**t*) as well as *c*S = argmax*θ**t**ξ*(*θ**t*) and *c*R = argmax*θ**t**ρ*(*θ**t*).[4](#fn4) Having observed *θ**t* and the corresponding values, S transmits a signal *σ**t* to R. Based on the signal, R then decides whether to hire the current candidate in round *t* or not. Every decision is final and cannot be revoked later on. If R decides to hire, the process ends. Otherwise, round *t* + 1 starts and the next candidate arrives. The process ends by the end of round *n* at the latest.\nFollowing we assume that there is commitment power, i.e., S shall commit a-priori on a signaling strategy *ϕ*, mapping each partially revealed vector of states of nature (*θ*1, …, *θ**t*) to a signal *σ**t*, for all *t* ∈ [*n*]. The order of events is (1) S commits to a signaling strategy *ϕ*, (2) first candidate arrives, (3) S learns *θ*1 and sends signal *ϕ*(*θ*1) to R, (4) R decides to hire or not, (5) repeat from step (2) with *θ*2 and *ϕ*(*θ*1, *θ*2) etc., until a candidate gets hired or all *n* candidates arrived. In a scenario *without disclosure*, the sender’s signals (*ϕ*(*θ*1), …, *ϕ*(*θ*1, …, *θ**t*)) are the only information for R in the beginning of round *t* (in addition to the a-priori information). In a scenario *with disclosure*, R also gets informed about the rejected candidates (*θ*1, …, *θ**t* − 1) in the beginning of round *t*. Hence, with disclosure R knows the set of remaining candidates that will arrive in rounds *t*, …, *n*.\nThe goal of both S and R is to maximize their individual objective. We consider two variants, for each S and R: The cardinal objective, where a player strives to maximize the *expected utility*; and the ordinal objective, where the sender (receiver) wants to maximize the *success probability*, i.e., the probability that the hired candidate is the optimal *c*S (*c*R), resp. All probabilities/expectations are with respect to input randomization and internal randomization of the mechanism. Our mechanisms for S are truthful-in-expectation mechanisms, where we strive to maximize the sender’s objective.\nApplying a revelation-principle style argument , S can restrict herself to a signaling strategy *ϕ* that is *direct* and *persuasive*. For direct signals, S directly recommends one candidate for hire, i.e., *σ**t* ∈ {HIRE, NOT}. A direct mechanism *ϕ* is persuasive if R in every round maximizes her expected utility (from the candidate she hires eventually) by being obedient, i.e., always hiring upon receiving the signal HIRE and never hiring upon receiving the signal NOT. We assume that ties are broken in favor of S, i.e., only when R can strictly increase her success probability/expected utility, there is an incentive to deviate from the signals of *ϕ*.\nResults and Techniques\n----------------------\nOur paper provides a comprehensive analysis of sixteen variants of the recommendation problem. For exposure, we here focus on two canonical cases in which S and R either both have cardinal utilities (cardinal case) or both have ordinal utilities (ordinal case). Results for the remaining scenarios (e.g., cardinal sender and ordinal receiver) follow along the same lines. Throughout the paper, all asymptotics are in *n*, the number of candidates.\n#### Cardinal Case\nOur baseline is the *basic scenario*, where we propose the Pareto mechanism (based on the Pareto procedure in Algorithm [algo:Pareto]), an optimal persuasive mechanism (Proposition [prop:Pareto]) that determines an optimal signal based on the Pareto curve of the candidate set. We use the optimal sender utility (denoted by OPT) in this scenario as a benchmark for the other variants. OPT can differ substantially from *ξ*max, can depend on the complete candidate set, and has no immediate closed-form expression. Perhaps interestingly, in the Pareto mechanism, if the sender-optimal candidate *c*S offers at least $\\mu^{{\\ensuremath{\\mathcal{R}}}} = \\frac{1}{n} \\sum\\_i {\\rho}\\_i$ to R, then it gets recommended deterministically. Otherwise, the mechanism will compose a sender-optimal signal that has expected value *μ*R for R, i.e., does not offer useful information.\nIn the *secretary scenario*, S faces a problem with the main characteristics of the secretary problem, i.e., unknown candidate values and random-order arrival. The values are only revealed to S once the candidate arrives. Observe that this scenario strictly generalizes the standard secretary problem. In particular, if *ξ**i* = *ρ**i* for every candidate *i* ∈ [*n*], the incentives of both players are perfectly aligned. In the basic scenario, when S knows all candidates upfront, she would simply choose *c*S = *c*R and recommend it for hire. Thus, the benchmark in this case is expected utility OPT = *ξ*max. It is easy to see that in such instances no persuasive mechanism in the secretary scenario can beat the 1/*e* − *o*(1) guarantee from the classic optimal algorithm .\nWhen the incentives are not perfectly aligned, our mechanisms depart substantially from this classic template. Here, every persuasive mechanism must guarantee an expected utility for the receiver of *μ*R. The Pareto mechanism optimizes the sender utility under this constraint. In the secretary scenario, we apply this technique adaptively to the set of arrived candidates. Perhaps surprisingly, this suffices to generate a distribution of HIRE signals, which leaves R with no additional information beyond the guarantee of expected utility *μ*R. We obtain a persuasive mechanism and a $(1/(3\\sqrt{3})-o(1))$-approximation of OPT (Theorem [theo:senderCardinalReceiverCardinalSecretaryNoDisclosure]).\nIn the basic scenario with disclosure, we give a characterization of the optimal mechanism using an exponential family of nested linear programs. In terms of polynomial-time algorithms, we design another adaptive version of the Pareto mechanism w.r.t. the shrinking set of non-arrived candidates. It obtains an expected utility for S of at least 1/3 − *o*(1) times OPT (Theorem [theo:senderCardinalReceiverCardinalKnownUtilDisclosure]).\nFor the most challenging scenario with disclosure, we show that there are cases in which no persuasive mechanism can obtain an expected utility of more than 1/*n* – while in the benchmark scenario the expected utility is 1/2. Thus, in terms of approximation of OPT, no persuasive mechanism can have a better approximation ratio than 2/*n* (Theorem [theo:senderOrdinalReceiverCardinalSecretaryDisclosure] and Corollary [cor:senderCardinalReceiverCardinalSecretaryDisclosure]) – while 1/*n* is always achieved by the trivial strategy of recommending a candidate uniformly at random.\n#### Ordinal Case\nOur results for the ordinal case are closer to existing results and techniques for the secretary problem. Our baseline is again the basic scenario. For ordinal utility, the receiver’s only interest is to find her single best candidate. Without the sender signal this is quite hopeless, especially when *n* grows large. Hence, it is not surprising that we can obtain a success probability of 1 − *o*(1) for the sender. The benchmark is simply the standard benchmark of the secretary problem – the best candidate *c*S for S. The intuition applies even in the basic scenario with disclosure – we again obtain success probability of 1 − *o*(1) for the sender (Proposition [prop:receiverOrdinalKnownUtilDisclosure]).\nIn the secretary scenario, we apply a variation of the classic template – we mix the optimal algorithm for S and for R and decide randomly upfront which version is applied. Although intuitive, it requires some effort to show that the mix retains persuasiveness for R even when it strongly favors S. The mechanism yields a success probability of 1/*e* − *o*(1) (Theorem [theo:receiverOrdinalSecretaryNoDisclosure]).\nIn contrast, in the secretary scenario with disclosure, such an approach is not sufficient to incentivize R to follow the mechanism. The first round, in which a best-so-far candidate for R is rejected and revealed, R learns that the sender-optimal version is used. This can give R an incentive to ignore any upcoming HIRE signal. Instead, we run the two optimal algorithms in parallel and hire the first candidate that either algorithm would hire (first-come first-serve). We term this the First-Opt algorithm, and the success probability for S becomes at least 1/4 − *o*(1). Our main insight is that for negatively correlated utilities, this algorithm and the guarantee are indeed optimal, which we prove via a generalized dynamic programming technique (Theorems [theo:receiverOrdinalSecretaryDisclosure] and [thm:BeckmannUB]).\nc||c|c||c|c & & \n& Ordinal S & Cardinal S & Ordinal S & Cardinal S \nBasic & & **1/2** (Thm. [theo:senderOrdinalReceiverCardinalKnownUtilDisclosureUB], [theo:senderOrdinalReceiverCardinalKnownUtilDisclosure]) & 1/3 (Thm. [theo:senderCardinalReceiverCardinalKnownUtilDisclosure]) \nSecretary & 1/4 (Thm. [theo:senderOrdinalReceiverCardinalSecretaryNoDisclosure]) & $1/(3\\sqrt{3})$ (Thm. [theo:senderCardinalReceiverCardinalSecretaryNoDisclosure]) & **\\Theta(1/n)** (Thm. [theo:senderOrdinalReceiverCardinalSecretaryDisclosure]) & **\\Theta(1/n)** (Cor. [cor:senderCardinalReceiverCardinalSecretaryDisclosure])\n[table:lowerBoundsReceiverCard]\nc||c|c||c|c & & \n& Ordinal S & Cardinal S & Ordinal S & Cardinal S \nBasic & **1** (Prop. [prop:receiverOrdinalKnownUtilNoDisclosure]) & **1** (Prop. [prop:receiverOrdinalKnownUtilNoDisclosure]) & **1** (Prop. [prop:receiverOrdinalKnownUtilDisclosure]) & **1** (Prop. [prop:receiverOrdinalKnownUtilDisclosure]) \nSecretary & **1/e** (Thm. [theo:receiverOrdinalSecretaryNoDisclosure]) & **1/e** (Thm. [theo:receiverOrdinalSecretaryNoDisclosure]) & **1/4** (Thm. [theo:receiverOrdinalSecretaryDisclosure], [thm:BeckmannUB]) & **1/4** (Thm. [theo:receiverOrdinalSecretaryDisclosure], [thm:BeckmannUB]) \n[table:lowerBoundsReceiverOrd]\nOverview\n--------\nIn Table [table:lowerBoundsReceiverCard], we summarize the approximation guarantees when R has cardinal utility, in Table [table:lowerBoundsReceiverOrd] the ones when R has ordinal utility. We view the basic scenario without disclosure as our benchmark scenario. Hence, all other entries in both tables represent approximation ratios with respect to the optimal value for S that can be obtained in this case. In particular, for ordinal sender utility OPT = 1 − *o*(1), the optimal success probability in the benchmark scenario. Horizontal comparisons imply whether or not disclosure of rejected candidates to R comes at a cost for S. Vertical comparisons yield the loss in performance due to S not knowing valuation pairs upfront. Observe that except for two variants (secretary with disclosure and cardinal receiver), all bounds are reasonable constants. Hence, when S does not know the future or rejected candidates are revealed, it usually implies just a limited loss in her expected utility.\nWe complement the majority of our ratios with asymptotically tight upper bounds. Some of these upper bounds are new, such as 1/2 in the basic scenario with disclosure, or 1/4 and *O*(1/*n*) in the secretary scenario with disclosure. These bounds imply structural differences between our signaling scenarios and the standard variant of the secretary problem.\nThe subsequent technical parts are organized as follows. Section [sec:cardinal] focuses on the case where the receiver maximizes her expected utility. The techniques in this section depart from the classical secretary problem. The section discusses the four scenarios of the problem: basic vs. secretary and disclosure vs. non-disclosure. Each scenario is analyzed for ordinal and cardinal sender utility. We open the section with the benchmark case (basic, non-disclosure) and conclude with the most complex one (secretary with disclosure). Each scenario is analyzed for cardinal and ordinal sender utility, respectively. For ordinal receiver utility, we proceed similarly in Appendix [sec:ordinal].\nCardinal Utility for R\n======================\nWe consider cardinal receiver utility, i.e., R wants to maximize the expected utility of the hired candidate. To capture the sender’s value (i.e., her success probability/expected utility), we use the Pareto frontier, a geometric interpretation of the candidate set. It consists of the upper boundary of the convex hull of the candidates’ valuations. Hence, for increasing receiver-values, the Pareto frontier is non-increasing in the sender-values. This means that the value for S of the Pareto curve at *ρ* = *μ*R is an obvious upper bound on the sender’s expected utility.\n[t] Scale and normalize the candidate values, set *μ*R ← ∑*i* = 1*n**ρ**i*/*n* Let P = {(*ρ**i*, *ξ**i*) ∣ *i* ∈ {1, …, *n*}} and *c**o**n**v*(P) be the convex hull of P Let *P**C*(P) be the Pareto frontier of *c**o**n**v*(P) If *ρ**c*S ≥ *μ*R, then set *a* = *b* = *c*S; otherwise, find *a*, *b* ∈ {1, …, *n*} s.t. (*a*, *b*) is the segment of *P**C*(P) that intersects with line *ρ* = *μ*R (see Fig. [fig:mechanism] for an illustration) `// a=b possible` Determine probability for candidate *a*: \nIf *ρ**a* = *ρ**b* then *α* ← 1; if *ρ**a* ≠ *ρ**b* and *ξ**a* = *ξ**b*, then *α* ← 0; else $\\alpha \\leftarrow \\frac{\\mu^{{\\ensuremath{\\mathcal{R}}}} - {\\rho}\\_b}{{\\rho}\\_a - {\\rho}\\_b}$. Draw *x* uniform from [0, 1]. If *x* ≤ *α* then *c* ← *a*, else *c* ← *b*. candidate *c*\nBenchmark: Basic Scenario without Disclosure\n--------------------------------------------\nIn the benchmark scenario we assume that the point set P = {(*ρ**i*, *ξ**i*) ∣ *i* ∈ [*n*]} is known a-priori to both S and R. Only upon hiring, the identity of the hired candidate becomes known to R. The identity of rejected candidates is not revealed. Consider the *Pareto mechanism*. It is based on the Pareto procedure in Algorithm [algo:Pareto] that computes the best linear combination of points (denoted by OPT) within the convex hull of P that guarantees at least the average value *μ*R for R. W.l.o.g. this is a linear combination of at most two points on the Pareto frontier. We interpret the linear combination as the probability distribution to signal for the corresponding candidates (see Fig. [fig:mechanism] for an example). The procedure returns the chosen candidate *c*. The Pareto mechanism then signals *σ**t* = HIRE exactly in the round *t* in which candidate *θ**t* = *c* arrives.\nWe assume that internally the Pareto procedure works with a scaled and normalized version of its input. First, it scales all values in ${\\bm{{\\xi}}}$ such that *ξ*max = 1, and all values in ${\\bm{{\\rho}}}$ such that *ρ*max = 1. Second, in case of ordinal sender utility, it sets *ξ**i*ʹ = 0 for all other candidates *i*ʹ ≠ *c*S. In this way, the procedure can be applied to both instances with ordinal or cardinal sender utility, and its output is based on a lottery over candidates that maximizes success probability or expected utility for S, resp. Observe that in the scenarios discussed below, the procedure is applied with different subsets of candidate values as input. In this case, the procedure applies the internal adjustment w.r.t. the subset of candidates provided in the input.\nc||c|c|c|c|c|c|c|c||c|c & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & *μ* & OPT \n*ρ* & 1 & 3 & 6 & 7 & 12 & 13 & 14 & 16 & 9 & 9 \n*ξ* & 8 & 1 & 10 & 4 & 8 & 5 & 6 & 2 & 6.5 & 9\n[baseline=2cm, scale =.35] (0,0) – (17,0) node[below] *ρ*; (0,0) – (0,11) node[left] *ξ*;\n(0,10) – (6,10) – (12,8) – (14,6) – (16,2) – (16,0);\nat (1,8) (1) × ; at (3,1) (2) × ; at (6,10) (3) × ; at (5.7,11) (a) *a*; at (7,4) (4) × ; at (12,8) (5) × ; at (12.3,9) (b) *b*; at (13,5) (6) × ; at (14,6) (7) × ; at (16,2) (8) × ;\nat (9, -1) (mur) *μ*R; at (9,9) (opt) $\\blacksquare$; at (9.7,10) (opt) OPT;\n(9,0) – (9,8.5);\n[prop:Pareto] The Pareto mechanism is an optimal persuasive mechanism.\nConsider the Pareto mechanism and the event that R gets a signal *σ**t* = HIRE in round *t*. We denote by ${\\bm{x}}$ the distribution over candidates in round *t* conditioned on this event. Clearly ${\\bm{x}}$ is the same distribution, no matter in which round *t* R gets a signal to hire. Now denote by ${\\bm{y}}$ the distribution over candidates in round *t*ʹ conditioned on a HIRE signal in round *t* ≠ *t*ʹ. Clearly, ${\\bm{y}}$ is the same distribution, for any *t*, *t*ʹ with *t* ≠ *t*ʹ.\nFor R, the expected utility of following a signal is ${\\bm{x}}^T {\\bm{{\\rho}}}\\ge \\mu^{{\\ensuremath{\\mathcal{R}}}}$ by construction of the mechanism. By simply hiring in round 1 deterministically, R gets a value of *μ*R and a signal with probability 1/*n*. Hence, $({\\bm{x}}+ (n-1) {\\bm{y}})^T {\\bm{{\\rho}}}= n \\mu^{{\\ensuremath{\\mathcal{R}}}}$. Consequently, the expected utility of R for hiring upon a signal is ${\\bm{y}}^T {\\bm{{\\rho}}}\\le \\mu^{{\\ensuremath{\\mathcal{R}}}}$. Thus, the mechanism is persuasive.\nNow consider any persuasive mechanism *ϕ* used by S. Let *x**i* denote the probability that (over random arrival and randomization in *ϕ*) candidate *i* ∈ [*n*] is the first one with signal HIRE. We use ${\\bm{x}}$ to denote the vector. Truthfulness of *ϕ* implies the following constraints: (1) ${\\bm{x}}^T {\\bm{{\\rho}}}\\ge \\mu^{{\\ensuremath{\\mathcal{R}}}}$, since this expected utility can be achieved by R simply by hiring in any fixed round; (2) $\\| {\\bm{x}}\\|\\_1 \\le 1$, since the mechanism can be assumed to recommend at most one candidate; (2) $\\| {\\bm{x}}\\|\\_1 \\ge 1$, since all *ρ**i* ≥ 0 at least one candidate must be recommended (otherwise, R will deviate from *ϕ* by hiring in the last round). Hence, the distribution ${\\bm{x}}$ resulting from the optimal persuasive mechanism is a feasible solution to the following maximization problem for the expected sender utility\n \n$$\\label{eq:basicLP}\n \\begin{array}{lrcll}\n \\renewcommand{{1.66}}{0.5}\n \\mbox{Max.} & \\multicolumn{2}{l}{\\bm{x}^T {\\bm{{\\xi}}}} \\\\\n \\mbox{s.t.} & {\\bm{x}}^T {\\bm{{\\rho}}}& \\ge & \\mu^{{\\ensuremath{\\mathcal{R}}}} \\\\\n & \\|{\\bm{x}}\\|\\_1 & = & 1\\\\\n & x\\_i & \\ge & 0 &\\hspace{0.5cm} \\mbox{for all } i=1,\\ldots,n\n \\end{array}$$ \nIt is straightforward to see that the Pareto mechanism computes a distribution that represents an optimum solution to the above LP. Hence, it is an optimal persuasive mechanism.\nSecretary Scenario without Disclosure\n-------------------------------------\nThe secretary scenario differs from the previous one by S a-priori lacking knowledge of the valuation pairs. To cope with this challenge, we apply the Pareto procedure adaptively. Our mechanism *ϕ*(*s*) first samples a number of rounds and signals *σ**t* = NOT for rounds *t* = 1, …, *s*. Then, in each round *t* = *s* + 1, …, *n* − 1 let *A**t* be the set of all previously arrived candidates, including the one currently under consideration in round *t*. The mechanism invokes the Pareto procedure on the set *A**t*. It signals *σ**t* = HIRE if and only if the candidate chosen by the Pareto procedure has arrived in the current round *t*.\n[t] *A*0 ← ∅ Signal HIRE on *n*-th candidate.\nWhen *σ**t* = HIRE and R deviates by refusing to hire, the mechanism signals in every subsequent round. If *t* = *n* and the mechanism has not signaled so far, it sets *σ**n* = HIRE deterministically in the last round. Due to the dependence on a growing candidate set *A**t*, we term this mechanism the *Growing Pareto mechanism* (see Algorithm [algo:GrowingPareto]).\nThe main results about the Growing Pareto mechanism are as follows. We first show that the mechanism is persuasive. We then prove the approximation guarantee of 1/4 − *o*(1) for the ordinal sender utility and $1/(3\\sqrt{3}) - o(1)$ for the cardinal sender utility.\nLet us first consider persuasiveness. Consider round *t*. We follow and rephrase the generation of the candidate arriving in round *t* as follows: First choose the subset *A**t* of candidates that arrived in rounds 1, …, *t* uniformly at random from [*n*], then pick the candidate arriving in round *t* uniformly at random from *A**t*.\n[lem:signalT] Consider a given round *t* and a given subset of candidates *A**t* that arrived up to round *t*. In the Growing Pareto mechanism \n$$\\Pr\\left[\\bigwedge\\_{i=1}^{t-1} {\\sigma}\\_i = {\\text{NOT}}{\\mathrel{}\\middle|\\mathrel{}}A\\_{t-1} \\right] = \\begin{cases} 1 & t = 2,\\ldots,s+1\\\\\n \\frac{s}{t-1} & t = s+2,\\ldots,n\\end{cases}$$ \n and \n$$\\Pr[{\\sigma}\\_t = {\\text{HIRE}}\\mid A\\_t] = \\begin{cases} 0 & t = 1,\\ldots,s\\\\\n \\frac 1t \\cdot \\frac{s-1}{t-1} & t = s+1,\\ldots,n-1\\\\\n \\frac{s}{n-1} & t = n\\end{cases}.$$ \nGiven that the set *A**t* of candidates arrived up to round *t*, we draw all possible arrival sequences of *A**t* in the first *t* rounds in a reverse fashion. For *A**t*, the Pareto procedure singles out the candidates *a* and *b*. The candidate *i**t* ∈ [*n*] that arrives in round *t* is chosen uniformly at random from *A**t*. Thus, the probability is $\\frac 1t$ that the candidate is in {*a*, *b*} and that in addition the Pareto procedure applied in round *t* would return this candidate. For a signal *σ**t* = HIRE of the Growing Pareto mechanism, however, it also needs to hold that *σ**i* = NOT for all *i* ≤ *t* − 1.\nGiven candidate set *A**t* arrives in the first *t* rounds and candidate *i**t* in round *t*, consider the signal in round *t* − 1. Now given the set *A**t* − 1 = *A**t* \\ {*i**t*}, the Pareto procedure singles out some candidates *a* and *b*. Since the candidate *i**t* − 1 arriving in round *t* − 1 is chosen uniformly at random from *A**t* − 1, the probability that the Pareto procedure in round *t* − 1 results in signal is $\\frac{t-2}{t-1}$.\nSuppose in round *t* − 1 a candidate *i**t* − 1 arrives and the Pareto procedure results in signal *σ**t* − 1 = NOT. Then, we can apply the same argument for round *t* − 2 and candidate set *A**t* − 2 = *A**t* \\ {*i**t*, *i**t* − 1}. Since we need a signal in all previous rounds *i* ∈ [*t* − 1], we continue to apply the argument pointwise for all subsets *A**i*. Note that for *i* ≤ *s* the probability Pr[*σ**i* = NOT] = 1 always.\nHence, for every set *A**t* − 1 we obtain \n$$\\Pr\\left[\\bigwedge\\_{i=1}^{t-1} {\\sigma}\\_i = {\\text{NOT}}{\\mathrel{}\\middle|\\mathrel{}}A\\_{t-1} \\right]= \\frac{t-2}{t-1} \\cdot \\frac{t-3}{t-2} \\cdots \\frac{s}{s+1} = \\frac{s}{t-1}$$ \n for rounds *t* = *s* + 2, …, *n*. Thus, for rounds *t* = *s* + 1, …, *n* − 1 we see \n$$\\Pr[{\\sigma}\\_t = {\\text{HIRE}}\\mid A\\_t] = {\\mathbb{E}}\\_{i\\_t} \\left[\\Pr[\\sigma\\_t = {\\text{HIRE}}\\mid A\\_t, i\\_t] \\cdot \\Pr\\left[\\bigwedge\\_{i=1}^{t-1} {\\sigma}\\_i = {\\text{NOT}}{\\mathrel{}\\middle|\\mathrel{}}A\\_t \\setminus \\{i\\_t\\}\\right]\\right] = \\frac{1}{t} \\cdot \\frac{s-1}{t-1}$$ \n and, analogously, $ \\Pr[{\\sigma}\\_t = {\\text{HIRE}}\\mid A\\_t] = 1\\cdot \\frac{s}{n-1}$ for *t* = *n*.\n[lem:GrowingParetoIC] The Growing Pareto mechanism is persuasive in the secretary scenario without disclosure.\nSuppose the Growing Pareto mechanism sets *σ**t* = HIRE in some round *t* ∈ {*s* + 1, …, *n*} and *σ**t*ʹ = NOT for all *t*ʹ ∈ [*t* − 1]. Note *A**t* is a subset chosen uniformly at random. For every *i* ∈ *A**t*, the probability to receive only signals for *A**t* \\ {*i*} in rounds 1, …, *t* − 1 is the same. Hence, by following the signal, R obtains an expected utility of \n$${\\mathbb{E}}[{\\rho}\\_t \\mid {\\sigma}\\_t = {\\text{HIRE}}] \\quad \\ge \\quad {\\mathbb{E}}\\_{A\\_t}\\left[\\sum\\_{i \\in A\\_t} \\frac{{\\rho}\\_i}{t}\\right] \\quad = \\quad \\mu^{{\\ensuremath{\\mathcal{R}}}}.$$ \n Consider the event that R sees only NOT in all rounds up to *t* − 1. By Lemma [lem:signalT], this event has the same probability for all subsets *A**t* − 1. Hence, even conditioned on this event, the candidate arriving in round *t* is uniformly distributed. Thus, $ {\\mathbb{E}}\\left[{\\rho}\\_t {\\mathrel{}\\middle|\\mathrel{}}\\bigwedge\\_{t'=1}^{t-1} {\\sigma}\\_{t'} = {\\text{NOT}}\\right] = \\mu^{{\\ensuremath{\\mathcal{R}}}}$. Therefore, the expected utility of hiring when is received also in round *t* is ${\\mathbb{E}}\\left[{\\rho}\\_t {\\mathrel{}\\middle|\\mathrel{}}\\bigwedge\\_{t'=1}^{t} {\\sigma}\\_{t'} = {\\text{NOT}}\\right] \\le \\mu^{{\\ensuremath{\\mathcal{R}}}}$.\nNow consider the event that R observes *σ**t* = HIRE. For all possible subsets *A**t* this event has the same probability by Lemma [lem:signalT]. Hence, for any round *r* > *t*, candidate *θ**r* is uniformly distributed and gives expected value of *μ*R. Thus, no *r* > *t* yields a profitable deviation. The lemma follows.\nWe now prove the approximation result, first for the case when S has cardinal utility and aims to maximize her expected utility. The Growing Pareto mechanism provides a constant-factor approximation to the optimum in the corresponding benchmark scenario.\n[theo:senderCardinalReceiverCardinalSecretaryNoDisclosure] In the secretary scenario without disclosure, the Growing Pareto mechanism with $s= \\lfloor n/\\sqrt{3} \\rfloor$ yields a $\\left(\\frac{1}{3\\sqrt{3}} - o(1)\\right)$-approximation of the optimal expected utility in the corresponding benchmark scenario.\nl0.45\n[xscale =.5, yscale=0.32] (0,0) – (9,0) node[below] *ρ*; (0,0) – (0,9) node[left] *ξ*; (0,7.2) – (4.2,7.2) – (7.4,3.4) – (8,0) ; (0,6.25) – (5,6.25) – (8,0); at (5, -0.75) (mur) *μ*R; at (4.2,7.6) (a) *a*; at (7.8,3.4) (b) *b*; at (6.2,6.25) (opt) OPT; at (5,6.25) (markeropt) $\\blacksquare$;\nThe proof of this theorem relies on some auxiliary observations. Let *L* = {*i* ∈ [*n*] ∣ *ρ**i* ≤ *μ*R} and *H* = {*i* ∈ [*n*] ∣ *ρ**i* > *μ*R} = [*n*] \\ *L* be a partition of the candidates into the ones with low receiver utility and high receiver utility, respectively. Let *d* = ∣*L*∣ denote the cardinality of *L*. For the sake of analysis, we assume that *ρ*max = 1 and *ξ*max = 1.\nWe first concentrate on the case with cardinal sender utility. Consider the Pareto mechanism in the benchmark scenario. If the sender-optimal candidate has receiver utility *ρ**c*S ≥ *μ*R, then *a* = *b*. The mechanism will wait for this particular candidate and recommend HIRE deterministically. Otherwise, *a* ≠ *b*, and OPT is both expected utility for S and success probability for hiring *c*S.\nFor any subset of candidates *M* ⊆ [*n*], we use OPT− *M* to denote the expected utility for S in the benchmark scenario with candidate set [*n*] \\ *M*. Similar to OPT− {*i*}, we define $\\mu\\_{-i}^{{\\ensuremath{\\mathcal{R}}}} = \\frac{1}{n-1}\\left(\\sum\\_{j=1}^n {\\rho}\\_j - {\\rho}\\_i\\right)$ for *i* ∈ [*n*].\n[lem:lossPerRoundCard] Consider an instance of the benchmark scenario with candidate set [*n*]. Let *a* and *b* be the candidates as determined in the Pareto procedure. It holds that \n∑*i* ≠ *a*, *b*OPT− {*i*} ≥ (*n* − 3)OPT. \nConsider the signaling scheme computed by the Pareto mechanism. If we adapt the instance and set *ξ**j* = 0 for all *j* ≠ *a*, *b*, this does not change OPT, but it can only worsen the optima OPT− {*i*}. Hence, in the new instance, the Pareto frontier of the convex hull of P consists only of the (at most) two points for candidates *a*, *b*, and a point for candidate (max*i**ρ**i*, 0) and the point (0, max*i**ξ**i*) = (0, *ξ**a*) (see Fig. [fig:reducedFrontier]). As described above, we partition the candidate set into the sets *L* and *H* and set *d* = ∣*L*∣. The resulting value OPT− {*i*} is different depending on the candidate *i* that gets removed from the pool. We underestimate the Pareto frontier with a line to (*ρ*max, 0) (see the dashed line in Fig. [fig:reducedFrontier]). More formally, to estimate the loss in expected utility, use the slope $\\frac{{\\ensuremath{\\operatorname{OPT}}}}{{\\rho}\\_{\\max} - \\mu^{{\\ensuremath{\\mathcal{R}}}}}$ from the point (*μ*R, OPT) representing the optimum to the point with maximal receiver value. The loss in sender utility caused by the change $\\mu^{{\\ensuremath{\\mathcal{R}}}}\\_{-i} - \\mu^{{\\ensuremath{\\mathcal{R}}}} = \\sum\\_{j \\in [n] \\setminus \\{i\\}} \\frac{{\\rho}\\_j}{n-1} - \\mu^{{\\ensuremath{\\mathcal{R}}}}$ can be bounded by\n \n$$\\begin{aligned}\n &\\sum\\_{i \\in L \\setminus\\{a\\}}{\\ensuremath{\\operatorname{OPT}}}- {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} \\quad \\le \\quad \\frac{{\\ensuremath{\\operatorname{OPT}}}}{{\\rho}\\_{\\max}-\\mu^{\\ensuremath{\\mathcal{R}}}} \\sum\\_{i \\in L} \\left[ \\sum\\_{j \\in [n] \\setminus \\{i\\}} \\frac{{\\rho}\\_j}{n-1} - \\mu^{{\\ensuremath{\\mathcal{R}}}}\\right]\\\\\n &\\quad = \\quad \\frac{1}{n-1}\\cdot \\frac{{\\ensuremath{\\operatorname{OPT}}}}{{\\rho}\\_{\\max}-\\mu^{{\\ensuremath{\\mathcal{R}}}}} \\sum\\_{i \\in L} \\left[ \\sum\\_{j \\in L \\setminus \\{i\\}} {\\rho}\\_j + \\sum\\_{j \\in H} {\\rho}\\_j - (n-1)\\mu^{{\\ensuremath{\\mathcal{R}}}} \\right] \\\\\n &\\quad = \\quad \\frac{{\\ensuremath{\\operatorname{OPT}}}}{(n-1)({\\rho}\\_{\\max}-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{j \\in H} \\left({\\rho}\\_j - \\mu^{{\\ensuremath{\\mathcal{R}}}} \\right) \\\\\n &\\quad \\le \\quad \\frac{{\\ensuremath{\\operatorname{OPT}}}}{(n-1)({\\rho}\\_{\\max}-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{j \\in H} \\left({\\rho}\\_{\\max} - \\mu^{{\\ensuremath{\\mathcal{R}}}} \\right) \\\\\n &\\quad = \\quad \\frac{n-d}{(n-1)} {\\ensuremath{\\operatorname{OPT}}}\\quad \\le \\quad {\\ensuremath{\\operatorname{OPT}}}\\enspace.\n \\end{aligned}$$ \nIf a candidate *i* ∈ *H* arrives, *μ*− *i*R ≤ *μ*R, and thus the expected payoff for S can only increase. We can lower bound the payoff by OPT, c.f. Fig. [fig:reducedFrontier], i.e. \n∑*i* ∈ *H* \\ {*b*}OPT − OPT− {*i*} ≤ 0. \n Overall, this implies \n(*n* − 2) ⋅ OPT − ∑*i* ∈ [*n*] \\ {*a*, *b*}OPT− {*i*} ≤ OPT \n which implies the lemma.\nLet OPT*t* be the expected value of the Pareto mechanism when applied to the benchmark scenario composed of the random subset *A**t*. Note that OPT*n* = OPT, the optimum in the benchmark scenario.\n[cor:valueRoundTCard] For *t* ≥ 3 it holds that \n$${\\ensuremath{\\operatorname{OPT}}}\\_t \\ge \\prod\\_{k=t+1}^{n} \\left(1 - \\frac{3}{k}\\right) {\\ensuremath{\\operatorname{OPT}}}= \\frac{t(t-1)(t-2)}{n(n-1)(n-2)}{\\ensuremath{\\operatorname{OPT}}}.$$ \nWe can generate the random set *A**t* by starting with [*n*] and iteratively removing a random candidate. Note that for *t* = *n* − 1 we have by Lemma [lem:lossPerRoundCard] \n$${\\ensuremath{\\operatorname{OPT}}}\\_{n-1} = \\frac{1}{n} \\sum\\_{i \\in [n]} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} \\ge \\frac{1}{n} \\sum\\_{i \\in [n] \\setminus \\{a,b\\}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} \\ge \\frac{n-3}{n} {\\ensuremath{\\operatorname{OPT}}}.$$ \n The result for *t* < *n* − 1 follows by repeated application.\nFinally, we prove Theorem [theo:senderCardinalReceiverCardinalSecretaryNoDisclosure], the approximation result for cardinal sender utility.\n[Proof of Theorem [theo:senderCardinalReceiverCardinalSecretaryNoDisclosure]] By combining the insights of Corollary [cor:valueRoundTCard] and Lemma [lem:signalT], we see that in a given round *t* = *s* + 1, …, *n* − 1 the Growing Pareto mechanism obtains an expected utility for S of at least Pr[*σ**t* = HIRE] ⋅ OPT*t*. For simplicity, we underestimate the utility in the last round *t* = *n* by 0. For the expected utility of S, we use linearity of expectation over all rounds and set *s* = ⌊*c* ⋅ *n*⌋ for a constant *c*:\n \n$$\\begin{aligned}\n &\\sum\\_{t=s+1}^{n-1} \\frac{1}{t} \\cdot \\frac{s}{t-1} \\cdot \\frac{t(t-1)(t-2)}{n(n-1)(n-2)} {\\ensuremath{\\operatorname{OPT}}}\\quad\n = \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\; \\frac{s}{n(n-1)(n-2)} \\sum\\_{t=s+1}^{n-1} (t-2) \\\\\n &\\quad = \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\; \\frac{s}{n(n-1)(n-2)} \\cdot \\left( \\frac{n(n-1)}{s} - \\frac{s(s+1)}{2} - 2(n-1-s) \\right) \\\\\n &\\quad = \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\left(\\frac{s}{n-2} - \\frac{s^2(s+1)}{2n(n-1)(n-2)} - \\frac{2s(n-1-s)}{n(n-1)(n-2)} \\right) \\\\\n &\\quad = \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\; (c - c^3 - o(1))\\enspace.\n \\end{aligned}$$ \nThe last expression is maximized at $c = \\frac{1}{\\sqrt{3}}$, so we set $s = \\lfloor n/\\sqrt{3}\\rfloor$. The theorem follows.\nWe now proceed to the approximation result for ordinal sender utility.\n[theo:senderOrdinalReceiverCardinalSecretaryNoDisclosure] In the secretary scenario without disclosure, the Growing Pareto mechanism with *s* = ⌊*n*/2⌋ yields a success probability of at least $\\left(\\frac{1}{4}-o(1)\\right)$ times the success probability in the corresponding benchmark scenario.\nWe use the same notation as for the cardinal case above. In addition, let *ρ*2*n**d* denote the second highest utility of any candidate for R. For the Growing Pareto mechanism, the Pareto procedure requires cardinal values for all candidates in *A**t*. Since we assume ordinal sender utility, we set the sender utility to 1 for the best known candidate for S in *A**t*, and 0 otherwise. Note that then OPT becomes both – expected utility for S and success probability for hiring *c*S. In step 1, the Pareto procedure then applies internal scaling and normalization to *A**t* (as discussed above in Section [sec:cardinal]). It scales the receiver utilities from the input such that the best known candidate for R in *A**t* is normalized to receiver utility 1.\nThe next Lemma [lem:lossPerRoundOrd] represents an improved version of Lemma [lem:lossPerRoundCard]. Throughout the following analysis, we use OPT− *M* to denote the probability of hiring the best candidate for S in [*n*] \\ *M*.\n[lem:lossPerRoundOrd] Let OPT and OPT− {*i*} denote the expected utility in the benchmark scenario for candidate sets [*n*] and [*n*] \\ {*i*}, respectively. Then, the following holds: \n$$\\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}\\cdot {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} \\ge {\\ensuremath{\\operatorname{OPT}}}\\left(n-2 - \\frac{1}{n-1}\\right)\\enspace.$$ \nFor the first case, we assume that the sender-optimal candidate *c*S has receiver utility *ρ**c*S ≤ *μ*R ≤ *ρ*2*n**d*. In this case, the Pareto procedure sets *a* = *c*S and *b* = *c*R, since *c*S is the only candidate with non-zero utility for S and has receiver utility below *μ*R. Since all other candidates have sender utility 0, the convex hull is composed of the segment between sender- and receiver-optimal candidates *c*S and *c*R.\nThe resulting value OPT− {*i*} is different depending on the candidate that gets removed from the pool. If a candidate *i* ∈ *L* with *ρ**i* ≤ *μ*R is removed, then *μ*− *i*R ≥ *μ*R, which implies that OPT− {*i*} ≤ OPT. Note that upon removal of *b*, we have OPT− {*b*} = 1 if *μ*− *b*R ≤ *ρ**a*. Otherwise, the new optimum point is located at *μ*− *b*R and has value ${\\ensuremath{\\operatorname{OPT}}}\\_{-\\{b\\}} = \\frac{{\\rho}\\_{2nd} - \\mu\\_{-b}^{{\\ensuremath{\\mathcal{R}}}}}{\\mu\\_{2nd} - {\\rho}\\_a}$. Overall, we see that\n \n$$\\begin{aligned}\n &\\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}\\cdot {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} \\\\\n &= \\sum\\_{i \\in L\\setminus \\{a\\}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + \\sum\\_{i \\in H \\setminus \\{b\\}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}\\cdot \\min \\left\\{1, \\frac{{\\rho}\\_{2nd} - \\mu\\_{-b}^{{\\ensuremath{\\mathcal{R}}}}}{{\\rho}\\_{2nd}-{\\rho}\\_a}\\right\\} \\\\\n &= {\\ensuremath{\\operatorname{OPT}}}\\left[(d-1) - \\frac{1}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{i\\in L \\setminus \\{a\\}} \\left( \\mu^{{\\ensuremath{\\mathcal{R}}}} - {\\rho}\\_i \\right) + (n-d-1) \\right.\\\\\\*\n &\\hspace{45pt} + \\left. \\sum\\_{i \\in H \\setminus \\{b\\}} \\min \\left\\{\\frac{{\\rho}\\_i-\\mu^{{\\ensuremath{\\mathcal{R}}}}}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})},\\frac{\\mu^{{\\ensuremath{\\mathcal{R}}}}-{\\rho}\\_a}{1-\\mu^{{\\ensuremath{\\mathcal{R}}}}}\\right\\} + \\frac{{\\rho}\\_{2nd} - \\mu\\_{-b}^{{\\ensuremath{\\mathcal{R}}}}}{{\\rho}\\_{2nd}-{\\rho}\\_a} \\right] \\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left[(n-2) - \\frac{1}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{i\\in L} \\left( \\mu^{{\\ensuremath{\\mathcal{R}}}} - {\\rho}\\_i \\right) \\right.\\\\\\*\n &\\hspace{45pt} + \\left. \\sum\\_{i \\in H \\setminus \\{b\\}} \\min \\left\\{\\frac{{\\rho}\\_i-\\mu^{{\\ensuremath{\\mathcal{R}}}}}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})},\\frac{\\mu^{{\\ensuremath{\\mathcal{R}}}}-{\\rho}\\_a}{1-\\mu^{{\\ensuremath{\\mathcal{R}}}}}\\right\\} + \\frac{{\\rho}\\_{2nd} - \\mu^{{\\ensuremath{\\mathcal{R}}}}}{{\\rho}\\_{2nd}-{\\rho}\\_a} \\right] \\\\\n &= {\\ensuremath{\\operatorname{OPT}}}\\left[(n-2) + \\frac{{\\rho}\\_{2nd} - \\mu^{{\\ensuremath{\\mathcal{R}}}}}{{\\rho}\\_{2nd}-{\\rho}\\_a} - \\frac{1}{n-1} \\right.\\\\\\*\n &\\hspace{45pt} - \\left. \\frac{1}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{i\\in H \\setminus \\{b\\}} \\left( \\left({\\rho}\\_i - \\mu^{{\\ensuremath{\\mathcal{R}}}}\\right) - \\min \\left\\{{\\rho}\\_i-\\mu^{{\\ensuremath{\\mathcal{R}}}},(\\mu^{{\\ensuremath{\\mathcal{R}}}}-{\\rho}\\_a)(n-1)\\right\\} \\right) \\right] \\enspace.\\\\\n \\end{aligned}$$ \nWe consider two subcases.\nSubcase $\\mu^{{\\ensuremath{\\mathcal{R}}}} < \\frac{{\\rho}\\_{2nd} + (n-1) {\\rho}\\_a}{n}$:\nHere, we see\n \n$$\\begin{aligned}\n &\\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}\\cdot {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}}\\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left[(n-2) + \\frac{{\\rho}\\_{2nd} - \\mu^{{\\ensuremath{\\mathcal{R}}}}}{{\\rho}\\_{2nd}-{\\rho}\\_a} - \\frac{1}{n-1} \\right.\\\\\n &\\hspace{45pt} - \\left. \\frac{1}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{i\\in H \\setminus \\{b\\}} \\left( \\left({\\rho}\\_i - \\mu^{{\\ensuremath{\\mathcal{R}}}}\\right) - \\min \\left\\{{\\rho}\\_i-\\mu^{{\\ensuremath{\\mathcal{R}}}},(\\mu^{{\\ensuremath{\\mathcal{R}}}}-{\\rho}\\_a)(n-1)\\right\\} \\right) \\right] \\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left[(n-2) + 1 - \\frac{\\mu^{{\\ensuremath{\\mathcal{R}}}}-{\\rho}\\_a}{{\\rho}\\_{2nd}-{\\rho}\\_a} - \\frac{1}{n-1} - \\frac{n-2}{n-1} \\right] \\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left[n-2 - \\frac{1}{n} \\right]\\enspace.\n \\end{aligned}$$ \nFor the penultimate line we used that *ρ**i* ≤ 1 and the min is at least 0. In the last inequality, we used the assumption that $\\mu^{{\\ensuremath{\\mathcal{R}}}} < \\frac{{\\rho}\\_{2nd} + (n-1) {\\rho}\\_a}{n}$ and thus \n$$- \\frac{\\mu^{{\\ensuremath{\\mathcal{R}}}}-{\\rho}\\_a}{{\\rho}\\_{2nd} - {\\rho}\\_a} \\ge - \\frac{{\\rho}\\_{2nd}-{\\rho}\\_a}{n({\\rho}\\_{2nd}-{\\rho}\\_a)} = -\\frac{1}{n}\\enspace.$$ \nSubcase $\\mu^{{\\ensuremath{\\mathcal{R}}}} \\ge \\frac{{\\rho}\\_{2nd} + (n-1) {\\rho}\\_a}{n}$:\nIn this case, we see \n∑*i* ∈ *H* \\ {*b*}((*ρ**i* − *μ*R) − min{*ρ**i* − *μ*R, (*μ*R − *ρ**a*)(*n* − 1)}) = 0 \n as *ρ**i* ≤ *ρ*2*n**d* for all *i* ∈ *H* \\ {*b*} and *ρ*2*n**d* − *μ*R ≤ (*μ*R − *ρ**a*)(*n* − 1). Thus,\n \n$$\\begin{aligned}\n &\\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}\\cdot {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} \\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left[(n-2) + \\frac{{\\rho}\\_{2nd} - \\mu^{{\\ensuremath{\\mathcal{R}}}}}{{\\rho}\\_{2nd}-{\\rho}\\_a} - \\frac{1}{n-1} \\right.\\\\\n &\\hspace{45pt} - \\left. \\frac{1}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{i\\in H \\setminus \\{b\\}} \\left( \\left({\\rho}\\_i - \\mu^{{\\ensuremath{\\mathcal{R}}}}\\right) - \\min \\left\\{{\\rho}\\_i-\\mu^{{\\ensuremath{\\mathcal{R}}}},(\\mu^{{\\ensuremath{\\mathcal{R}}}}-{\\rho}\\_a)(n-1)\\right\\} \\right) \\right] \\\\\n &= {\\ensuremath{\\operatorname{OPT}}}\\left[(n-2) + \\frac{{\\rho}\\_{2nd} - \\mu^{{\\ensuremath{\\mathcal{R}}}}}{{\\rho}\\_{2nd}-{\\rho}\\_a} - \\frac{1}{n-1} \\right] \\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left[n-2 - \\frac{1}{n-1} \\right]\\enspace.\n \\end{aligned}$$ \nNow, consider the second case in which the sender-optimal candidate *c*S has receiver utility *ρ**c*S, *ρ*2*n**d* ≤ *μ*R. Again, in this case the Pareto procedure chooses *a* = *c*S and *b* = *c*R. We can bound as follows\n \n$$\\begin{aligned}\n &\\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}\\cdot {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} = \\sum\\_{i \\in L \\setminus \\{a\\}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}\\cdot {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} \\\\\n &\\ge (n-2){\\ensuremath{\\operatorname{OPT}}}- \\frac{1}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{i \\in L } \\left( \\mu^{{\\ensuremath{\\mathcal{R}}}} - {\\rho}\\_i \\right) \\\\\n &= \\left(n-2-\\frac{1}{n-1}\\right){\\ensuremath{\\operatorname{OPT}}}\\enspace.\n \\end{aligned}$$ \nFinally, consider the third case, *c*S has *ρ**c*S > *μ*R. Then, the Pareto procedure chooses *a* = *b* = *c*S, and OPT− *c*R = 1. We can bound as follows\n \n$$\\begin{aligned}\n &\\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}= \\sum\\_{i \\in L} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + \\sum\\_{i \\in H \\setminus \\{c\\_{\\ensuremath{\\mathcal{S}}}, c\\_{\\ensuremath{\\mathcal{R}}}\\}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + {\\ensuremath{\\operatorname{OPT}}}\\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left(d - \\frac{1}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})} \\sum\\_{i \\in L} \\left(\\mu^{{\\ensuremath{\\mathcal{R}}}}-{\\rho}\\_i\\right) \\right) + \\sum\\_{i \\in H \\setminus \\{c\\_{\\ensuremath{\\mathcal{S}}}, c\\_{\\ensuremath{\\mathcal{R}}}\\}} {\\ensuremath{\\operatorname{OPT}}}+ {\\ensuremath{\\operatorname{OPT}}}\\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left(n-1 - \\frac{1}{(n-1)(1-\\mu^{{\\ensuremath{\\mathcal{R}}}})} (n-d) \\left(1 - \\mu^{{\\ensuremath{\\mathcal{R}}}}\\right) \\right) \\\\\n &\\ge (n-2){\\ensuremath{\\operatorname{OPT}}}\\quad \\ge \\quad \\left(n-2-\\frac{1}{n-1}\\right){\\ensuremath{\\operatorname{OPT}}}\\enspace.\n \\end{aligned}$$ \nLet OPT*t* be the success probability of the Pareto mechanism when applied to the benchmark scenario with random subset *A**t*. We can repeatedly apply Lemma [lem:lossPerRoundOrd] in the same way as Lemma [lem:lossPerRoundCard] in Corollary [cor:valueRoundTCard] above.\n[cor:valueRoundTOrd] For all *t* ∈ [*n*] it holds that ${\\ensuremath{\\operatorname{OPT}}}\\_t \\ge \\frac{(t-2)(t-1)}{(n-2)(n-1)}{\\ensuremath{\\operatorname{OPT}}}. $\nThe random subset *A**t* can be generated by starting from the candidate set [*n*] and iteratively removing a candidate picked uniformly at random. In a single step\n \n$$\\begin{aligned}\n{\\ensuremath{\\operatorname{OPT}}}\\_{n-1} \\quad &= \\quad \\frac{1}{n} \\sum\\_{i \\in [n]} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} \\quad\n = \\quad \\frac{1}{n} \\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + \\frac{1}{n} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} \\\\ \n &\\ge \\quad \\frac{1}{n} \\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + \\frac{1}{n} {\\ensuremath{\\operatorname{OPT}}}\\cdot {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} \\quad\n \\ge \\quad \\frac{n-2-\\frac{1}{n-1}}{n} \\cdot {\\ensuremath{\\operatorname{OPT}}}\\end{aligned}$$ \ndue to OPT− {*c*S} = 0, OPT ≤ 1, and Lemma [lem:lossPerRoundOrd]. Repeated application of this property implies\n \n$$\\begin{aligned}\n {\\ensuremath{\\operatorname{OPT}}}\\_t \\quad &\\ge \\quad {\\ensuremath{\\operatorname{OPT}}}\\prod\\_{k=t+1}^n \\frac{k-2-\\frac{1}{k-1}}{k} \\quad \n \\ge \\quad {\\ensuremath{\\operatorname{OPT}}}\\prod\\_{k=t+1}^n \\frac{k-3}{k-1} \\\\\n &= \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\frac{t-2}{t} \\cdot \\frac{t-1}{t+1} \\cdot \\frac{t}{t+2} \\cdot \\dots \\cdot \\frac{n-4}{n-2} \\cdot \\frac{n-3}{n-1} \\quad\n = \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\frac{(t-2)(t-1)}{(n-2)(n-1)}\\enspace.\n \\end{aligned}$$ \nNow, using Lemma [lem:lossPerRoundOrd] and Corollary [cor:valueRoundTOrd], Theorem [theo:senderOrdinalReceiverCardinalSecretaryNoDisclosure] can be proved similar as Theorem [theo:senderCardinalReceiverCardinalSecretaryNoDisclosure] above.\n[Proof of Theorem [theo:senderOrdinalReceiverCardinalSecretaryNoDisclosure]] In a given round *t* = *s* + 1, …, *n* − 1, the Growing Pareto mechanism obtains a success probability of at least Pr[*σ**t* = HIRE] ⋅ OPT*t*. We underestimate the success probability in round *n* by 0. Using linearity of expectation and choosing a sample size *s* = ⌊*c* ⋅ *n*⌋ with a constant *c*, we obtain\n \n$$\\begin{aligned}\n &\\sum\\_{t= s+1}^{n-1} \\frac{1}{t} \\cdot \\frac{s}{t-1} \\cdot \\frac{(t-1)(t-2)}{(n-1)(n-2)} {\\ensuremath{\\operatorname{OPT}}}\\quad = \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\frac{s}{(n-1)(n-2)} \\sum\\_{t=s+1}^{n-1} \\frac{t-2}{t} \\\\\n &\\quad = \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\left(\\frac{s}{n-2} - \\frac{s^2}{(n-1)(n-2)} - \\frac{2s \\cdot (H\\_{n-1} - H\\_{s+1})}{(n-1)(n-2)} \\right) \n \\quad = \\quad {\\ensuremath{\\operatorname{OPT}}}\\cdot \\left(c - c^2 - o(1)\\right)\\enspace.\n \\end{aligned}$$ \nHere, *H**k* = ∑*i* = 1*k*1/*i* is the *k*-th harmonic number. The last expression is maximized at *c* = 1/2, therefore we choose *s* = ⌊*n*/2⌋. The theorem follows.\nBasic Scenario with Disclosure\n------------------------------\nIn this section, we consider the basic scenario in which rejected candidates are revealed to R. We give a characterization of the optimal mechanism using backwards induction. We then show that there are instances in which no online mechanism can achieve more than a fraction of 1/2 of OPT, i.e., the success probability/expected utility obtained in the benchmark scenario. We describe and analyze the polynomial-time persuasive Shrinking Pareto mechanism. It obtains a constant-factor approximation of OPT with a ratio of 1/2 − *o*(1) (hence, asymptotically optimal) for ordinal sender utility. For cardinal sender utility the ratio is 1/3 − *o*(1).\nLet us start by deriving an optimal mechanism via backwards induction. Again, we utilize the results obtained in the case of cardinal sender utility to express bounds for the ordinal sender case. By setting *ξ*max = 1 and *ξ**i* = 0 for all sender-suboptimal candidates *i* ≠ *c*S, success probability and expected utility are the same value. Hence, the following result holds for both sender objectives. For simplicity, we state and prove the theorem for cardinal sender utility.\nAn optimal mechanism for the scenario can be computed by backward induction and solution of a series of linear programs. These programs express the optimal decision given a subset *C* ⊆ [*n*] of candidates and given the optimal decision policy computed for each subset *C* \\ {*i*}, for every *i* ∈ *C*. It is an interesting open problem whether there is a more intricate, polynomial-time algorithm to compute the optimal persuasive mechanism in this scenario.\n[thm:disclosureLPs] An optimal persuasive mechanism for the sender’s expected utility in the basic scenario with disclosure can be computed by solving 2*n* linear programs.\nSuppose we reach the beginning of round *t* with a set *C* of *n* − *t* + 1 remaining candidates. For every candidate *i* ∈ *C*, suppose the sender sees candidate *i* in round *t* and has computed the signaling policy for an optimal persuasive mechanism for rounds *t* + 1, …, *n* and subset *C* \\ {*i*}.\nClearly, in round *t* = *n* the sender should set *σ**n* = HIRE with probability 1. This is the only persuasive signaling scheme. Now suppose we are in round *t* < *n*. We denote by \n*x**i**C* = Pr[*σ**t* = HIRE ∣ *i* ∈ *C*arrives in round *t*]. \n In case *i* is not hired although S signaled, S will never signal again. Let *u**C* \\ {*i*}S be the expected utility of S from the optimal persuasive mechanism applied from round *t* + 1 onwards with candidate set *C* \\ {*i*}. Similarly, we define *u**C* \\ {*i*}R for R. Assuming the signaling scheme *ϕ* determined by *x* is persuasive, then the expected utility obtained by S \n$$\\frac{1}{|C|} \\sum\\_{i \\in C} (x\\_i^{C} {\\xi}\\_i + (1-x\\_i^{C})\\cdot u^{{\\ensuremath{\\mathcal{S}}}}\\_{C \\setminus \\{i\\}})$$ \n is the objective function of a linear program. In this LP we have the obvious constraints *x**i**C* ∈ [0, 1], and two constraints on *x* that ensure persuasiveness. Suppose R gets signal. This happens with total probability Pr[*σ**t* = HIRE ∣ *C*] = ∑*i* ∈ *C**x**i**C*/∣*C*∣. We assume w.l.o.g. that this probability is positive, otherwise there is nothing to prove. The probability that the recommended candidate is *i* is $p^h\\_i = \\frac{x\\_i^C}{\\sum\\_{j \\in C} x\\_j^C}$. Upon compliance with the signal, R gets a utility of ∑*i* ∈ *C**p**i**h**ρ**i*. Upon deviation, the candidate is rejected and S stops to signal. Thus, the expected utility of R becomes $\\sum\\_{i \\in C} p\\_i^h \\cdot \\frac{1}{|C|-1}\\sum\\_{j \\in C \\setminus \\{i\\}} {\\rho}\\_j$. Hence, persuasiveness requires\n \n$$\\label{eq:optRev}\n \\begin{array}{rrcl}\n & {\\displaystyle}\\sum\\_{i \\in C} p^h\\_i {\\rho}\\_i &\\ge& {\\displaystyle}\\sum\\_{i \\in C}p^h\\_i \\cdot \\frac{1}{|C|-1} \\sum\\_{j \\in C \\setminus \\{i\\}} {\\rho}\\_j\\\\\n \\Leftrightarrow \\; & {\\displaystyle}\\sum\\_{i \\in C} x\\_i^C {\\rho}\\_i &\\ge& {\\displaystyle}\\sum\\_{i \\in C}x\\_i^C \\cdot \\frac{1}{|C|-1} \\sum\\_{j \\in C \\setminus \\{i\\}} {\\rho}\\_j\\\\\n \\Leftrightarrow \\; & {\\displaystyle}\\sum\\_{i \\in C} x\\_i^C \\left({\\rho}\\_i - \\frac{1}{|C| - 1} \\sum\\_{j \\in C \\setminus\\{i\\}} {\\rho}\\_j\\right) &\\ge& 0 \\\\\n \\end{array}$$ \nSimilarly, R should not have an incentive to accept a candidate upon the event of *σ**t* = NOT. We again assume that the probability of this event is positive. The probability that upon a signal the sender sees candidate *i* is $p^{nh}\\_i = \\frac{1-x\\_i^C}{\\sum\\_{j \\in C} (1-x\\_j^C)}$. Upon deviation and hiring the candidate, R gets utility ∑*i* ∈ *C**p**i**n**h**ρ**i*. Upon following the signal, R gets a utility of ∑*i* ∈ *C**p**i**n**h**u**C* \\ {*i*}R. Hence, persuasiveness requires\n \n$$\\label{eq:optRev2}\n \\begin{array}{rrcl}\n & {\\displaystyle}\\sum\\_{i \\in C} p^{nh}\\_i u^{{\\ensuremath{\\mathcal{R}}}}\\_{C \\setminus\\{i\\}} &\\ge& {\\displaystyle}\\sum\\_{i \\in C}p^{nh}\\_i \\cdot {\\rho}\\_i\\\\\n \\Leftrightarrow \\; & {\\displaystyle}\\sum\\_{i \\in C} (1-x\\_i^C) u^{{\\ensuremath{\\mathcal{R}}}}\\_{C \\setminus\\{i\\}} &\\ge& {\\displaystyle}\\sum\\_{i \\in C} (1-x\\_i^C) \\cdot {\\rho}\\_i\n \\end{array}$$ \n[claim:redundant] Constraint is redundant and implied by .\nWe show the claim below. Thus, we get the following family of linear programs to determine *u**C*S:\nx i C (xiC i + (1-xiC)uSC {i})[LP:utilitynested][LP:templatenested]\n[Proof of Claim [claim:redundant]] Every solution $\\bm{x}$ that satisfies for some values for *u**C* \\ {*i*}R also fulfills it for pointwise larger values. This is clear since with expected utility in subsequent rounds becoming larger, R does not get incentive to deviate from a signal and hire in round *t*.\nRecall that R can always ignore all signals and hire a random candidate. Hence, clearly, $u^{{\\ensuremath{\\mathcal{R}}}}\\_{C \\setminus \\{i\\}} \\ge \\frac{1}{|C|-1}\\sum\\_{j\\in C\\setminus\\{i\\}} {\\rho}\\_j$. The pointwise smallest values $u^{{\\ensuremath{\\mathcal{R}}}}\\_{C \\setminus \\{i\\}} = \\frac{1}{|C|-1}\\sum\\_{j\\in C\\setminus\\{i\\}} {\\rho}\\_j$ imply for the constraint\n \n$$\\label{eq:optRev3}\n \\begin{array}{rrcl}\n & {\\displaystyle}\\sum\\_{i \\in C} (1-x\\_i^C) \\cdot \\frac{1}{|C|-1}\\sum\\_{j\\in C\\setminus\\{i\\}} {\\rho}\\_j &\\ge& {\\displaystyle}\\sum\\_{i \\in C} (1-x\\_i^C) \\cdot {\\rho}\\_i \\\\\n \\Leftrightarrow \\; & {\\displaystyle}\\sum\\_{i \\in C} x\\_i^C \\left({\\rho}\\_i - \\frac{1}{|C|-1} \\sum\\_{j \\in C\\setminus\\{i\\}} {\\rho}\\_j \\right) &\\ge& 0\n \\end{array}$$ \nWhat reduction of utility does S suffer when rejected candidates are announced to R? The following theorem shows an upper bound of $\\frac 12$.\n[theo:senderOrdinalReceiverCardinalKnownUtilDisclosureUB] For every *ɛ* > 0, there is an instance such that every persuasive mechanism in the basic scenario with disclosure guarantees at most a fraction of $\\left(\\frac{1}{2} + \\varepsilon\\right)$ of the optimum in the benchmark scenario. This holds for both cardinal as well as ordinal sender utility.\nConsider the following class of instances. Candidate 1 has value $({\\rho}\\_1,{\\xi}\\_1) = (\\frac{n-2}{n-1},1)$, candidate 2 has value (*ρ*2, *ξ*2) = (0, 0), and candidates *i* = 3, …, *n* have values (*ρ**i*, *ξ**i*) = (1, 0). Hence, this can be seen as an instance for ordinal as well as cardinal sender utility. In the remainder of the proof, we will only use “expected utility”.\nIn the benchmark case, the Pareto mechanism would pick *a* = *b* = 1 and signal if and only if candidate 1 arrives. This would yield an expected utility of 1 for S and $\\frac{n-2}{n-1}$ for R.\nFor the announcement case, constraint implies that *x*2*C* = 0 is optimal for all *C* ∋ 2, i.e., an optimal mechanism never recommends to hire 2. S has no value for candidates other than 1, so w.l.o.g. the optimal mechanism will always try to recommend 1 with *x*1*C* = 1 for every *C* ∋ 1. For every subset $C \\not\\ni 1$, the utility is 0, and we can assume that *x**i**C* = 1 for all *i* ∈ *C*, *i* ≠ 2.\nLet $k = \\sqrt{n}$, and consider the event *E* that the following two conditions are satisfied: (1) at least one of candidates {1, 2} arrives within the first *n* − *k* rounds, and (2) candidate 2 arrives before candidate 1. Suppose we draw two arrival rounds for candidates 1 and 2 uniformly at random. The probability that we draw both rounds from the last *k* rounds is $p = \\frac{k}{n}\\cdot\\frac{k-1}{n-1}$. For any set of two rounds drawn uniformly at random, the probability that 2 arrives in the earlier one is 1/2. Thus, \n$$\\Pr[E] = \\frac{1}{2} - \\frac{k(k-1)}{2n(n-1)}\\enspace.$$ \n Suppose candidate 2 arrives in round *t*. Since 2 is always rejected, consider any subsequent round *t* + 1. The set *C* of remaining candidates consists only of candidate 1 and a subset of candidates 3, …, *n*. Since the latter are symmetric, the optimal mechanism uses the same value *x**i**C* = *x**j**C* = *x*− 1*C* for all candidates *i*, *j* ∈ *C* \\ {1}. The optimal mechanism needs to satisfy \n$$(|C|-1) \\cdot x\\_{-1}^C \\cdot \\left(1 - \\frac{1}{|C|-1} \\cdot \\left(|C|-2 + \\frac{n-2}{n-1}\\right)\\right) + x\\_1^C \\cdot \\left(\\frac{n-2}{n-1} - \\frac{1}{|C|-1} \\cdot (|C|-1)\\right) \\ge 0\\enspace.$$ \n Using the assumption that *x*1*C* = 1 and solving for *x*− 1*C* implies \n$$x\\_{-1}^C \\ge 1\\enspace.$$ \n Hence, after candidate 2 is rejected in round *t*, there is an optimal mechanism that signals with probability 1 in round *t* + 1. The probability that candidate 1 is hired in this round is only 1/∣*C*∣ ≤ 1/(*k* − 1). Note that this upper bound holds even conditioned on event *E*, since the round in which 1 arrives remains uniformly distributed among the remaining ones. Thus, the overall expected utility of the optimal mechanism in the disclosure case is upper bounded by \n$$(1-\\Pr[E])\\cdot 1 + \\Pr[E] \\cdot \\frac{1}{k-1} \\le \\frac{1}{2} + \\frac{k(k-1)}{2n(n-1)} + \\frac{1}{2(k-1)} - \\frac{k}{2n(n-1)} = \\frac{1}{2} + o(1)$$ \n since $k = \\sqrt{n}$.\nWe now turn to our polynomial-time approximation mechanism for this scenario. It is another variant of the Pareto mechanism, termed Shrinking Pareto mechanism (see Algorithm [algo:ShrinkingPareto]). The Shrinking Pareto mechanism applies the Pareto procedure adaptively to the set of remaining candidates *R**t*. The mechanism signals HIRE in round *t* if and only if the candidate chosen by the Pareto procedure on *R**t* is the one arriving in round *t*.\n[t] *R*1 ← [*n*]\nWe establish three properties of the Shrinking Pareto mechanism. We show that the mechanism is persuasive, it yields a (1/3 − *o*(1))-approximation for cardinal sender utility, and a (1/2 − *o*(1))-approximation for ordinal sender utility.\n[lem:shrinkingParetoIC] The Shrinking Pareto mechanism is persuasive in the basic scenario with disclosure.\nRecall from our analysis above that the probabilities for a signal *σ**t* = HIRE resulting from the Pareto procedure are chosen as a linear combination, which yields at least an expected utility of *μ*R to R. Formally, consider the decision of R in round *t* conditioned on a subset *R**t* of remaining candidates. Suppose the mechanism sends. Then, the expected utility for R is \n$${\\mathbb{E}}[ {\\rho}({\\theta}\\_t) \\mid \\sigma\\_t = {\\text{HIRE}}\\wedge R\\_t ] \\ge \\frac{1}{|R\\_t|} \\sum\\_{i \\in R\\_t} {\\rho}\\_i\\enspace.$$ \n We use *x**i**R**t* = Pr[*σ**t* = HIRE ∣ *R**t* ∧ (*θ**t* = *i* ∈ *R**t*)], i.e., the probability that the set *R**t* remains to arrive in rounds *t*, …, *n* and *i* ∈ *R**t* arrives in round *t*. For the Pareto procedure we have ∑*i* ∈ *R**t**x**i**R**t* = 1, thus \n$$\\frac{1}{|R\\_t|} \\sum\\_{i \\in R\\_t} {\\rho}\\_i = \\frac{1}{|R\\_t|} \\sum\\_{i \\in R\\_t} x\\_i^{R\\_t} \\sum\\_{j \\in R\\_t} {\\rho}\\_j = \\frac{1}{|R\\_t|} \\sum\\_{i \\in R\\_t} x\\_i^{R\\_t} \\sum\\_{j \\in R\\_t \\setminus\\{i\\}} {\\rho}\\_j + \\frac{1}{|R\\_t|}\\sum\\_{i \\in R\\_t} x\\_i^{R\\_t} {\\rho}\\_i\\enspace.$$ \n Therefore \n$$\\sum\\_{i \\in R\\_t} x\\_i^{R\\_t} {\\rho}\\_i \\ge \\frac{1}{|R\\_t|-1} \\sum\\_{i \\in R\\_t} x\\_i^{R\\_t} \\sum\\_{j \\in R\\_t \\setminus \\{i\\}} {\\rho}\\_j\\enspace.$$ \n The signaling probabilities *x**i**R**t* of the Shrinking Pareto mechanism represent a feasible solution for every linear program . Hence, the mechanism is persuasive.\nThe following theorem is our main result for cardinal sender utility. The Shrinking Pareto mechanism guarantees a constant-factor approximation of the optimal utility achieved in the benchmark scenario. It is an interesting open question to prove a tight bound in this scenario.\n[theo:senderCardinalReceiverCardinalKnownUtilDisclosure] In the basic scenario with disclosure, the Shrinking Pareto mechanism scenario obtains a $\\left(\\frac{1}{3} - o(1)\\right)$-approximation of the optimum in the corresponding benchmark scenario.\nConsider an instance of the benchmark scenario with *n* candidates. We use OPT to denote the optimal expected utility of S achieved by the Pareto mechanism. Consider the same instance in the public announcement scenario, and let SP be the expected utility of S achieved by the Shrinking Pareto mechanism.\nWe denote by *θ**t* the candidate arriving in round *t*. Consider the first round *t* = 1 and the application of the Pareto procedure. Obviously, *σ*1 = HIRE with probability $\\frac{1}{n}$, and the expected utility E[*ξ**θ*1 ∣ *σ*1 = HIRE] = OPT.\nNow suppose *σ*1 = NOT. For every subset *S* ⊆ [*n*], let OPT− *S* and SP− *S* be the expected utility of S achieved by the (Shrinking) Pareto mechanism on the benchmark (disclosure) instance with candidates [*n*] \\ *S*, respectively. For the expected value of the Shrinking Pareto mechanism we can derive the following recursive lower bound\n \n$$\\begin{aligned}\n &\\frac{1}{n} \\sum\\_{i \\in [n]} \\Pr[{\\theta}\\_1 = i, \\sigma\\_1 = {\\text{HIRE}}] \\cdot {\\xi}\\_i + \\Pr[{\\theta}\\_1 = i, \\sigma\\_1 = {\\text{NOT}}] \\cdot {\\ensuremath{\\operatorname{SP}}}\\_{-\\{i\\}} \\\\\n &= \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\frac{1}{n} \\left((1-\\alpha){\\ensuremath{\\operatorname{SP}}}\\_{-\\{a\\}} + \\alpha {\\ensuremath{\\operatorname{SP}}}\\_{-\\{b\\}} + \\sum\\_{i \\neq a,b} {\\ensuremath{\\operatorname{SP}}}\\_{-\\{i\\}} \\right)\\\\\n &\\ge \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\frac{1}{n} \\sum\\_{i \\neq a,b} {\\ensuremath{\\operatorname{SP}}}\\_{-\\{i\\}}\\\\\n &\\ge \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\frac{1}{n} \\sum\\_{i \\neq a,b} \\left(\\frac{1}{n-1} \\sum\\_{j \\in [n] \\setminus\\{i\\}} \\Pr[{\\theta}\\_2 = j, \\sigma\\_2 = {\\text{HIRE}}] \\cdot {\\xi}\\_j + \\Pr[{\\theta}\\_2 = j, \\sigma\\_2 = {\\text{NOT}}] \\cdot {\\ensuremath{\\operatorname{SP}}}\\_{-\\{i,j\\}}\\right)\\\\\n &\\ge \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\frac{1}{n} \\left(\\sum\\_{i \\neq a,b} \\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-i}}{n-1} + \\frac{1}{n-1} \\sum\\_{i \\in [n]\\setminus\\{a,b\\}} \\sum\\_{j \\in [n]\\setminus\\{i\\}} \\Pr[{\\theta}\\_2 = j, \\sigma\\_2 = {\\text{NOT}}] \\cdot {\\ensuremath{\\operatorname{SP}}}\\_{-\\{i,j\\}} \n \\right)\\\\\n &\\ge \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\sum\\_{i \\neq a,b} \\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}}}{n(n-1)} + \\sum\\_{i \\in [n]\\setminus\\{a,b\\}} \\sum\\_{\\substack{j \\in [n]\\setminus\\{i\\} \\\\ j \\neq a\\_{-i},b\\_{-i}}} \\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i,j\\}}}{n(n-1)(n-2)} + \\ldots \n \\end{aligned}$$ \nwhere *a*− *i*, *b*− *i* are the candidates *a* and *b* identified by the Pareto procedure when applied to the candidate set [*n*] \\ {*i*}. Applying the lower bound on ∑*i* ≠ *a*, *b*OPT− *i* from Lemma [lem:lossPerRoundCard] repeatedly in the formula above, we obtain\n \n$$\\begin{aligned}\n &\\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\sum\\_{i \\neq a,b} \\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}}}{n(n-1)} + \\sum\\_{i \\in [n]\\setminus\\{a,b\\}} \\sum\\_{\\substack{j \\in [n]\\setminus\\{i\\} \\\\ j \\neq a\\_{-i},b\\_{-i}}} \\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i,j\\}}}{n(n-1)(n-2)} + \\ldots \\\\\n &\\ge \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\frac{n-3}{n(n-1)}{\\ensuremath{\\operatorname{OPT}}}+ \\sum\\_{i \\in [n]\\setminus\\{a,b\\}} \\frac{n-4}{n(n-1)(n-2)} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + \\ldots\\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\cdot \\frac{1}{n(n-1)(n-2)} \\left(\\sum\\_{i=1}^{n-3} (n-i)(n-i-1)\\right)\\\\\n &= {\\ensuremath{\\operatorname{OPT}}}\\cdot \\left(\\frac{1}{3} - \\frac{2}{n^3 - 3n^2 + 2n}\\right)\\enspace.\n \\end{aligned}$$ \nFor the ordinal case, we show a bound that asymptotically matches the upper bound shown in Theorem [theo:senderOrdinalReceiverCardinalKnownUtilDisclosureUB]. In contrast to the cardinal case, this implies that the approximation guarantee of our mechanism is asymptotically optimal. The proof is conceptually similar to the previous theorem, but instead of Lemma [lem:lossPerRoundCard] relies on the improved guarantee in Lemma [lem:lossPerRoundOrd].\n[theo:senderOrdinalReceiverCardinalKnownUtilDisclosure] The Shrinking Pareto mechanism yields a success probability of at least $\\left(\\frac{1}{2}-o(1)\\right)$ in the basic scenario with disclosure.\nLet *S**P* be the success probability from the Shrinking Pareto mechanism. We show that \n$$SP \\ge {\\ensuremath{\\operatorname{OPT}}}\\cdot \\left(\\frac{1}{2} - \\frac{1}{2n}\\right)\\enspace.$$ \n The utility obtained in a round depends on the current candidate. With probability $\\frac{1}{n}$, the current candidate is *a*. The signal HIRE is sent with probability *α* = OPT. With probability 1 − OPT, NOT is sent. Analogously, if the current candidate is *b*, NOT is sent with probability OPT while the signal is HIRE with probability 1 − OPT. Any other candidate is guaranteed to get a signal NOT. In the following, we denote by *S**P*− *M* for any subset *M* of candidates the expected utility from the Shrinking Pareto mechanism for the instance [*n*] \\ *M*. We apply Lemma [lem:lossPerRoundOrd] recursively and obtain\n \n$$\\begin{aligned}\n SP &= \\frac{1}{n}{\\ensuremath{\\operatorname{OPT}}}+ \\frac{1}{n}{\\ensuremath{\\operatorname{OPT}}}\\cdot SP\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} + \\frac{1}{n} \\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} SP\\_{-\\{i\\}} \\\\\n &= \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} \\left[ \\frac{1}{n-1}{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}} + \\frac{1}{n-1}{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}}\\cdot SP\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}},c\\_{\\ensuremath{\\mathcal{R}}}'\\}} + \\frac{1}{n-1} \\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}},c\\_{\\ensuremath{\\mathcal{R}}}'} SP\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}},i\\}} \\right] \\\\\\*\n &\\hspace{25pt} + \\frac{1}{n} \\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} \\left[\\frac{1}{n-1}{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} + \\frac{1}{n-1} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} SP\\_{-\\{i,c\\_{\\ensuremath{\\mathcal{R}}}\\}} + \\frac{1}{n-1} \\sum\\_{j \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}},i} SP\\_{-\\{i,j\\}}\\right] \\\\\n &= \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\underbrace{\\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} \\cdot \\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}}}{n-1} + \\frac{1}{n}\\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} \\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}}}{n-1}}\\_{\\ge \\frac{1}{n(n-1)} \\cdot {\\ensuremath{\\operatorname{OPT}}}\\cdot \\left(n-2-\\frac{1}{n-1}\\right)} + \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n}\\cdot\\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}}}{n-1} \\cdot SP\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}},c\\_{\\ensuremath{\\mathcal{R}}}'\\}} \\\\\n &\\hspace{25pt} + \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n}\\frac{1}{n-1} \\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}},c'\\_{\\ensuremath{\\mathcal{R}}}} SP\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}},i\\}} + \\frac{1}{n} \\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} \\frac{1}{n-1} {\\ensuremath{\\operatorname{OPT}}}\\_{-\\{i\\}} \\cdot SP\\_{-\\{i,c\\_{\\ensuremath{\\mathcal{R}}}\\}} \\\\\n &\\hspace{25pt} + \\frac{1}{n}\\frac{1}{n-1} \\sum\\_{i \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}}} \\sum\\_{j \\ne c\\_{\\ensuremath{\\mathcal{S}}},c\\_{\\ensuremath{\\mathcal{R}}},i} SP\\_{-\\{i,j\\}} \\\\\n &\\ge \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n} + \\frac{1}{n}\\frac{1}{n-1} \\cdot \\left(n-2-\\frac{1}{n-1}\\right)\\cdot {\\ensuremath{\\operatorname{OPT}}}+ \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n}\\cdot\\frac{{\\ensuremath{\\operatorname{OPT}}}\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}}\\}}}{n-1} \\cdot SP\\_{-\\{c\\_{\\ensuremath{\\mathcal{R}}},c\\_{\\ensuremath{\\mathcal{R}}}'\\}} \\\\\n &\\ge \\frac{{\\ensuremath{\\operatorname{OPT}}}}{n}\\left[1 + \\frac{1}{n-1} \\left(n-2-\\frac{1}{n-1}\\right) + \\frac{1}{n-1}\\frac{1}{n-2} \\cdot \\left( n-2-\\frac{1}{n-1}\\right)\\left( n-3-\\frac{1}{n-2}\\right) + \\dots \\right] + \\dots \\\\\n &\\ge {\\ensuremath{\\operatorname{OPT}}}\\left(\\frac{1}{2} - \\frac{1}{2n}\\right) \\enspace. \\end{aligned}$$ \nSecretary Scenario with Disclosure\n----------------------------------\nFinally, consider the secretary scenario with disclosure and cardinal receiver utility. In this scenario, S does not have any information on the valuation pairs. Moreover, the rejected candidates are revealed to R. Initially, R has the same information as usual (knows all valuation pairs, knows they arrive in random order, knows mechanism *ϕ*). At the end of each round with a rejection, R also learns the valuation pair of the candidate she just rejected.\nObviously, S can apply the *trivial mechanism*: Recommending a candidate chosen uniformly at random. This mechanism is persuasive. It achieves a success probability 1/*n* and an expected utility of 1/*n* ⋅ *ξ*max, a trivial lower bound.\nWe complement this insight with a strong upper bound for a sender with ordinal utility. The subsequent Corollary [cor:senderCardinalReceiverCardinalSecretaryDisclosure] shows a similar bound when the sender has cardinal utility.\n[theo:senderOrdinalReceiverCardinalSecretaryDisclosure] In the secretary scenario with disclosure, there is no persuasive mechanism that guarantees S a success probability greater than $\\frac{2}{n} \\cdot {\\ensuremath{\\operatorname{OPT}}}$, where OPT is the success probability in the corresponding benchmark instance.\nThe lower bound follows from the following two different instances. In both instances, there is one candidate *a* with value-pair (*ρ**a*, *ξ**a*) = (0, 1) and *n* − 2 indistinguishable candidates (each termed *c*) with value-pair (1/2, 0). In instance I, there is one additional candidate *b* with value-pair (1, 0). In instance II, there is just another *c*-candidate with (1/2, 0).\nHence, the only uncertainty for S in this set of instances is about the existence of *b*.\nNow consider an arbitrary persuasive mechanism. Clearly, we can assume that the mechanism issues exactly one HIRE signal in exactly one of the rounds *t* ∈ [*n*]. If this signal is not followed, it just issues NOT signals. This can only increase the incentive for R to follow the HIRE signal. We always assume that the mechanism issues a HIRE signal in the last round if it has not done so before. This is just a standard assumption. Note that in the last round, the arriving candidate is perfectly known to R.\nNow consider round *t* = 3, …, *n* − 1. Suppose the mechanism has only sent NOT signals so far. Based on the set of candidates *A**t* − 1 that arrived up to and including round *t* − 1 (and whose arrival is now known to both S and R), we distinguish eight cases.\nl||c|c|c|c|c|c|c|c *θ**t* & *a* & *a* & *b* & *b* & *c* & *c* & *c* & *c* \nCandidates in *A**t* − 1 & only *c* & *b*, *c* & only *c* & *a*, *c* & only *c* & *a*, *c* & *b*, *c* & *a*, *b*, *c* \nPr[*σ**t* = HIRE ∣ *A**t* − 1, *θ**t*] & *p**a*, *c**t* & *p**a*, *b**t* & *p**b*, *c**t* & *p**b*, *a**t* & *p**c*, *c**t* & *p**c*, *a**t* & *p**c*, *b**t* & 1\nNote that, obviously, in the last case when both *a* and *b* arrived and were rejected, it is clear to both S and R that the underlying instance is instance I and there are only candidates *c* to come in the subsequent rounds. Hence, in this case we can w.l.o.g. assume that S issues a HIRE signal directly. In round *t* = 2, this case does not exist:\nl||c|c|c|c|c|c|c *θ*2 & *a* & *a* & *b* & *b* & *c* & *c* & *c* \nCandidate *θ*1 & *c* & *b* & *c* & *a* & *c* & *a* & *b* \nPr[*σ*2 = HIRE ∣ *θ*1, *θ*2] & *p**a*, *c*2 & *p**a*, *b*2 & *p**b*, *c*2 & *p**b*, *a*2 & *p**c*, *c*2 & *p**c*, *a*2 & *p**c*, *b*2\nFinally, for round *t* = 1, there are only three cases depending on whether *θ*1 is *a*, *b* or *c*. We denote the probabilities for a HIRE signal by *p**a*1, *p**b*1 and *p**c*1, respectively.\nIn the next lemma, we show necessary constraints for persuasiveness.\n[lem:IncentiveConstraints] For every mechanism that is persuasive in both instances I and II, it must hold for round *t* = 1 \n$$p\\_b^1 \\ge p\\_a^1 \\hspace{1cm} \\text{and} \\hspace{1cm} p\\_c^1 \\ge p\\_a^1$$ \n and for every round *t* = 2, …, *n* − 1 \n$$p\\_{b,c}^t \\ge p\\_{a,c}^t \\hspace{1cm} p\\_{b,a}^t \\ge p\\_{c,a}^t \\hspace{1cm} p\\_{c,b}^t \\ge p\\_{a,b}^t \\hspace{1cm} p\\_{c,c}^t \\ge p\\_{a,c}^t\\enspace.$$ \nSuppose there is a HIRE signal in round *t*. R must find it in her interest to follow the signal. More precisely, in what follows we condition on *σ**t* = HIRE. Now, for persuasiveness the expected utility from *θ**t* must exceed the expected utility from *θ**t* + 1.\nFirst, consider round *t* = 1:\nInstance I:\nWe must have E[*ρ**θ*1 ∣ *σ*1 = HIRE] ≥ E[*ρ**θ*2 ∣ *σ*1 = HIRE], which implies\n \n$$\\begin{aligned}\n &p\\_a^1 \\cdot 0 + p\\_b^1 \\cdot 1 + (n-2) \\cdot p\\_c^1 \\cdot 1/2\\\\\n &\\quad\\ge \\quad p\\_a^1 \\cdot \\frac{(n-2)\\cdot 1/2 + 1}{n-1} + p\\_b^1 \\cdot \\frac{(n-2)\\cdot 1/2}{n-1} + (n-2)\\cdot p\\_c^1 \\cdot \\frac{(n-3)\\cdot 1/2 + 1}{n-1}\n \\end{aligned}$$ \nThe constraint becomes *p**b*1 ≥ *p**a*1.\nInstance II:\nWe must have E[*ρ**θ*1 ∣ *σ*1 = HIRE] ≥ E[*ρ**θ*2 ∣ *σ*1 = HIRE], which implies\n \n$$\\begin{aligned}\n &p\\_a^1 \\cdot 0 + (n-1) \\cdot p\\_c^1 \\cdot 1/2 \\quad\\ge \\quad p\\_a^1 \\cdot 1/2 + (n-1)\\cdot p\\_c^1 \\cdot \\frac{(n-2)\\cdot 1/2}{n-1}\n \\end{aligned}$$ \nThe constraint becomes *p**c*1 ≥ *p**a*1.\nHence, the mechanism must be more likely to signal in round 1 for each *c* and *b* than for *a*. Technically, these two constraints also hold for round *t* = *n*, since the probability to signal is 1 for every candidate. The two constraints also emerge in the remaining rounds *t* = 2, …, *n* − 1:\nInstance I, *A**t* − 1 contains only *c*:\nE[*ρ**θ**t* ∣ *σ**t* = HIRE] ≥ E[*ρ**θ**t* + 1 ∣ *σ**t* = HIRE] implies\n \n$$\\begin{aligned}\n & p\\_{a,c}^t \\cdot 0 + p\\_{b,c}^t \\cdot 1 + (n-t-1)\\cdot p\\_{c,c}^t \\cdot 1/2\\\\\n &\\quad\\ge p\\_{a,c}^t \\cdot \\frac{(n-t-1)\\cdot 1/2 + 1}{n-t} + p\\_{b,c}^t \\cdot \\frac{(n-t-1)\\cdot 1/2}{n-t} + (n-t-1) \\cdot p\\_{c,c}^t \\cdot \\frac{(n-t-2)\\cdot 1/2 + 1}{n-t}\n \\end{aligned}$$ \nObserve that the *p**c*, *c**t* terms cancel, and the constraint becomes *p**b*, *c**t* ≥ *p**a*, *c**t*.\nInstance I, *A**t* − 1 contains *a*:\nE[*ρ**θ**t* ∣ *σ**t* = HIRE] ≥ E[*ρ**θ**t* + 1 ∣ *σ**t* = HIRE] implies\n \n$$\\begin{aligned}\n &p\\_{b,a}^t \\cdot 1 + (n-t)\\cdot p\\_{c,a}^t \\cdot 1/2 \\quad\\ge \\quad p\\_{b,a}^t \\cdot 1/2 + (n-t) \\cdot p\\_{c,a}^t \\cdot \\frac{(n-t-1)\\cdot 1/2 + 1}{n-t}\n \\end{aligned}$$ \nThe constraint becomes *p**b*, *a**t* ≥ *p**c*, *a**t*.\nInstance I, *A**t* − 1 contains *b*:\nE[*ρ**θ**t* ∣ *σ**t* = HIRE] ≥ E[*ρ**θ**t* + 1 ∣ *σ**t* = HIRE] implies\n \n$$\\begin{aligned}\n &p\\_{a,b}^t \\cdot 0 + (n-t)\\cdot p\\_{c,b}^t \\cdot 1/2 \\quad\\ge \\quad p\\_{a,b}^t \\cdot 1/2 + (n-t) \\cdot p\\_{c,b}^t \\cdot \\frac{(n-t-1)\\cdot 1/2}{n-t}\n \\end{aligned}$$ \nThe constraint becomes *p**c*, *b**t* ≥ *p**a*, *b**t*.\nInstance I, *A**t* − 1 contains *a*, and *b*:\nFor *t* = 2 this case does not occur. For *t* ≥ 3, both players are aware that there are only *c*-candidates left. R follows any signaling strategy that guarantees a HIRE signal in the remaining rounds.\nInstance II, *A**t* − 1 contains only *c*:\nE[*ρ**θ**t* ∣ *σ**t* = HIRE] ≥ E[*ρ**θ**t* + 1 ∣ *σ**t* = HIRE] implies\n \n$$\\begin{aligned}\n &p\\_{a,c}^t \\cdot 0 + (n-t)\\cdot p\\_{c,c}^t \\cdot 1/2 \\quad\\ge \\quad p\\_{a,c}^t \\cdot 1/2 + (n-t) \\cdot p\\_{c,c}^t \\cdot \\frac{(n-t-1)\\cdot 1/2}{n-t}\n \\end{aligned}$$ \nThe constraint becomes *p**c*, *c**t* ≥ *p**a*, *c**t*.\nInstance II, *A**t* − 1 contains *a*:\nIn this case R knows that there are only *c*-candidates left. R follows any signaling strategy that guarantees a HIRE signal in the remaining rounds.\nThe proof of the theorem now follows by analyzing the approximation ratio when applying such a mechanism in instance I.\n[Proof of Theorem [theo:senderOrdinalReceiverCardinalSecretaryDisclosure]] Consider instance I. Instance I in the benchmark scenario yields an optimal success probability of 1/2 (recommend *a* or *b* each with probability 1/2). Let us analyze the performance of a mechanism on instance I that satisfies Lemma [lem:IncentiveConstraints]. Such a mechanism is much more careful to recommend *a*. If candidate *b* would not exist, there would be a lack of interest of R (that knows instance II is present) to accept a recommendation of *a*. This risk arising from the potential presence of instance II is captured by the constraints in Lemma [lem:IncentiveConstraints].\nFormally, if the mechanism reaches round *n* without a previous HIRE signal, then, obviously, it is optimal to signal HIRE in round *n* with probability 1. We will prove by induction that there is an optimal mechanism *ϕ* with the following property. If *ϕ* reaches any round *t* without a previous HIRE signal, it is optimal to hire in round *t* with probability 1. Thus, there is an optimal mechanism that hires in round 1 with probability 1. Obviously, this mechanism has success probability 1/*n*.\nSuppose the inductive assumption is true for rounds *t* + 1, …, *n*, i.e., if the mechanism reaches round *t* + 1 without a previous HIRE signal, it is optimal to hire any candidate in that round with probability 1.\nNow consider round *t* ≥ 2.\n*A**t* − 1 contains only *c*:\nIf *a* arrives in round *t*, then the success probability is *p**a*, *c**t*. If *b* arrives, the success probability is $(1-p\\_{b,c}^t)\\cdot\\frac{1}{n-t}$, since by assumption the mechanism hires every candidate in the next round, which due to random arrival is *a* with uniform probability. If *c* arrives, the success probability is $(1-p\\_{c,c}^t)\\cdot\\frac{1}{n-t}$ by similar arguments. Overall, we want to maximize \n$$p\\_{a,c}^t + \\frac{1-p\\_{b,c}^t}{n-t} + (n-t-1)\\cdot \\frac{1-p\\_{c,c}^t}{n-t}\\enspace.$$ \n Due to Lemma [lem:IncentiveConstraints] *p**b*, *c**t* ≥ *p**a*, *c**t* and *p**c*, *c**t* ≥ *p**a*, *c**t*. Hence, the expression is maximized if both constraints hold with equality, in which case it becomes \n$$p\\_{a,c}^t + \\frac{1-p\\_{a,c}^t}{n-t} + (n-t-1)\\cdot \\frac{1-p\\_{a,c}^t}{n-t} = 1\\enspace,$$ \n i.e., *independent* of the value of *p**a*, *c**t*. Thus, *p**a*, *c**t* = *p**b*, *c**t* = *p**c*, *c**t* = 1 is an optimal choice.\n*A**t* − 1 contains *a*:\nIf *a* has arrived and been rejected, the success probability is 0. Setting *p**b*, *a* = *p**c*, *a* = 1 is an optimal choice.\n*A**t* − 1 contains *b*:\nIf *a* arrives in round *t*, then the success probability is *p**a*, *b**t*. If *c* arrives, the success probability is $(1-p\\_{c,b}^t)\\cdot\\frac{1}{n-t}$, since by assumption the mechanism hires every candidate in the next round, which due to random arrival is *a* with uniform probability. Overall, we want to maximize \n$$p\\_{a,b}^t + (n-t)\\cdot \\frac{1-p\\_{c,b}^t}{n-t}\\enspace.$$ \n Due to Lemma [lem:IncentiveConstraints] *p**c*, *b**t* ≥ *p**a*, *b**t*. Hence, the expression is maximized when the constraints holds with equality, in which case it becomes \n$$p\\_{a,b}^t + (n-t)\\cdot \\frac{1-p\\_{a,b}^t}{n-t} = 1\\enspace.$$ \n Thus, *p**a*, *b**t* = *p**c*, *b**t* = 1 is an optimal choice.\n*A**t* − 1 contains *a*, and *b*:\nIf *a* has arrived and been rejected, the success probability is 0. We already observed above that in this case we can directly hire in round *t* with probability 1.\nFor round *t* = 1 we make a final similar observation. The success probability is *p**a*1 if *a* arrives, $(1-p\\_b^1) \\cdot \\frac{1}{n-1}$ if *b* arrives and $(1-p\\_c^1) \\cdot \\frac{1}{n-1}$ if *c* arrives. Overall, we want to maximize \n$$p\\_a^1 + \\frac{1-p\\_b^1}{n-1} + (n-2)\\cdot \\frac{1-p\\_c^1}{n-1}$$ \n Due to Lemma [lem:IncentiveConstraints] *p**b*1 ≥ *p**a*1 and *p**c*1 ≥ *p**a*1. Hence, the expression is maximized when both constraints hold with equality, in which case it becomes \n$$p\\_a^1 + (n-1)\\cdot \\frac{1-p\\_a^1}{n-1} = 1\\enspace.$$ \n Thus, *p**a*1 = *p**b*1 = *p**c*1 = 1 is an optimal choice.\nThis proves the theorem.\nThe proof can be applied literally by simply replacing “success probability” with “expected utility” for S, since in both instances I and II the best candidate *a* is the only one that yields positive utility for S. This implies the following corollary.\n[cor:senderCardinalReceiverCardinalSecretaryDisclosure] There is no persuasive mechanism that guarantees S an expected utility of more than $\\frac{2}{n}\\cdot{\\ensuremath{\\operatorname{OPT}}}$, where OPT is the optimal expected utility in the corresponding benchmark instance.\nAcknowledgements\n----------------\nNiklas Hahn and Martin Hoefer are funded by the German-Israel Foundation grant I-1419-118.4/2017. Rann Smorodinsky is funded by the joint United States-Israel Binational Science Foundation and National Science Foundation grant 2016734, German-Israel Foundation grant I-1419-118.4/2017, Israel Ministry of Science and Technology grant 19400214, Technion VPR grants, and the Bernard M. Gordon Center for Systems Engineering at the Technion.\nOrdinal Utility for R\n=====================\nLet us now consider the problem when R is only interested in hiring her (unique) best candidate, i.e., the case of ordinal receiver utility.\nBasic Scenario without Disclosure\n---------------------------------\nIn this scenario, the set of valuation pairs is known a-priori to both parties. While S observes the candidates, R only observes the signals sent by S. R is indifferent about all candidates which are not her best. As such, for a persuasive mechanism for S, we can restrict to signal HIRE for exactly one of the two optimal candidates *c*R and *c*S.\nIn the Elementary mechanism, we decide upfront whether to signal HIRE for either *c*R or *c*S: Draw *x* ∼ *U**n**i**f*[0, 1]. If *x* ≤ 1/*n*, signal HIRE upon arrival of *c*R, otherwise, signal HIRE upon arrival of *c*S. Signal NOT for any other candidate.\n[prop:receiverOrdinalKnownUtilNoDisclosure] The Elementary mechanism is persuasive in the basic scenario without disclosure. It yields a success probability of (1 − *o*(1)) and an expected utility of (1 − *o*(1)) ⋅ *ξ*max for S.\nFirst, we prove persuasiveness.\nIn round *t* with *σ**t* = HIRE we have $\\Pr[\\theta\\_t = c\\_{\\ensuremath{\\mathcal{R}}}] = \\frac{1/n}{1/n+(n-1)/n} = 1/n$. Suppose R deviates to hire in a later round *r* > *t*. Since S will not signal HIRE again, R must choose *r* without additional information. Then Pr[*θ**r* = *c*R] = (1 − 1/*n*) ⋅ 1/(*n* − 1) = 1/*n*. Thus, it is optimal for R to hire *θ**t*.\nIf *σ**t*ʹ = NOT for all *t*ʹ ∈ [*t*], then $\\Pr[\\theta\\_t = c\\_{{\\ensuremath{\\mathcal{R}}}}] = \\frac{\\frac{n-1}{n}}{n-2+\\frac{1}{n}+\\frac{n-1}{n}} = \\frac{1}{n}$. Hence, it is optimal for *c*R to wait for the round with signal HIRE.\nAdditionally, the Elementary mechanism clearly implies the sender-optimal candidate *c*S is hired with probability 1 − *o*(1) and yields an expected utility of (1 − *o*(1))*ξ*max for S.\nSecretary Scenario without Disclosure\n-------------------------------------\nIn this case, we assume that R has the same information as in the basic scenario. Now the valuation pairs are unknown to S. In this scenario, she can use the following mechanism that relies on the classic secretary algorithm due to Dynkin .\nThe Simple Secretary mechanism decides once in the beginning whether to run the classic algorithm based either on *ξ*-values or *ρ*-values. In the classic algorithm, we sample the first *s* = ⌊*n*/*e*⌋ candidates. Then we signal HIRE for the first candidate that is the best one so far (in terms of either *ξ*- or *ρ*-values, depending on the variant that is used). The classic algorithm hires the best candidate with probability 1/*e* − *o*(1). For our mechanism, with probability 1 − *e*/*n*, we run the classic algorithm using the *ξ*-values of S, with probability *e*/*n* the *ρ*-values of R. Note that *e*/*n* = *o*(1), so as the number of candidates increases, the mechanism tends to run the classic algorithm almost exclusively in the sender-optimal version. By assumption, if the mechanism decided to signal *σ**t* = NOT in all rounds *t* = 1, …, *n* − 1, it sets *σ**n* = HIRE.\n[theo:receiverOrdinalSecretaryNoDisclosure] The Simple Secretary mechanism is persuasive in the secretary scenario without disclosure. It yields a success probability of 1/*e* − *o*(1) and an expected utility of (1/*e* − *o*(1)) ⋅ *ξ*max for S.\nWe begin the proof by showing persuasiveness.\nDue to random order and the symmetric structure of the secretary algorithm, it produces the same distribution of HIRE signals, both for the sender-optimal or the receiver-optimal version of the secretary algorithm. In particular, for every round *t* > ⌊*n*/*e*⌋, and for any set *A**t* of arrived candidates in the rounds 1, …, *t*, the mechanism sends a hire signal in round *t* when the best candidate from *A**t* arrives in round *t* and the second best arrived in the sample phase. Hence, $\\Pr[\\sigma\\_t = {\\text{HIRE}}\\mid A\\_t] = \\frac{1}{t} \\cdot \\frac{s}{t-1}$, which is the same for every set *A**t* and for both the sender-optimal or receiver-optimal variant. There is no disclosure, so R cannot distinguish which variant is used.\nWe first show persuasiveness for negatively correlated utilities, i.e., when the *i*-th best candidate for S is the (*n* − *i* + 1)-th best candidate for R.\nConsider a round *t* ∈ [*n* − 1].\nCase *σ**t* = HIRE:\nWith probability *e*/*n* the receiver-optimal version is run and the candidate is the best so far in terms of *ρ*. The probability that this candidate is actually *c*R is *t*/*n*. If the sender-optimal version is run, there is no HIRE signal for *c*R. Hence, overall in this case Pr[*θ**t* = *c*R] = *e**t*/*n*2.\nIf R decides to deviate to a later round, she does not receive additional information from S. Due to random order arrival and the independence of the HIRE signal of the set of arrived candidates, in any given round *r* > *t* we have $\\Pr[\\theta\\_r = c\\_{\\ensuremath{\\mathcal{R}}}] = \\frac{n-t}{n} \\cdot \\frac{1}{n-t} = \\frac{1}{n}$. Since *t* ≥ *s* + 1 > *n*/*e* we have *e**t*/*n*2 > 1/*n*. Hence, following the HIRE signal is optimal.\nCase *t* > *s* and *σ**i* = NOT for all *i* ≤ *t*:\nWith probability *e*/*n* the receiver-optimal version is run and the current candidate in round *t* is not the best so far in terms of *ρ*. Then it cannot be *c*R. With probability 1 − *e*/*n* the sender-optimal version is run and the candidate in round *t* is not the best so far in terms of *ξ*. Then, *c*R has probability *t*/*n* to be among the set *A**t* of the first *t* candidates. Given that *c*R ∈ *A**t*, it arrives in round *t* with probability $\\frac{1}{t-1}$ – there have been only NOT signals, so the best candidate from *A**t* must be in the sample phase. Hence, overall in this case $\\Pr[\\theta\\_t = c\\_{\\ensuremath{\\mathcal{R}}}] = \\left(1-\\frac{e}{n}\\right) \\cdot \\frac{t}{n(t-1)} = \\frac{(n-e)t}{n^2(t-1)}$.\nIf instead R follows the mechanism, then in round *r* > *t* the candidate is best so far with probability $\\frac{1}{r} \\cdot \\frac{s}{r-1} = \\frac{s}{r(r-1)}$. Note that in this case, we condition on NOT signals in all rounds ≤ *t*. Hence, candidate *θ**r* is best so far with probability $\\frac{s/(r(r-1))}{s/t}$. This candidate is *c*R with probability *e**r*/*n*2 as noted above.\nMoreover, in the last round, there is always a HIRE signal. The probability that no candidate after the sample phase is ever the best so far is $\\frac{s}{n}$. Since we condition on NOT signals in all rounds ≤ *t*, this gives a probability of $\\frac{s/n}{s/t}$. This candidate can be *c*R only when we run the sender-optimal variant. Hence, the probability that this candidate is *c*R is $\\left(1-\\frac{e}{n}\\right) \\cdot \\frac{1}{n-1}$. Hence, in this case, following the mechanism R obtains *c*R with probability\n \n$$\\label{eq:conditionedProp}\n \\sum\\_{r=t+1}^n \\frac{s/(r(r-1))}{s/t} \\cdot \\frac{er}{n^2} + \\frac{s/n}{s/t} \\cdot \\left(1-\\frac{e}{n}\\right)\\cdot \\frac{1}{n-1} = \\frac{t}{n^2} \\left( \\frac{n-e}{n-1} + \\sum\\_{r=t+1}^n \\frac{e}{r-1} \\right)$$ \nThus, deviation by hiring in round *t* is unprofitable if \n$$\\frac{n-e}{t-1} \\le \\frac{n-e}{n-1} + e\\cdot \\sum\\_{r=t+1}^n \\frac{1}{r-1}$$ \n or, equivalently \n$$\\frac{1}{t-1} - \\frac{1}{n-1} \\le \\frac{e}{n} \\left(\\frac{1}{t-1} - \\frac{1}{n-1} + \\sum\\_{r=t+1}^n \\frac{1}{r-1} \\right)$$ \n and, thus,\n \n$$\\begin{aligned}\n \\label{eq:crazy} \n \\frac{n-t}{(n-1)(t-1)\\left(\\sum\\_{r=t-1}^{n-2} \\frac{1}{r} \\right)} \\le \\frac{e}{n}\\enspace.\n \\end{aligned}$$ \n[cl:monotone] The lower bound in equation is monotonically decreasing in *t*. Hence, the strongest lower bound arises in round *t* = *s* + 1.\nWe want to prove that \n$$\\frac{n-t}{(n-1)(t-1)\\left(\\sum\\_{"},"data_id":{"kind":"string","value":"arxiv_0183117"}}},{"rowIdx":45795,"cells":{"content":{"kind":"string","value":"High magnetic field theory for the local density of states in graphene with smooth arbitrary potential landscapes\n=================================================================================================================\nWe study theoretically the energy and spatially resolved local density of states (LDoS) in graphene at high perpendicular magnetic field. For this purpose, we extend from the Schrödinger to the Dirac case a semicoherent-state Green’s-function formalism, devised to obtain in a quantitative way the lifting of the Landau-level degeneracy in the presence of smooth confinement and smooth disordered potentials. Our general technique, which rigorously describes quantum-mechanical motion in a magnetic field beyond the semi-classical guiding center picture of vanishing magnetic length (both for the ordinary two-dimensional electron gas and graphene), is connected to the deformation (Weyl) quantization theory in phase space developed in mathematical physics. For generic quadratic potentials of either scalar (i.e., electrostatic) or mass (i.e., associated with coupling to the substrate) types, we exactly solve the regime of large magnetic field (yet at finite magnetic length - formally, this amounts to considering an infinite Fermi velocity) where Landau-level mixing becomes negligible. Hence, we obtain a closed-form expression for the graphene Green’s function in this regime, providing analytically the discrete energy spectra for both cases of scalar and mass parabolic confinement. Furthermore, the coherent-state representation is shown to display a hierarchy of local energy scales ordered by powers of the magnetic length and successive spatial derivatives of the local potential, which allows one to devise controlled approximation schemes at finite temperature for *arbitrary* and possibly disordered potential landscapes. As an application, we derive general analytical non-perturbative expressions for the LDoS, which may serve as a good starting point for interpreting experimental studies. For instance, we are able to account for many puzzling features of the LDoS recently observed by high magnetic field scanning tunneling spectroscopy experiments on graphene, such as a roughly $\\sqrt{m}$-increase in the *m*th Landau-level linewidth in the LDoS peaks at low temperatures, together with a flattening of the spatial variations in the Landau-level effective energies at increasing *m*.\nIntroduction\n============\nQuantum-Hall effect in graphene\n-------------------------------\nThe observation of an anomalous quantization of the Hall resistance in graphene at high magnetic fields, related to the massless, relativistic-like spectrum of low-energy electrons on the two-dimensional honeycomb lattice, has triggered much excitation in recent years, see Ref. for a review. Indeed, the experimentally measured Hall resistance follows the Landau-level structure expected for massless Dirac electrons, $E\\_{m}=\\pm \\sqrt{m} \\hbar \\Omega\\_{c}$ in the clean case, with *m* a positive integer and $\\Omega\\_{c}=\\sqrt{2} v\\_{F}/l\\_{B}$ the graphene characteristic frequency given in terms of the Fermi velocity *v**F* and of the magnetic length $l\\_{B}=\\sqrt{\\hbar c/|e|B}$ (here *e* = − ∣*e*∣ is the electron charge, *c* the speed of light, and *B* the magnetic field strength). The $\\sqrt{B}$ dependence of the characteristic frequency Ω*c* in graphene, to be contrasted with the linear dependence of the cyclotron frequency *ω**c* = ∣*e*∣*B*/(*m*\\**c*) of more standard two-dimensional electron gases (2DEGs) based on semiconducting heterostructures (in this case, *m*\\* is the electronic effective mass) described by Schrödinger equation, constitutes one of the main signatures used so far in experiments to exhibit the relativistic-like character of the massless charge carriers.\nAlso quite remarkable is that graphene displays a surface opened to the outside world, providing a direct window to its electronic excitations. This is a clear experimental advantage of graphene compared to 2DEGs based on semiconducting heterostructures, where the 2DEG is buried deep inside the structure (typically 100 nm or more). Graphene thus offers the opportunity to obtain precise insights into local physical properties of quantum-Hall systems, such as the local density of states (LDoS) via scanning tunneling spectroscopy (STS) measurements. In contrast, such local probes experiments have very poor spatial resolution in ordinary heterostructures, although some progress has been made recently, see Ref.. This technical advantage will be certainly important in the future to elucidate the relation between microscopic inhomogeneities induced by various disorder types and macroscopic transport properties of large samples. Various open questions in this respect are the nature of the universal plateau to plateau quantum phase transition, or on a more quantitative level the precise formation of wide Hall plateaus. To pursue this goal, STS is one of the interesting available experimental techniques, and first experiments in graphene at high magnetic field have been performed recently. Since this spectroscopic method gives direct information on the local electronic states, a better understanding of the LDoS, specific to the case of graphene at high magnetic fields and in arbitrary potential landscapes (without proceeding to disorder averaging), needs to be achieved. This is the main aim of the present paper. A second important aspect of our work is to obtain analytical solutions for a large class of parabolic confinement models, and as a motivation we now discuss the different types of potentials that can be involved in the two-dimensional Dirac Hamiltonian.\nDisorder types for graphene\n---------------------------\nBecause of the multicomponent structure of the wave function for graphene, several types of disorder can occur, which we introduce here. The quasiparticle dispersion for graphene has two Dirac cones (two “valleys”) at low energies. For a given valley, the Hamiltonian in the presence of a perpendicular magnetic field has a matrix structure and is written as\n \n$$H\\_{0}= v\\_{F} {\\bm \\sigma} \\cdot \\hat{\\bm \\Pi},\n\\label{Hgraphene}$$ \nwhere *v**F* is the Fermi velocity, ${\\bm \\sigma}$ is a vector whose components are the Pauli matrices *σ**x* and *σ**y* in the pseudospin space, and the momentum operator is\n \n$$\\hat{{\\bm \\Pi} }=-i \\hbar {\\bm \\nabla}\\_{{\\bf r}}-\\frac{e}{c}{\\bf A}({\\bf r}).$$ \nThe vector potential ${\\bf A}$ is related to the uniform transverse magnetic field ${\\bf B}$ via the relation ${\\bm \\nabla} \\times {\\bf A}={\\bf B}=B \\hat{{\\bf z}}$. For convenience, we will omit both physical spin and valley indices, thus assuming that the two valleys of graphene remain completely decoupled from each other and can be studied separately. \nQuite generally, potential terms appear as either a random scalar potential, a random Dirac mass or a random vector potential. The Hamiltonian in presence of these potentials is given by\n \n$$H=H\\_{0}+ V({\\bf r}), \\label{prob}$$ \nwhere the function $V({\\bf r})$ takes the general form\n \n$$V({\\bf r}) =\\sum\\_{p=s,x,y,z} \\sigma\\_{p} V\\_{p}({\\bf r})\n\\label{prob2}$$ \nwith *σ**s* the identity matrix in the pseudospin space, associated to the scalar potential term $V\\_{s}({\\bf r})$. This contribution may have many different physical origins: electrostatic confinement potential, impurity random potential, and/or Hartree potential resulting from the mean-field mutual Coulomb interaction between the electrons. The diagonal but antisymmetric term $V\\_{z}({\\bf r})$, associated to the *σ**z* Pauli matrix, describes the so-called random mass potential. This contribution might be introduced by the underlying substrate in single-layer graphene, while in bilayer graphene, such a term can be produced in a controllable way by introducing different electrostatic potentials in the two layers. The off-diagonal contributions coming as ${\\bf V}({\\bf r})=\\left[V\\_{x}({\\bf r}),\nV\\_{y}({\\bf r})\\right]$ can be associated with a random vector potential, coming from the spatial distortion of the graphene sheet in the third dimension by ripples. In what follows, all three possible types of disorder will be considered within the high magnetic field regime.\nExisting theoretical results for graphene in various potential types\n--------------------------------------------------------------------\nLet us first discuss various toy models of potentials (in a magnetic field) that were studied in the recent graphene literature. Quite generally, within the Dirac equation fewer models can be solved exactly than within its non-relativistic counterpart. For instance, the classic one-dimensional parabolic confinement model, as well as the circular parabolic confinement model, are seemingly not analytically tractable. For the 2DEG, the former is the well-known model to introduce the edge states and explain the quantized conductance in Hall bars. The latter is the basic model for quantum dots and leads under magnetic field to the Fock-Darwin states with discrete energies. Thus, only much simpler models can be solved analytically for graphene, such as the uniform electric field. Progress can be achieved for circular hard-wall confinement with either scalar or mass potentials, but only a solution in terms of special functions is then possible. For parabolic and more complex potentials, fully numerical methods have to be used, e.g., see Ref.. We will show in this paper that the limit of negligible Landau level mixing allows one to solve analytically a large class of parabolic models, providing new insights in the high magnetic field regime.\nComing to the more complex question of disorder, even less is actually known. Recent work devoted to the quantum-Hall effect in graphene has proposed to take into account disorder phenomenologically in the expression of Green’s function by adding a constant imaginary part *i*Γ in the self-energy, but Hall quantization obtains only in the limit where the energy rate Γ → 0. The LDoS in the vicinity of a single pointlike impurity and in the presence of a strong magnetic field has been studied recently. Various types of disorders were also considered in Refs. and within the self-consistent Born approximation. While this method may be justified for short-range scatterers, it turns out to be inappropriate for a smooth potential in high magnetic fields. Because a quasi-local picture takes place in the high magnetic field regime, our calculation will be able to provide accurate expression for the LDoS in smooth *arbitrary* potentials.\nHigh magnetic field regime\n--------------------------\nThe strategy to follow is best explained by starting to discuss the specific nature of disorder for 2DEGs at high magnetic field. For very clean heterostructures, the disordered potential seen by the electrons is mostly smooth on large length scales (several tens of nanometers), as the majority of impurities sit far away from the 2DEG. In contrast to the low magnetic field regime, where the electrons explore ergodically macroscopic regions of the sample, the high field regime is characterized by cyclotron motion close to equipotential lines of potential landscape $V(\\bf r)$ with a narrow transverse spread proportional to the magnetic length (which is smaller than 10 nm at several tesla). The disorder landscape felt by the electronic wave functions is therefore very smooth in that situation. We note that in graphene additional sharper potential variations (such as atomic vacancies of the carbon layer, or local imperfections from the nearby substrate) can occur, although these tend to be detrimental to quantum-Hall physics by increasing the mixing of Landau levels. The coupling to the substrate can however be removed by suspending graphene flakes or with a decoupled layer in epitaxial graphene, resulting in very high mobility samples. In fact, for both non-relativistic 2DEGs and graphene, the essence of the quantum Hall effect lies already by considering smooth potential variations only, which is the case to be followed from now on.\nTheoretically, this smooth disorder regime was shown to be problematic at high magnetic field for standard quantum-mechanical methods based on perturbative expansions in potential strength. In that case, the high magnetic field regime is the correct starting point, and is characterized by two different dimensionless small parameters: (i) *l**B*/*ξ* associated to the transverse spread of the wave function along the classical guiding center ${\\bf R}$, with *l**B* the magnetic length and *ξ* the typical length scale related to local variations in the potential; (ii) $l\\_B |{\\bm \\nabla} V|/\\hbar \\omega\\_c \\simeq l\\_B \\delta V/ \\xi \\hbar \\omega\\_c$ associated to Landau level mixing by local gradients $|{\\bm \\nabla}V|$ of the potential, introducing *δ**V* the typical amplitude variations in the potential on the scale *ξ*, and the cyclotron frequency *ω**c* in the 2DEG case.\nClearly, quantum mechanics calls for non-zero *l**B*/*ξ*, otherwise the so-called semiclassical guiding center picture at *l**B* = 0 emerges, giving at best a qualitative picture, and missing important quantum effects such as level quantization, tunneling, or interferences effects due to the potential energy $V({\\bf R})$. The second parameter $l\\_B |{\\bm \\nabla} V|/\\hbar \\omega\\_c$ controls the degree of Landau level mixing, so that Landau levels strictly decouple at infinite *ω**c*. Most previous works have considered either limits separately (either *l**B* → 0 or *ω**c* → ∞), and the necessary formalism to incorporate both non-zero *l**B* and finite *ω**c* was developed for the standard 2DEG by the authors in Refs., which will be extended in the present paper to the case of graphene. This mathematical construction shows that a local picture of the high magnetic field physics emerges in terms of semicoherent-state Green’s function, with a hierarchy of local energy scales ordered by powers of the magnetic length and successive spatial derivatives of the confinement or disordered potential.\nIn the simplified, yet fully quantum limit of infinite cyclotron frequency and non-zero *l**B*, initial progress was made by other authors in Refs. and for the 2DEG case, where it was shown that Schrödinger equation acquires a uni-dimensional character, offering an analysis for toy models of confinement or tunneling in the lowest Landau level. The general structure of this limit was clarified in further developments in the Green’s-function formalism, and this will be also examined in detail for graphene in the present paper. Our methodology is based on the exclusive use of Green’s functions, not wave functions, for the simple reason that we project the quantum dynamics onto a semi-coherent representation with nonorthogonal states, forcing us *de facto* to give up the wave-functions picture. Noticeably, because the overcomplete character of the chosen representation allows one to get rid of the Hilbert-space formulation inherent to the traditional operator formulation of quantum mechanics, a unification of closed and open systems quantum mechanics is made possible here, i.e., one can get and treat quantization and lifetime effects on an equal footing. An important application is the possibility to write down in the 2DEG case a unique Green’s-function expression which holds for all cases of quadratic potentials. This derivation has clearly proved that the appearance of lifetimes (expressing the presence of decaying states, i.e., an intrinsic time asymmetry) has for physical origin the instability of the dynamics occurring at saddle points of the potential landscape; see Ref. for a thorough discussion of this point.\nIn the graphene case, our calculation at large characteristic frequency Ω*c* (or equivalently at large Fermi velocity) brings important information, because, in contrast to standard 2DEGs, even simple models of parabolic confinement for graphene do not possess an analytic solution at finite Ω*c*. However, we will show that the limit Ω*c* → + ∞ is exactly solvable for most quadratic potentials, allowing us to extract the explicit discrete energy spectrum in case of several parabolic confinement models, and also, in principle, the transmission coefficients in case of tunneling near saddle points. Going beyond these toy models, our general formalism also allows us to calculate in a controlled way the LDoS in an *arbitrary* and possibly disordered potential landscape. Our results will be discussed with respect to recent experimental findings. \nStructure of the paper and summary of results\n---------------------------------------------\nFirst, in Sec. [sec:free], we shall investigate the free Dirac Hamiltonian in a transverse magnetic field, and introduce the graphene vortex states, which are the building blocks of the whole theory developed here. These states form an overcomplete family of semicoherent states, strongly localized around arbitrary guiding center positions ${\\bf R}$, and encode the cyclotron motion quantum mechanically.\nIn subsequent Sec. [sec:formalism], we introduce the Green’s function for graphene vortex states and derive its general equation of motion, Eq. ([Dysonfinal]), including Landau-level mixing processes. The general connection to the real-space Green’s function is also explicitly made in Eq. ([passage2]), allowing one to calculate, in principle, any physical observable.\nIn Sec. [sec:high], we show that the problem simplifies greatly in the limit of negligible Landau level mixing. First, for locally flat potentials (away from saddle points or bottom of potential wells), we find that the *m*th Landau level acquires a dependence on the position ${\\bf R}$, according to the simple formula,\n \n$$\\begin{aligned}\n\\xi\\_{m,\\pm}({\\bf R})= \\tilde{v}\\_{m}^{+}({\\bf R}) \\pm\n\\sqrt{(\\hbar \\Omega\\_c \\sqrt{m})^{2}+\\left[\\tilde{v}\\_{m}^{-}({\\bf R}) \\right]^{2}}.\n\\label{effpot}\\end{aligned}$$ \nHere $\\tilde{v}\\_{m}^{+}({\\bf R})$ and $\\tilde{v}\\_{m}^{-}({\\bf R})$ are renormalized effective potentials that are simple functionals of the bare scalar and mass potentials ; for their definitions in terms of *V**s* and *V**z*, see Eq. ([tildev]) and the associated discussion in Sec. [Matrix]. Second, when curvature of the potential is included, we find that simple analytic solutions for several parabolic models can be obtained. In particular, for circular parabolic *scalar* potential $V\\_s({\\bf r}) =\n(1/2) U\\_0 (x^2+y^2)$, the discrete energy spectrum (in terms of Landau level index *m* ≥ 1 and an extra quantum number *n*, which is a positive integer ≥ 1) reads:\n \n$$E\\_{m,n} = \\pm \\hbar \\Omega\\_c \\sqrt{m} + l\\_B^2 U\\_0 (m+n+1/2)\n\\label{spectrum1}$$ \n(we have assumed Ω*c* ≫ *l**B*2*U*0 above). Apart from the well-known anomalous Landau-level quantization with respect to quantum number *m*, this result is very reminiscent of Fock-Darwin states for standard 2DEGs with respect to the linear dependence in the integer *n*. More interestingly, for circular parabolic *mass* potential $V\\_z({\\bf r}) =\n(1/2) U\\_0 (x^2+y^2)$, the discrete energy spectrum displays now an anharmonic form\n \n$$E\\_{m,n} = -\\frac{l\\_{B}^{2}U\\_{0}}{2} \\pm \\sqrt{(\\hbar \\Omega\\_c \\sqrt{m})^2\n+ (l\\_B^2 U\\_0 [m+n+1/2])^2}\n\\label{spectrum2}$$ \nthat was not obtained to our knowledge. Generalization to non-circularly symmetric parabolic potentials is also readily obtained, as well as for the combination of uniform scalar and parabolic mass terms (and vice versa), with detailed calculations appearing in several appendices. However, we show that potentials that combine sizeable spatial variations in both scalar and mass terms are in general not analytically tractable, even in the high magnetic field regime, except for the lowest Landau level.\nIn Sec. [sec:deformation], we make explicit the connection of our formalism to the so-called deformation (or Weyl) quantization, which corresponds to the proper way of quantizing the dynamics in phase space. For two-dimensional problems in a magnetic field, the vortex state formalism is in fact performing a mixed representation of phase space in terms of the two-dimensional coordinates of the center of mass together with a discrete quantum number associated to Landau levels while standard Weyl quantization would introduce a four-dimensional description in terms of positions and momenta of the electron. This latter choice is however unpractical for the high magnetic field regime and this shows that the vortex states are most robust in this regime. As should be expected, in the limit of infinite frequency (*ω**c* → ∞ for the 2DEG or Ω*c* → ∞ for graphene), Landau levels become fully decoupled, and the quantum dynamics reduces to a unidimensional one in terms of the two vortex coordinates, acting as conjugate variables. An effective one-dimensional picture of motion is thus rigorously obtained, overcoming certain regularization problems of the path-integral technique. \nFinally, in Sec. [sec:local], we provide generic expressions for the LDoS in an *arbitrary* scalar or mass potential that can be described *locally* up to its first-order derivatives (generalized graphene drift states). Regarding recent experimental findings, we show that:\n* positions, amplitudes, and widths of the LDoS peaks qualitatively depend on the dominant type (scalar or mass) of local potential, see, e.g., Eq. ([effective]);\n* as the tip scans the surface, the LDoS peak energy of the *m*th Landau level follows the effective potential given in Eq. ([effpot]), see Fig. [figVeff], so that the resulting energy variations *shrink* with increasing *m*, in agreement with the experimental findings for graphene and standard 2DEG;\n* on the contrary, the width of the LDoS peaks at *fixed* tip position *grows* with increasing *m* (roughly as $\\sqrt{m}$), as observed in Ref. for graphene. Such a dependence is also expected for the ordinary 2DEG.\nFree Hamiltonian : Vortex states of graphene\n============================================\nVortex states for the standard 2DEG\n-----------------------------------\nBefore investigating the case of graphene under magnetic field, we briefly recall the vortex states for the case of the non-relativistic 2DEG. This introduction will be useful to show that many physical and technical aspects of the 2DEG can be directly transposed to the case of graphene (studied in the next subsection).\nA single free electron of effective mass *m*\\* confined in a (*x**y*) two-dimensional plane and subjected to a uniform magnetic field pointing in the perpendicular direction ${\\bf B}=B \\hat{{\\bf z}}$ is described by the Hamiltonian\n \n$$H\\_{2DEG}=\\frac{\\hat{{\\bm \\Pi}}^{2}}{2 m^{\\ast}}=\\frac{\\hat{\\Pi}\\_{x}^{2}+\\hat{\\Pi}\\_{y}^{2}}{2 m^{\\ast}}.\n\\label{Ham2D}$$ \nThen, the eigenvalue problem *H*2*D**E**G*Ψ = *ɛ*Ψ leads to the well-known quantization of the kinetic energy into Landau levels,\n \n$$\\varepsilon\\_{m}=\\left( m + \\frac{1}{2} \\right) \\hbar \\omega\\_{c}\n\\label{Sol1}$$ \nwith the cyclotron pulsation *ω**c* = ∣*e*∣*B*/*m*\\**c* = ℏ/*m*\\**l**B*2 and *m* ≥ 0 a positive integer (here $l\\_{B}=\\sqrt{\\hbar\nc/|e|B}$ is the magnetic length). It is important to note here the large degeneracy of the Landau energy levels *ɛ**m*. Indeed, for the motion of an electron in the two-dimensional plane, one expects at least two quantum numbers since there are two degrees of freedom. The degeneracy means that there is a great freedom in the choice of the second (degeneracy) quantum number, or equivalently, in the choice of a basis of eigenstates Ψ. Consequently, there exist in the literature different ways to derive the energy quantization, Eq. ([Sol1]). Eigenstates characterized by a peculiar symmetry of the (gauge-invariant) probability density ∣Ψ∣2 are preferentially chosen in many contexts. For instance, the Landau states, with a conserved momentum as the degeneracy quantum number, are translationally invariant in one direction. Circular eigenstates characterized by a rotation invariance around the origin are also well known and often used. It is worth stressing that the real difference between the Landau states and the circular states is not the gauge because both kinds of states can be obtained in any gauge. The real difference is in the choice of the gauge-invariant quantum numbers, which are intimately related to the symmetry of the probability density ∣Ψ∣2.\nImportantly, both Landau and circular eigenstates do not reflect the symmetry of the cyclotron motion around an arbitrary point ${\\bf R}=(X,Y)$ in the (*x*, *y*) plane so that the consideration of the classical limit with these sets of states is rather tricky. Since they do not correspond to the classical picture of the motion, it is difficult to appreciate the wave-particle duality. By imposing that the probability density ∣Ψ∣2 of the eigenstates has the same symmetry as the cyclotron motion, i.e., is a function of $\\left|{\\bf\nr}-{\\bf R}\\right|$ only, we get the so-called vortex states, given in the symmetrical gauge (${\\bf A}=B \\hat{{\\bf z}} \\times {\\bf r}/2$) by\n \n$$\\begin{aligned}\n\\Psi\\_{m,{\\bf R}}({\\bf r}) =\n\\frac{1}{l\\_{B}\\sqrt{2 \\pi m!}}\n\\left[\n\\frac{x-X+i(y-Y)}{\\sqrt{2}l\\_{B}}\n\\right]^{m} \\nonumber \\hspace\\*{1cm} \\\\\n\\times\n\\exp\\left[\n-\\frac{(x-X)^{2}+(y-Y)^{2}+2i(yX-xY)}{4 l\\_{B}^{2}}\n\\right]\n.\n\\label{Sol2}\\end{aligned}$$ \nFor practical convenience, we shall now use the Dirac bracket notation by writing $\\Psi\\_{m,{\\bf R}}({\\bf r})=\\langle {\\bf r}| m,{\\bf R} \\rangle$. Eigenstates ([Sol2]) of Hamiltonian ([Ham2D]), associated with energy quantization ([Sol1]), are characterized by the set of quantum numbers $|m,{\\bf R} \\rangle$, where *m* is a positive integer related to the quantization of the circulation around the vortex and ${\\bf R}=(X,Y)$ is a continuous quantum number corresponding to the vortex location in the plane [note with Eq. ([Sol2]) the “vortex”-like phase singularity at ${\\bf\nr}={\\bf R}$ for *m* ≥ 1, which justifies the chosen denomination for the set of states]. These localized wave functions clearly encode the classical cyclotron motion around the guiding center ${\\bf R}$ quantum mechanically. The vortex states form a semiorthogonal basis, with the overlap\n \n$$\\begin{aligned}\n\\langle m\\_{1}, {\\bf R}\\_{1}|m\\_{2},{\\bf R}\\_{2} \\rangle\n=\n\\delta\\_{m\\_{1},m\\_{2}}\n\\langle {\\bf R}\\_{1}| {\\bf R}\\_{2} \\rangle, \\label{overlap}\\end{aligned}$$ \nwhere\n \n$$\\langle {\\bf R}\\_{1}| {\\bf R}\\_{2} \\rangle\n=\\exp\\left[ -\\frac{({\\bf R}\\_{1}-{\\bf R}\\_{2})^{2}-2i \\hat{{\\bf z}} \\cdot ({\\bf R}\\_{1} \\times {\\bf R}\\_{2})}{4 l\\_{B}^{2}}\\right].\n\\label{overlapR}$$ \nAn important property is that the states ([Sol2]) present the coherent character with respect to the degeneracy quantum number ${\\bf R}$, i.e., they satisfy coherent states algebra. Note that these states are however eigenstates of the free Hamiltonian associated to the Landau-level index *m*, and form more precisely a semicoherent basis with respect to the quantum numbers (*m*,${\\bf R})$. In particular, they also obey the following completeness relation\n \n$$\\begin{aligned}\n\\int\n \\!\\!\\! \\frac{d^{2} {\\bf R}}{2 \\pi l\\_{B}^{2}} \\sum\\_{m=0}^{+ \\infty} | m,{\\bf R} \\rangle \\langle m,{\\bf R}\n|\n=1. \\label{compvortex}\\end{aligned}$$ \nAccording to this relation ([compvortex]) and general unicity properties of the decomposition onto coherent states, it is possible to expand arbitrary states or operators in the vortex state representation. Hence, despite being nonorthogonal, the set of states $|m,{\\bf R} \\rangle$ with *m* ≥ 0 does form a basis of eigenstates, as the Landau and the circular states.\nBesides providing a clear quantum mechanical dual of the classical cyclotron motion, there are several good reasons to prefer specifically the vortex states over an orthogonal set of eigenstates to study the process of lifting of the Landau level degeneracy in the presence of a smooth *arbitrary* potential. First, in contrast to the Landau states or circular states, the vortex states do not impose a symmetry to the degeneracy quantum number, and thus permit a great adaptability to the spatial variations in the local electric fields, coming from either random impurity donors, confinement potentials, or macroscopic voltage drops (in a nonequilibrium regime). This property leads to advantages in terms of computability since it is possible in the vortex representation to calculate and classify Landau-level mixing processes in a simple and natural manner (this will be illustrated in Sec. [Matrix]). Second, at a more fundamental level, the vortex states are expected to be quite insensitive to any kind of smooth perturbations, since the quantum number *m* has a purely topological origin in the vortex representation (for the Landau states or circular states, the quantization of the kinetic energy comes either partially or entirely from the condition of vanishing of the wave function at infinity, what makes them much less robust to perturbations as a result of their nonlocality). Owing to this quantum robustness, the vortex states are thus naturally selected by the dynamics in the presence of a smooth potential with an arbitrary spatial dependence. They appear to be much more stable than their superpositions (for instance, the Landau states) since they are the only states surviving under the action of such an interaction potential without any internal symmetry. Interestingly, the vortex states are also the best states to describe the transition from quantum to classical. Despite being fully quantum, they thus encode de facto classicality properties and insensitivity to openness of the system. Therefore, they provide the best playground to understand the mechanisms of irreversibility, decoherence and dissipation in high magnetic fields. We will comment on this point in more detail later, in Sec. [hier].\nGraphene vortex states\n----------------------\nWe now come for good to graphene, which is described in the absence of potential by Hamiltonian ([Hgraphene]). By searching the wave functions under the spinorial form\n \n$$\\tilde{\\Psi} =\\left(\\begin{array}{c}\nu \\\\w\n\\end{array}\n \\right)$$ \nwith\n \n$$H\\_{0} \\tilde{\\Psi}=E \\tilde{\\Psi},$$ \nwe get the following equations:\n \n$$\\begin{aligned}\n\\left(\\hat{\\Pi}\\_{x}-i \\hat{\\Pi}\\_{y} \\right) w &= & \\frac{E}{v\\_{F}} u, \\label{S1}\n\\\\\n\\left(\\hat{\\Pi}\\_{x}+i \\hat{\\Pi}\\_{y} \\right) u &= & \\frac{E}{v\\_{F}} w \\label{S2}\\end{aligned}$$ \nwith *E* the energy eigenvalue. Getting rid of the component *u* we get the Schrödinger-type equation for the component *w*,\n \n$$\\begin{aligned}\n\\left(\\hat{\\Pi}\\_{x}+i \\hat{\\Pi}\\_{y} \\right)\\left(\\hat{\\Pi}\\_{x}-i \\hat{\\Pi}\\_{y}\n\\right) w &= & \\left(\\frac{E}{v\\_{F}}\\right)^{2} w. \\label{v}\\end{aligned}$$ \nUsing that\n \n$$\\left[ \\hat{\\Pi}\\_{x}, \\hat{\\Pi}\\_{y}\n\\right]\n= -i \\hbar \\frac{|e|B}{c}=-i \\frac{\\hbar^{2}}{ l\\_{B}^{2}},$$ \nwe find that Eq. ([v]) reads\n \n$$\\hat{{\\bm \\Pi} }^{2}w\n= \\tilde{E} w \\label{E1}$$ \nwith\n \n$$\\tilde{E}= \\left(\\frac{E}{v\\_{F}} \\right)^{2} + \\frac{\\hbar^{2}}{ l\\_{B}^{2}}.$$ \nBy posing *Ẽ* = 2*m*\\**ɛ* in Eq. ([E1]), where *ɛ* has the dimension of an energy, we directly recognize the eigenproblem for a free 2DEG under magnetic fields discussed in the former section. This mapping shows that there is also a great freedom to choose a basis of eigenstates in the case of graphene. In the following, we introduce the analog of vortex states, Eq. ([Sol2]), for graphene.\nFrom Eq. ([Sol1]), we directly deduce that\n \n*Ẽ* = (2*m* + 1)ℏ2/*l**B*2. \nTherefore, we get that the energy eigenvalues of the graphene Hamiltonian are\n \n$$E\\_{m,\\lambda}=\\lambda \\sqrt{m} \\, \\hbar \\sqrt{2}\\frac{v\\_{F}}{l\\_{B}}\n=\\lambda \\sqrt{m} \\, \\hbar \\Omega\\_{c}, \\label{Em}$$ \nwhere *λ* is a band index, which is equal to ± 1 if *m* ≥ 1, and 0 if *m* = 0. We see that the energy levels are no more equidistant in energy and that the characteristic energy for graphene reads $\\hbar \\Omega\\_{c} \\propto \\sqrt{B}$ instead of ℏ*ω**c* ∝ *B* for 2DEGs. The component *u* of the spinorial wave function $\\tilde{\\Psi}$ is straightforwardly obtained from the knowledge of the component *w* by using Eq. ([S1]). The corresponding normalized graphene vortex states are thus\n \n$$\\tilde{\\Psi}\\_{m,{\\bf R},\\lambda}({\\bf r})=\\frac{1}{\\sqrt{1+|\\lambda|}} \\left(\n\\begin{array}{c}\n\\lambda \\Psi\\_{m-1,{\\bf R}}({\\bf r}) \\\\\n i \\Psi\\_{m,{\\bf R}}({\\bf r})\n\\end{array}\n \\right).$$ \nWithin the Dirac notation, the set of vortex quantum numbers we shall consider for graphene takes therefore the form\n \n$$|m,{\\bf R},\\lambda \\rangle\n=\n\\frac{1}{\\sqrt{1+|\\lambda|}}\n\\left(\n\\begin{array}{c}\n\\lambda |m-1,{\\bf R}\\rangle\n\\\\\ni |m,{\\bf R} \\rangle\n\\end{array}\n\\right)\n.\n\\label{Dirac}$$ \nThe label *λ* which characterizes the spinorial structure of the eigenvectors appears here as an additional quantum number with respect to the 2DEG.\nUsing the semiorthogonality property, Eq. ([overlap]), of the vortex states, we can easily check that the graphene vortex states present the same property as their “non-relativistic” counterparts. Indeed, we have\n \n$$\\begin{aligned}\n\\langle m\\_{1}, {\\bf R}\\_{1},\\lambda\\_{1} |m\\_{2},{\\bf R}\\_{2},\\lambda\\_{2} \\rangle\n= \\frac{1}{\\sqrt{1+|\\lambda\\_{1}|}}\\frac{1}{\\sqrt{1+|\\lambda\\_{2}|}} \\hspace\\*{0.8cm} \\nonumber \\\\\n\\times\n\\left(\n\\begin{array}{c}\n\\lambda\\_{1} \\langle m\\_{1}-1,{\\bf R}\\_{1} |\n\\\\\n- i \\langle m\\_{1},{\\bf R}\\_{1} |\n\\end{array}\n \\right)\n\\cdot \\left( \\begin{array}{c}\n\\lambda\\_{2} |m\\_{2}-1,{\\bf R}\\_{2}\\rangle\n\\\\\n i |m\\_{2},{\\bf R}\\_{2}\\rangle\n\\end{array}\\right)\n\\nonumber\n\\\\\n= \\delta\\_{m\\_{1},m\\_{2}} \\langle {\\bf R}\\_{1}| {\\bf R}\\_{2}\n\\rangle\n\\left(\n\\frac{\\lambda\\_{1}\\lambda\\_{2}+1}{\\sqrt{1+|\\lambda\\_{1}|}\\sqrt{1+|\\lambda\\_{2}|}}\n\\right) \\hspace\\*{0.7cm} \\nonumber\n \\\\ =\n \\delta\\_{m\\_{1},m\\_{2}} \\langle {\\bf R}\\_{1}| {\\bf R}\\_{2} \\rangle \\delta\\_{\\lambda\\_{1},\\lambda\\_{2}}. \\hspace\\*{3.5cm}\\end{aligned}$$ \nFor convenience, in the next section we shall condense the full set of quantum numbers $|m,{\\bf R},\\lambda \\rangle$ into the single notation ∣*ν*⟩. Therefore the sum over quantum numbers *ν* will stand for\n \n$$\\begin{aligned}\n\\sum\\_{\\nu}=\\int \\!\\!\\! \\frac{d^{2} {\\bf R}}{2 \\pi l\\_{B}^{2}}\n\\sum\\_{m=0}^{+\\infty} \\sum\\_{\\lambda}.\\end{aligned}$$ \nIt is finally straightforward to prove that the set of graphene vortex states $|m,{\\bf R},\\lambda \\rangle$ obeys a completeness relation, which reads\n \n$$\\begin{aligned}\n\\lefteqn{\n\\sum\\_\\nu |\\nu \\rangle \\langle \\nu| =\n \\int \\!\\!\\! \\frac{d^{2} {\\bf R}}{2 \\pi l\\_{B}^{2}}\n\\sum\\_{m=0}^{+ \\infty}\n\\sum\\_{\\lambda} \\frac{1}{1+|\\lambda|} }\n\\nonumber\n\\\\ & \\times\n\\left(\n\\begin{array}{cc}\n\\lambda^{2} |m-1,{\\bf R} \\rangle \\langle m-1,{\\bf R} | & - i \\lambda |m-1,{\\bf R} \\rangle \\langle m,{\\bf R} | \\\\\ni \\lambda |m,{\\bf R} \\rangle \\langle m-1,{\\bf R} | & |m,{\\bf R} \\rangle \\langle m,{\\bf R} |\n\\end{array}\n\\right) \\nonumber \\\\\n& = \\displaystyle{\n\\int \\!\\!\\! \\frac{d^{2} {\\bf R}}{2 \\pi l\\_{B}^{2}}} \\sum\\_{m=0}^{+ \\infty}\n\\left(\n\\begin{array}{cc}\n|m,{\\bf R} \\rangle \\langle m,{\\bf R} | & 0 \\\\\n0 & |m,{\\bf R} \\rangle \\langle m,{\\bf R} |\n\\end{array}\n\\right)\n \\nonumber\n\\\\\n& =\n\\left(\n\\begin{array}{cc}\n1 & 0 \\\\\n0 & 1\n\\end{array}\n\\right), \\hspace\\*{5.3cm}\n \\label{comp}\\end{aligned}$$ \nwhere we have used the completeness relation ([compvortex]) satisfied by the vortex states.\nGeneral formalism for a smooth potential\n========================================\nMatrix elements of the potential\n--------------------------------\nIn order to investigate the effect of a smooth potential under magnetic field, we shall naturally project the different contributions of Hamiltonian ([prob]) in the graphene vortex representation. Although being basic, this projection sheds already interesting light on the different processes at play and shows the essential differences between the different kinds of potentials that may be encountered in graphene, see Eq. ([prob2]). Using Eq. ([Dirac]), the matrix elements of the diagonal part of the potential (i.e., associated to scalar and mass potentials) can be written as\n \n$$\\begin{aligned}\n\\lefteqn{\\langle \\nu\\_{1} | V\\_{\\mathrm{diag}} |\\nu\\_{2} \\rangle\n= \\left[\n(1+|\\lambda\\_{1})|(1+|\\lambda\\_{2}|)\n\\right]^{-1/2}\n\\nonumber} \\hspace\\*{1cm}\n\\\\&\n\\times \\left\\{\n\\lambda\\_{1} \\lambda\\_{2} \\langle m\\_{1}-1,{\\bf R}\\_{1}|V\\_{s}+V\\_{z}|m\\_{2}-1,{\\bf R}\\_{2}\n\\rangle\n\\right. \\nonumber \\hspace\\*{0.5cm} \\\\\n&\n\\left.\n+ \\langle m\\_{1},{\\bf R}\\_{1}|V\\_{s}-V\\_{z}|m\\_{2},{\\bf R}\\_{2}\n\\rangle\n\\right\\}\n. \\hspace\\*{1.3cm}\n\\label{mat}\\end{aligned}$$ \nThe off-diagonal terms of the potential (i.e., the random vector potential contribution) give rise to the following matrix elements:\n \n$$\\begin{aligned}\n\\lefteqn{\\langle \\nu\\_{1}| V\\_{\\mathrm{off}} |\\nu\\_{2} \\rangle\n=\ni \\left[\n(1+|\\lambda\\_{1}|)(1+|\\lambda\\_{2})\n\\right]^{-1/2}} \\hspace\\*{1cm}\n\\nonumber \\\\\n& \\times\n\\left\\{\n \\lambda\\_{1} \\langle m\\_{1}-1,{\\bf R}\\_{1}|V\\_{x}-iV\\_{y}|m\\_{2},{\\bf R}\\_{2}\n\\rangle \\nonumber \\hspace\\*{1.3cm}\n\\right.\n\\\\\n&\n\\left.\n- \\lambda\\_{2} \\langle m\\_{1},{\\bf R}\\_{1}|V\\_{x}+ iV\\_{y}|m\\_{2}-1,{\\bf R}\\_{2}\n\\rangle\n\\right\\}\n.\n\\label{matoff} \\hspace\\*{0.4cm}\\end{aligned}$$ \nWe have shown in Ref. that it is possible to evaluate exactly the matrix elements of a smooth function $V({\\bf r})$ in the vortex representation [provided that *V*(*x*, *y*) is an analytic function of both *x* and *y*] and write them as a series in powers of the magnetic length *l**B*,\n \n$$\\begin{aligned}\n\\langle m\\_{1},{\\bf R}\\_{1}|V|m\\_{2},{\\bf R}\\_{2}\n\\rangle\n=\\langle {\\bf R}\\_{1} | {\\bf R}\\_{2}\\rangle \\,\nv\\_{m\\_{1};m\\_{2}}\\left( {\\bf R}\\_{12}\\right)\n\\label{smallv}\\end{aligned}$$ \nwith ${\\bf R}\\_{12}=\\left[\n{\\bf R}\\_{1}+{\\bf R}\\_{2}+i ({\\bf R}\\_{2}-{\\bf R}\\_{1}) \\times \\hat{{\\bf z}}\n\\right]/2$ and\n \n$$\\begin{aligned}\n\\label{general}\nv\\_{m\\_{1};m\\_{2}}({\\bf R}) &=& \\int d^2 {\\bm \\eta} \\,\n\\Psi\\_{m\\_1,{\\bf R}}^{\\ast}({\\bm \\eta}) \\Psi\\_{m\\_2,{\\bf R}}({\\bm \\eta})\nV({\\bm \\eta}) \\\\\n&=&\n\\sum\\_{j=0}^{+ \\infty}\n\\left(\\frac{l\\_{B}}{\\sqrt{2}} \\right)^{j} v^{(j)}\\_{m\\_{1},m\\_{2}}({\\bf R}),\n\\label{series}\n\\\\\nv^{(j)}\\_{m\\_{1};m\\_{2}}({\\bf R})\n&=&\n\\sum\\_{k=0}^{j}\n\\frac{(m\\_{1}+k)!}{\\sqrt{m\\_{1}!m\\_{2}!}}\n\\frac{\\delta\\_{m\\_{1}+k,m\\_{2}+j-k}}{k!(j-k)!}\n\\label{space} \\\\\n&& \\times\n\\left( \\partial\\_{X}+i\\partial\\_{Y}\\right)^{k}\n\\left( \\partial\\_{X}-i\\partial\\_{Y}\\right)^{j-k}\nV({\\bf R}). \\nonumber\\end{aligned}$$ \nClearly, the use of an analytical expansion around the complex point ${\\bf R}\\_{12}$ in Eq. ([smallv]) puts some constraints on the types of potential that can be considered in the present formalism. We emphasize that relation ([smallv]) holds for any *physical* potentials *V* (which are necessarily smooth functions of the space variables). In contrast, pointlike (i.e., zero-range) potentials involving Dirac delta functions which represent toy models simulating short-range potentials can not be treated within the present formalism. If the magnetic length *l**B* corresponds to the shortest length scale [here, basically, *l**B* has to be compared with the characteristic length scale of spatial variations in the function $V({\\bf R})$, see Eqs. ([series]) and ([space])], we see that we have naturally ordered the different contributions to the matrix elements by their order of magnitude in high magnetic fields.\nAt leading order (*l**B* → 0), we get from Eqs. ([mat])-([space]) for coinciding vortex positions ${\\bf R}\\_{1}={\\bf R}\\_{2}={\\bf R}$,\n \n$$\\begin{aligned}\n\\langle \\nu\\_{1}| V\\_{\\mathrm{diag}} |\\nu\\_{2} \\rangle\n \\approx\n\\delta\\_{m\\_{1},m\\_{2}} \\left[ \\delta\\_{\\lambda\\_{1},\\lambda\\_{2}}\n\\, V\\_{s}\\left( {\\bf R} \\right)\n-\\delta\\_{\\lambda\\_{1},-\\lambda\\_{2}} \\, V\\_{z}\\left( {\\bf R} \\right)\n\\right]\n. \\nonumber \\\\\n\\label{Vdiag0}\\end{aligned}$$ \nWe remark that in the limit *l**B* → 0 the diagonal elements *V**s* and *V**z* of *V* do not introduce a mixing between Landau levels. For smooth functions *V**x* and *V**y* we get in the same limit *l**B* → 0,\n \n$$\\begin{aligned}\n \\lefteqn{ \\langle \\nu\\_{1} | V\\_{\\mathrm{off}} |\\nu\\_{2} \\rangle\n\\approx - i \\left[\n(1+|\\lambda\\_{1}|)\n(1+|\\lambda\\_{2}|)\n \\right]^{-1/2} \\nonumber } \\hspace\\*{1cm} \\\\\n&\n\\times\n\\left\\{\n\\lambda\\_{2}\\delta\\_{m\\_{1},m\\_{2}-1}\\left[V\\_{x}({\\bf R})+iV\\_{y}({\\bf R}) \\right]\n\\nonumber \\right. \\hspace\\*{2cm} \\\\\n& \\hspace\\*{0.1cm}\n\\left.\n-\\lambda\\_{1} \\delta\\_{m\\_{1}-1,m\\_{2}}\\left[V\\_{x}({\\bf R})-iV\\_{y}({\\bf R}) \\right]\n\\right\\}. \\hspace\\*{1cm}\n\\label{Voff0}\\end{aligned}$$ \nWe note with Eq. ([Voff0]) that the off-diagonal elements *V**x* and *V**y* do mix adjacent Landau levels already at leading order in *l**B*, in contrast to the diagonal elements *V**s* and *V**z* of *V*. This difference clearly calls for a different treatment of the diagonal and off-diagonal parts of the total potential *V*. Off-diagonal contributions can be treated perturbatively at high magnetic field by assuming that *V**x* and *V**y* are small in amplitude in addition of being smooth functions at the scale *l**B*. Such a constraint on the amplitude can be relaxed in the treatment of the diagonal contributions of *V*.\nThe next (sub-dominant) contributions of order *l**B* to the matrix elements of *V*diag are proportional to\n \n$$\\begin{aligned}\n\\delta\\_{m\\_{1}+1,m\\_{2}}\n (\\partial\\_{X}+i\\partial\\_{Y})\n\\left\\{\n\\sqrt{m\\_{2}} \\left[ V\\_{s}({\\bf R})-V\\_{z}({\\bf R})\\right]\n\\nonumber \\right. \\hspace\\*{0.5cm}\n\\\\\n\\left.\n+\\lambda\\_{1}\\lambda\\_{2} \\sqrt{m\\_{1}}\\left[ V\\_{s}({\\bf R})+V\\_{z}({\\bf R})\\right]\n\\right\\}\n+ c.c. (1 \\leftrightarrow 2)\n,\n\\label{v1}\\end{aligned}$$ \nwhere the notation *c*.*c*.(1 ↔ 2) means taking the complex conjugate and exchanging the indexes 1 and 2 of the former expression. This contribution induces a mixing between both adjacent Landau levels and band indices *λ*. Moreover, the mixture of positive- and negative-energy components stems from both components *V**s* and *V**z* of the potential energy. It is interesting to note that for a large Landau-level index, the mixture arising purely from *V**s* (i.e., taking the mass term *V**z* = 0) gets negligible when *λ*1*λ*2 = − 1. For instance, when *m*1 and *m*2 ≫ 1, we have\n \n$$\\sqrt{m\\_{1}+1}+\\lambda\\_{1}\\lambda\\_{2} \\sqrt{m\\_{1}} \\approx \\sqrt{m\\_{1}}\n\\left( 1+\\lambda\\_{1}\\lambda\\_{2}\n\\right)$$ \nfor the component *δ**m*1 + 1, *m*2 of the matrix elements, Eq. ([v1]), associated with *V**s*. On the other hand, the band mixing becomes significant for *m*1 and *m*2 close to 0. Specific signatures resulting from this interband mixing, such as Zitterbewegung (or trembling motion) in a magnetic field, have been discussed in the literature. By looking at next-order contributions in *l**B*2 for the matrix elements, we note that interband mixing occurs also with the second derivatives of a pure scalar potential *V**s* without mixing the Landau levels. These mixing processes will be analyzed further in Sec. [sec:high].\nGreen’s-function formalism\n--------------------------\nThe nonorthogonality of the graphene vortex states preventing us to build a wave-function perturbation theory, we shall instead use a Green’s-function formalism to get a more quantitative insight on the effect of a smooth potential, following Refs. and. Although the derivation of the equations of motion for the graphene Green’s function is very similar to that for the 2DEG Green’s function, we shall nevertheless describe the principal steps with some detail here, in order to make this paper self-contained (we shall, however, not reproduce the very technical details).\nRetarded and advanced Green’s functions are, respectively, defined as\n \n$$\\begin{aligned}\nG^{R}(x\\_{1},x\\_{2}) &=& - i \\theta(t\\_{1}-t\\_{2}) \\langle\n\\left\\{ \\mathcal{\\psi}(x\\_{1}), \\mathcal{\\psi}^{\\dagger}(x\\_{2})\\right\\} \\rangle, \\\\\nG^{A}(x\\_{1},x\\_{2}) &=& i \\theta(t\\_{2}-t\\_{1}) \\langle\n\\left\\{ \\mathcal{\\psi}(x\\_{1}), \\mathcal{\\psi}^{\\dagger}(x\\_{2})\\right\\} \\rangle,\\end{aligned}$$ \nwhere {, } means the anti-commutator, and *θ* the Heaviside step function [i.e., *θ*(*t*) = 0 for *t* < 0 and *θ*(*t*) = 1 for *t* > 0]. The averages are evaluated in the grand canonical ensemble. The Green’s functions relate the field operator \\psi(*x*1) of the particle at one point $x\\_{1}=({\\bf r}\\_{1},t\\_{1})$ in space-time to the conjugate field operator \\psi†(*x*2) at another point $x\\_{2}=({\\bf r}\\_{2},t\\_{2})$. The field operators \\psi(*x*1) and \\psi†(*x*2) are expressed in terms of the eigenfunctions $\\tilde{\\Psi}\\_{\\nu}({\\bf r})$ and eigenvalues *E**ν* as\n \n$$\\begin{aligned}\n\\mathcal{\\psi}(x\\_{1}) &=& \\sum\\_{\\nu} c\\_{\\nu} \\tilde{\\Psi}\\_{\\nu}({\\bf r}\\_{1}) \\, e^{-iE\\_{\\nu}t\\_{1}/\\hbar},\n\\\\\n\\mathcal{\\psi}^{\\dagger}(x\\_{2}) &=& \\sum\\_{\\nu} c^{\\dagger}\\_{\\nu} \\tilde{\\Psi}^{\\dagger}\\_{\\nu}({\\bf r}\\_{2}) \\, e^{iE\\_{\\nu}t\\_{2}/\\hbar},\\end{aligned}$$ \nwhere *c**ν*† and *c**ν* are, respectively, the creation and destruction operators.\nAs a basis of states, we shall then use the graphene vortex states $|\\nu \\rangle\n=|m,{\\bf R},\\lambda \\rangle$ which are eigenstates of Hamiltonian *H*0 [Eq. ([Hgraphene])]. It is worth noting that, although these states ∣*ν*⟩ are nonorthogonal, the associated creation and destruction operators *c**ν*† and *c**ν* obey the usual algebra with the anti-commutation rules {*c**ν*1†, *c**ν*2†} = {*c**ν*1, *c**ν*2} = 0 and {*c**ν*1, *c**ν*2†} = *δ**ν*1, *ν*2.\nCompleteness relation ([comp]) allows us to express the Green’s function in the graphene vortex representation, which we note $G^{R,A}(\\nu\\_{1},t\\_{1};\\nu\\_{2},t\\_{2})=G^{R,A}(m\\_{1},{\\bf\nR}\\_{1},\\lambda\\_{1},t\\_{1} ;m\\_{2},{\\bf R}\\_{2},\\lambda\\_{2},t\\_{2})$. Transposing its definition originally made in terms of the electronic coordinates $({\\bf r},t)$ into the vortex language, the latter Green’s function gives the probability amplitude for a vortex with circulation *m*1 and band index *λ*1 that is initially at position ${\\bf R}\\_{1}$ at time *t*1 to be at point ${\\bf\nR}\\_{2}$ at time *t*2 with a new circulation *m*2 and a band index *λ*2. After Fourier transformation with respect to the time difference *t* = *t*1 − *t*2, the Green’s function (denoted by *G*0) corresponding to Hamiltonian *H*0 [i.e., Hamiltonian ([prob]) with *V* = 0] are written in the energy (*ω*) representation as\n \n$$\\begin{aligned}\nG\\_{0}^{R,A}(\\nu\\_{1};\\nu\\_{2})=\\frac{\\delta\\_{m\\_{1},m\\_{2}} \\,\n\\delta\\_{\\lambda\\_{1},\\lambda\\_{2}} \\,\\langle {\\bf R}\\_{1} | {\\bf R}\\_{2}\n\\rangle}{\\omega-E\\_{m\\_{1},\\lambda\\_{1}} \\pm i 0^{+}}.\\end{aligned}$$ \nRetarded and advanced Green’s function in the presence of the smooth potential *V* are obtained from Dyson equation, which takes the following form in the *ν* representation (we again considered the Fourier transform of Green’s function with respect to time difference)\n \n$$\\begin{aligned}\n\\left(\n\\omega - E\\_{m\\_{1},\\lambda\\_{1}} \\pm i 0^{+}\n\\right) G^{R,A}(\\nu\\_{1};\\nu\\_{2})\n=\\langle \\nu\\_{1}| \\nu\\_{2} \\rangle \\hspace\\*{1cm} \\nonumber \\\\\n+\\sum\\_{\\nu\\_{3}} V\\_{\\nu\\_{1};\\nu\\_{3}} G^{R,A}(\\nu\\_{3};\\nu\\_{2})\n. \\label{Dyson}\\end{aligned}$$ \nHere the general matrix elements $V\\_{\\nu\\_{1};\\nu\\_{2}}=\\langle \\nu\\_{1} |V|\\nu\\_{2}\n\\rangle= \\langle {\\bf R}\\_{1} | {\\bf R}\\_{2} \\rangle \\,\nv\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R}\\_{12}) $ are given by expressions ([mat])-([space]). For *V* ≠ 0, the graphene vortex Green’s function is generally no more diagonal with respect to the quantum numbers *m* and *λ*, and the mixing between the different quantum numbers depend on the characteristic properties of the potential *V*. However, it turns out that, as a result of the coherent states character with respect to vortex position ${\\bf\nR}$ encompassed within overlap ([overlapR]), the propagation of the graphene vortex Green’s function with respect to vortex positions ${\\bf R}\\_{1}$ and ${\\bf\nR}\\_{2}$ is constrained to necessarily take the form\n \n$$\\begin{aligned}\nG(\\nu\\_{1};\\nu\\_{2})= \\langle {\\bf R}\\_{1} |{\\bf R}\\_{2} \\rangle \\,\ng\\_{m\\_{1},\\lambda\\_{1} ;m\\_{2},\\lambda\\_{2}}\n\\left(\n{\\bf R}\\_{12}\n\\right)\n,\n \\label{Gvortexnonlocal}\\end{aligned}$$ \nsimilarly to the matrix elements of the potential [see Eq. ([smallv])]. Such exact dependence, Eq. ([Gvortexnonlocal]), can be derived from Dyson Eq. ([Dyson]) in the same way as done in Ref.. Remarkably, it implies that the nonlocal graphene Green’s function *G*(*ν*1; *ν*2) will be entirely determined once it is known at coinciding vortex positions ${\\bf R}\\_{1}={\\bf R}\\_{2} \\equiv {\\bf R}$, and this result holds irrespective of the potential *V*. It is then sufficient to consider Eq. ([Dyson]) for coinciding vortex positions. Because the derivation is the same as for the 2DEG, we briefly outline here the last step leading to the final equation of motion governing the function $g\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})$ and refer the reader to Sec. II of Ref. for the mathematical details. The nonlocal dependencies of the functions *G*(*ν*3; *ν*2) and *V**ν*1; *ν*3 on the vortex positions which are known according to relations ([Gvortexnonlocal]) and ([smallv]) are exploited to evaluate the integral over the continuous variable ${\\bf R}\\_{3}$ on the right-hand side of Eq. ([Dyson]). This integral then transforms into a series expansion in powers of *l**B*. We obtain that Dyson equation for the retarded graphene vortex Green’s function $g({\\bf R})$ (from now on, we drop the *R* upperscript associated to retarded) corresponding to Hamiltonian ([prob]) reads\n \n$$\\begin{aligned}\n(\\omega-E\\_{m\\_{1},\\lambda\\_{1}}+ i 0^{+})\ng\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})\n= \\delta\\_{m\\_{1},m\\_{2}} \\, \\delta\\_{\\lambda\\_{1},\\lambda\\_{2}}\n\\nonumber\n\\\\\n+\n\\sum\\_{k=0}^{+\\infty}\\left(\\frac{l\\_{B}}{\\sqrt{2}} \\right)^{2k} \\frac{1}{k!}\n\\sum\\_{m\\_{3},\\lambda\\_{3}}\n\\left(\\partial\\_{X}-i\\partial\\_{Y } \\right)^{k}\nv\\_{m\\_{1},\\lambda\\_{1};m\\_{3},\\lambda\\_{3}}({\\bf R})\n\\nonumber\n\\\\\n\\times\n\\left(\\partial\\_{X}+i\\partial\\_{Y } \\right)^{k}\ng\\_{m\\_{3},\\lambda\\_{2};m\\_{2},\\lambda\\_{2}} ({\\bf R}) \\hspace\\*{0.5cm}\n\\label{Dyson4}\\end{aligned}$$ \nwith $E\\_{m,\\lambda}=\\lambda E\\_{m}=\\lambda \\hbar \\sqrt{2m} v\\_{F}/l\\_{B}$.\nAnother important aspect of the change in function ([Gvortexnonlocal]), which appears clearly with the form ([Dyson4]) of Dyson equation and with expressions ([series]) and ([space]) for the matrix elements of the potential taken at coinciding vortex positions, is that the nonanalytic dependence of the nonlocal graphene vortex Green’s function *G*(*ν*1; *ν*2) on the magnetic *l**B* has been entirely extracted [in formula ([Gvortexnonlocal]), this nonanalytic dependence is only contained in the overlap $\\langle {\\bf R}\\_{1} | {\\bf R}\\_{2} \\rangle$]. In other terms, the function $g({\\bf R})$ is obviously analytic in *l**B* and thus well behaves in the semiclassical limit of zero magnetic length (*l**B* → 0). This property can be used to solve Eq. ([Dyson4]) order by order in powers of *l**B* and thus to provide a semiclassical expansion of the graphene vortex Green’s function *g**m*1, *λ*1; *m*2, *λ*2 as\n \n$$g\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}=\\sum\\_{j=0}^{+\\infty}\n\\left(\\frac{l\\_{B}}{\\sqrt{2}} \\right)^{j} g^{(j)}\\_{m\\_{1},\\lambda\\_{1}; m\\_{2},\\lambda\\_{2}}.\n\\label{expg}$$ \nBecause the series, Eq. ([expg]), is then only asymptotic in nature (the obtained solution holds in the limit *l**B* → 0, but is not controlled at finite *l**B*), we aim here at solving directly and non-perturbatively in *l**B* Dyson Eq. ([Dyson4]).\nFor this purpose, we have found in Ref. that it is very convenient to introduce the simultaneous changes in functions,\n \n$$\\begin{aligned}\n\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R}) &=& e^{-(l\\_{B}^{2}/4) \\Delta\\_{{\\bf R}}} g\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R}), \\label{changeg} \\\\\n\\tilde{v}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R}) &=& e^{-(l\\_{B}^{2}/4) \\Delta\\_{{\\bf R}}} v\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R}), \\label{changev}\\end{aligned}$$ \nwhere the symbol $\\Delta\\_{{\\bf R}}$ means the Laplacian operator taken with respect to the vortex position ${\\bf R}$. After substitution of these expressions ([changeg]) and ([changev]) into Eq. ([Dyson4]), we get a new equation for the unknown function $\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})$ with a higher-order differential operator than the one appearing on the right-hand side of Eq. ([Dyson4])\n \n$$\\begin{aligned}\n(\\omega-E\\_{m\\_{1},\\lambda\\_{1}}+ i 0^{+})\n\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})\n=\n \\delta\\_{\\lambda\\_{1},\\lambda\\_{2}} \\delta\\_{m\\_{1},m\\_{2}} \\nonumber \\\\\n+\n\\sum\\_{m\\_{3},\\lambda\\_{3}}\n\\tilde{v}\\_{m\\_{1},\\lambda\\_{1};m\\_{3},\\lambda\\_{3}}({\\bf R})\n\\star\n\\tilde{g}\\_{m\\_{3},\\lambda\\_{3};m\\_{2},\\lambda\\_{2}} ({\\bf R}), \\hspace\\*{0.5cm}\n\\label{Dysonfinal}\\end{aligned}$$ \nwhere the symbol ⋆ stands for the bi-differential operator defined by\n \n$$\\begin{aligned}\n\\star=\\exp\\left[i \\frac{l\\_{B}^{2}}{2} \\left( \\overleftarrow{\\partial}\\_{X} \\overrightarrow{ \\partial}\\_{Y}\n- \\overleftarrow{\\partial}\\_{Y} \\overrightarrow{ \\partial}\\_{X} \\right) \\right].\n\\label{star}\\end{aligned}$$ \nThe arrow above the partial derivatives indicates to which side the derivative acts. Note that the passage from Eq. ([Dyson4]) to Eq. ([Dysonfinal]) is more straightforward by going to Fourier space (see Appendix A of Ref. ). It is worth mentioning that, by starting from the other Dyson equation (i.e., formally *G* = *G*0 + *G**V**G*0) and following the same steps as detailed previously, we can derive a second equation satisfied by the function *g̃*,\n \n$$\\begin{aligned}\n(\\omega-E\\_{m\\_{2},\\lambda\\_{2}}+ i 0^{+})\n\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})\n=\n \\delta\\_{\\lambda\\_{1},\\lambda\\_{2}} \\delta\\_{m\\_{1},m\\_{2}} \\nonumber \\\\\n+\n\\sum\\_{m\\_{3},\\lambda\\_{3}}\n\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{3},\\lambda\\_{3}}({\\bf R})\n\\star\n\\tilde{v}\\_{m\\_{3},\\lambda\\_{3};m\\_{2},\\lambda\\_{2}} ({\\bf R}). \\hspace\\*{0.5cm}\n\\label{Dysonfinalbis}\\end{aligned}$$ \nThe particular form of exact Eqs. ([Dysonfinal])-([Dysonfinalbis]), reminiscent of the so-called star-product, will be further used and commented in Secs. [sec:high] and [sec:deformation].\nIn order to compute local physical observables such as the local density of states, we need to express Green’s function in terms of the electronic positions ${\\bf r}$. The electronic Green’s function is a 2 x 2 matrix in the pseudo-spin space, and is defined as $\\hat{G}({\\bf r},{\\bf r}')=\\langle {\\bf r}\n|\\hat{G} |{\\bf r}'\\rangle$. At a practical level, it is useful to directly relate the nonlocal electronic Green’s function to the *local* graphene vortex Green’s function $g\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})$ (at coinciding vortex positions) or alternatively to the modified vortex Green’s function $\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})$. First, the electronic Green’s function can be straightforwardly linked to the nonlocal graphene vortex Green’s function *G*(*ν*1; *ν*2) through a change in representation which is performed by using twice completeness relation ([comp]). Then, using Eq. ([Gvortexnonlocal]) and following the calculations made in Ref. for the 2DEG, we get the following relation\n \n$$\\begin{aligned}\n\\hat{G}({\\bf r},{\\bf r}',\\omega)\n= \\int \\!\\!\\!\n\\frac{d^{2}{\\bf R}}{2 \\pi l\\_{B}^{2}}\n\\sum\\_{m\\_{1},\\lambda\\_{1}} \\sum\\_{m\\_{2},\\lambda\\_{2}}\n\\left(\n\\begin{array}{cc}\n\\lambda\\_{1}\\lambda\\_{2}\n\\Psi^{\\ast}\\_{m\\_{2}-1,{\\bf R}}\n({\\bf r}')\n\\Psi\\_{m\\_{1}-1,{\\bf R}}\n({\\bf r}) & - i \\lambda\\_{1} \\Psi^{\\ast}\\_{m\\_{2},{\\bf R}}\n({\\bf r}')\n\\Psi\\_{m\\_{1}-1,{\\bf R}}\n({\\bf r})\n\\\\\ni \\lambda\\_{2} \\Psi^{\\ast}\\_{m\\_{2}-1,{\\bf R}}\n({\\bf r}')\n\\Psi\\_{m\\_{1},{\\bf R}}\n({\\bf r})\n&\n\\Psi^{\\ast}\\_{m\\_{2},{\\bf R}}\n({\\bf r}')\n\\Psi\\_{m\\_{1},{\\bf R}}\n({\\bf r})\n\\end{array}\n\\right)\n\\nonumber \\\\\n\\times\ne^{ -(l\\_{B}^{2}/2) \\Delta\\_{{\\bf R}}} \\left[\n\\frac{g\\_{m\\_{1}, \\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})}\n{\n\\sqrt{1+|\\lambda\\_{1}|}\n\\sqrt{1+|\\lambda\\_{2}|}\n} \\right]\n,\n\\label{passage}\\end{aligned}$$ \nwhere the functions $\\Psi\\_{m,{\\bf R}}({\\bf r})$ correspond to the so-called vortex wave functions written in Eq. ([Sol2]). Inverting expression ([changeg]), i.e., writing $g\\_{m\\_{1}, \\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf\nR}) = e^{(l\\_{B}^{2}/4) \\Delta\\_{\\bf R} } \\,\\tilde{g}\\_{m\\_{1},\n\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})$ and inserting this result into Eq. ([passage]), we get after integrations by parts (so that the operator involving the Laplacian acts on the product of wave functions rather on the local vortex Green’s function)\n \n$$\\begin{aligned}\n\\hat{G}({\\bf r},{\\bf r}',\\omega)\n= \\int \\!\\!\\!\n\\frac{d^{2}{\\bf R}}{2 \\pi l\\_{B}^{2}}\n\\sum\\_{m\\_{1},\\lambda\\_{1}} \\sum\\_{m\\_{2},\\lambda\\_{2}}\ne^{ - (l\\_{B}^{2}/4) \\Delta\\_{{\\bf R}}}\n\\left(\n\\begin{array}{cc}\n\\lambda\\_{1}\\lambda\\_{2}\n\\Psi^{\\ast}\\_{m\\_{2}-1,{\\bf R}}\n({\\bf r}')\n\\Psi\\_{m\\_{1}-1,{\\bf R}}\n({\\bf r}) & - i \\lambda\\_{1} \\Psi^{\\ast}\\_{m\\_{2},{\\bf R}}\n({\\bf r}')\n\\Psi\\_{m\\_{1}-1,{\\bf R}}\n({\\bf r})\n\\\\\ni \\lambda\\_{2} \\Psi^{\\ast}\\_{m\\_{2}-1,{\\bf R}}\n({\\bf r}')\n\\Psi\\_{m\\_{1},{\\bf R}}\n({\\bf r})\n&\n\\Psi^{\\ast}\\_{m\\_{2},{\\bf R}}\n({\\bf r}')\n\\Psi\\_{m\\_{1},{\\bf R}}\n({\\bf r})\n\\end{array}\n\\right)\n\\nonumber \\\\\n\\times\n\\frac{\\tilde{g}\\_{m\\_{1}, \\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})}\n{\n\\sqrt{1+|\\lambda\\_{1}|}\n\\sqrt{1+|\\lambda\\_{2}|}\n}\n.\n\\label{passage2}\\end{aligned}$$ \nBecause the functions *g̃**m*1, *λ*1; *m*2, *λ*2 may depend on *λ*1 and *λ*2, the electronic Green’s function $\\hat{G}({\\bf r},{\\bf r}')$ possesses, in general, off-diagonal elements. The above equation is a central one, because it shows that any physical observable can be computed from the knowledge of the *local* vortex Green’s function $\\tilde{g}\\_{m\\_{1}, \\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})$.\nHigh magnetic field regime [high]\n=================================\nRegime of negligible Landau-level mixing\n----------------------------------------\nWhile Eqs. ([Dysonfinal])-([Dysonfinalbis]) can, in principle, be considered for any magnetic fields, we shall investigate here the regime of high magnetic field only, for which Landau level mixing can be safely neglected. This regime can be reached under reasonable conditions (i.e., for fields on the order of 1 T or higher) provided that the potential landscape is sufficiently smooth. Indeed, Landau-level mixing processes are described within Eq. ([Dysonfinal]) by the matrix elements *ṽ**m*1, *λ*1; *m*3, *λ*3 with *m*1 ≠ *m*3. From the expressions of the matrix elements of the potential coupling adjacent Landau levels calculated in the vortex representation in Sec. [Matrix], we can formulate a clear quantitative criterion for neglecting Landau-level mixing due to the diagonal contributions of the potential *V* in graphene,\n \n$$l\\_{B} \\left|{\\bm \\nabla}\\_{{\\bf R}}V({\\bf R})\\right| \\ll\n\\left(\\sqrt{m+1}-\\sqrt{m} \\right)\\hbar \\Omega\\_{c}.\n\\label{ineq1}$$ \nIn graphene and for a field of 5 T, we have $\\hbar \\Omega\\_{c}=\\hbar \\sqrt{2}v\\_{F}/l\\_{B} \\approx 80$ meV and *l**B* ≃ 11 nm. Recent experimental STS measurements of the spatial dispersion of Landau levels in epitaxial graphene give at most typical linear variations in *δ**V* ≃ 5 meV on length scales *ξ* ≃ 20 nm. Thus $l\\_B |{\\bm \\nabla} V|/\\hbar \\Omega\\_{c} \\lesssim l\\_B \\delta V/ \\xi \\hbar\n\\Omega\\_c \\simeq 0.03$, a very small number indeed, so that the limit of negligible Landau level mixing is well obeyed. We shall furthermore suppose that the Landau level mixing processes due to the off-diagonal part of *V* are small. According to Eq. ([Voff0]), this implies\n \n$$\\begin{aligned}\n\\left| V\\_{x,y}({\\bf R})\\right| \\ll \\left(\\sqrt{m+1}-\\sqrt{m} \\right) \\hbar \\Omega\\_{c}.\n\\label{ineq2}\\end{aligned}$$ \nUnder inequalities ([ineq1]) and ([ineq2]), Landau-level mixing processes due to the spatial variations in the scalar potential *V**s* and of the random mass *V**z* or to the spatial fluctuations (*V**x* and *V**y*) of the vector potential are small and can be accounted for perturbatively on the basis of Eq. ([Dysonfinal]).\nHenceforth, we shall concentrate on the main relevant processes occurring at high magnetic field in a smooth potential. In this regime, the Landau-level degeneracy is principally lifted by the presence of both the potentials *V**s* and *V**z*, which give rise for *m* ≥ 1 to the following diagonal (*m*1 = *m*2 = *m*) matrix elements in the vortex representation\n \n$$\\begin{aligned}\nv\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R})\n&=\n&\n\\delta\\_{m\\_{1},m\\_{2}} v\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})\n\\nonumber \\\\\n&=&\n\\delta\\_{\\lambda\\_{1},\\lambda\\_{2}} \\, v\\_{m}^{+}({\\bf R})+\\delta\\_{\\lambda\\_{1},-\\lambda\\_{2}}\n\\, v\\_{m}^{-} ({\\bf R})\n, \\hspace\\*{0.5cm} \\label{matdia}\\end{aligned}$$ \nwhere the diagonal and off-diagonal components of the potential matrix elements in pseudospin space, respectively, read\n \n$$\\begin{aligned}\n\\nonumber\n v\\_{m}^{\\pm}({\\bf R}) &=&\n\\frac{1}{2} \\int d^2 {\\bm \\eta} \\left[ \\,\n|\\Psi\\_{m,{\\bf R}}({\\bm \\eta})|^2 \n(V\\_s({\\bm \\eta})-V\\_z({\\bm \\eta})) \\right. \\\\\n\\label{vplus}\n&& \\left. \\pm \n|\\Psi\\_{m-1,{\\bf R}}({\\bm \\eta})|^2\n(V\\_s({\\bm \\eta})+V\\_z({\\bm \\eta})) \\right] \\\\\n&=& \\frac{1}{2}\n\\sum\\_{j=0}^{+\\infty}\n \\frac{(m+j)!}{m! (j!)^{2}} \\left(\\frac{l\\_{B}^{2}}{2} \\Delta\\_{{\\bf R}} \\right)^{j}\n\\Big\\{ \\frac{}{} V\\_{s}({\\bf R})-V\\_{z}({\\bf R}) \\nonumber \\\\\n&& \\pm \\frac{m}{m+j}\\left( V\\_{s}({\\bf R})+V\\_{z}({\\bf R}) \\right)\n\\Big\\}.\n\\label{vplusexpand}\\end{aligned}$$ \nTo write down expressions ([matdia])-([vplusexpand]), we have used Eqs. ([mat]) and ([general])-([space]). We notice that even a *scalar* potential *V**s* introduces a coupling between the bands *λ* = ± for *m* ≥ 1 through its nonlocal differential contributions arising with *j* > 1. For instance, a quadratic scalar potential generically mixes the positive and negative energy components, even in the absence of a mass term (*V**z* = 0). The case *m* = 0 has to be treated as a special case since there is only one band (*λ* = 0 necessarily). The matrix elements for the lowest Landau level *m* = 0 read\n \n$$\\begin{aligned}\nv\\_{0}({\\bf R}) &=& \n\\int d^2 {\\bm \\eta} \n|\\Psi\\_{0,{\\bf R}}({\\bm \\eta})|^2 \n\\left[V\\_s({\\bm \\eta})-V\\_z({\\bm \\eta}) \\right] \\\\\n& = & \\sum\\_{j=0}^{+\\infty}\n \\frac{1}{j!} \\left(\\frac{l\\_{B}^{2}}{2} \\Delta\\_{{\\bf R}} \\right)^{j}\n[V\\_{s}({\\bf R})-V\\_{z}({\\bf R})]\n. \\label{v0}\\end{aligned}$$ \nWe have seen previously that Dyson Eq. ([Dysonfinal]) is greatly simplified when considering modified matrix elements $\\tilde{v}\\_{m,\\lambda\\_{1};m,\\lambda\\_{2}}({\\bf R}) = e^{-(l\\_{B}^{2}/4)\n\\Delta\\_{{\\bf R}}} v\\_{m,\\lambda\\_{1};m,\\lambda\\_{2}}({\\bf R})$, which constitute the *effective potential* in Landau level *m*. Using results given in the Appendix B of Ref., we get the action of the exponential differential operator onto the product of two vortex functions with identical Landau level *m* and positions ${\\bf r}$:\n \n$$\\begin{aligned}\n K\\_{m}({\\bf R}-{\\bf r}) &\\equiv&\ne^{-(l\\_{B}^{2}/4) \\Delta\\_{{\\bf R}}} |\\Psi\\_{m,{\\bf R}}({\\bf r})|^2\n\\\\\n&=& \\frac{1}{\\pi m! l\\_{B}^{2}}\n \\frac{\\partial^{m}}{\\partial s^{m}}\n\\left. \\frac{e^{-A\\_{s} ({\\bf R}-{\\bf r})^{2}/l\\_{B}^{2}}}{1+s} \\right|\\_{s=0},\n\\label{Km}\\end{aligned}$$ \nwith *A**s* = (1 − *s*)/(1 + *s*). Thus, the diagonal and off-diagonal effective potentials (in pseudospin space) read for *m* ≥ 1,\n \n$$\\begin{aligned}\n\\nonumber\n\\tilde{v}\\_{m}^{\\pm}({\\bf R}) &=&\n\\frac{1}{2} \\int d^2 {\\bm \\eta} \\left[ \\,\nK\\_m({\\bf R}-{\\bm \\eta})\n(V\\_s({\\bm \\eta})-V\\_z({\\bm \\eta})) \\right. \\\\\n&& \\left. \\pm \nK\\_{m-1}({{\\bf R}-\\bm \\eta})\n(V\\_s({\\bm \\eta})+V\\_z({\\bm \\eta})) \\right].\n\\label{tildev}\\end{aligned}$$ \nWe emphasize that formula ([tildev]) is non-perturbative in *l**B* and possibly applies for potentials *V**s* and *V**z* with sizeable variations at the scale of *l**B*. The effective potential in the lowest Landau level is also readily obtained as:\n \n$$\\begin{aligned}\n\\tilde{v}\\_{0}({\\bf R}) &=& \n\\int d^2 {\\bm \\eta} \nK\\_0({\\bf R}-{\\bm \\eta})\n\\left[ V\\_s({\\bm \\eta})-V\\_z({\\bm \\eta}) \\right]. \n\\label{tildev0}\\end{aligned}$$ \nObviously, we find that the modified Green’s function becomes also diagonal with respect to the Landau-level quantum number at large magnetic field (yet at finite magnetic field),\n \n$$\\begin{aligned}\n\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})= \\delta\\_{m\\_{1},m\\_{2}}\n\\, \\tilde{g}\\_{m\\_{1};\\lambda\\_{1};\\lambda\\_{2}}({\\bf R})\n,\\end{aligned}$$ \nand is determined for *m* ≥ 1 by Dyson equation,\n \n$$\\begin{aligned}\n\\lefteqn{(\\omega-E\\_{m,\\lambda\\_{1}}+ i 0^{+})\n\\tilde{g}\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R})\n=\n \\delta\\_{\\lambda\\_{1},\\lambda\\_{2}}\n \\nonumber} \\\\\n&&\n+ \\tilde{v}\\_{m}^{+}({\\bf R}) \\star \\tilde{g}\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R})\n+\n\\tilde{v}\\_{m}^{-}({\\bf R})\n\\star\n\\tilde{g}\\_{m;-\\lambda\\_{1};\\lambda\\_{2}} ({\\bf R}) \\hspace\\*{0.5cm}\n\\label{Dysondiag}\\end{aligned}$$ \nand for *m* = 0 by\n \n$$\\begin{aligned}\n(\\omega+ i 0^{+})\n\\tilde{g}\\_{0}({\\bf R})\n&=&\n1\n+\n\\tilde{v}\\_{0}({\\bf R})\n\\star\n\\tilde{g}\\_{0} ({\\bf R}).\n\\label{Dyson0}\\end{aligned}$$ \nThe other Dyson Eq. ([Dysonfinalbis]) generates the different equation,\n \n$$\\begin{aligned}\n\\lefteqn{(\\omega-E\\_{m,\\lambda\\_{2}}+ i 0^{+})\n\\tilde{g}\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R})\n=\n \\delta\\_{\\lambda\\_{1},\\lambda\\_{2}}}\n \\nonumber \\\\&&\n+ \\tilde{g}\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R}) \\star \\tilde{v}\\_{m}^{+} ({\\bf R})\n+\n\\tilde{g}\\_{m;\\lambda\\_{1};-\\lambda\\_{2}} ({\\bf R})\n\\star \\tilde{v}\\_{m}^{-}({\\bf R}) \\hspace\\*{0.5cm}\n\\label{Dysondiagbis}\\end{aligned}$$ \nfor *m* ≥ 1, and\n \n$$\\begin{aligned}\n(\\omega+ i 0^{+})\n\\tilde{g}\\_{0}({\\bf R})\n&=&\n1\n+\n\\tilde{g}\\_{0}({\\bf R})\n\\star\n\\tilde{v}\\_{0} ({\\bf R})\n\\label{Dyson0bis}\\end{aligned}$$ \nfor the lowest Landau level *m* = 0.\nLocally flat potentials\n-----------------------\nNow, we aim at solving Eqs. ([Dysondiag])-([Dyson0bis]) at leading order, which is vindicated when the potential is *locally* flat, i.e., when potential curvature is small. This calculation includes the case of one-dimensional potentials (i.e., *globally* flat potentials), for which the solution presented below is exact. Indeed, as is clear from its explicit expression ([star]), the ⋆ -bidifferential operator involves derivatives in two orthogonal positions. In case where the potentials $V\\_{s}({\\bf R})$ and $V\\_{z}({\\bf R})$ are purely one-dimensional potentials depending on the same coordinate, the function $\\tilde{g}({\\bf R})$ will also only depend on the same and unique variable, so that the ⋆ product between the functions *ṽ* and *g̃* reduces to the standard product of functions. In case of arbitrary spatial varying two-dimensional potentials, this constitutes a good approximation as long as temperature is higher than the energy scales associated to local curvature terms, see Sec. [hier] for a general discussion. Dyson equation then is trivially solved, as the system of differential Eqs. ([Dysondiag])-([Dyson0bis]) transforms into a system of purely algebraic equations. Taking the difference of Eqs. ([Dysondiag]) and ([Dysondiagbis]), we get for *m* ≥ 1 the relations between the different components of *g̃*,\n \n$$\\begin{aligned}\n\\tilde{g}\\_{m;+;-}({\\bf R})&=&\\tilde{g}\\_{m;-;+}({\\bf R}) \\label{R1} \\\\\n\\tilde{g}\\_{m;-;-}({\\bf R})&=&\\tilde{g}\\_{m;+;+}({\\bf R})-2 \\frac{E\\_{m;+}}\n{\\tilde{v}\\_{m}^{-}({\\bf R})} \\, \\tilde{g}\\_{m;-;+} ({\\bf R}). \\hspace\\*{0.5cm} \\label{R2}\\end{aligned}$$ \nAfter simple algebra, we directly obtain the solution\n \n$$\\begin{aligned}\n\\!\\tilde{g}\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R}) &=&\n\\!\\frac{1}{ \\left[\n\\omega-\\xi\\_{m,+}({\\bf R})+i0^{+}\n\\right]\n\\left[\n\\omega-\\xi\\_{m,-}({\\bf R})+i0^{+}\n\\right]\n}\n\\nonumber \\\\\n&& \\hspace{-2cm} \\times \\left\\{ \\left[\\omega-\\tilde{v}\\_{m}^{+}({\\bf R})+E\\_{m,\\lambda\\_{1}} \\right]\n\\delta\\_{\\lambda\\_{1},\\lambda\\_{2}}\n+\\tilde{v}\\_{m}^{-}({\\bf R}) \\delta\\_{\\lambda\\_{1},-\\lambda\\_{2}} \\right\\}\n\\label{g1Dm}\\end{aligned}$$ \nwith the poles (corresponding to the renormalized Landau levels) giving the effective energies,\n \n$$\\begin{aligned}\n\\xi\\_{m,\\pm}({\\bf R})=\n\\tilde{v}\\_{m}^{+}({\\bf R})\\pm \\sqrt{E\\_{m}^{2}+\\left[\\tilde{v}\\_{m}^{-}({\\bf R}) \\right]^{2}}.\n\\label{effective}\\end{aligned}$$ \nFor *m* = 0, the Green’s function is characterized by a single pole and reads\n \n$$\\begin{aligned}\n\\tilde{g}\\_{0}({\\bf R})&=& \\frac{1}\n{\n\\omega-\\xi\\_{0}({\\bf R})+i0^{+}\n}\n, \\label{g1D0}\\end{aligned}$$ \nwhere $\\xi\\_{0}({\\bf R})=\\tilde{v}\\_{0}({\\bf R})$.\nEquation ([effective]), with the explicit expression for the renormalized potentials given in Eqs. ([tildev]), provides the leading result for the local Landau-level energy in arbitrary potentials of diagonal type (i.e., scalar or mass-like). This expression of course includes the case of a purely unidimensional (i.e., globally flat) potential as an exact particular solution, but is a very good approximation for smooth disordered potentials, which can be used to analyze experimental STS results, as we discuss in Sec. [sec:local].\nLocally curved potentials\n-------------------------\nThis section presents the resolution of Dyson equation at next to leading order, by extending the above calculation of the local vortex Green’s function to the incorporation of the effects of geometrical curvature in the potential landscape. It has therefore a two-fold purpose. First, it provides a crucial refinement of the previous expression ([g1Dm]), that includes important quantum effects such as quantization of energy levels or tunneling associated to the potentials *V**s* and *V**z*, which are clearly missed in the leading order guiding center Green’s function. Smaller energy scales associated to these physical processes are now accessible, and the final expression will apply to arbitrary smooth potentials that are *locally* curved. These important aspects are discussed in more detail in Sec. [hier]. Second, in the special case of purely quadratic potentials (which thus have a *global* constant curvature), the calculation provides essentially the exact Green’s function, from which one can gain interesting insights on the physics of confinement or tunneling in graphene. We thus obtain analytically the quantization spectra of parabolic quantum dots and show that the structure of energy levels qualitatively depends on the type of confinement (electrostatic or mass type). We henceforth assume that the diagonal potentials are *locally* well described up to their second-order spatial derivatives.\n### Lowest Landau level: Solution with both curved scalar and mass potentials\nWe start by considering the lowest Landau level *m* = 0, which is the simplest case to solve, as band indices are not involved. In that situation both locally curved $V\\_{s}({\\bf R})$ and $V\\_{z}({\\bf R})$ can be solved altogether (this is not the case for higher *m* ≥ 1 states, as will be discussed in the next paragraphs). Actually, Dyson Eq. ([Dyson0]) for the lowest Landau level is formally equivalent to the equations obtained for the 2DEG, as the electrostatic potential $V({\\bf R})$ for the 2DEG is just formally replaced by the combination $V\\_{s}({\\bf R})-V\\_{z}({\\bf R})$ for graphene. Working in the next to leading order, i.e., keeping local curvature terms of order *l**B*4 in the ⋆ -bidifferential operator, Eq. ([star]), we can directly transpose the solution of Ref. to the graphene case (for the method, see also Appendix [appA] of the present paper), which reads\n \n$$\\begin{aligned}\n\\tilde{g}\\_{0}({\\bf R})\n= - i \\int\\_{0}^{+ \\infty} \\!\\!\\! \\!\\!\\! dt \\frac{\ne^{i [\\eta\\_0({\\bf R})/\\gamma\\_0({\\bf R})] \\left[t - \\tau\\_0(t) \\right]}\n }{\\cos (\\sqrt{\\gamma\\_0({\\bf R}) }t)}\ne^{i t [\\omega-\\xi\\_{0}({\\bf R})+i0^{+}]}\n\\nonumber\n\\\\\n\\label{g0}\\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n\\tau\\_0(t) &=& \\frac{1}{\\sqrt{\\gamma\\_0({\\bf R})}} \\tan(\\sqrt{\\gamma\\_0({\\bf R})}t).\n\\label{tau}\\end{aligned}$$ \nThe parameters $\\gamma\\_0({\\bf R})$ and $\\eta\\_0({\\bf R})$ in Eqs. ([g0]) and ([tau]) are geometric coefficients characterizing the local effective potential landscape $\\tilde{v}\\_0({\\bf R})$ in the lowest Landau level:\n \n$$\\begin{aligned}\n\\gamma\\_0({\\bf R}) &=& \\frac{l\\_{B}^{4}}{4}\n\\left[\n(\\partial\\_{X}^{2}\\tilde{v}\\_0)(\\partial\\_{Y}^{2}\\tilde{v}\\_0)-(\\partial\\_{X}\\partial\\_{Y}\n\\tilde{v}\\_0 )^{2}\n\\right]\\_{{\\bf R}},\n\\label{gamma} \\\\\n\\eta\\_0({\\bf R}) &=& \\frac{l\\_{B}^{4}}{8}\n\\left[\n(\\partial\\_{X}^{2}\\tilde{v}\\_0)(\\partial\\_{Y}\\tilde{v}\\_0)^{2}\n+(\\partial\\_{Y}^{2}\\tilde{v}\\_0)(\\partial\\_{X}\\tilde{v}\\_0)^{2} \\right. \\nonumber \\\\\n&& \\left. \\hspace\\*{0.7cm} -2(\\partial\\_{X}\\partial\\_{Y} \\tilde{v}\\_0 )(\\partial\\_{X}\n\\tilde{v}\\_0)(\\partial\\_{Y} \\tilde{v}\\_0) \\right]\\_{{\\bf R}}.\n\\label{eta}\\end{aligned}$$ \nThe coefficient $\\gamma\\_0({\\bf R})$ is directly proportional to the Gaussian curvature of the surface defined in the three-dimensional “space” *X**Y**Z* by the equation *Z* = *ṽ*0(*X*, *Y*). Its sign reflects the local topology of the effective potential: $\\gamma\\_0({\\bf R}) >0$ indicates a locally elliptic potential with the presence of a local extremum (maximum or minimum), while $\\gamma\\_0({\\bf R}) <0$ corresponds to a locally hyperbolic (or saddle-shaped) potential. At the borders between the regions with curvatures with opposite signs, the potential is locally parabolic (the lines where the Gaussian curvature is zero are consequently called parabolic lines). For a complex disordered effective potential landscape, one expects that surface regions with positive and negative Gaussian curvature alternate. Note that both cosine and tangent trigonometric functions in Eqs. ([g0]) and ([tau]) transform into their hyperbolic counterparts in the case $\\gamma\\_0({\\bf R}) <0$. Equation ([g0]) thus provides a general approximation scheme in the lowest Landau level in the presence of arbitrary scalar and mass potentials that are locally well described by local curvature coefficients, Eqs. ([gamma]) and ([eta]).\nNow, in the *particular* case of purely quadratic scalar and mass potentials, i.e., $V\\_s({\\bf R})-V\\_z({\\bf R})=V\\_{s0}-V\\_{z0}\n+ \\frac{1}{2} \\left[({\\bf R}-{\\bf R}\\_{0}) \\cdot {\\bm \\nabla}\\_{{\\bf R}}\n\\right]^{2}(V\\_s-V\\_z) $, with ${\\bf R}\\_{0}$ chosen as the single point where the potential gradient vanishes, expression ([g0]) yields the *exact* Green’s function of the problem. In that situation, the parameter *γ*0 in Eq. ([gamma]) becomes ${\\bf R}$-independent,\n \n$$\\gamma\\_0 = \\frac{l\\_{B}^{4}}{4} \\left\\{\n\\partial\\_{X}^{2}(V\\_s-V\\_z)\\partial\\_{Y}^{2}(V\\_s-V\\_z)\n-\\left[\\partial\\_{X}\\partial\\_Y (V\\_s-V\\_z)\\right]^{2}\n\\right\\},$$ \nand describes the uniform (global) curvature of the potential, while the ${\\bf R}$-independent part of the effective potential results from the simple relation $ \\tilde{v}\\_0 \\equiv \\tilde{v}\\_{0}({\\bf R}\\_{0}) = \n\\tilde{v}\\_0({\\bf R})-\\eta\\_0({\\bf R})/\\gamma\\_0=\nV\\_{s0}-V\\_{z0} +(l\\_B^2/2)\\Delta\\_{\\bf R}(V\\_s-V\\_z)$. For a confining potential, i.e., when *γ*0 > 0, *τ*0(*t*) is a $2\\pi/\\sqrt{\\gamma\\_0}$ periodic function of time *t*. Direct Fourier analysis of expression ([g0]) using the above relations can be done and shows (see Appendix [appC]) that the entire energy spectrum necessarily decomposes onto discrete modes:\n \n$$\\begin{aligned}\n\\label{spectrumg0}\nE\\_{0,n} &=& \\tilde{v}\\_0 + \\mathrm{sgn}(\\eta\\_0) \\sqrt{\\gamma\\_0} (2n+1)\\end{aligned}$$ \nwith *n* ≥ 0 a positive integer, yielding a harmonic-oscillatorlike spectrum for the parabolic quantum-dot model (in the large magnetic field regime considered here). The general form of this spectrum will be discussed in the next section. In contrast, for *γ*0 < 0, the vortex Green’s function expressed in the time representation is no more periodic but decays on a time scale $1/\\sqrt{-\\gamma\\_0}$, due to the cutoff function $1/\\cosh(\\sqrt{-\\gamma\\_0}t)$. These lifetime effects associated to negative Gaussian curvature are clear manifestations of quantum tunneling in saddle-point potentials, and will be considered in a future publication where transport properties in high magnetic field will be considered.\n### Arbitrary Landau level: Solution for a curved scalar potential combined with a flat mass potential\nFor *m* ≥ 1, the structure of Dyson Eq. ([Dysondiag]) for graphene differs from that for the 2DEG case because of the possible coupling between positive- and negative-energy bands. Two kinds of processes are actually at work here. First, non-zero mass potential *V**z* directly couples the two bands, as is clearly seen from the leading order Green’s function in Eq. ([g1Dm]). Second and less obviously, higher-order *scalar* processes can also induce band-mixing. Indeed, the effective off-diagonal potential in Eq. ([vplus]) reads in the small *l**B*-expansion: $\\tilde{v}\\_{m}^{-}({\\bf R}) = -\nV\\_{z}({\\bf R})-m \\frac{l\\_{B}^{2}}{2} \\Delta\\_{{\\bf R}}\nV\\_{z}({\\bf R})+\\frac{l\\_{B}^{2}}{4} \\Delta\\_{{\\bf R}} V\\_{s}({\\bf\nR})+\\mathcal{O}(l\\_B^4)$. Thus, even for an identically zero mass term (*V**z* = 0), positive-and negative-energy bands are necessarily coupled by the second derivatives of the scalar potential.\nFor reasons mentioned previously, one cannot analytically progress for the *m* ≥ 1 Landau levels in case where both scalar and mass potentials are strongly spatially dependent. In this section we therefore assume that the scalar potential varies in space with sizeable local parabolic dispersion, while the mass potential has much smoother spatial variations, so that local derivatives of the mass term are associated to tiny energy scales (the reversed situation, where the mass potential variations dominate the ones of the scalar potential, is considered below in Sec. [parabolicmass]). Since the calculation leading to the Green’s function for graphene is largely inspired from the 2DEG’s derivation, details are produced in Appendix [appA]. The solution reads\n \n$$\\begin{aligned}\n\\nonumber\n\\tilde{g}\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R})\n&=&\\!\\! -\\frac{i}{2}\n\\int\\_{0}^{+ \\infty} \\!\\!\\!\\! \\! dt\n \\frac{\ne^{i(\\eta\\_m^+({\\bf R})/\\gamma\\_m^+({\\bf R}))[t-\\tau\\_m^+(t)]}}\n{ \\cos\\Big(\\sqrt{\\gamma\\_m^+({\\bf R})} t \\Big)} \\\\\n\\nonumber\n&& \\times \\sum\\_{\\epsilon=\\pm}\ne^{i t (\\omega -\\xi\\_{m,\\epsilon}({\\bf R})+i0^{+})} \\\\\n&& \\hspace{-2cm} \\times [ (1+\\epsilon \\lambda\\_{1} \\alpha\\_{m}({\\bf R}))\n\\delta\\_{\\lambda\\_{1},\\lambda\\_{2}}\n+ \\epsilon \\beta\\_{m}({\\bf R}) \\delta\\_{-\\lambda\\_{1},\\lambda\\_{2}} ]\n\\label{solution}\\end{aligned}$$ \nwhere the effective energy $\\xi\\_{m,\\pm}({\\bf R})$ is given by Eq. ([effective]), and\n \n$$\\begin{aligned}\n\\alpha\\_{m}({\\bf R}) &=& \\frac{E\\_{m}}{\\sqrt{E\\_{m}^{2}+\\left[\\tilde{v}\\_{m}^{-}({\\bf R})\n\\right]^{2}}}, \\label{alpha} \\\\\n\\beta\\_{m}({\\bf R}) &=& \\frac{\\tilde{v}\\_{m}^{-}({\\bf R})}{\\sqrt{E\\_{m}^{2}+\n\\left[\\tilde{v}\\_{m}^{-}({\\bf R}) \\right]^{2}}}.\n\\label{beta}\\end{aligned}$$ \nThe geometric parameters $\\gamma\\_m^+({\\bf R})$ and $\\eta\\_m^+({\\bf R})$ have the same definitions as in Eqs. ([gamma]) and ([eta]), where $\\tilde{v}\\_0({\\bf R})$ is simply replaced by the effective potential $\\tilde{v}\\_m^+({\\bf R})$. The function *τ**m*+(*t*) has also a similar definition as in Eq. ([tau]) now in terms of $\\gamma\\_m^+({\\bf R})$. Again, the above expression ([solution]) is quite general, and can be used to describe arbitrary disordered (yet smooth) scalar potentials. A mass contribution may be present, but only with negligible spatial variations for the approximation to be valid.\nNow, in the *particular* case where the bare scalar potential is globally quadratic (i.e., has uniform curvature) and the mass potential is globally uniform, this expression provides the *exact* Green’s function. A possible parametrization of such potentials reads $V\\_s({\\bf R})=V\\_{s0}\n+ \\frac{1}{2} \\left[({\\bf R}-{\\bf R}\\_{0}) \\cdot {\\bm \\nabla}\\_{{\\bf R}}\n\\right]^{2}V\\_s $ (with ${\\bf R}\\_{0}$ chosen as the point where the scalar potential gradient vanishes) and $V\\_z({\\bf R}) = V\\_{z0}$. The Gaussian curvature of the scalar potential becomes then constant and independent of *m*,\n \n$$\\label{gammaplus}\n\\gamma^+ = \\frac{l\\_{B}^{4}}{4} \\left[\n\\partial\\_{X}^{2}V\\_s\\partial\\_{Y}^{2}V\\_s\n-(\\partial\\_{X}\\partial\\_Y V\\_s)^{2}\n\\right]$$ \nwhile the ${\\bf R}$-independent parts of the effective potentials read $\\tilde{v}\\_{m}^{+} \\equiv \\tilde{v}\\_{m}^{+}({\\bf R}\\_{0})=V\\_{s0}+m (l\\_{B}^{2}/2) \n\\Delta\\_{{\\bf R}} V\\_{s}$ and $\\tilde{v}\\_m^{-} = - V\\_{z0}+(l\\_{B}^{2}/4) \\Delta\\_{{\\bf R}} V\\_{s}$. Fourier analysis as done in Appendix [appC] provides a spectrum of purely discrete energy levels in the presence of 2D-parabolic scalar potential,\n \n$$E\\_{m,n} = \\tilde{v}\\_{m}^{+} \\pm \\sqrt{E\\_m^2+(\\tilde{v}\\_m^-)^2} +\n\\mathrm{sgn}(\\eta^+)\\sqrt{\\gamma^+} (2n+1).\n\\label{energy1}$$ \nThis form of quantization is quite reminiscent of the Fock-Darwin spectrum for the non-relativistic 2DEG: besides the renormalization of Landau levels (labeled by the integer *m*) due to the ${\\bf R}$-independent part of the effective potentials *ṽ**m*+ and *ṽ**m*−, the linear dependence in the second discrete number *n* provides an additional harmonic-oscillatorlike contribution. As a specific illustration for the case of a circular parabolic scalar potential $V\\_s({\\bf r}) = V\\_{s0} + (1/2) U\\_0 (x^2+y^2)$ together with a zero mass term, one gets the following energy spectrum:\n \n$$E\\_{m,n} = V\\_{s0}+ l\\_B^2 U\\_0(m+n+\\frac{1}{2}) \\pm \n\\sqrt{(\\hbar \\Omega\\_c \\sqrt{m})^2 +(l\\_B^2 U\\_0/2)^2},$$ \nthat we have already quoted in Eq. ([spectrum1]) in the large Ω*c* limit.\n### Arbitrary Landau level: Solution for a flat scalar potential combined with a curved mass potential\nWe now consider the alternative solvable case of locally flat scalar potential, together with a spatially dependent mass potential that can be locally well described by a quadratic expansion. Solution of Dyson Eq. ([Dysondiag]) can then similarly be achieved, leading to the Green’s function for *m* ≥ 1,\n \n$$\\begin{aligned}\n\\nonumber\n\\tilde{g}\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R}) &=& -i\n\\int\\_{0}^{+ \\infty} \\!\\!\\! \\! ds \\frac{e^{i [\\kappa\\_{m}({\\bf R})+i0^+]s}}\n{ \\cos \\left(\\sqrt{\\gamma\\_m^-({\\bf R})}s\\right)}\n\\Big\\{ \\cos[\\theta\\_m(s)] \\\\\n&&\\hspace{-2.3cm}\n\\times \\frac{\\omega-\\tilde{v}^{+}\\_{m}({\\bf R})+\\lambda\\_{1}E\\_{m}}{\\kappa\\_m({\\bf R})}\n\\delta\\_{\\lambda\\_{1},\\lambda\\_{2}} +i \\sin[\\theta\\_m(s)]\\delta\\_{\\lambda\\_{1},-\\lambda\\_{2}} \\Big\\}\n \\label{gVzquadra}\\end{aligned}$$ \nwith $\\theta\\_m(s) = [\\eta\\_m^-({\\bf R})/\\gamma\\_m^-({\\bf R})][\\tau\\_m^-(s)-s ]\n- s \\tilde{v}^{-}\\_{m}({\\bf R})$ and\n \n$$\\begin{aligned}\n\\kappa\\_{m}({\\bf R})=\\mathrm{sgn}\\left[\\omega-\\tilde{v}^{+}\\_{m}({\\bf R})\\right] \\,\n\\left|\\left(\\omega-\\tilde{v}^{+}\\_{m}({\\bf R})\\right)^{2}-E\\_{m}^{2} \\right|^{1/2}\n\\label{kappa1}\\end{aligned}$$ \nif $|\\omega-\\tilde{v}^{+}\\_{m}({\\bf R})| \\geq E\\_{m}$, and\n \n$$\\begin{aligned}\n\\kappa\\_{m}({\\bf R})=i\\left|\\left[\\omega-\\tilde{v}^{+}\\_{m}({\\bf R})\\right]^{2}-E\\_{m}^{2}\n\\right|^{1/2}\n\\label{kappa2}\\end{aligned}$$ \nif $|\\omega-\\tilde{v}^{+}\\_{m}({\\bf R})| < E\\_{m}$. Details for the derivation of result ([gVzquadra]) can be found in Appendix [appB]. The geometric parameters $\\gamma\\_m^-({\\bf R})$ and $\\eta\\_m^-({\\bf R})$ in formula ([gVzquadra]) are defined as in Eqs. ([gamma]) and ([eta]) with $\\tilde{v}\\_0({\\bf R})$ replaced by $-\\tilde{v}\\_m^-({\\bf R})$. Taking the imaginary part of expression ([gVzquadra]), we get that the local density of states vanishes when $|\\omega-\\tilde{v}^{+}\\_{m}({\\bf R})| < E\\_{m}$, meaning that there are no states within this energy interval.\nNow, in the *particular* case where the bare scalar potential is globally uniform and the mass potential is globally quadratic, this expression provides the *exact* Green’s function in the absence of Landau-level mixing. A possible parametrization of such potentials reads $V\\_s({\\bf R})=V\\_{s0}$ and $V\\_z({\\bf R}) = V\\_{z0} + \\frac{1}{2} \\left[({\\bf R}-{\\bf R}\\_{0}) \\cdot\n{\\bm \\nabla}\\_{{\\bf R}} \\right]^{2}V\\_z $ (with ${\\bf R}\\_{0}$ chosen as the point where the mass potential gradient vanishes). The Gaussian curvature of the mass potential becomes then constant,\n \n$$\\gamma^- = \\frac{l\\_{B}^{4}}{4} \\left[\n\\partial\\_{X}^{2}V\\_z\\partial\\_{Y}^{2}V\\_z\n-(\\partial\\_{X}\\partial\\_Y V\\_z)^{2}\n\\right]$$ \nwhile the ${\\bf R}$-independent parts of the effective potentials read $\\tilde{v}\\_{m}^{+} = V\\_{s0}- (l\\_{B}^{2}/4) \\Delta\\_{{\\bf R}} V\\_{z}$ and $\\tilde{v}\\_m^{-} \\equiv \\tilde{v}\\_m^{-}({\\bf R}\\_{0})= - V\\_{z0}+m(l\\_{B}^{2}/2) \\Delta\\_{{\\bf R}} V\\_{z}$. Fourier analysis as done before implies that the eigenenergies *ω* = *E**m*, *n* are determined by the implicit equation $\\kappa\\_m-\\tilde{v}\\_m^- =\n\\mathrm{sgn}(\\eta^{-})\\sqrt{\\gamma^-}(2n+1) $ [we remind that the dependence on *ω* is contained in *κ**m*, see Eq. [kappa1]], leading to the following discrete energy level spectrum in the presence of a parabolic mass potential,\n \n$$E\\_{m,n} = \\tilde{v}\\_{m}^{+} \\pm \\sqrt{E\\_m^2+\\left[\\tilde{v}\\_m^-\n+\\mathrm{sgn}(\\eta^-)\\sqrt{\\gamma^{-}}(2n+1)\\right]^2}.\n\\label{energy2}$$ \nThe energy dependence with respect to the second discrete number *n* is now quite different from the previous Fock-Darwin-type spectrum in a scalar 2D-parabolic potential, Eq. ([energy1]). As a specific illustration for the case of a circular parabolic mass potential $V\\_z({\\bf r}) = (1/2) U\\_0 (x^2+y^2)$ together with a zero scalar term *V**s* = 0, the discrete energy levels are clearly anharmonic with respect to *n*,\n \n$$E\\_{m,n} = -\\frac{l\\_B^2}{2} U\\_0 \\pm \\sqrt{(\\hbar \\Omega\\_c \\sqrt{m})^2\n+ [l\\_B^2 U\\_0 \\left(m+n+1/2\\right)]^2},$$ \nan expression which was already quoted in Eq. ([spectrum2]).\nDiscussion for arbitrary smooth potentials: A hierarchy of local energy scales\n------------------------------------------------------------------------------\nIt is worth emphasizing that for arbitrary two-dimensional potentials $V\\_{s}({\\bf R})$ and $V\\_{z}({\\bf R})$ that are smooth at the scale of the magnetic length *l**B*, the present vortex formalism turns out to be extremely useful because it explicitly puts forward the existence of a hierarchy of local energy scales. Such a hierarchy can then be exploited to devise successive approximation schemes, leading to controlled expressions for all physical observables at finite temperature. This has already been proved with the concrete example of the temperature-broadened STS local density of states for the 2DEG (see Sec. IV of Ref. ), and the same mechanism holds also in the case of graphene studied here.\nTo understand qualitatively the origin of this hierarchy of local energy scales, it is useful to rewrite the ⋆ -bidifferential operator, Eq. ([star]), under the equivalent form,\n \n$$\\begin{aligned}\n\\star &=& \\sum\\_{p=0}^{+ \\infty} \\frac{1}{p!}\\left(i\\frac{l\\_{B}^{2}}{2} \\right)^{p}\n\\hat{C}^{p} \\label{op1}\\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n\\hat{C}=\\left(\\overleftarrow{\\partial}\\_{X}\\overrightarrow{\\partial}\\_{Y}\n-\\overleftarrow{\\partial}\\_{Y}\\overrightarrow{\\partial}\\_{X} \\right).\\end{aligned}$$ \nThe arbitrary large number of derivatives in expression ([op1]) is clearly an indication of the nonlocal nature of quantum mechanics. However, and remarkably here, we realize that nonlocality manifests itself through *quasilocality* in the vortex representation. This is due to the fact that the nonlocal electronic Green’s function $\\hat{G}({\\bf r},{\\bf r}')$ can entirely be determined from the knowledge of the *local* vortex function $\\tilde{g}\\_{m}({\\bf R})$, see connection formula ([passage2]). This quasilocality property (which holds independently of the form of the potential landscape and thus can be seen as resulting uniquely from the coherent character of the vortex states), allows one to have a quasilocal quantization view. Clearly, the local Green’s function $\\tilde{g}\\_{m}({\\bf R})$ depends on the potential matrix elements $\\tilde{v}\\_{m}({\\bf R})$ via the action of the ⋆ product, see Eqs. ([Dysondiag])-([Dyson0bis]). As obvious from expression ([op1]), each power of the bidifferential operator *Ĉ* acting on the functions $\\tilde{v}\\_{m}({\\bf R})$ and $\\tilde{g}\\_{m}({\\bf R})$ generates higher and higher derivatives $l\\_{B}^{p} \\partial\\_{{\\bf R}}^{p}\n\\tilde{g}\\_{m}({\\bf R})$ of the local Green’s function associated with hierarchy of energy scales of the type $l\\_{B}^{p} \\partial\\_{{\\bf R}}^{p}\n\\tilde{v}\\_{m}({\\bf R})$. These energy scales get smaller and smaller at increasing *p* in the case of a potential smooth at the magnetic length scale, allowing one to control systematically the calculation.\nFor instance, leading order expressions ([g1D0]) and ([g1Dm]) for the vortex Green’s function were derived assuming that one can neglect potential curvature terms (associated to the geometric invariants involving second-order spatial derivatives of the potential). This type of approximation is in fact *controlled* as long as temperature exceeds the local energy scales appearing at next to leading order, respectively $[\\gamma({\\bf R})]^{1/2}$ of Eq. ([gamma]) and $[\\eta({\\bf R})]^{1/3}$ of Eq. ([eta]). In that case, quantum effects such as quantization and tunneling are certainly missing, yet this basic approximation already encodes the structure of the delocalized edge states far from the regions where the potential is strongly curved.\nWe have seen in Sec. [curved] that it is possible to go one step further by including the curvature contributions [term *p* = 2 in Eq. ([op1])], and this reintroduces quantization and tunneling in case of confined or open potentials, respectively. Again, one expects that the refined expressions obtained for the vortex Green’s function [Eqs. ([g0]), ([solution]), and ([gVzquadra]) depending on the dominant type of scatterers] encode correctly the quantum dynamics down to further and even smaller energy scales associated to geometrical invariants involving third order spatial derivatives of the potential.\nThese considerations show the existence of a hierarchy of local energy scales formed by the successive spatial derivatives of the potential and hint that the passage from purely local physics (which is the hallmark of classical mechanics) to highly nonlocal quantum-mechanical physics (which is the apanage of highly unstable quantum states) is worked out *gradually* when the temperature is progressively decreased. Therefore, at least in the large magnetic field regime, it is not needed to diagonalize numerically the random Schrödinger or Dirac equation in order to calculate precisely physical quantities, since temperature down to the Kelvin range in real experiments is not likely to be very small compared to the tiny energy scales at order *l**B*2 (for smooth potentials). What is neglected in our approximation scheme are contributions of some highly non-local quantum states superpositions, which are irrelevant in realistic experiments at *finite* temperature.\nConnection with the deformation quantization theory\n===================================================\nDeformation quantization theory in classical phase space\n--------------------------------------------------------\nBefore exploiting the expressions for the Green’s functions derived in Sec. [high], we would like to make important comments on the structure of the dynamical equations obeyed by the Green’s functions $\\tilde{g}\\_{m}({\\bf R})$ [general Dyson Eq. ([Dysonfinal]) at any magnetic field or Eq. ([Dysondiag]) in the absence of Landau-level mixing at high magnetic field]. After completion of Ref., we have indeed realized that the ⋆ -bidifferential operator, Eq. ([star]), involved in these latter equations has a form analogous to the so-called star product, which has been the subject of intense research in mathematical and in high-energy physics because of its fundamental role in the principles themselves of quantum mechanics. More precisely, there have been many attempts to formulate quantum mechanics from a classical point of view, i.e., as a theory of functions on phase space, and one suggestion was to understand quantization as a deformation of the structure of (Poisson-Lie) algebra of classical observables. The ℏ-deformation theory of the classical mechanics relies on the introduction of a star product,\n \n$$\\begin{aligned}\n\\star\\_{\\hbar} =\\exp\\left[i \\frac{\\hbar}{2} \\left( \\overleftarrow{\\partial}\\_{x}\n\\overrightarrow{ \\partial}\\_{p\\_{x}}\n- \\overleftarrow{\\partial}\\_{p\\_{x}} \\overrightarrow{ \\partial}\\_{x} \\right) \\right]\n\\label{star2}\\end{aligned}$$ \nin place of the usual product between phase-space functions. Here, *x* and *p**x* are, respectively, the position and momentum which are canonically conjugate variables. We discuss first here the quantization for a particle in one dimension in the absence of a magnetic field (in two dimensions, classical phase space is four dimensional, see discussion in Sec. [mixed]). As a key principle, the entire quantum dynamics is encapsulated in the noncommutative operator, Eq. ([star2]), which turns out to be the unique associative pseudodifferential deformation of the ordinary product. Within the deformation quantization theory, the Poisson brackets of classical mechanics between two phase-space functions *f*(*x*, *p**x*) and *g*(*x*, *p**x*) are replaced by the Moyal brackets defined as commutators (in the star-product sense) [*f*, *g*]*M* = (*f* ⋆ ℏ*g* − *g* ⋆ ℏ*f*)/*i*ℏ. Obviously, Moyal brackets are ℏ-dependent brackets which reduce smoothly to the Poisson brackets in the limit ℏ → 0 (hence the origin of the “deformation” picture).\nThe deformation quantization approach appears as a generalization of original ideas put forward by Weyl, Wigner and Moyal (for a short historical account, see paper and references therein), which were aimed at getting a sound insight into the correspondence principle between classical and quantum mechanics. The deformation quantization formulation has acquired a clearer mathematical status 30 years ago with the work of Bayen *et al.*, where its autonomous and alternative character with respect to other formulations of quantum mechanics, such as the conventional Hilbert space and path integral formulations, has been proved (for the recent status of the theory, see Refs. and ). Because the basic *continuous* structure of the classical phase space is conceptually kept in the deformation quantization theory, classical mechanics is easily and transparently recovered via a smooth transformation, in full contrast to the conventional operatorial approach of quantum mechanics formulated in a Hilbert space (spanned by a *countable* basis of square integrable states) where the emergence of a classical character from the quantum substrate appears singular and rather challenging. For this reason, it has been underlined that the deformation view is presumably the right way to look at quantization.\nVortex Green’s functions as a mixed phase-space formulation of quantum mechanics\n--------------------------------------------------------------------------------\nNow considering explicitly two-dimensional electronic quantum dynamics in the ordinary 2DEG, the standard deformation quantization theory introduces electronic coordinates (*x*, *y*) and momenta (*p**x*, *p**y*) as natural variables in a four-dimensional phase space. In a large magnetic field however, the electronic classical dynamics consists of a fast cyclotron motion, which is centered around a slowly moving guiding center ${\\bf R}=(X,Y)$. In the popular operatorial language of quantum mechanics, these two relevant degrees of freedom are introduced by decomposing the electronic coordinate operator $\\hat{{\\bf r} }=\\hat{{\\bf R}} +\\hat{{\\bm \\eta}}$ into a relative position $\\hat{{\\bm \\eta}}$ linked to cyclotron orbits and a guiding center position $\\hat{{\\bf R}} =(\\hat{X},\\hat{Y})$. It is well known that the guiding center coordinate operators obey the commutation rule [*X̂*, *Ŷ*] = *i**l**B*2, showing analogy with the canonical quantization rule between the position *x̂* and the conjugate momentum *p̂**x*. Therefore, the square of the magnetic length, *l**B*2, plays the role of an effective magnetic field-dependent Planck’s constant. Moreover, cyclotron motion associated to the relative circular orbits $\\hat{{\\bm \\eta}}$ leads to quantized Landau levels and at very large magnetic fields completely decouples from the guiding center dynamics.\nThis physical discussion shows that the canonical description of phase space in terms of electronic coordinates (*x*, *y*) and momenta (*p**x*, *p**y*) becomes awkward in a magnetic field. Quantum mechanically, this is reflected by the property that states that are coherent with respect to both positions and momenta cannot be eigenstates of the kinetic part of the Hamiltonian associated to cyclotron motion, contrary to the vortex states. With the benefit of hindsight, the program that we have followed in the string of recent papers is precisely the formulation of deformation quantization in a *mixed* phase space associated with the combination of discrete Landau levels *m* and two-dimensional guiding center coordinates (*X*, *Y*), which correspond to physical space. For the 2DEG, this decomposition is naturally encoded within the vortex states $\\Psi\\_{m,{\\bf R}}$ of Eq. ([Sol2]), whose coherent character with respect to the guiding center ${\\bf R}$ brings a doubly continuous parametrization of phase space, while the discrete quantum number is associated to a standard quantization of cyclotron motion.\nThe general equation of motion at any magnetic field for graphene is then given by Eq. ([Dysonfinal]), and simplifies into a dynamics in two-dimensional phase space (*X*, *Y*) given by Dyson Eq. ([Dysondiag]) in the large magnetic field regime, as cylotron motion giving rise to Landau levels exactly decouples from the vortex (or guiding center) motion. In that case, Dyson equation has precisely the form of a star product, see the obvious connection between the ⋆ operator, Eq. ([star]), of the vortex formalism and the ⋆ ℏ product, Eq. ([star2]), of the deformation quantization theory. High magnetic field dynamics is thus isomorphous to a one-dimensional Schrödinger (for the ordinary 2DEG) or Dirac equation (for graphene) with conjugate variables *X* and *Y*. More specifically, if we consider the lowest Landau level (allowing one to forget the spinorial structure proper to graphene), Dyson Eq. ([Dysondiag]) is equivalent to the standard operatorial formulation with the Hamiltonian *H* = *ṽ*0(*X̂*, *Ŷ*), where the effective potential $\\tilde{v}\\_0({\\bf R})$ is given by Eq. ([tildev0]). In that case, dynamics results from the commutation rule [*X̂*, *Ŷ*] = *i**l**B*2 so that kinetic-like energy terms emerge from the identification of the conjugate momentum to *X̂* with *P̂**X* = ℏ*Ŷ*/*l**B*2. We emphasize that this derivation is free of the ambiguities found in the path integral formulation and reproduces the lowest Landau projection method pioneered for the 2DEG by Girvin and Jach. The vortex formulation of phase space is however more general, because it allows to consider not only the projection onto arbitrary Landau levels at infinite magnetic field, but also the coupling between them for arbitrary magnetic field.\nTherefore, the semicoherent character of the vortex representation offers a *local* quantization view in high magnetic field because phase space reduces to the *physical* space of guiding center coordinates ${\\bf R}$. When considering the motion in complicated potential landscapes, this leads to the existence of a hierarchy of local energy scales, allowing one to describe smoothly the crossover from the semiclassical guiding center motion at high temperature to the fully quantum dynamics at very low temperature, as discussed in Sec. [hier].\nLocal density of states\n=======================\nGeneralities\n------------\nWe now use the formalism developed in the previous sections and the resulting expressions for the graphene Green’s function to investigate the characteristic features of the local density of states (LDoS). The goal of Sec. [sec:local] is to show that a lot of information concerning the different potentials at play in graphene can be extracted from the widths and shapes of the LDoS peaks in a high magnetic field.\nThe LDoS is related to the electronic Green’s function via the formula\n \n$$\\begin{aligned}\n\\rho({\\bf r},\\omega) &=&- \\frac{1}{\\pi} \\mathrm{Im}\\, \\mathrm{Tr} \\, \\hat{G}({\\bf r},{\\bf r},\\omega).\n\\label{defdos}\\end{aligned}$$ \nNote that with Eq. ([passage2]), we can directly write the LDoS in terms of the modified local Green’s function $\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})$. In the case where the modified Green’s function is diagonal with respect to the Landau-level quantum number, i.e., $\\tilde{g}\\_{m\\_{1},\\lambda\\_{1};m\\_{2},\\lambda\\_{2}}({\\bf R})=\n\\delta\\_{m\\_{1},m\\_{2}} \\, \\tilde{g}\\_{m\\_{1};\\lambda\\_{1};\\lambda\\_{2}} ({\\bf R})$, we have to evaluate the action of the exponential differential operator onto the product of two vortex functions with identical Landau level, as done in Eq. ([Km]). We therefore find that the LDoS [Eq. ([defdos])] can quite generally be written in the absence of Landau level mixing as\n \n$$\\begin{aligned}\n\\rho({\\bf r},\\omega) &=& \n- \\frac{4}{ \\pi} \\mathrm{Im}\n\\int \\!\\!\\!\n\\frac{d^{2}{\\bf R}}{2 \\pi l\\_{B}^{2}}\n\\Big[K\\_{0}({\\bf R}-{\\bf r}) \\tilde{g}\\_{0}({\\bf R})\n+\\frac{1}{2}\n\\sum\\_{m=1}^{+ \\infty} \\sum\\_{\\lambda\\_{1},\\lambda\\_{2}}\n\\nonumber \\\\\n&& \\hspace{-1cm} \n\\times\n\\left\\{\n\\lambda\\_{1} \\lambda\\_{2} K\\_{m-1}({\\bf R}-{\\bf r}) + K\\_{m}({\\bf R}-{\\bf r})\n\\right\\}\n\\tilde{g}\\_{m;\\lambda\\_{1};\\lambda\\_{2}}({\\bf R}) \\frac{}{}\n\\Big],\n\\nonumber\n\\\\\n\\label{rhosimpler}\\end{aligned}$$ \nwhere the kernel $K\\_m({\\bf R})$ has been previously obtained in Eq. ([Km]). We have also taken into account here the spin and valley degeneracies, which provide an overall prefactor of 4 when evaluating the trace in formula ([defdos]).\nIn actual experimental conditions, one never has a direct access to the zero-temperature LDoS, due to an extrinsic smearing occasioned by the finite temperature *T*. The STS spectra at fixed energy *ɛ* are proportional to the temperature broadened LDoS\n \n$$\\begin{aligned}\n\\rho^{STS}({\\bf r},\\varepsilon,T)\n= -\\int d \\omega \\rho({\\bf r},\\omega) n\\_F'(\\omega-\\varepsilon),\n\\label{rhoSTS}\\end{aligned}$$ \nwhere *n**F*ʹ(*ω*) = − 1/[4*T*cosh2(*ω*/2*T*)] is the derivative of the Fermi-Dirac function.\nLDoS for locally flat potentials\n--------------------------------\n### General expression\nThe leading order result for the vortex Green’s function, Eqs. ([g1Dm]) and ([g1D0]), applies when the disorder potential is locally flat on the scale *l**B*. Mathematically, this approximation is controlled for temperatures larger than the smaller energy scales associated to local Gaussian curvature, such as Eq. ([gamma]). In that case, using previous formulas ([rhosimpler]) and ([rhoSTS]), we get\n \n$$\\begin{aligned}\n\\nonumber\n\\rho^{STS}({\\bf r},\\varepsilon,T)\n= -4\n\\int \\!\\!\\! \\frac{d^{2}{\\bf R}}{2 \\pi l\\_{B}^{2}}\n\\Big[n\\_F'(\\varepsilon-\\xi\\_{0}({\\bf R}+{\\bf r})) \\,\nK\\_{0}({\\bf R})\n\\nonumber \n\\\\\n +\\frac{1}{2} \\sum\\_{m=1}^{+ \\infty} \\sum\\_{\\epsilon=\\pm} n\\_F'(\\varepsilon-\\xi\\_{m,\\epsilon}({\\bf R}+{\\bf r}))\n\\left\\{ \\left(1+\\epsilon \\beta\\_{m}({\\bf R}+{\\bf r}) \\right)\n\\right. \\nonumber \n\\\\\n\\left.\n\\times\nK\\_{m}({\\bf R}) +\\left(1-\\epsilon \\beta\\_{m}({\\bf R}+{\\bf r})\\right)\n K\\_{m-1}({\\bf R})\n\\right\\} \n\\frac{}{}\\Big], \\hspace\\*{0.5cm}\n\\label{rhoFlat}\\end{aligned}$$ \nwhere the effective energy $\\xi\\_{m,\\pm}({\\bf R})$, the electron-hole asymmetry parameter $\\beta\\_m({\\bf R})$ and the kernel $K\\_m({\\bf R})$ are given respectively by Eqs. ([effective]), ([beta]) and ([Km]). The kernels $K\\_m({\\bf R})$ are oscillating yet normalized functions that are localized around ${\\bf R}={\\bf 0}$ on a characteristic length scale $L\\_{m} = l\\_B \\sqrt{2m+1}$, which one associates with the cyclotron radius. Only for the lowest Landau level *m* = 0 does this length reduce to the magnetic length *l**B*.\nIn principle, one cannot strictly set the temperature to zero in Eq. ([rhoFlat]) unless the effective potentials $\\tilde{v}\\_{m}^{+}({\\bf R})$ and $\\tilde{v}\\_{m}^{-}({\\bf\nR})$ which compose the function $\\xi\\_{m,\\epsilon}({\\bf R})$ are *globally* flat. Indeed, for arbitrary potentials $\\tilde{v}\\_{m}^{+}({\\bf R})$ and $\\tilde{v}\\_{m}^{-}({\\bf R})$, it is important to have in mind that expression ([rhoFlat]) overlooks the fine structure of the zero-temperature local density of states, which requires to take into account all existing spatial derivatives of these potentials $\\tilde{v}\\_{m}^{\\pm}({\\bf R})$ [see Eq. ([Dysondiag])]. Nevertheless, it captures accurately the shape of the LDoS when the temperature exceeds the (smaller) energy scales involving second and higher derivatives (in orthogonal directions) of the potentials $\\tilde{v}\\_{m}^{\\pm}({\\bf R})$ associated to curvature. Basically, under the inequalities $L\\_{m}|{\\bm \\nabla} \\xi\\_{m,\\epsilon}({\\bf r})|\n> T \\gg \\sqrt{|\\gamma\\_{m}^{\\pm}({\\bf R})|}$, one expects that the temperature gives a small contribution to the smearing of the LDoS in comparison to the intrinsic smearing generated by the spatial dispersion of the function $\\xi\\_{m,\\epsilon}({\\bf R}+{\\bf r})$, (i.e., by the potential gradients) when performing the integration over the vortex position ${\\bf R}$ in Eq. ([rhoFlat]).\n### High-temperature regime\nAt very high temperatures such that $T\\gg L\\_{m}\n|{\\bm \\nabla} \\xi\\_{m,\\epsilon}({\\bf r})|$, the spatial dependence on the vortex position ${\\bf R}$ inside the Fermi derivative function can be neglected [here, we also disregard the ${\\bf R}$ dependence of the smooth function $\\beta\\_{m}({\\bf R}+{\\bf r})$], so that expression ([rhoFlat]) simplifies into:\n \n$$\\begin{aligned}\n\\label{rhoHT}\n\\rho^{STS}({\\bf r},\\varepsilon,T) & = & \n\\frac{(-4)}{2\\pi l\\_B^2}\\\\\n\\nonumber\n&&\\hspace{-1cm}\\times \\left[ n\\_F'(\\varepsilon-\\xi\\_{0}({\\bf r})) +\n\\sum\\_{m=1}^{+ \\infty} \\sum\\_{\\epsilon=\\pm}\nn\\_F'(\\varepsilon-\\xi\\_{m,\\epsilon}({\\bf r}))\n\\right].\\end{aligned}$$ \nThis semiclassical expression provides LDoS peaks of width 2*T* that are centered around the effective Landau-level energies $\\xi\\_{m,\\pm}({\\bf r})$ given by Eq. ([effective]). In this regime, the thermal broadening of the LDoS peaks is thus independent of the Landau-level index, and the electron and hole peaks are characterized by the same heights. At lower temperatures, we now show that different linewidths, line shapes and particle-hole asymmetries are generated in the LDoS spectra, providing additional insight into the underlying scalar and mass potentials.\n### Low-temperature regime for potentials smooth on the cyclotron radius *L**m*\nIn case when $T\\leq L\\_{m} |{\\bm \\nabla} \\xi\\_{m,\\epsilon}({\\bf r})|$, the spatial dependence of the Fermi function derivative must be kept. We first assume here that the potential is well approximated by its first-order gradient on the *whole* cyclotron orbit of radius *L**m*, i.e., $\\xi\\_{m,\\epsilon}({\\bf R}+{\\bf r})\\simeq \\xi\\_{m,\\epsilon}({\\bf r}) +{\\bf R}.{\\bm \\nabla}\n\\xi\\_{m,\\epsilon}({\\bf r})$. We can then perform analytically the Gaussian integral over ${\\bf R}$ in Eq. ([rhoFlat]) and obtain the intuitive result for the zero-temperature LDoS (see Appendix [appD]),\n \n$$\\begin{aligned}\n\\rho({\\bf r},\\omega) \\simeq\n\\frac{1}{2 \\pi l\\_{B}^{2}} \\frac{4}{\\sqrt{\\pi}} \n\\left[ \n\\frac{1}{\\Gamma\\_0^{loc}({\\bf r})}\n\\exp\\left\\{-\\left(\\frac{\\omega-\\xi\\_{0}({\\bf r})}{\\Gamma\\_{0}^{loc}({\\bf r})}\n\\right)^{2}\\right\\}\n+ \\frac{1}{2} \\sum\\_{m=1}^{+ \\infty} \\sum\\_{\\epsilon=\\pm} \n\\frac{1}{\\Gamma\\_{m,\\epsilon}^{loc}({\\bf r})} \n\\left\\{ \n\\frac{1+\\epsilon \\beta\\_{m}}{2^m m!}\nH\\_m^2\n\\left[\n\\frac{\\omega-\\xi\\_{m,\\epsilon}({\\bf r}) }{\\Gamma\\_{m,\\epsilon}^{loc}({\\bf r})} \\right]\n \\nonumber \\right. \\right. \\\\\n\\left. \\left.\n+ \\frac{1-\\epsilon \\beta\\_{m}}{2^{m-1} (m-1)!}\nH\\_{m-1}^2\n\\left[\n\\frac{\\omega-\\xi\\_{m,\\epsilon}({\\bf r}) }{\\Gamma\\_{m,\\epsilon}^{loc}({\\bf r})} \\right]\n\\right\\}\n\\exp\\left\\{-\\left(\\frac{\\omega-\\xi\\_{m,\\epsilon}({\\bf r})}\n{\\Gamma\\_{m,\\epsilon}^{loc}({\\bf r})} \\right)^{2}\\right\\}\n\\right],\n\\label{dosgradient}\\end{aligned}$$ \nwith $\\Gamma\\_{m,\\epsilon}^{loc}({\\bf r})=l\\_{B} |{\\bm \\nabla} \\xi\\_{m,\\epsilon}({\\bf r})|$ the *local* energy scale associated to the drift motion and *H**m*(*x*) the *m*th Hermite polynomial. In order to keep the above expression compact, we have only written the zero-temperature local density of states, but the STS local density of states is readily obtained from Eq. ([rhoSTS]). The above expression is quite reminiscent of the expression that"},"data_id":{"kind":"string","value":"arxiv_0183118"}}},{"rowIdx":45796,"cells":{"content":{"kind":"string","value":". This is independent of the diagnostics used.\nFinally, we discussed scenarios for the unusual explosion site of ASASSN-14jb. We conclude that, although the probablity of ejection from the disk is low from theory and the specific mechanism to initiate the star formation at this heights is uncertain, both scenarios require a low mass progenitor with a ZAMS mass of ≈ 10 − 12 *M*⊙. This estimate is consistent with the physical parameters derived for the explosion.\nNM acknowledges the Insitute of Astrophysics at PUC, the Millennium Institute of Astrophysics (MAS), and the Astronomy Nucleus for University Diego Portales (UDP) for supporting this research. We thank Luc Dessart, Enrique Perez-Montero, Ondrej Pejcha, Franz Bauer, Melina Bersten, Lin Xiao, and Antonia Bevan for valuable discussions.\nSupport for NM and JLP was provided in part by FONDECYT through the grant 1151445. Support for NM, JLP, and AC was also provided by the Ministry of Economy, Development, and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. The authors thank Las Cumbres Observatory and its staff for their continued support of ASAS-SN. ASAS-SN is supported by the Gordon and Betty Moore Foundation through grant GBMF5490 to the Ohio State University and NSF grant AST-1814440. Development of ASAS-SN has been supported by NSF grant AST-0908816, the Center for Cosmology and AstroParticle Physics at the Ohio State University, the Mt. Cuba Astronomical Foundation, the Chinese Academy of Sciences South America Center for Astronomy (CASSACA), and by George Skestos.\nThis research has made use of data from the Public ESO Spectroscopic Survey of Transient Objects and *Spitzer* IRAC data from 2015-09-13 (program ID 11053), 2016-08-17 (program ID 12099) and 2014-09-05 (program ID 10139). This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics. We acknowledge the use of the HyperLeda database (http://leda.univ-lyon1.fr). 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G., et al. 2017,, 601, A29 Zhang, J., & Wang, X. 2014, The Astronomer’s Telegram, 6601, Zinn, P.-C., Grunden, P., & Bomans, D. J. 2011,, 536, A103 de Jaeger, T., González-Gaitán, S., Anderson, J. P., et al. 2015,, 815, 121 de Jaeger, T., González-Gaitán, S., Hamuy, M., et al. 2017,, 835, 166\nPhotometry\n==========\nHere we describe the photometric pipeline used to derive the magnitudes of ASASSN-14jb in the standard system. For each frame previously reduced we do the following steps to obtain the aperture photometry for the supernova:\n* We run SExtractor to obtain an initial estimate for the FWHM of the stars in the image, using bright stars far from the edges of the frame, and global background level.\n* We re-run SExtractor using the initial estimates and fixing the aperture to 1.6 times the median FWHM of the stars. After cross-matching this catalog with our local standard star catalog we reject the standard objects that have photometric uncertainties $\\sigma\\_{\\rm phot}^{i} > 0.1$.\n* With the zero-point and uncertainty for each local standard star (no color term) $(\\Delta m^{i},\\sigma\\_{\\rm phot}^{i})$ we calculate the mean zero-point of the image with its final error including both statistical photometric uncertainties and the rms of the zero-point. The typical value of the light curves deviations were $\\lesssim 0.02$ mag.\n \n$$\\begin{aligned}\n\\left<\\Delta m \\right> &=& \\frac{\\sum{\\Delta m^{i}}}{N\\_{star}} \\nonumber \\\\\nVar(\\left<\\Delta m \\right> ) = rms(\\Delta m)^2 &+& \\sum{\\left(Var\\_{phot}^{i}+ Var\\_{catalog}^i\\right)}\\end{aligned}$$ \nSo with this the final photometry of the supernova and its variance are:\n \n$$\\begin{aligned}\nm\\_{SN}^i &=& m\\_{phot}^i + \\left<\\Delta m \\right> \\nonumber \\\\\nVar (m\\_{SN}^i) &=& \\sigma\\_{phot}^2 + Var(\\left<\\Delta m \\right>)\\end{aligned}$$\n* Regarding color terms, as it is showed in Valenti et al. (in prep)[10](#fn10), the only significant color term for LCOGT cameras in observations obtained in the *B**V**g**r**i* filters are on the *g*-band. We apply this correction using the *g* − *r* color but with the *r*-band magnitudes only corrected trough a mean zero-point term. That is, we follow the equations: \n \n$$\\begin{aligned}\ng &=& g\\_{phot} + \\left<\\Delta g\\right> + C(g-r) \\nonumber \\\\\nr &=& r\\_{phot} + \\left<\\Delta r\\right>\\end{aligned}$$ \nwith the corresponding error propagation. The *g*-band color term correction obtained was important only at early times when the SN is bluest.\nTo verify our results from aperture photometry, we ran PSF fitting photometry through our pipeline using the softwares DOPHOT and DAOPHOT in IRAF for selected frames, and in both cases we obtained consistent results. We chose the SExtractor aperture photometry because it was the most robust method. The PSF photometry of DOPHOT was limited because of the fixed gaussian profile shape and DAOPHOT required too much tuning of the parameters to obtain good results on frames that had significant sky variations or were noiser than average.\nP-Cygni profile fitting\n=======================\nExpansion velocities were estimated using the velocity shift at maximum abpsorption in the, continuum normalized, P-Cygni profiles. For a given supernova we took each individual spectra *F**λ* and, after masking the regions for the strong lines H*α* and H*β* and NaI *λ**λ*5890, 5896 a black body is fitted. If the spectra is after 20 days past explosion the range *λ* < 5100 Å is also removed to avoid the contamination from line blanketing. Also a simple power law is fitted and if the residuals in this case are better this fit is adopted as the continuum. after estimating the continuum *F**C* the spectra and wavelength is normalized as $ y = \\frac{F\\_\\lambda}{F\\_C}(x = \\frac{\\lambda -\\lambda\\_0}{\\lambda\\_0})$, where the 0 subscript refers to the line position (e.g. *λ*0 = 6563 Å for H*α*). \nTo find a minima of a profile throughout a polinomial fitting, a function written in Python code was created. If no initial guess is given for the maximum abpsorption velocity *v*\\* (or equivalently *λ*\\* = *λ*0 + *λ*0*v*0/*c*) first the data is smoothed with a high degree polinomial to find a guess. Then a new fitting window is defined, centered on the this guess, and a new low degree polinomial (from 2 up to 8) is fitted to find a new minima. In this final fit a weighting is applied to the data to avoid problems on the defined edges of the fitting window and the adopted guess. This weighting goes on the residuals as :\n \n$$\\begin{aligned}\n \\chi^2 = \\sum\\_{i} w\\_i(y\\_i - y\\_{model}(x\\_i))^2 \\nonumber \\\\ \n w\\_i = \\exp{(-|x-x\\_\\*|/s\\_x)}\\end{aligned}$$ \nThe e-fold scale *s**x* is used to only consider the points more closed to the assumed profile minima. For the sample of this work a systematic error was considered using different values for *s**x* of 10,50 and 100 Å, at least. This procedure effectively measures how biased is the chosen minima *x*\\*.\nWe estimate a global statistical error, for each *s**x*, using a Jacknife procedure, where each data point is masked before fitting and a mean and standard deviation are obtained from the distribution.\n![Example of a P-Cygni profile fit. The x-axis is in units of 1000 km/s. The blue dots are the data points for the normalized spectra and the red points are the points actually used for the fitting procedure. The vertical line marks the position of the final velocity adopted and the cyan vertical indicates zero velocity at the line position. [fig:pcygnifit]](pcygni_fit.png)Example of a P-Cygni profile fit. The x-axis is in units of 1000 km/s. The blue dots are the data points for the normalized spectra and the red points are the points actually used for the fitting procedure. The vertical line marks the position of the final velocity adopted and the cyan vertical indicates zero velocity at the line position. [fig:pcygnifit]\nDistance Measurement\n====================\nThe methods for Type II SNe distance measurement can be categorized in theoretically based, such as the Expanding Photosphere Method or the spectra-fitting expanding atmosphere method, and purely empirical such as the Standard Candle Method, the Photospheric Magnitude Method or the Photometric Color Method.\nWe will compute distances to ASSASN-14bj using the three empirical ones. We will not apply *K*-corrections since the low redshift of the source makes them very small (e.g., less than 0.01 mag for the *V*-band).\nPhotospheric Magnitude Method\n-----------------------------\nThe PMM is a method to measure distances to Type II SNe that relies on a time-variable standardization of the photospheric magnitude evolution. This method can be viewed as a generalization for various epochs of the SCM () in the sense that both rely on the physical connection between the expansion velocity (a measure of the explosion energy) and the plateau luminosity, but the PMM has a different approach in the calibration. For a distance to be measured we need a good constraint on the shock-breakout epoch, magnitudes corrected from both host and local extinction and corrected by the *K*-correction, and expansion velocities inferred from the minima of the P-Cygni profiles (usually from the Fe II lines) in the photospheric phase. Explicitly we have,\n \n$$\\begin{aligned}\n\\mu\\_{\\lambda} &=& m\\_{\\lambda}^{corr} - \\mathcal{M}\\_{\\lambda} - \\mathcal{R} \\nonumber \\\\\n \\mathcal{R} &=& -5log\\left(R\\_{ph}\\right)+5 = -5log\\left(v\\_{ph}(t-t\\_0)\\right)+5\\end{aligned}$$ \nwhere *m**λ**c**o**r**r* is the rest-frame and extinction corrected magnitude at the wavelength *λ*, M*λ* is the photospheric magnitude, R is the factor that considers the surface area of the photosphere and depends on the photospheric radius, which for homologous expansion is simply the product of photospheric velocity *v**p**h* and the time since the beginning of the expansion *t* − *t*0 (see R14 for details). Although we have named *t*0 the `explosion epoch`, this reference time is actually associated with the shock breakout. As our constraint on *t*0 is of the order of the shock propagation time to reach a RSG surface after core-collapse we do not make an explicit distinction. We have measured the velocities of ASASSN-14jb from the Fe II *λ*5169 line (see Figure [fig:compvels]) and the extinction corrected magnitudes in the *V*-band without applying a *K*-correction. Averaging the distances obtained at the three epochs with Fe II *λ*5169 velocities, we obtain an average distance modulus of $\\left<\\mu\\right>\\_{PMM} = 31.94 \\pm 0.06 $ mag.\nTo make a direct comparison of the distance measurements of other methods we need consider that the PMM was calibrated using local distance ladder measurements, so the method carries an implicit value of the Hubble constant. As the value used by R14 is $H\\_0 = 68\\, \\rm km/s/Mpc$ for the *V*-band calibration, we will take this value as reference in the following analysis. We also use now the CMB frame redshift of *z**C**M**B* = 0.00505 for the ASASSN-14jb host..\nStandard Candle Method\n----------------------\nThe most common method to measure Type IIP like distances is the SCM which relies on the tight correlation between brightness and the expansion velocity at a given epoch in the plateau phase. Similar to the Type Ia SNe calibrations, the SCM parameters are fitted directly in the Hubble diagram following the model,\n \n*M**λ*1 = − *α**l**o**g*(*v**p**h*) + *β*(*M**λ*2 − *M**λ*1) \nwhere *M**λ**k* is the absolute magnitude in the *k*-band, *v**p**h* is the photospheric velocity, usually at 50 days, and (*α*, *β*) are the parameters to fit using a statistically significant sample. The distance modulus in the DJ17 prescription takes the form,\n \n$$\\begin{aligned}\n\\mu\\_{\\lambda\\_1} &=& m\\_{\\lambda\\_1} + \\alpha log\\left(\\frac{v(H{\\beta})}{<v(H{\\beta})>}\\right) - \\beta (m\\_{\\lambda\\_2}-m\\_{\\lambda\\_1}) \\nonumber \\\\\n &-& 5\\log(H\\_0) - \\mathcal{M}\\_{SCM} + 25\\end{aligned}$$ \n. We use the parameters obtained it DJ17 for the SDSS *i*, (*r* − *i*) color combination[11](#fn11). With the interpolated velocity in *H**β* at 40 days, using a power law model, of *v*(*H**β*) = 4181 ± 146 *k**m*/*s* and the color at 45 days (*r* − *i*)0 = 0.03 ± 0.13mag, the distance modulus is *μ**S**C**M* = 32.17 ± 0.20 mag.\nPhotometric Colour Method\n-------------------------\nThe PCM, presented in, exploits the correlation between the light curve decline rate at the plateau, *s*2, and the luminosity. In this method the distance modulus is obtained via,\n \n$$\\begin{aligned}\n\\mu\\_{PCM} &=& i - \\alpha\\_{PCM} s\\_{2,i}+\\beta\\_{PCM} (r-i)\\_0 + \\mathcal{M}\\_{PCM} \\nonumber \\\\\n&+& 5log(H\\_0) - 25 \\end{aligned}$$ \nwhere the values of the parameters taken from DJ17 are *α**P**C**M* = 0.39 ± 0.08, *β**P**C**M* = 0.8 ± 0.48, M*P**C**M* = − 0.84 ± 0.08. Measuring the slope in the *i*-band light curve we find *s*2, *i* = − 0.365 ± 0.187. With the corresponding color (*r* − *i*)0 = 0.027 ± 0.16 we obtain *μ**P**C**M* = 32.4 ± 0.2 mag.\ncccccccc & *α*(*d**e**g*) & *δ*(*d**e**g*) & B & V & g & r & i \n1 & 335.744 & − 28.968 & 15.274 (0.020) & 14.416 (0.010) & 14.840 (0.015) & 14.194 (0.012) & 13.947 (0.026) \n2 & 335.773 & − 28.917 & 15.742 (0.062) & 15.154 (0.016) & 15.437 (0.022) & 15.024 (0.000) & 14.967 (0.111) \n3 & 335.805 & − 28.912 & 16.232 (0.104) & 15.562 (0.082) & 15.849 (0.037) & 15.380 (0.016) & 15.309 (0.104) \n4 & 335.808 & − 29.040 & 17.149 (0.213) & 16.056 (0.000) & 16.523 (0.045) & 15.998 (0.098) & 15.823 (0.026) \n5 & 335.829 & − 29.009 & 15.577 (0.037) & 14.658 (0.024) & 15.064 (0.014) & 14.389 (0.014) & 14.146 (0.056) \n6 & 335.838 & − 28.927 & 15.838 (0.018) & 14.829 (0.042) & 15.325 (0.011) & 14.527 (0.030) & 14.262 (0.036) \n7 & 335.844 & − 28.936 & 17.087 (0.043) & 15.579 (0.067) & 16.356 (0.006) & 14.986 (0.039) & 14.083 (0.041) \n8 & 335.852 & − 29.013 & 14.603 (0.029) & 13.906 (0.012) & 14.229 (0.000) & 13.712 (0.022) & 13.573 (0.046) \n9 & 335.863 & − 29.016 & 14.755 (0.017) & 14.084 (0.041) & 14.369 (0.010) & 13.945 (0.025) & 13.824 (0.046) \n10 & 335.870 & − 28.995 & 15.693 (0.099) & 14.964 (0.053) & 15.272 (0.022) & 14.794 (0.026) & 14.655 (0.136) \n[tab:std]\nlcccccl UT Date & MJD & (*t* − *t*0)( \\* ) & V (1*σ* error) & Telescope \n& (days) & (days) & (mag) & \n2014-10-13 & 56943.098 & -3.485 & > 18.343 & ASAS-SN \n2014-10-19 & 56949.093 & 2.510 &16.861 (0.129) & ASAS-SN \n2014-10-20 & 56950.060 & 3.477 &16.284 (0.069) & ASAS-SN \n2014-10-22 & 56952.058 & 5.475 &16.154 (0.065) & ASAS-SN \n2014-10-23 & 56953.051 & 6.468 &16.072 (0.065) & ASAS-SN \n2014-10-29 & 56959.084 & 12.501 &16.139 (0.069) & ASAS-SN \n2014-11-04 & 56965.117 & 18.534 &16.040 (0.109) & ASAS-SN \n2014-11-07 & 56968.064 & 21.481 &16.026 (0.112) & ASAS-SN \n2014-11-11 & 56972.059 & 25.476 &16.071 (0.063) & ASAS-SN \n2014-11-12 & 56973.144 & 26.561 &15.907 (0.063) & ASAS-SN \n2014-11-16 & 56977.056 & 30.473 &16.118 (0.078) & ASAS-SN \n2014-11-21 & 56982.058 & 35.475 &16.089 (0.068) & ASAS-SN \n2014-11-24 & 56985.046 & 38.463 &16.106 (0.068) & ASAS-SN \n2014-12-01 & 56992.053 & 45.470 &16.125 (0.088) & ASAS-SN \n2014-12-04 & 56995.031 & 48.448 &16.230 (0.103) & ASAS-SN \n2014-12-08 & 56999.035 & 52.452 &16.213 (0.084) & ASAS-SN \n2014-12-09 & 57000.034 & 53.451 &15.995 (0.081) & ASAS-SN \n2014-12-14 & 57005.026 & 58.443 &15.984 (0.108) & ASAS-SN \n2014-12-19 & 57010.044 & 63.460 &16.030 (0.062) & ASAS-SN \n2015-04-10 & 57122.421 & 175.838 & > 16.596 & ASAS-SN \n[tab:asassn]\n ccccccccc UT Date & MJD & (*t* − *t*0)( \\* ) & *B* & *V* & *g* & *r* & *i* & Telescope \n2014-10-19 & 56949.785 & 3.202 & …& …&16.077 (0.023) & 16.402 (0.019) & …&LCOGT 1m/SAAO \n2014-10-19 & 56949.901 & 3.318 & …& …&16.072 (0.020) & 16.376 (0.014) & …&LCOGT 1m/SAAO \n2014-10-20 & 56950.393 & 3.810 & …& …&15.986 (0.022) & 16.237 (0.015) & …&LCOGT 1m/SSO \n2014-10-20 & 56950.802 & 4.219 & …& …&15.960 (0.021) & 16.159 (0.014) & …&LCOGT 1m/SAAO \n2014-10-21 & 56951.536 & 4.953 &16.084 (0.036) & 16.016 (0.020) & 15.973 (0.021) & 16.080 (0.021) & 16.231 (0.033) & LCOGT 1m/SSO \n2014-10-22 & 56952.532 & 5.949 &16.055 (0.025) & 16.022 (0.016) & 15.948 (0.021) & 16.055 (0.013) & 16.217 (0.021) & LCOGT 1m/SSO \n2014-10-23 & 56953.527 & 6.944 &15.992 (0.025) & 15.970 (0.017) & 15.944 (0.018) & 15.984 (0.012) & 16.116 (0.023) & LCOGT 1m/SSO \n2014-10-24 & 56954.141 & 7.558 &15.995 (0.027) & 15.941 (0.016) & 15.913 (0.022) & 15.982 (0.014) & 16.195 (0.027) & LCOGT 1m/MDO \n2014-10-24 & 56954.543 & 7.960 &16.021 (0.033) & 15.960 (0.016) & 15.927 (0.020) & 15.945 (0.011) & 16.091 (0.022) & LCOGT 1m/SSO \n2014-10-26 & 56956.830 & 10.247 &16.026 (0.023) & 15.974 (0.016) & …&15.948 (0.012) & 16.128 (0.020) & LCOGT 1m/SAAO \n2014-10-29 & 56959.546 & 12.963 &16.097 (0.023) & 15.980 (0.015) & 15.967 (0.021) & 15.911 (0.011) & 16.084 (0.020) & LCOGT 1m/SSO \n2014-11-01 & 56962.518 & 15.935 &16.166 (0.036) & 16.020 (0.026) & 16.064 (0.026) & 15.910 (0.017) & 16.096 (0.042) & LCOGT 1m/SSO \n2014-11-04 & 56965.576 & 18.993 &16.211 (0.028) & 15.955 (0.019) & 16.023 (0.030) & 15.861 (0.014) & 15.991 (0.023) & LCOGT 1m/SSO \n2014-11-07 & 56968.505 & 21.921 &16.309 (0.032) & 15.984 (0.024) & 16.046 (0.029) & 15.860 (0.023) & 15.934 (0.037) & LCOGT 1m/SSO \n2014-11-10 & 56971.505 & 24.921 &16.348 (0.029) & 15.949 (0.020) & 16.127 (0.024) & 15.833 (0.016) & 15.948 (0.027) & LCOGT 1m/SSO \n2014-11-13 & 56974.763 & 28.180 &16.440 (0.028) & 16.018 (0.017) & 16.193 (0.022) & 15.866 (0.012) & 15.980 (0.021) & LCOGT 1m/SAAO \n2014-11-17 & 56978.766 & 32.183 &16.510 (0.032) & 16.023 (0.019) & 16.212 (0.023) & 15.867 (0.014) & 15.939 (0.023) & LCOGT 1m/SAAO \n2014-11-21 & 56982.777 & 36.194 &16.566 (0.034) & 15.998 (0.019) & 16.223 (0.024) & 15.853 (0.014) & 15.861 (0.029) & LCOGT 1m/SAAO \n2014-11-25 & 56986.781 & 40.198 &16.582 (0.026) & 15.980 (0.017) & 16.274 (0.022) & 15.848 (0.013) & 15.902 (0.021) & LCOGT 1m/SAAO \n2014-11-28 & 56989.854 & 43.271 &16.633 (0.031) & 15.972 (0.021) & 16.259 (0.024) & 15.839 (0.014) & 15.830 (0.022) & LCOGT 1m/SAAO \n2014-12-01 & 56992.776 & 46.193 &16.679 (0.050) & 15.979 (0.025) & 16.255 (0.025) & 15.847 (0.023) & 15.805 (0.033) & LCOGT 1m/SAAO \n2014-12-04 & 56995.836 & 49.253 &16.660 (0.059) & 15.961 (0.026) & 16.259 (0.030) & 15.784 (0.021) & 15.751 (0.028) & LCOGT 1m/SAAO \n2014-12-08 & 56999.102 & 52.519 &16.681 (0.029) & 15.962 (0.016) & 16.262 (0.021) & 15.802 (0.012) & 15.797 (0.019) & LCOGT 1m/CTIO \n2014-12-13 & 57004.784 & 58.201 &16.669 (0.027) & 15.955 (0.015) & 16.239 (0.020) & 15.785 (0.011) & 15.788 (0.019) & LCOGT 1m/SAAO \n2014-12-16 & 57007.794 & 61.211 &16.678 (0.026) & 15.954 (0.016) & 16.273 (0.020) & 15.815 (0.011) & 15.802 (0.020) & LCOGT 1m/SAAO \n2014-12-26 & 57017.040 & 70.457 &16.750 (0.032) & 15.974 (0.017) & 16.304 (0.020) & 15.825 (0.012) & 15.807 (0.020) & LCOGT 1m/CTIO \n2014-12-29 & 57020.430 & 73.847 &16.765 (0.035) & 15.997 (0.016) & 16.320 (0.023) & 15.843 (0.012) & 15.822 (0.021) & LCOGT 1m/SSO \n2015-04-11 & 57123.408 & 176.824 &20.600 (0.109) & 19.114 (0.034) & …& …& …&ESO-NTT \n2015-10-11 & 57306.204 & 359.621 &21.938 (0.042) & 21.108 (0.021) & …& …& …&ESO-NTT \n2015-10-11 & 57306.237 & 359.654 &22.035 (0.032) & …& …& …& …&ESO-NTT \n2015-12-11 & 57367.058 & 420.475 &22.752 (0.069) & …& …& …& …&ESO-NTT \n2015-12-11 & 57367.091 & 420.508 & …&21.827 (0.063) & …& …& …&ESO-NTT \n[tab:allphot]\nccccc UT Date & MJD & (*t* − *t*0)( \\* ) & Airmass & Telescope \n2014-10-20 & 56950.15 & 3.57 & 2.03 & FLWO 1.5m \n2014-10-21 & 56951.19 & 4.61 & 2.11 & FLWO 1.5m \n2014-10-22 & 56952.16 & 5.58 & 2.04 & FLWO 1.5m \n2014-10-23 & 56953.17 & 6.59 & 2.05 & FLWO 1.5m \n2014-10-24 & 56954.17 & 7.59 & 2.08 & FLWO 1.5m \n2014-10-25 & 56955.18 & 8.59 & 2.11 & FLWO 1.5m \n2014-10-27 & 56957.13 & 10.55 & 2.03 & FLWO 1.5m \n2014-11-20 & 56981.09 & 34.51 & 2.05 & FLWO 1.5m \n2014-12-20 & 57011.08 & 64.50 & 2.66 & FLWO 1.5m \n2015-01-04 & 57026.55 & 79.96 & 2.20 & Baade 6.5m \n2015-11-14 & 57340.10 & 393.52 & 1.28 & VLT-UT4 8.4m \n[tab:spectra]\nccccccc & MJD & *t* − *t*0 & *H**β* & Fe II *λ*5018 & Fe II *λ*5169 & H*α* \n2014-10-20 & 56950.15 & 3.57 &7900 (148) &…&…&10784 ( 386) \n2014-10-21 & 56951.19 & 4.61 &8624 (54) &…&…&… \n2014-10-22 & 56952.16 & 5.58 &8631 (17) &…&…&9324 (310) \n2014-10-23 & 56953.17 & 6.59 &8313 (43) &…&…&9146 (256) \n2014-10-24 & 56954.17 & 7.59 &8038 (16) &…&…&8667 (316) \n2014-10-25 & 56955.18 & 8.59 &7721 (127) &…&…&8573 (231) \n2014-10-27 & 56957.13 & 10.55 &7432 (41) &…&…&8205 (189) \n2014-11-20 & 56981.09 & 34.51 &4826 (21) &3052 (112) &3221 (88) &6354 (110) \n2014-12-20 & 57011.08 & 64.50 &3300 (65) &2296 (47) &2263 (110) &4605 (115) \n2015-01-04 & 57026.05 & 79.46 &2803 (72) &2024 (20) &1976 (32) &4211 (24) \n| SN Name | *t*0 | *E*(*B* − *V*) | Host | *z*( \\* ) | *μ* | References( + ) |\n| --- | --- | --- | --- | --- | --- | --- |\n| SN1999em | 2451476.44 | 0.100 | NGC 1637 | 0.00239 | 30.360 | 1,2,3,12 |\n| SN1999gi | 2451518.32 | 0.210 | NGC 3184 | 0.00198 | 30.830 | 4,12 |\n| SN2005cs | 2453548.92 | 0.050 | M51 | 0.00200 | 29.630 | 5,12 |\n| SN2007pk | 2454412.32 | 0.100 | NGC 3953 | 0.01665 | 34.230 | 6 |\n| SN2009N | 2454845.46 | 0.132 | NGC 4487 | 0.00450 | 31.840 | 7 |\n| SN2012aw | 2456002.50 | 0.086 | M95 | 0.00260 | 30.150 | 8,9,12 |\n| SNLSQ13fn | 2456293.98 | 0.054 | GALEXASCJ115117.64-293638.0 | 0.06300 | 37.100 | 10 |\n| SN2013ab | 2456340.00 | 0.044 | NGC 5669 | 0.00456 | 31.920 | 11 |\n \n(\\*): Heliocentric redshift from NED. (+): (1),(2),(3),(4),(5), (6),(7),(8),(9),(10),(11), (12) Distance from \ncccc Filter/band & Magnitude & System & Reference \nGalex *F**U**V* & 17.27 ± 0.05 & AB & 1 \nGalex *N**U**V* & 15.24 ± 0.05 & AB & 1 \n*B* & 14.45 ± 0.05 & Vega & this work \n*V* & 14.10 ± 0.05 & Vega & this work \n*g* & 14.23 ± 0.05 & AB & this work \n*r* & 13.92 ± 0.05 & AB & this work \n*i* & 13.50 ± 0.05 & AB & this work \n*K**s* & 12.89 ± 0.10 & Vega & this work \nSpitzer 3.6 *μ*m & 14.90 ± 0.05 & AB & 2 \nSpitzer 4.5 *μ*m & 15.33 ± 0.05 & AB & 2 \n(1), (2) \n---\n1. IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.[↩](#fnref1)\n2. <http://archive.eso.org/eso/eso_archive_main.html>[↩](#fnref2)\n3. <http://sha.ipac.caltech.edu/applications/Spitzer/SHA/>[↩](#fnref3)\n4. <http://www.mpe.mpg.de/~ott/dpuser/qfitsview.html>[↩](#fnref4)\n5. We note that this equation is a corrected version of Equation 33 in RW11. The original one is *R*\\* = 0.7 × *f**ρ*0.1[*T**c**o**l*/(*f**T*/1.2)]4*t*51.9 × 1012[*c**m*], which is inconsistent with Equation 13 of the same paper (our Equation [eq:Rph]).[↩](#fnref5)\n6. Assuming $\\frac{\\beta\\_{6300,6364}}{\\beta\\_{5577}} = 0.5-1$ as in J14 and estimating the fluxes by fitting a skewed Gaussian to the line profiles.[↩](#fnref6)\n7. Image obtained with SOFI at the NTT by the PESSTO program (ID 191.D-0935) with SOFI mounted on the ESO/NTT[↩](#fnref7)\n8. *M**H**I* = 2.356 × 105 × *D*2∫*S**ν**d**ν* M⊙, D = 23.5 Mpc[↩](#fnref8)\n9. http://leda.univ-lyon1.fr/[↩](#fnref9)\n10. See <http://wiki.pessto.org/pessto-operation-groups/pessto-targets-alerts/lcogt-1m-telescope-time-for-pessto>.[↩](#fnref10)\n11. *α**S**C**M* = 3.18 ± 0.41, *β**S**C**M* = 0.97 ± 0.25, M*S**C**M* = − 1.13 ± 0.04[↩](#fnref11)\n\nThe Extraplanar Type II Supernova ASASSN-14jb in the Nearby Edge-on Galaxy ESO 467-G051\n=======================================================================================\n### Received / Accepted\nIntroduction\n============\nOriginally classified based on the absence (Type I) or presence (Type II) of hydrogen lines in their optical spectra, supernovae (SNe) represent the explosive ending of a star. Decades of research have added a considerable degree of complexity to the simple scheme of Minkowski. For more detail see or. The great diversity of core-collapse supernovae (CCSNe) is understood as the result of a rich variety of parent systems. Initial differences in mass, radius, metallicity or rotation, and evolutionary differences in mass lost to stellar winds or interacting binary companions, would result in a wide distribution of envelope masses when the progenitor stars reach the time of core collapse. According to the chemical composition of the outer layers at the time of explosion the spectroscopic display will be of Type II, or Type IIb, Ib or Ic, with little or no presence of hydrogen. The latter are collectively called “stripped envelope SNe”. Finally, according to the total mass of hydrogen in the envelope, a bona fide Type II SN will show a slower or faster rate of decline after maximum and will be named “plateau” (IIP) or “linear”. The convention has stuck although we know now that there is a continuous distribution of decline rates and that Type IIP and IIL have similar progenitors. Extreme mass loss shortly before explosion may lead to the formation of a dense circumstellar medium (CSM). The interaction of SN ejecta with the CSM would produce narrow emission lines of hydrogen and the SN is named Type IIn in these cases.\nProgenitors of CCSNe have been identified in pre-explosion images, and found to be red supergiants (RSG) with Zero Age Main Sequence (ZAMS) masses between ∼ 8 and ∼ 17 M⊙ for Type II SNe or objects with masses between ∼ 4 and ∼ 10 M⊙ at the time of explosion after a complex interacting binary evolution for Type IIb and Ib SNe.\nAs CCSNe progenitors are massive stars with relatively short lifetimes of ∼ 4 − 50 Myr, it is expected that the SNe are still associated with their birth places, namely spiral arms or H II regions, although CCSNe in early-type galaxies with residual star formation have been reported in the literature.\nThe correlation between CCSNe and star forming regions is expected to decrease toward the lower-mass progenitors and get diluted by progenitors resulting from binary evolution which can result in longer timescales before explosion. Recent work has found that stripped envelope SNe are more closely associated with star forming regions than hydrogen rich SNe, as expected from the increasing progenitor mass of the sequence *I**a* → *I**I* → *I**I**b*/*I**I**n* → *I**b*/*c*.\nObservations of the SN host metallicity, either global or local, and the spatial distribution of CCSNe, also provide constrains on the different progenitors. Chemical abundance studies generally associate stripped envelope SNe with higher metallicity hosts as expected from the probable increase of mass loss with metallicity. Studies of the radial distribution of SNe find that Type Ib/c SNe are more centrally concentrated in their hosts, which is again consistent with the higher metal enrichment towards the center of galaxies. And studies of the height distribution of SNe in edge-on disk galaxies, find that CCSNe are nearly twice as much concentrated towards the disk than thermonuclear SNe, a result consistent with the height scale of the stellar populations where their progenitors are expected to originate.\nSome CCSNe, however, defy the common sense implicit in the previous description by appearing far from any identifiable birthplace. One striking example is SN 2009ip located ∼ 5 kpc from NGC 7259, the nearest spiral galaxy. SN 2009ip was first identified as a SNe impostor in 2009 three years before exploding as a Type IIn supernova. Late time HST data rules out the presence of star forming regions comparable to Carinae or the Orion Nebula at the explosion site. The possibility that the progenitor was a runaway star is also rejected. The peculiar velocity of the SN is smaller than 400 km/s and the high mass of the progenitor implies a lifetime shorter than the travel time from the nearest star forming site located at ∼ 1.5 kpc.\nThis paper introduces another example of this sort, the Type IIP like supernova ASASSN-14jb in the edge-on disk galaxy ESO 467-G051. The SN exploded at 2.5 kpc from the center and 2.1 kpc above the galactic disk where no significant star forming region is detected. Also, it is interesting that ESO 467-G051 and NGC 7259 (the host of SN 2009ip) form an interacting pair. The SNe are separated by 2.4’ in the sky ( ∼ 18 kpc).\nWe present here photometric and spectroscopic observations of ASASSN-14jb and Integral Field Spectroscopy (IFS) of its explosion site. We perform a thorough analysis to estimate physical parameters of the SN, the progenitor star, and the parent galaxy. The paper is organized as follows. In Section [sec:data] we describe the photometric and spectroscopic observations. Section [sec:analysis] contains our comparative analysis of the SN, including the photometric and spectroscopic evolution of the ASASSN-14jb. Section [sec:parameters] contains the estimates of basic physical parameters of the SN, Section [sec:host] includes the analysis of the host galaxy and its H II regions. Finally, Section [sec:discusion] includes our discussion and Section [sec:conclusion] our conclusions.\nData\n====\nDiscovery and Explosion Time\n----------------------------\nASASSN-14jb was discovered on UT 2014-10-19.09 (MJD = 2456949.09, $V\\_{\\rm disc} = 16.9$ mag, ) at RA = 22:23:16.12, DEC = -28:58:30.78 (J2000.0) by the ongoing All Sky Automated Survey for SuperNovae from the ``Cassius“ 4-telescope unit at the Cerro Tololo Inter-American Observatory (CTIO), in Chile, hosted by the Las Cumbres Observatory. It was spectroscopically classified as a young Type II on UT 2014-10-20. The last non-detection was reported to be on MJD = 2456943.098 with a 3*σ* magnitude upper limit of *V* > 18.5 mag, 6 days before the time of discovery. We take the midpoint between this last non-detection and the discovery time, *t*0 = 2456946.1 ± 3 (MJD), as the ``explosion” time. The uncertainty is estimated as half the interval between the last non-detection and the discovery epoch.\n[fig:fieldimage]\nPhotometry\n----------\nOptical photometric observations of ASASSN-14jb were obtained with the ASAS-SN unit ``Cassius\" at CTIO in the *V*-band and the LCOGTN 1-meter telescopes at CTIO, the Siding Spring Observatory (SSO), the South African Astronomical Observatory (SAAO), and the McDonald Observatory (MDO) in the *B**V* and the Sloan Digital Sky Survey (SDSS) *g**r**i* filters. All the ASAS-SN images are processed in an automated pipeline using the ISIS image subtraction package, with further details given in. Using the IRAF[1](#fn1) *apphot* package, we performed aperture photometry on the subtracted images and then calibrated the results using the AAVSO Photometric All-Sky Survey. The ASAS-SN photometry is presented in Table [tab:asassn].\nThe fully reduced LCOGTN 1m images (bias/overscan subtracted and flat-fielded) were retrieved from the LCOGTN data archive. We solved the astrometry of each CCD frame using the *astrometry.net* software package. We obtained calibrated *B**V**g**r**i* magnitudes for local standard stars in the field from the APASS catalog. In Figure [fig:fieldimage] we display one of the LCOGT images showing the position of the SN, the local standards, and the nearby SN 2009ip. The data of the local standard sequence is given in Table [tab:std]. ASASSN-14jb was recorded with high signal-to-noise ratio (SNR) in the images of LCOGTN. This, together with the negligible background from the host galaxy at the SN site, prompted us to measure the SN flux using aperture photometry. The procedure is explained in detail in Appendix A.\nAs our observations extended only ∼ 80 days after explosion, we searched the data in public domain for images of the field which could have recorded ASASSN-14jb at later times. We found some images in the ESO archive[2](#fn2). We retrieved ESO/NTT EFOSC images in *B**V* filters that contained ASASSN-14jb’s explosion site at three different epochs during 2015, obtained by the Public ESO Spectroscopic Survey of Transient Objects. The CCD frames were overscan subtracted and flat-fielded with calibration frames obtained the same day using standard IRAF routines. Since the SN is fainter in these late-time images, we measured the SN brightness using PSF fitting photometry as implemented in IRAF DAOPHOT package. We calibrated the magnitudes using local standards from. The early LCOGTN and late ESO/NTT photometry are presented in Table [tab:allphot].\nWe supplemented our ground-based optical photometry with data from the *Swift* and *Spitzer* space-born observatories. Repeated optical and near-UV observations of ASASSN-14jb were obtained by *Swift*/UVOT in the passbands *w2,m1 w1,u,b,v* during the first ∼ 30 days after explosion. We retrieved them from the *Swift* Optical/Ultraviolet Supernova Archive. Also, the field of ASASSN-14jb and its host have been observed nine times since 2010 in the 3.6 *μ*m and 4.5 *μ*m bands by the *Spitzer* IRAC instrument. The observations obtained on 2015-09-13 (program ID 11053) and 2016-08-17 (program ID 12099) were taken when the SN was ∼ 330 and ∼ 670 days past explosion, respectively. We retrieved these images from the data archive[3](#fn3) and also images taken on 2014-09-05 (program ID 10139) to use as templates for image subtraction. We used HOTPANTS for difference imaging and measured aperture photometry on the difference images. The SN is detected on 2015-09-13 and undetected on 2016-08-17.\nOur near-UV and optical photometry is displayed in Figures [fig:lcearly] and [fig:lclate]. The first shows the near-UV and optical light curves for the first ∼ 80 days after explosion, and the second the complete light curves in *B* and *V* bands. The *Spitzer* mid-infrared photometry is discussed in § [ss:dust].\nASAS-SN *V*, LCOGTN *B**V**g**r**i*, and *Swift*-UVOT *w*2, *m*1, *w*1, *u**b**v*, light curves of ASASSN-14jb. Each band has been shifted to the value showed in the label for presentation purposes. The observed time is given in days since the estimated explosion epoch *t*0 = 56946.1 (MJD) in the bottom x-axis and in MJD in the top axis. The black squares are the non-detection derived from the ASAS-SN *V*-band images.[fig:lcearly]\n[ht!] \nSpectroscopy\n------------\nA total of ten single-slit, low-resolution optical spectra of ASASSN-14jb were obtained with the FAST spectrograph mounted on the Fred L. Whipple Observatory Tillinghast 1.5-m telescope (3500 − 7400 Å, *R* ∼ 1700) and with the Inamori-Magellan Areal Camera and Spectrograph mounted on the Magellan Baade 6.5m telescope at Las Campanas Observatory (3800 − 9800 Å, *R* ∼ 1000). The CCD images were reduced and the 1D spectra were extracted and calibrated using standard routines in the IRAF packages *ccdproc* and *onedspec*, respectively. The epochs of the ten spectra obtained in the photospheric phase are presented in Table [tab:spectra]. A montage of the spectra obtained in the photospheric phase is shown in Figure [fig:spectra].\n[ht!] \nWe also observed the field of ASASSN-14jb as part of the All-weather MUse Supernova Integral field Nearby Galaxies (AMUSING; ) with the Multi Unit Spectroscopic Explorer (MUSE, ) on ESO’s Very Large Telescope UT4 (Yepun). MUSE is a state-of-the-art integral field spectrograph with a field of view of 1 arcmin2 and $0\\farcs 2$ spaxels. It covers the spectral range 4800 − 9300 Å with a resolving power *R* ≃ 1800 − 3000. Our MUSE data were obtained on 2015-11-14 and consisted of two different pointings of four dithered exposures each with an integration time of 698 sec. The sky conditions were clear, and we measure a full-with-half-maximum for the stellar point-spread function of 1 at 6600Å. We reduced the MUSE spectroscopy with version 1.2.1 of the EO(fpipeline, but also checked and corrected the astrometric zeropoint using LCOGTN images (for further detail on MUSE data reduction see ).\nAlthough the MUSE observations were obtained ∼ 394 days after explosion (see Table [tab:spectra]) when ASASSN-14jb was faint, the quality and depth of the exposures allow us to easily detect the SN and extract a nebular phase spectrum. We used a circular aperture with a radius of $1\\farcs 4$ to extract the spectrum at the position of the SN with QFitsView[4](#fn4). To take care of any offset in the flux scale we estimated the *V* magnitude of the SN at the time of the spectrum from the late time light curve (*V* = 21.51) and compared it with the synthetic *V*-band magnitude obtained from the nebular spectrum (*V* = 21.86). We then applied an offset of 0.35 mag (factor of 1.38 in flux) to obtain an approximate absolute flux calibration. The nebular spectrum of ASASSN-14jb is displayed in Figure [fig:Neball].\nComparative Analysis\n====================\nTo put ASASSN-14jb in context we assembled a comparison sample representative of the wide range of properties of Type II SNe. These include the luminous Type IIP SN 2007pk, normal Type IIP like SN 1999em, and the subluminous Type II SN 2005cs. We chose objects with data in the public domain, good coverage of the photospheric phase, ideally including *g**r**i* photometry (8 SNe), and optical nebular spectra at ∼ 400 days after explosion (6 SNe). Further details of the comparison set are given in Table [tab:compTable].\nEstimates of the distance and total foreground reddening are needed in order to compare ASASSN-14jb with the rest. We use a distance modulus of *μ* = 32.0 mag, to be justified below.\nWe estimate extinction by dust in the Galaxy using the maps of. Taking the average in a circle with 5 arcminutes of diameter centered on the SN position we obtain *E*(*B* − *V*) = 0.0154 ± 0.001 mag. Extinction by foreground dust in the host galaxy is expected to be low, as the SN exploded at a significant distance from the host galaxy disk. Consistent with this, the narrow Na I D doublet at the host galaxy redshift are not detected in our spectra, which is a clear sign of low reddening. In addition, the MUSE data to be discussed below shows no significant star forming region near the explosion site. We conclude, hence, that ASASSN-14jb is affected only by Galactic extinction. Assuming a standard reddening law with a ratio of total to selective absorption *R**V* = 3.1, we estimate *A**V* = 0.05 mag.\nPhotometric Evolution\n---------------------\nFigure [fig:Vcomp] shows the absolute *V*-band light curve of ASASSN-14jb together with those of the comparison set. The peak at *M**V*, *m**a**x* = − 16.04 ± 0.18 mag is within the usual range of Type IIP SNe, although lower than the average *M**V*, *m**a**x* = − 16.71 ± 1.01 mag in the sample of Type II SNe of. ASASSN-14jb is dimmer than the classic, well-studied Type II-SN 1999em by 0.80 ± 0.18 mag, but it shows a similar behavior in its *V*-band plateau, including the early double peak. Up to ∼ 80 days since explosion the photometry shows the behavior typical of Type IIP like SNe.\nAbsolute *V*-band magnitude evolution for ASASSN-14jb (black filled stars) and the comparison sample of Type II SNe as a function of days after their estimated time of explosion. See Table [tab:compTable] for references of each SNe, including distance and reddening adopted.[fig:Vcomp]\nExtinction corrected *B* − *V* colors for ASASSN-14jb (black filled stars) and the comparison sample of Type II SNe. Reddening values used are in Table [tab:compTable].[fig:BV]\nExtinction corrected *V* − *i*, *g* − *i*, *r* − *i* and *g* − *r* colors for ASASSN-14jb (black filled stars) and the Type II SNe comparison sample. Reddening values used are in Table [tab:compTable].[fig:gri]\nThe measured slope of the plateau (35 − 70 days after explosion) in the *V*-band for ASASSN-14jb is *s*2, *V* = − 0.15 ± 0.02 mag per 100 days, lower than the average *V*-band plateau slope in the A14 sample (1.27 ± 0.93 mag per 100 days). The slow increase in brightness of the plateau is consistent with the correlation between plateau slope and absolute *V*-band magnitude found by A14 and later work. It is also an effect seen in bolometric light curves of CCSNe models resulting from the explosion of low mass progenitors.\nIn Figures [fig:BV] and [fig:gri] we present the extinction corrected color evolution of ASASSN-14jb and the comparison set. The (*B* − *V*)0 color of ASASSN-14jb is bluer than average at 50 days after explosion, being comparable only to LSQ 13fn and SN 2007pk at 80 days after explosion. The change in color from the onset of the plateau up to 80 days is Δ(*B* − *V*) = 0.20 ± 0.04 mag, low when compared with the comparison sample. The color evolution of ASASSN-14jb in (*g* − *r*)0, (*g* − *i*)0 and (*r* − *i*)0 also indicates a comparatively bluer continuum, except in the (*V* − *i*)0 color. This behavior may be due to a true difference in the temperature evolution or it may be an effect of lower line-blanketing in the blue part of the spectrum, resulting from low-metallicity of the progenitor. The similarity in the color evolution of ASASSN-14jb and LSQ 13fn, a Type II SN that probably comes from a low-metallicity progenitor, might suggest that a similar physical mechanism is at play.\nSpectroscopic Evolution in the Plateau phase\n--------------------------------------------\nAs shown in Figure [fig:spectra], the spectral evolution of ASASSN-14jb resembles the evolution of other hydrogen-rich Type II SNe. At very early times the ejecta are optically thick and the spectra show a hot blue continuum with relatively weak, but broad, Balmer lines without P-Cygni absorption troughs. In the first spectrum at ∼ 4 days after explosion, we detect broad Balmer lines in emission and the high ionization He II *λ*4686 line. The FWHM of the Balmer lines and the He II lines in this earliest spectrum are ∼ 11000 − 13000 km/s and ∼ 8000 km/s, respectively, and all the emission peaks are clearly blueshifted by ∼ 2000 − 4000 km/s. The strength of the He II line decreases substantially in the spectrum obtained a day later, at 5 days after explosion, and the Balmer lines start to develop P-Cygni profiles. We also detect the He I *λ*5875 line in the early spectra. At 6 days after explosion, the He II line has weakened further and it is undetected in the spectrum obtained at 7 days after explosion.\nAt phases later than $\\gtrsim 30$ days, the ejecta cools and the hydrogen lines develop strong P-Cygni profiles. The blueshift of the peak of the Balmer lines decreases to ∼ 1000 km/s. This is a common feature in Type II SNe during the photospheric epochs and it has been attributed to the shallow density profile of the ejecta compared to a typical stellar wind. Lines from transitions in metals such as Fe II, Sc II, and Ti II also start to appear and significantly contribute to line blanketing. The Na I D doublet at *λ**λ*5890, 5896 appears now in the wavelength range where He I *λ*5875 was detected at early times. Comparison of the photospheric phase spectra of ASASSN-14jb with those of the comparison sample at ∼ 60 days past explosion (see Figure [fig:speccomp]) shows that the absorption features of the former in the range 5300 − 6300 Å are relatively weak.\nIn Figure [fig:lines] we present the time evolution of the H*α*, H*β* and Fe II *λ*5169 lines. The last two spectra we obtained in the plateau phase, at 64 and 80 days after explosion, show clear absorption features on the blue side of both the H*α* and H*β* main absorption troughs. These high velocity (HV) features are at ∼ 7000 km/s, while the main absorption features are at ∼ 4000 km/s. The HV absorption lines cannot be explained by Si II *λ*6355 or Ba II absorption features and have been observed in 59% of the Type II SNe studied by.\nComparison of rest-frame photospheric phase spectra at ≈ 50 days after explosion. The spectra are not corrected for extinction. [fig:speccomp]\n [fig:lines]\nExpansion velocities of the H*β* (top), Fe II *λ*5169 (middle), and H*α* (bottom) lines as a function of time after explosion in the plateau phase, measured from the P-cygni profile minima of the Balmer lines $\\mathrm{H}{\\alpha},\\mathrm{H{}\\beta}$ and the Fe II *λ*5169 line for ASASSN-14jb (black filled stars) and the Type II SNe comparison sample. [fig:compvels]\nExpansion velocities\n--------------------\nThe ejecta of the Type II SNe achieve near homologous expansion after a few days and at early times the hydrogen atmosphere is opaque and the line forming region is in the outer layers at high velocities. As the temperature drops, the opacity decreases and the photosphere recedes in mass (Lagrangian) coordinate, and appears at lower velocities. The photosphere is tightly bound to the opacity drop caused by hydrogen recombination. In practice one estimates the photospheric velocity by measuring the blueshift of maximum absorption of an specific P-Cygni profile. This can over or under estimate the true photospheric velocity. Other methods used are the cross-correlation with library spectra, the comparison with detailed NLTE codes like CMFGEN or PHOENIX or comparison with more simplified, parametrized LTE spectral modeling (e.g., SYNOW, ).\nThe expansion velocity measured from optical spectra is an important physical parameter directly related to the explosion energy and the plateau luminosity. These are also fundamental for distance measurements. Strong Balmer lines and Fe II lines velocities were measured in our spectra. The spectra were first normalized to a global continuum fitting a black body or a local power law. Then, when possible, low degree polynomials were fitted to the P-Cygni profile absorptions and the minimum was taken as a proxy for the expansion velocity. We did not apply corrections by reddening or peculiar velocities. Details of the P-cygni profile fitting are in Appendix B. In Figure [fig:compvels] we show the expansion velocities of ASASSN-14jb together with those of the comparison set. For LSQ 13fn we only present the velocities after 30 days due to the low signal to noise of the spectra. The velocities of SN 2013ab are directly taken from. The measured velocities are also collected in Table [tab:vel] together with the average velocity of our comparison set and those of.\nTo compare with well known correlations, we interpolated the velocities of our sample at 50 days after explosion using a Monte-Carlo approach. For each SN we generated a re-sampling of each velocity using the measured errors and following a non-linear least squares procedure, using the curve fit function from the NumPy library, we fit a power law model each time. For this procedure we select only velocities after 20 days, as early velocities do not follow the same decay rate as in the plateau. We then take the average and standard deviation of the re-samplings as the 50 days velocity and error, respectively. For all the lines measured the velocities for ASASSN-14jb are slightly under average. The Fe II *λ*5169 velocity interpolated at 50 days is measured to be 2774 ± 69 km/s, compared to the average of 3442 ± 977 km/s of our comparison sample. Our sample average velocity is in agreement with the value obtained with a larger sample in, of 3537 ± 851 km/s at 53 days after explosion. The photospheric velocities inferred are in agreement with the velocity-luminosity correlations (see Section 3.2 below).\nAs is common in Type II SNe distance measurements, we fitted a power law to interpolate the velocities. The Fe II *λ*5169 logarithmic decay presents an average of − 0.55 ± 0.16 in our sample, compared to the value obtained for ASASSN-14jb, of − 0.62 ± 1.80. We observe a strong anti-correlation between the velocity decay slope and the *V*-band absolute magnitude at 50 days. This is expected as there is an internal correlation between both parameters of the power law and the velocity scale correlates with the luminosity.\nIn conclusion, ASASSN-14jb presents expansion velocities consistent, both from the scale and velocity decay, with below average luminosity Type II SNe.\n[ht!] \nNebular Spectrum\n----------------\nThe nebular spectrum of ASASSN-14jb is shown in Figure [fig:Neball] together with a sample of Type II SNe. It shows the typical emission lines of Type IIP like SNe at nebular phases: H*α*, [Ca II] *λ**λ*7291, 7324, [O I] *λ**λ*6300, 6364, Ca II *λ**λ**λ*8498, 8542, 8662, [Fe II] *λ**λ*7155, 7172, with a boxy profile, Na I *λ**λ*5890, 5896 and [O I] *λ*5577, in order of observed intensity. Also, a strong [C I] *λ*8727 line is present, which has been detected in other Type IIP like SNe such as SN 2004dj. A weak He I line at *λ*7065 is observed, but with no He I counterpart at *λ*6678. The clear separation observed in different multiplets like [OI], [CaII] and Ca II + [CI] is a sign of a low explosion energy for ASASSN-14jb.\nThe blueshift with respect to rest-frame of ∼ 200 km/s observed in the peaks of the strongest lines is particularly interesting. It cannot be caused by the same physical effect as in the plateau because in this optically-thin phase the outer density profile is less relevant to the emission. Other hypotheses to explain this blueshift may be: 1) a shift in velocity relative to the host due to dynamical effects or an intrinsically high-velocity of the progenitor star; 2) dust production; 3) Some lines are still optically thick at 400 days past explosion; or 4) an asymmetry in the explosion itself. Notable is the double peak observed in *H**α* which is not observed in other lines, although this is also seen in the nebular spectra of other Type IIP like SNe.\nDistance and Physical parameters of ASSASN-14jb\n===============================================\nProgenitors of Type IIP like SNe have been constrained by pre-explosion images to be RSGs with ZAMS masses of ≈ 8 − 17 M⊙. The discrepancy between these initial mass constraints and the predictions from evolutionary codes has been called the ``RSG problem\". The data set we have compiled allow us to constrain the basic physical parameters of the progenitor of ASSASN-14jb, compare them with the expected values for SN Type IIP like progenitors, and contribute to this discussion. But doing so requires a good estimate of the distance to the SN.\nDistance\n--------\nWe measure the distance towards ASSASN-14jb using some of the most recent empirical methods: The Photospheric Magnitude Method, the Standard Candle Method, and the Photometric Color Method,. Further details of our hypothesis and calculations are given in Appendix C.\nThe three different estimates of distance are shown in Figure [fig:distances]. They are consistent with each other within each method’s intrinsic dispersion. They are also consistent with the distance obtained from the Virgo-inflow corrected recession velocity of the host galaxy and the Hubble law, assuming $H\\_0 = 68\\, \\rm km/s/Mpc$. Given this, we simply take as the final distance the weighted average, considering the statistical errors only, of *μ* = 32.00 ± 0.18 mag (*D* = 25.1 Mpc).\nDistance modulus estimates for ASASSN-14jb using the PMM, SCM and PCM methods (green dots). The black horizontal line is the distance modulus from the host galaxy redshift and a Hubble law with *H*0 = 68 km/s/Mpc, and the grey area shows a dispersion of 200 km/s as a reference error for the CMB frame redshift. The thick green line is the average from the Tully-Fisher measurements obtained from the NED database and the green area represent the standard deviation of the measurements, all normalized to *H*0 = 68 km/s/Mpc. The thick blue line is the adopted distance modulus in this work, *μ* = 32.00 ± 0.18, obtained as the weighted average of the PMM, SCM and PCM methods and the standard deviation is showed as a blue fill. We also show as a reference the distance measured to SN 2009ip in NGC 7259.[fig:distances]\nTemperature Evolution and the Progenitor Radius\n-----------------------------------------------\nThe early emission, a few days after explosion, for a Type II SN comes from the adiabatic cooling of the outer envelope and depends mainly on the pre-supernova radius. Observationally, the pre-supernova radius has been constrained from early UV and optical photometry, using semi-analytic models, on individual Type II SNe, and on larger samples to make statistic studies.\nWe build the spectral energy distribution (SED) for ASASSN-14jb at early times, from ∼ 4 to 26 days after explosion to derive an estimate of the progenitor radius. Using the *Swift/UVOT* near-UV and optical photometry, we build the SED of ASASSN-14jb at each epoch, after correcting for extinction assuming an standard Galactic extinction law with total to selective extinction ratio *R**V* = 3.1. We fit a blackbody function to the SED and obtain the photospheric temperature and radius evolution, assuming the distance to the host galaxy obtained in Section 3.2.\nThe results are presented in Figure [fig:temp]. The observed photospheric temperatures are in the range 5-21 × 103 *K* and the radii are in the range 2-18 × 103 R⊙. We observe a change in the evolution ∼ 10 days after explosion. The observed temperature logarithmic decay increases from − 0.726 ± 0.055 to − 0.850 ± 0.019 and the photospheric radius growth rate increases from 0.890 ± 0.012 to 1.45 ± 0.06.\n developed analytic expressions for the early evolution of the photospheric radius *R**p**h* and temperature *T**p**h*, expected to be valid before hydrogen recombination is significant. The expressions depend on progenitor properties at the time of explosion such as radius *R*\\*, opacity *κ*, and density profile. They also depend on the explosion energy *E* and ejecta mass *M**e**j*.\n \n$$\\begin{aligned}\n\\frac{R\\_{ph}}{[cm]} &=& 3.3 f\\_{\\rho}^{-0.062} E\\_{51}^{0.41}M\\_{ej}^{-0.31}\\kappa\\_{0.34}^{0.093} t\\_5^{0.81} \\times 10^{14} \\label{eq:Rph}\\\\\n\\frac{T\\_{ph}}{[eV]} &=& 1.6 f\\_{\\rho}^{-0.037}\nE\\_{51}^{0.0027}M\\_{ej}^{-0.054}R\\_{\\*,13}^{1/4}\\kappa\\_{0.34}^{-0.28} t\\_5^{-0.45} \\label{eq:Tph}\\end{aligned}$$ \nwhere 0.079 < *f**ρ* < 0.13 for RSGs, depends on the progenitor density profile, *M**e**j* is the ejected mass in solar masses, *E*51 is the explosion energy in units of 1051 erg, *κ*34 is the opacity normalized by 0.34, *R*\\* , 13 is the progenitor radius in units of 1013 cm and *t*5 is the time since shock breakout in units of 105 seconds.\nIt is well known that the emission from Type II SNe differs from a pure black body. When the atmosphere is highly ionized, the electron scattering opacity dominates and the photon thermalization layer differs from the photosphere, the former being underneath at higher temperatures. Because of this the observed temperature (or color temperature *T**c*, as it is usually called) can differ significantly from the photospheric temperature *T**p**h*. This effect, together with the increasing line blanketing and departures from the plane parallel atmosphere approximation, can make the observed SED significantly different from a black body. This effect can be corrected using a theoretically estimated *dilution factor* *ζ*, so that the emergent flux is defined as,\n \n$$\\label{eq:zeta}\nF\\_{\\lambda} = \\theta^2\\ \\pi B\\_{\\lambda}(T\\_c) = \\zeta\\_{\\lambda}^2 \\left(\\frac{R\\_{ph}}{d\\_L}\\right)^2\\pi B\\_{\\lambda}(T\\_c)$$ \nwhere *d**L* is the luminosity distance and *B**λ*(*T**c*) is the Planck function. The dilution factor is in general wavelength dependent and directly affects the angular diameter distance estimate to a supernova. However, as we take the luminosity distance from empirical methods independent of [eq:zeta], the dilution factor only affects the photospheric radius estimation and not the temperature. Note, however, that in the RW11 scheme *ζ* : = 1/*f**T*2 ≈ 0.694, which translates in a factor of 1/*ζ* ≈ 1.44 increase in the photospheric radius (see equation 38 in RW11). In our case we choose a representative value of *ζ**λ* = 0.5 for this high temperature regime, as obtained in.\nTo test the effect of the deviations from a black body, we used a semi-empirical approach. We first selected the 15 *M*⊙ progenitor mass supernova explosion model from that best fits the observed near-UV colors (named m15z8m8). Then, at each epoch we fit the SED using,\n \n$$F\\_{\\lambda}^{data} = \\theta^2 \\left(\\pi B\\_{\\lambda}(T\\_c)\\right)^w \\times \\left(\\frac{F\\_{\\lambda}^{model}}{max(F\\_{\\lambda}^{model})}\\right)^{(1-w)}$$ \na model built from the weighted geometric mean of a black body and the normalized explosion model. The normalization ensures that the model’s intrinsic brightness does not affect the fitted parameters. In principle, the weighting parameter *w* is unconstrained but to ensure a smooth correction we fixed its value to be between 0.6 − 0.9 and selected the model with the minimum residuals. Figure [fig:temp] show our results for both approaches. We see that both temperature and radius evolution become smoother when including the models, probably because the hybrid approach accounts better for line blanketing. Since the available models start from ≈ 10 days past explosion, all the SEDs before that time were fitted with a single black body. The derived slope in the temperature changes to − 0.703 ± 0.017 after 10 days. The photospheric radius follows roughly a linear expansion with a logarithmic slope of 1.167 ± 0.068 during the same epochs.\nNow we apply the RW11 models using the modified black body. We fitted the temperature evolution assuming the fiducial values of *E*51 = 1, *κ*34 = 1, *f**ρ* = 0.1, and use only temperatures greater than ∼ 1 eV because RW11 inform that their models are more reliable in this range. The radius we obtain is 985 ± 49 R⊙. Noting the impact of the time since explosion in the result, we varied the shock-breakout (SBO) epoch and look for the one that minimizes the *χ*2 of the fit. We then obtain *R* = 580 ± 28 R⊙, for a SBO occurring 1.75 days earlier than our initial estimate (reduced *χ*2 = 3.62).\nInverting Equation [eq:Tph] is another form of understanding possible variations of *R*\\*:\n \n$$R\\_\\* \\approx 1.52 \\times f\\_{\\rho}^{0.148} \\left[\\frac{T\\_{col}(t)}{(f\\_T/1.2)}\\right]^4 t\\_5^{1.8} \\times 10^{12} [cm]. \\label{eq:R\\*}$$ \nwhere *f**T* = *T**c**o**l*/*T**p**h* is the ratio of the color temperature and true photospheric temperature. Following RW11 we take *f**T* = 1.2 as a good approximation at early times [5](#fn5).\nThe values obtained using Equation [eq:R\\*] with early time observations are consistent with *R*\\* ≈ 400 − 1000 R⊙. As expected, the radius decreases as the actual decay of the photospheric temperatures is faster than the model, and the *f**T* varies due to the changing opacity of the ejecta.\nWe choose as our best estimate of the progenitor radius, the RW11 fit with an SBO offset of − 1.75 days, which gives an estimate of *R*\\* = 580 ± 28 R⊙. This value is consistent with the color evolution of the mildly sub-solar (Z = 0.4 Z⊙) and compact (*R*\\* = 611 R⊙) RSG progenitor (m15z8m3) used in. As the photospheric radius growth strongly depends on the explosion energy to ejecta mass ratio (equation [eq:Rph]) we can estimate the ejecta mass as a function of the explosion energy. Fixing *κ*34 = 1,*f**ρ* = 0.1 as before we obtain *M**e**j* ≈ 36.0 × *E**e**x**p*1.32.\nBlack-body temperature (blue dots) and radius evolution (red dots) as inferred from the SED of *S**w**i**f**t* data. The SED was corrected by a reddening value of *E*(*B* − *V*) = 0.0154 ± 0.001 mag. The horizontal line indicates the 1 eV level, where recombination is expected to become relevant afterwards. The vertical black solid line indicates the assumed explosion epoch at MJD 56946.1 ± 3 and the vertical dashed lines is at the assumed offset (Δ*t*0 = − 1.75) in the explosion epoch, for the optimal cooling model from RW11. The red dashed lines shows a power-law fit to the photospheric radii (only using the data points with 1 − 1.2 × *T**p**h* < 1*e**V*) while the blues dashed lines shows the same for the Temperatures plus a line with a factor of 1.2 (i.e., the lines shows a fit to 1 − 1.2 × *T**p**h*). The black curved dashed lines are the fits assuming the RW11 for *T**p**h* and *R**p**h*.[fig:temp]\nBolometric Light Curve and the 56Ni Mass\n----------------------------------------\nWe constructed a bolometric light curve of ASASSN-14jb, by applying the semi-analytic bolometric corrections from for the extinction corrected *B* − *V* colors. Figure [fig:bolall] shows the result, together with the bolometric light curves of the comparison sample. The bolometric luminosity of ASSASN-14jb ∼ 50 days after explosion is log(*L*/*L*⊙) = 8.4 ± 0.18, which is under-luminous compared to SN 1999em, SN 2012aw or SN 2013ab.\nThe late-time bolometric light curve can be used to estimate the 56Ni and total masses ejected in the explosion. We assume a model of partial trapping of gamma rays, where the energy released by Cobalt decay is scaled by the transparency factor *I*(*t*) = (1 − exp( − (*T*0/*t*)2)), and the characteristic time is given by *T*0 = *C*(*η*, *κ**γ*)*M**E*− 0.5. We assumed a fixed density slope *η* = 10 and constant gamma-ray opacity *κ**γ* = 0.06 (which involves assuming an electron fraction per nucleon of *Y**e* = 0.03). The late-time light curve and fitted model are plotted in Figure [fig:Ni56]. The best fit provides *T*0 = 319 ± 33 days, which would correspond to an ejecta mass of 12 ± 1*M*⊙ for an explosion energy of 1051 erg (1 foe). The 56Ni mass results 0.0210 ± 0.0025 M⊙, which is slightly lower than the median of 0.031 M⊙ corresponding to the Type II SNe sample studied by.\nLate-time light curves that decay faster than Cobalt are expected in SNe with low mass ejecta, as in the case of stripped enveloped SNe and some fast declining Type II SNe. Given the low energetics shown in, for example, the expansion velocities, the ejected mass estimated above is consistent with the low mass range of Type IIP SNe progenitors.\nThe ejected mass estimated with the late-time light curve can be compared with that obtained in the previous section using the early light curve. The scaling in this case is *M**e**j* ≈ 12.0 × *E**e**x**p*0.5 M⊙ which implies an ``agreement\" pair (*M**e**j*, *E**e**x**p*) at (6 M⊙,0.25 foe).\n discusses the observations of higher deviation from the expected full trapping decay for the brightest Type II SNe in their sample (in the *V*-band) and they suggest that this would be explained by the expected more diluted (i.e., less massive ejecta or more extended) hydrogen associated with faster declining SNe. In our case, the ejecta mass is slightly below what would be needed for a relatively low energy explosion of ≈ 0.25 foe to have a high gamma ray opacity (*τ**γ* = (*T*0/*t*)2 ≥ 1) at ≈ 400 days. \nWe can also estimate the ejected 56Ni mass using its correlation with the FWHM of H*α*. A value of $\\rm FWHM \\simeq 35$ Å gives us a 56Ni mass of 0.0118 ± 0.0012 M⊙. This is lower than, but consistent with, the estimate obtained from the late-time bolometric light curve.\nThe low nickel mass log(*M*56*N**i*) ≈ − 1.5 agrees with the explosion parameters according to the classic explosion models for Type II SNe and observed correlations.\nComparison of bolometric light curves using the bolometric corrections of. Each SN was corrected for extinction. See Table [tab:compTable] for references of the comparison sample. [fig:bolall]\nBolometric light curves in the radioactive tail derived from the *B* and *V*-band photometry from EFOSC imaging, using the bolometric corrections of (PP15, blue dots) and (MH09, green squares). The decay is faster than the expected from full trapping of 56Co decay (showed as a dashed black line). The non-full trapping model from gives a 56Ni Mass of 0.0210 ± 0.0025 *M*⊙ for the bolometric corrections. The time scale for this fit is *T*0 = 319 ± 33 days, which corresponds to a ejecta mass of 12 *M*⊙ for an explosion energy of 1 foe.[fig:Ni56]\nNebular Spectra and the Progenitor Mass\n---------------------------------------\nIn the nebular phase, spectra of Type IIP like SNe show prominent H*α*, O I, Ca II and Fe II emission lines. The [O I] *λ**λ*6300, 6364 is used to estimate the Oxygen core mass and therefore the main sequence mass of the progenitor star. We compare our spectra with models from, and., in particular, presented radiative transfer models for Type II SNe using a grid of 15 M⊙ stellar models from MESA STAR, with varying mixing length, metallicity, and rotation. We choose the nebular spectra in their ``m15\" model, which reach up to 400 days past explosion (see Table 1 in D13 for details)., calculated models for 12, 15, 19 and 25 M⊙ stars evolved using the KEPLER code, assuming Solar metallicity and no rotation. We also included the recent model of 9 M⊙ from.\nIn Figure [fig:NebLuc] and [fig:NebJerk] we compare the data and the models for the [O I] *λ**λ*6300, 6364 and H*α* lines. In the case of D13 models m15z2m2 and m15z8m3 provide the best fit. They correspond to a progenitor metallicities of *Z* = 0.02 and 0.008, oxygen mass of *M**O* = 0.325 and 0.507 M⊙, and nickel masses of $M(^{56}{\\rm Ni}) = 0.05$ and 0.081 M⊙, respectively (for details see Table 1 in D13). In the case of the J14 and J18 models the 9 M⊙ and 12 M⊙ are closest to the data, although all the models over-predict the H*α* emission line. A problem with this direct comparison is that the explosion models presented in D13, J14 and J18 produce higher 56Ni masses than measured for ASASSN-14jb. If the spectra are normalized by the H*α* emission to approximately take care of this difference, the J14 models of 15 M ⊙ and 9 M⊙ become closest to the data. On the other hand, the D13 models reveal a high degeneracy given their differences in stellar evolution.\nOther approach we can use to alleviate the impact in the spectra of the 56Ni mass discrepancy between ASASSN-14jb and the models, is to normalize the flux in a given line by the 56Ni decay power. Doing this for the [O I] *λ**λ*6300, 6364 line, we obtain a ratio of ≈ 0.04. This leads to a surprising better consistency with the higher mass range of J14 models (15 − 19 M⊙), but again to a better fit of the 9 M⊙. According to this results both from the oxygen shell in the 9 M ⊙ model lying closer to the 56Ni and a significant contribution of Fe I to the [O I] *λ**λ*6300, 6364 doublet.\nAs a final estimate we calculate an upper limit to the emitting oxygen mass using the analytic formula provided by J14,\n \n$$max (M\\_{\\mathrm{O \\ I}}) = \\frac{L\\_{\\mathrm{[O \\ I]}}/\\beta\\_{\\mathrm{[O \\ I]}}}{9.7\\times{10^{41}}} \\times{\\exp\\left(\\frac{22720}{T}\\right)},$$ \nwhere *L*[O \\ I] is the luminosity of the [O I]*λ**λ*6300, 6364 doublet in erg/s, *β*[O \\ I] is its Sobolev escape probability, and *T* the equilibrium temperature in K. A temperature in the range 3900–4300 K was estimated from the ratio between the lines [O I] *λ**λ*6300, 6364 and [O I] *λ*5577[6](#fn6). The resulting values of *m**a**x*(*M*O \\ I) fall in the range 0.09–0.18 M⊙, which is 20 − 40% above that of the 9 M⊙ model of J18 and 1.5–3 times lower than that of the 14 M⊙ model of J14.\nWe conclude that the nebular spectra confidently points at a low mass progenitor for CCSNe standards, in the range ∼ 10 − 12 M⊙. This is supported by surveys of theoretical explosions spanning a wide grid of progenitor properties, which show that models above ∼ 12.5M⊙ are much more efficient at producing oxygen than below that ZAMS mass.\nComparison of the nebular spectra of ASASSN-14jb with the NLTE models of at ≈ 400 days past explosion. The vertical dashed lines shows the rest wavelength of H*α* and the [O I] *λ**λ*6300, 6364 doublet. [fig:NebLuc]\nComparison of the nebular spectra of ASASSN-14jb with the NLTE models of at ≈ 400 days past explosion. The vertical dashed lines shows the rest wavelength of H*α* and the [O I] *λ**λ*6300, 6364 doublet.[fig:NebJerk]\nWarm Dust in the Ejecta\n-----------------------\nASASSN-14jb is clearly detected in the 2015-09-13 *Spitzer* images, with *m*3.6 = 17.85 ± 0.03 mag and *m*4.5 = 16.38 ± 0.03 mag (magnitudes in the AB system), but undetected on 2016-08-17. The implied absolute magnitudes (*M*3.6 ≃ − 14.1, *M*4.5 ≃ − 15.6) and red color (*m*3.6 − *m*4.5 ≃ 1.5 mag) at ∼ 333 days after explosion are consistent with warm dust formation in other Type II SNe. We used a black body function with dust emissivity *Q**λ* ∝ *λ*− 1 to fit the mid-infrared SED and found *L* ∼ 1.1 × 106 L⊙, *T* ∼ 444 K, and *R* ∼ 839 AU. Using Equation 1 in (from ), we obtain a total dust mass of *M**d* ∼ 10− 4 M⊙. These estimates are consistent with models of newly formed dust in the SN ejecta and observations for other Type II SNe at a similar epoch after explosion.\nProgenitor Metallicity\n----------------------\nThe pseudo-equivalent widths (pEWs) of the Fe II *λ*5018 line during the photospheric epoch of Type IIP SNe are a proxy for the progenitor metallicity. This has been shown both by models (D14) and correlation of observed pWEs with metallicities of H II regions in the host galaxies. We measured the pEW evolution during the plateau phase of ASSASN-14jb and compare it with those of D13 models. The result is shown in Figure [fig:pEW]. The models span four progenitor metallicities (2 *Z*⊙, 1 *Z*⊙, 0.4 *Z*⊙ and 0.1 *Z*⊙) and four different mixing-length scales for the Solar composition model, which greatly change the progenitor radius. We see that the observed pEWs of the Fe II line are most consistent with the curve for 0.4 *Z*⊙, but the Solar metallicity model for the progenitor with largest radius is also close enough. If we linearly interpolate the models and, assume that the dispersion given by different radius at Solar metallicity is a fair estimate of the uncertainty, we obtain Z = 0.3 ± 0.1*Z*⊙ for ASASSN-14jb.\nMeasurements of the pseudo-equivalent width of the Fe II *λ*5018 line for ASASSN-14jb (black stars), with the correspondent epoch since explosion *t*0 = 56946.1 ± 3 (MJD). The solid lines show models for progenitors with different metallicities and the dashed line model progenitors with Solar metallicities and different initial radii. [fig:pEW]\nThe Host Galaxy of ASASSN-14jb\n==============================\nStatistical constrains on properties of the progenitor stars can be also establish through the study of the galactic environment of the SNe. Relevant to our discussion are recent integral field spectroscopy observations (IFS) which have provided metallicities and ages of nearby stellar populations.\nOf particular interest are the studies of H II regions, which rapidly fade away after the massive stars which provide the ionizing photons explode as SNe. Since typical lifetimes are up to a few tens of Myr they trace well the ongoing star formation. Typical sizes of giant H II regions are in the hundreds hundreds parsecs. Our MUSE observations of ASASSN-14jb were done with a seeing of 1 in the *R*-band, providing a resolution of ∼ 130 pc at the distance of the host.\nWe built 2D *H**α* emission maps of ESO 467-G051 from the MUSE datacube following the procedures of. In brief, we fitted the spectrum of each spaxel using a combination of the stellar continuum modeled with the stellar population synthesis code STARLIGHT, with Gaussian profiles at the central wavelengths of the emission lines detected with SNR greater than 3, and we corrected the line ratios using the Galactic reddening map of and an estimate of the internal extinction in ESO 467-G051 using the Balmer decrement. In addition, we used the H II-Explorer code to detect and analyze H II regions in the datacube.\nOverall Properties of ESO 467-G051\n----------------------------------\nThe host galaxy of ASASSN-14jb, ESO 467-G051, is an edge-on, Scd galaxy. Our observations, together with extensive data in the literature, make it possible to measure integrated fluxes from the UV up to the IR.\nTo do so, we (1) used co-added LCGOTN images with good seeing and measured the total *B**V**g**r**i* magnitudes using a large elliptical aperture in *ds9*; (2) used a near-infrared *K**s*-band image of the field from the ESO data archive[7](#fn7) to measure the magnitude of the host within an elliptical aperture calibrated to 2MASS stars in the field; (3) obtained archival ultraviolet magnitudes from GALEX and mid-infrared magnitudes from *Spitzer*’s S4G survey. The integrated apparent magnitudes are presented in Table [tab:hostmags]. Using our distance, the absolute magnitudes of ESO 467-G051, corrected by Galactic extinction, are *M**B* ≈ − 17.6 mag, *M**V* ≈ − 18.0 mag, and *M**K**s* ≈ − 19.1 mag.\nFigure [fig:histogramhost] shows a histogram with the distribution of host galaxy *M**K**s* magnitudes of Type II SNe discovered by ASAS-SN in 2013-2017 from. The host galaxy of ASASSN-14jb is in the lower *K**s* absolute magnitude (lower stellar mass) range of the Type II SNe discovered by ASAS-SN.\nThe stellar mass, age and SFR of ESO 467-G051 can be estimated from these data. We fitted stellar population synthesis (SPS) models to the *F**U**V*, *N**U**V*, *B**V**g**r**i*, *K**s*, 3.6 *μ*m, and 4.5 *μ*m magnitudes using FAST. We assumed a Galactic extinction law, a Salpeter IMF, an exponentially declining star formation law ($\\rm SFR \\propto e^{-t/\\tau}$, with *τ* = 1 Gyr), and the SPS models. The best-fit model provides the following parameters: *M*⋆ ≃ 1 × 109 M⊙, $\\rm age \\simeq 3.2$ Gyr, and $\\rm SFR \\simeq 0.07$ M⊙/yr, where the uncertainties are 1*σ*. The SFR can be also estimated from the H*α* extinction-corrected fluxes, which provides $\\rm SFR(H\\_{\\alpha})\\sim 0.07$ M⊙/yr. This is fully consistent with the SED SPS fit.\nNeutral hydrogen (HI 21 cm) observations suggest that the host galaxy of ASASSN-14jb is undergoing a direct encounter with 467-ESOG050 or NGC 7259, the host galaxy of SN 2009ip. The total HI flux is *S**H**I* = 23.1 ± 0.8 Jy km/s and the derived HI mass is *M**H**I* = 3.1 ± 0.8 [109M⊙] [8](#fn8). Using our distance estimate (25 ± 1 Mpc) the HI mass scales to *M**H**I* = 3.5 ± 0.9 [109M⊙]. This gives a gas fraction of *M**g**a**s*/(*M**g**a**s* + *M*\\*) ≈ 0.77.\nThe maximal HI rotational velocity is 60 ± 1 km/s (Hyperleda database[9](#fn9)). In our H*α* velocity map we observe this velocity limit more clearly on the East side of the galaxy.\n [fig:histogramhost]\nH*α* and Oxygen Abundance Maps\n------------------------------\nOur map of the H*α* dereddened emission line fluxes is given in Figure [fig:Halpha]. It includes H II regions, diffuse emission in the host, and the nebular emission from ASASSN-14jb. The position of the H II regions detected with the H II-Explorer code, and that of the SN, are marked. ASASSN-14jb is at a vertical distance of ≈ 2.1 kpc from the disk major axis. The H II region nearest to the SN is at of ≈ 1.4 kpc.\nThe spectra of the H II regions can be used to estimate the gas-phase oxygen abundance. We did so using the O3N2 and N2 strong emission line methods. The median obtained, with 99% confidence intervals, were $\\rm 12 + log(O/H) = 8.27^{+0.16}\\_{-0.20}$ and 8.19− 0.24+ 0.10, respectively. A new strong line calibration by, hereafter D16, which uses [S II], [N II], and H*α* lines, provides significantly lower values, with a median of $\\rm 12 + log(O/H) = 7.77^{+0.10}\\_{-0.24}$. The D16 and O3N2 maps are presented in Figure [fig:D12O3N2]. Median values of the maps are 12 + log(*O*/*H*) = 7.77− 0.10.19 and 12 + log(*O*/*H*) = 8.22− 0.140.11, for the spaxels with S/N ≥ 10.\nThe distributions of oxygen abundances obtained with the N2, O3N2 and D16 nebular emission line diagnostics is given in Figure [fig:Zhist]. All the mean values are sub-Solar (using as reference $\\rm 12+log(O/H)\\_{\\odot} = 8.69$, from ), and significantly below $\\rm 12+log(O/H)\\_{\\odot} = 8.54$, the average for H II regions near Type II SNe of the the PMAS/PPAK Integral-Field Supernova Host Compilation (PISCO). Our host galaxy *B*-band absolute magnitude, of *M**B* = − 17.6 mag, compares well with the mean *B*-band magnitude of *M**B**h**o**s**t* = − 17.66 mag in the recent sample of Type II SNe in low-metallicity, dwarf galaxies. Also, as showed in, lower luminosity galaxies hosts lower metallicity Type II SNe, filling the space between the 0.1 and 0.4 Z⊙ metallicity models. This is expected given the strong correlation between stellar mass (or galaxy luminosity) and metallicity. It is worthwhile to mention that the O3N2 map shows lower oxygen abundances in the cores of the H II regions. This could be a consequence of the unaccounted internal gradient of the ionization parameter of each H II region in the empirical strong line methods.\n[ht!]![H\\alpha MUSE map of the host galaxy and the supernova explosion site. Black circles marks the position of the H II regions detected with H II-Explorer. Coordinates are the right ascension and declination offsets from the center of ESO 467-G051, in arcminutes. The 1 kpc scale of the image is marked in the upper left.[fig:Halpha]](Ha_Blues.png \"fig:\")\n![D16 (top) and O3N2 (bottom) oxygen abundance maps of the host galaxy ESO467-G051 as seen by MUSE, shifting the D16 map by +0.4 dex to provide an easier comparison. Coordinates are the right ascension and declination offsets from the center of ESO 467-G051, in arc-minutes[fig:D12O3N2]. The 1 kpc scale of the image is marked in the upper left.](Metallicity_hot.png)D16 (top) and O3N2 (bottom) oxygen abundance maps of the host galaxy ESO467-G051 as seen by MUSE, shifting the D16 map by +0.4 dex to provide an easier comparison. Coordinates are the right ascension and declination offsets from the center of ESO 467-G051, in arc-minutes[fig:D12O3N2]. The 1 kpc scale of the image is marked in the upper left.\nD16 and the N/O ratio\n---------------------\nThe significant discrepancy of the D16 with other strong line oxygen abundance calibrations calls our attention.This abundance scale relies on a specific relation between the N/O and O/H ratios for which it is useful to consider a diagnostic independent on any a priori relation. developed a semi-empirical code, *H II-CHI-MISTRY*, that derives O/H, N/O and the ionization parameter U, using the reddening-corrected fluxes of [O II] *λ*3727, [O III] *λ*4363, *λ*5007, [N II] *λ*6584 and [S II] *λ**λ*6717,6731, relative-to- H*β*. When the O auroral lines are absent, as is our case, the code uses a limited grid of empirically constrained models to provide abundances that are consistent with the ``direct\" electron temperature measurement method. We applied *H II-CHI-MISTRY* v3.0 with the accessible lines [O I] *λ*5007, [N II] *λ*6584 and [S II] *λ**λ*6717,6731. The output N/O ratio is shown as a function of the oxygen abundances in Figure [fig:NOHU]. We observe a clear offset from the D16 calibration. The mean of the oxygen abundances is 8.53 ± 0.13 while the N/O ratio has a mean of − 1.63 ± 0.1. There is a trend with the ionization parameter so that higher oxygen abundances show lower N/O and U values (same with H*α* luminosity instead of U) and, vice versa, the lower abundances correspond to higher N/O ratios and also they are closer to the D16 prescription. The negative correlation between N/O and O/H, together with the trend with H*α* luminosity, has been observed in blue compact dwarf galaxies.\n![Distribution of the gas-phase oxygen abundances used for ESO467-G051 H II regions, based on the D16, O3N2 and N2 calibrations. The dashed lines represent the median for each distribution and the black vertical line indicates the average abundance for Type II SNe hosts from the PISCO compilation using the D16 calibration.[fig:Zhist]](O_H_dist.png)Distribution of the gas-phase oxygen abundances used for ESO467-G051 H II regions, based on the D16, O3N2 and N2 calibrations. The dashed lines represent the median for each distribution and the black vertical line indicates the average abundance for Type II SNe hosts from the PISCO compilation using the D16 calibration.[fig:Zhist]\nOxygen abundances versus N/O ratios obtained with the code developed in. Each point has the errorbars delivered by the code and each point is colored according to the ionization parameter U delivered. The solid orange line indicates the relation as assumed in the D16 calibration and the blue dots correspond to the observed values for our H II regions abundances using D16. The dashed lines indicate the asymptotes for orange line, at the very high and very low O/H ratios regimes.[fig:NOHU]\nDiscussion\n==========\nASASSN-14jb is a normal Type IIP SN located at a projected distance of ∼ 2.1 kpc from the mid-plane and ∼ 1.4 kpc from the closest, diffuse H II region of ESO 467-G051. Since Type II SNe come from stars with short life times, the unusual location begs for an explanation. Several hypothesis have been advanced for SNe with no clear parent galaxy. One is that the host is an undetected low-surface brightness (LSB) galaxy, or even a small fragment of a disrupted satellite galaxy. Other interesting possibility is that the progenitor of ASASSN-14jb traveled ≈ 1.4 kpc from its birthplace driven by gravitational interaction.\nType II SNe from Runaway Stars\n------------------------------\nDifferent scenarios have been proposed for SNe coming from runaway stars. One is the Binary Supernova Scenario (BSS) where they are pushed by the interaction with a massive binary companion that exploded as a supernova. Other is the Dynamical Ejection Scenario (DES), where the impulse comes from close encounters in clusters, including those with binaries, very massive stars, or intermediate or massive black-holes. Yet another is the hyper-velocity star (HVS) scenario, resulting from the so-called Hills Mechanism, where velocities higher than 100 km/s are provided by interaction with a supermassive black-hole at the center of the galaxy.\nThe approximate lifetime of a star with ZAMS mass of ∼ 10*M*⊙ in single stellar evolution is ∼ 30 Myr. Covering the ∼ 1.4 kpc from the nearest star forming region in this time would require a projected space velocity of ∼ 47 km/s. Considering the lifetime of the least massive stars that undergo core collapse in the models of, ∼ 48 Myr for ∼ 8 M⊙ progenitors, reduces the requirement to ∼ 30 km/s. Both are plausible velocities given the observed population of runaway stars in the Galaxy and the Magellanic Clouds. The velocity distribution of early type runaway stars in the"},"data_id":{"kind":"string","value":"arxiv_0183121"}}},{"rowIdx":45797,"cells":{"content":{"kind":"string","value":"Corona Decompositions for ParabolicUniformly Rectifiable Sets\n=============================================================\n[1](#fn1)\nWe prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together with our previous work, this allows us to conclude that if $\\Sigma\\subset\\mathbb R^{n+1}$ is parabolic Ahlfors-David regular, then the following statements are equivalent.\n1. Σ is parabolic uniformly rectifiable.\n2. Σ admits a corona decomposition with respect to regular Lip(1,1/2) graphs.\n3. Σ admits a bilateral corona decomposition with respect to regular Lip(1,1/2) graphs.\n4. Σ is big pieces squared of regular Lip(1,1/2) graphs.\nIntroduction\n============\nThe results proved in this paper are part of a program devoted to providing parabolic analogues of the influential work of G. David and S. Semmes concerning uniformly rectifiable sets. At the time was written, tools from the Calderón-Zygmund school of harmonic analysis were being rapidly adapted to less classical and rougher geometrical settings. A particularly important development was the proof of the *L*2 boundedness of the Cauchy integral on Lipschitz curves, and subsequent developments concerning the boundedness of singular integral operators of Calderón type on Lipschitz graphs and curves (see also ). In addition, the Calderón-Zygmund theory was extended beyond the Euclidean setting (see e.g. ).\nWhen restricting to the setting of *d*-dimensional subsets of *n*-dimensional Euclidean space, David and Semmes found necessary and sufficient conditions on these sets for the validity of the *L*2 boundedness of *d*-dimensional singular integral operators. David and Semmes coined the sets having this property *uniformly rectifiable*, and through a remarkable series of works they provided numerous characterizations of these sets. Of particular relevance to this paper is the following theorem stating a subset of the equivalent characterizations of uniformly rectifiable sets. We refer to for exact definitions of the different notions appearing in Theorem [DS1URCorona.thrm].\n[][DS1URCorona.thrm] Let *E* ⊂ R*n* + 1 be *n*-dimensional Ahlfors(-David) regular. Then the following statements are equivalent.\n1. *E* is uniformly rectifiable.\n2. *E* admits a corona decomposition with respect to Lipschitz graphs.\n3. All sufficiently regular convolution type Calderon-Zygmund operators with odd kernels are *L*2 bounded on *E*.\nThe purpose of this paper is to prove that the implication (a) $\\implies$ (b) holds in the context of *parabolic* uniformly rectifiable sets (see Definition [def1.UR]). To prove that the implication (c) $\\implies$ (b) holds in the parabolic setting is, as of writing, an interesting and open problem. Combining the main results established in this paper, see Theorem [main] and Theorem [main.bil.thrm] below, with the results of our previous work, we obtain the following.\n[thrm.char] Let Σ ⊂ R*n* + 1 be parabolic Ahlfors-David regular (see Definition [def1.ADR]). Then the following statements are equivalent.\n1. Σ is parabolic uniformly rectifiable.\n2. Σ admits a corona decomposition with respect to regular Lip(1,1/2) graphs.\n3. Σ admits a bilateral corona decomposition with respect to regular Lip(1,1/2) graphs.\n4. Σ is big pieces squared of regular Lip(1,1/2) graphs.\nAll implications stated in Theorem [thrm.char], but the implications (a) $\\implies$ (b) and (a) $\\implies$ (c), are contained[2](#fn2) in. We will prove[3](#fn3) directly that (a) and (b) imply (c) by essentially re-running the machine that we use to prove that (a) $\\implies$ (b), and by using additional information gleaned from the fact that (b) $\\implies$ (d). This means that the bulk of the paper is mainly devoted to the proof of the implication (a) $\\implies$ (b) and our proof will involve a very careful adaptation of the proof in, along with some additional arguments utilizing ideas from. The implication (a) $\\implies$ (b) is the content of Theorem [main] stated in precise form below. The precise statement of the result that (a) $\\implies$ (c) is given in Theorem [main.bil.thrm].\nThe study of parabolic uniformly rectifiable sets Σ ⊂ R*n* + 1 emanates in, where the third and fifth author, together with J. Lewis, introduced the notion of parabolic uniformly rectifiable sets. In the existence of big pieces of regular Lip(1,1/2) graphs, under the additional assumption that Σ is Reifenberg flat in the parabolic sense, is established. The results in, were the first of their kind in the context of parabolic problems and the studies, were motivated by the study of parabolic/caloric measure in rough time-varying domains.\nBy definition, a Lip(1,1/2) graph is the graph of a function which is Lipschitz with respect to the parabolic metric. A Lip(1,1/2) graph is a regular Lip(1,1/2) graph if, in addition, the 1/2-order time derivative of the function defining the graph is in (parabolic) BMO (see Definition [goodgraph.def]). Combining the results of Section 2 in with the developments in, it follows that a Lip(1,1/2) graph is parabolic uniformly rectifiable in the sense of, if and only if it is a regular Lip(1,1/2) graph. This explains the importance of regular Lip(1,1/2) graphs in the statement of Theorem [thrm.char], and in the statements of Theorem [main] and Theorem [main.bil.thrm]. As proved in,, the class of regular Lip(1,1/2) graphs is strictly smaller than the class of Lip(1,1/2) graphs.\nThe notions of regular Lip(1,1/2) graphs and parabolic uniform rectifiability are deeply rooted in the study of the Dirichlet problem for the heat equation in time-varying (graph) domains, and the solvability of the *L**p*-Dirichlet problem for the heat equation is intimately connected to quantitative mutual absolute continuity of the caloric measure and the surface measure. In particular, in it was proved that there are Lip(1,1/2) graph domains for which caloric measure and surface measure are mutually singular. Later, in their pioneering work, Lewis, Silver and Murray proved that for regular Lip(1,1/2) graph domains, the caloric measure and the surface measure are quantitatively related in the sense that they are mutual absolutely continuous, and the associated parabolic Poisson kernel satisfies a scale-invariant reverse Hölder inequality in *L**p* for some *p* ∈ (1, ∞). The importance and relevance of regular Lip(1,1/2) graph domains, from the perspective of parabolic singular integrals, layer potentials, boundary value problems and inverse problems, is emphasized through the works in. In particular, in the solvability of the *L*2-Dirichlet problem (and of the *L*2-Neumann problem) for the heat equation, using layer potentials, was obtained in the region above a regular Lip(1,1/2) graph under the restriction that the 1/2-order time derivative (measured in BMO) of the function defining the graph is small[4](#fn4). The highly influential work (see also ), devoted to parabolic operators with singular terms, should also be mentioned in this context. In particular, in the method of extrapolation of Carleson measure estimates was introduced, a method which was crucial in the resolution of the Kato conjecture, see,.\nIn, parts of the analysis on regular Lip(1,1/2) graph domains mentioned above was extended beyond the setting of graphs. However, other than, only a few notable works in rougher parabolic settings have appeared. This is in contrast to the corresponding elliptic contexts, see, for example,, and it is a long term goal of the authors to develop a corresponding parabolic theory. The present paper is one of the starting points of that effort. In particular, in forthcoming work we will show, using Theorem [thrm.char], that the main theorem in can be extended to the parabolic setting. In addition, in we have recently obtained, by expanding on, a flexible parabolic analogue of the work of David and Jerison. Still, to establish parabolic analogues of, for example, remain open problems which may require new methods and tools.\nAs mentioned, in the elliptic setting, some of the most significant recent progress in this area relies on methods which are currently unavailable in the parabolic setting. Thus, to prove parabolic versions of those results, one will require significantly new ideas. However, to prove the results of this paper, we are able to develop parabolic versions of the methods in. We prove two main results, whose statements may be found below. First, Theorem [main], whose elliptic analogue appears in, and we give the proof of Theorem [main] in Section [subsgraphconst], Section [pushsqfn] and Section [packsttime]. Second, Theorem [main.bil.thrm], whose elliptic analogue appears in, and we give the proof of Theorem [main.bil.thrm] in Section [s5]. Concerning Theorem [main], it turns out that we can follow quite closely the corresponding arguments in. The reader who is familiar with that work will certainly recognize substantial parts of the proof. The novelties in Section [subsgraphconst], Section [pushsqfn] and Section [packsttime] are in many respects mainly of technical nature: they arise here to treat the extension to the parabolic setting, and they rely in part on some ideas from. On the other hand, the corona decomposition is of central importance and utility in the theory of quantitative rectifiability, and we believe that this fact justifies a complete and careful treatment in the parabolic case. Finally, we note that the converse to Theorem [main], which we have established (in particular) in, does indeed rely on some new methods that are significantly different to those previously appearing in the literature.\nAs we have previously mentioned in, it is true that in, the author took on the ambitious challenge to develop the theory of parabolic uniformly rectifiable sets. Unfortunately though, at least in the papers in, the author either gives no proofs of his claims or he supplies proofs which have fundamental errors. In the present work we give a correct proof of the main result claimed in, and we shall explain momentarily two principal errors (as well as a third, more technical error) in the latter paper. The errors in are discussed in some detail in our paper, which gives a correct proof of those results.\nIn the author claims to prove that parabolic uniform rectifiability implies corona decompositions with respect to regular Lip(1,1/2) graphs. In, the author, as we do here, follows rather closely the outline of the proof of the corresponding elliptic result in, while also, as we do, employing some ideas from in order to adapt the arguments of to the parabolic case. It is in the latter context that the proof in breaks down, for the following reason: in Lemma 2.7 in, the constant *ĉ* cannot necessarily be taken small, given only the assumption of parabolic uniform rectifiability. In, the constant is small by virtue of the assumption of Reifenberg flatness used in that paper. Without smallness, the sets {*E**i*}, introduced in subsection 2.3.1 in, need not, as claimed, have bounded overlaps and therefore the estimate in display (20) in is not valid. This is a critical gap in the proof: it means that in there is no proof of the ``regularity\" (see Definition [goodgraph.def] below for the definition of a *regular* Lip(1,1/2) graph) of the approximating graphs, which is the essential feature required of the graphs in the parabolic setting (and which has no analogue in the elliptic case); thus, the proof in breaks down precisely at the point in the argument where one needs to do something new in the parabolic setting that has no counterpart in the corresponding elliptic proof. Below we deal with this problem by using a counting function introduced in, see Lemma [count] below.\nA second critical error in appears in the claimed proof of, which in turn relies[5](#fn5) on the claimed result. In fact, the latter is false. Indeed, in contrast to the situation in the elliptic setting, parabolic uniform rectifiability is *not* characterized by Carleson set conditions such as the ``Bilateral Weak Geometric Lemma\", even in the case of a Lip(1,1/2) graph; see for a discussion of this issue and a counter-example. Thus the proof of collapses.\nIn addition, we mention a technical (but still serious) error, in the crucial estimate in the display between displays (32) and (33) in. Here, the author claims a Poincaré inequality that need not hold if the function *ψ*2 depends on time, which it certainly does since the problem is parabolic; thus, again the error arises because the author in did not address the difference between the elliptic and parabolic settings. In this context, the correct estimates, based on a more refined inequality of Poincaré type appropriate to the parabolic setting, are given in our Lemma [l3] below.\nThe rest of the paper is organized as follows. Section [pre] is mainly of preliminary nature and we here provide notation and definitions. In Section [mr] we state our main result: Theorem [main] and Theorem [main.bil.thrm]. In Subsection [s4] we briefly discuss the outline of the proof of Theorem [main] to be presented in subsequent sections. The proof is divided into several parts and our argument adapts the corresponding arguments of, also utilizing some ideas from, to deal with the extension to the parabolic setting. The proof of Theorem [main] is given in Section [subsgraphconst], Section [pushsqfn] and Section [packsttime] and progresses from the construction of a Lip(1,1/2) graph with small constant for each stopping time regime, to the pushing of the geometric square function estimate (the Carleson measure estimate) to the graph, also verifying that the graph is actually a regular Lip(1,1/2) graph, and finally to the packing of the stopping time regimes. In Section [s5] we then prove Theorem [main.bil.thrm] by improving Theorem [main](4) to give a bilateral approximation as in Theorem [main.bil.thrm](4). The paper ends with two appendix containing a few technical estimates and observations used in the paper.\nPreliminaries\n=============\nIn this section we provide notation and definitions. We will for simplicity assume *n* ≥ 2. Our results also remain valid in the case *n* = 1, but in this case some notations used in the paper have to be adjusted. Points in Euclidean (*n* + 1)-space $ \\mathbb R^{n+1} $ are denoted by (*X*, *t*) = (*x*1, …, *x**n*, *t*), where $ X = ( x\\_1, \\dots,\nx\\_{n} ) \\in \\mathbb R^{n } $ are the spatial coordinates and *t* represents the time-coordinate. We set \n*d* : = *n* + 1. \n In general, *c* will denote a positive constant satisfying 1 ≤ *c* < ∞. We write $c\\_1\\lesssim c\\_2$, if *c*1/*c*2 is bounded from above by a positive constant depending on the structural constants of a lemma or theorem (e.g. *n*, *d*, the constant defining the Ahlfors-David regularity or the constants defining parabolic uniform rectifiability). We write *c*1 ∼ *c*2, if $c\\_1\\lesssim c\\_2$ and $c\\_2\\lesssim c\\_1$.\nWe let ⟨ ⋅ , ⋅ ⟩ denote the standard inner product on $ \\mathbb R^{n} $ and we let ∣*X*∣ = ⟨*X*, *X*⟩1/2 be the Euclidean norm of *X*. We let ∣∣(*X*, *t*)∣∣ : = ∣*X*∣ + ∣*t*∣1/2. Given $(X,t), (Y,s)\\in\\mathbb R^{n+1}$ we let \n*d**p*(*X*, *t*, *Y*, *s*) : = *d**p*((*X*, *t*), (*Y*, *s*)) = ∣*X* − *Y*∣ + ∣*t* − *s*∣1/2, \n and we define *d**p*(*X*, *t*, *E*) to equal the parabolic distance, defined with respect to *d**p*( ⋅ , ⋅ ), from $ (X,t) \\in \\mathbb R^{n+1} $ to $ E\\subset\\mathbb R^{n+1}$. We let diam(*E*) denote the parabolic diameter of *E*, that is, \ndiam(*E*) : = sup{*d**p*(*X*, *t*, *Y*, *s*) : (*X*, *t*), (*Y*, *s*) ∈ *E*}. \nWe let\n \n$$\\begin{aligned}\n\\label{cube1}\n C\\_r ( X, t ) \\, := \\, \\{ ( Y, s )\\in \\mathbb R^{n} \\times \\mathbb{R} : | y\\_i - x\\_i| < r, | t - s | < r^2 \\},\n \\end{aligned}$$ \nwhenever $(X,t)\\in\n\\mathbb R^{n+1}$, *r* > 0, and we will refer to *C**r*(*X*, *t*) as a parabolic cube of ``length\" *r* (in R*n* + 1). For (*x*, *t*) ∈ R*n*, we will use the notation\n \n$$\\begin{aligned}\n\\label{cube2} C'\\_r ( x, t ) \\, := \\, \\{ ( y, s )\\in \\mathbb R^{n-1} \\times \\mathbb{R} : | y\\_i - x\\_i| < r, | t - s | < r^2 \\},\\end{aligned}$$ \nto denote a parabolic cube in R*n*.\nWe let d*X*, d*x*, denote Lebesgue measure on $ \\mathbb R^{n}$, $ \\mathbb R^{n-1}$, respectively, and, given *η* > 0, we let $\\H^\\eta$ denote the (standard Euclidean) *η*-dimensional Hausdorff measure. To continue we need to define Hausdorff measure adapted to the parabolic setting. This is done in a straightforward way replacing the standard Euclidean diameter with its parabolic counterpart.\nFor *η*, *δ* > 0, *E* ⊆ R*n* + 1, we define \n$$\\H^\\eta\\_{p,\\delta}(E) = \\inf\\left\\{ \\sum \\operatorname{diam}(E\\_j)^\\eta: E \\subseteq \\cup\\_j E\\_j, \\operatorname{diam}(E\\_j) \\le \\delta \\right\\}.$$ \n We define the parabolic Hausdorff measure, of homogeneous dimension *η*, as\n \n$$\\begin{aligned}\n\\label{hme}\n\\H^\\eta\\_p(E) := \\lim\\_{\\delta \\to 0^+} \\H^\\eta\\_{p,\\delta}(E).\\end{aligned}$$ \nNote that the limit in exists (possibly as + ∞) since $\\H^\\eta\\_{p,\\delta}(E)$ is a decreasing function on (0, ∞).\nIn the sequel, $\\Sigma \\subset \\mathbb R^{n+1}$ will denote a closed set which will be clear from the context. Below we will, for (*X*, *t*) ∈ Σ and *r* > 0, use the notation Δ(*X*, *t*, *r*) = Δ*r*(*X*, *t*) : = Σ ∩ *C**r*(*X*, *t*) to denote a (parabolic) ``surface ball\". We define\n \n$$\\label{sigdef}\n\\sigma := \\sigma\\_{\\Sigma} := \\H^{n+1}\\_p|\\_\\Sigma,$$ \nto denote the parabolic surface measure on Σ.\nAlso other notions of surface measure can be considered in the parabolic context. Indeed, we let\n \n$$\\label{slicewise.def}\n \\mu ( E ) := \\int\\_{\\mathbb{R}}\\int\\_{{\\mathbb{R}^n}\\times \\{t\\}} \\mathbbm{1}\\_{E}(X,t) {\\, \\mathrm{d}}\\H^{n-1}(X) \\, {\\, \\mathrm{d}}\\H^{1}(t),$$ \nwhere $\\mathbbm{1}\\_{E}$ is the indicator function for the set *E*, denote a slice-wise or product-like measure of the set $E\\subset\\mathbb R^{n+1}$. This measure has been used in previous works concerning parabolic uniform rectifiability as in, the parabolic surface measure was defined as\n \n*σ**s* : = *μ*∣Σ. \nIn this paper we do not use the measure in, instead we use the measure defined in. We here collect a number of remarks concerning this choice. The non-trivial statements in Remark [r-measures], (i), (v), and (vi), are proved in Appendix [slicevspara.sect] below.\n[r-measures] \n* If $\\sigma^{\\bf s}$ (or for that matter *any* measure m defined on Σ) satisfies the parabolic Ahlfors-David Regularity (p-ADR) condition (see Definition [def1.ADR] below), then so does *σ*, and in that case the two measures are of course equivalent. This follows easily from the definition of the Hp*n* + 1 measure, and it is really just the same phenomenon that occurs in the classical (elliptic) case, see. We give the proof for $\\sigma^{\\bf s}$ in Appendix [slicevspara.sect].\n* Consequently, if Σ is a Lip(1,1/2) graph, then $\\sigma \\approx \\sigma^{\\bf s}$. In particular, on a hyperplane P ⊂ R*n* + 1 parallel to the *t*-axis, which we may identify with Euclidean space R*n*, we have that $\\H^n|\\_{\\mathcal{P}}\\approx {\\mathcal{H}^{n+1}\\_{\\text{p}}}|\\_{\\mathcal{P}}$, since the former is just *n*-dimensional Lebesgue measure on P, which is parabolic ADR on Σ = P.\n* If P is a hyperplane parallel to the *t*-axis, and if *π* is the orthogonal projection operator onto P, then Hp*n* + 1 measure does not increase under the action of *π*. In particular, by virtue of (ii), we have, for any Borel set *A*, that $\\H^n(\\pi(A)) = {\\mathcal{H}^{n+1}\\_{\\text{p}}}(\\pi(A)) \\leq {\\mathcal{H}^{n+1}\\_{\\text{p}}}(A)$.\n* If Σ is parabolic uniformly rectifiable (p-UR set; see Definition [def1.UR] below), defined with respect[6](#fn6) to $\\sigma^{\\bf s}$, then using (i) one deduces[7](#fn7) that Σ is also p-UR defined with respect to *σ*. Thus, *a priori* p-UR with respect to $\\sigma^{\\bf s}$ is a stronger notion, that is, it implies p-UR with respect to *σ*.\n* If Σ is parabolic uniformly rectifiable (p-UR; see Definition [def1.UR] below), with respect to *σ*, then the two measures $\\sigma^{\\bf s}$ and *σ* are equivalent and hence one deduces (in the same way as (iv)) that Σ is parabolic uniformly rectifiable, defined with respect to $\\sigma^{\\bf s}$. Thus, *a posteriori* the notions of p-UR are equivalent.\n* The measures are not equivalent in general, even in the p-ADR setting. In fact, $\\sigma^{\\bf s} \\leq c(n) \\sigma$, but the other direction does not need to hold.\n[tindplane.def] We say that a *n*-dimensional hyperplane *P* ⊂ R*n* + 1 is a *t*-independent plane if *P* contains a line in the *t*-direction. Equivalently, if *N⃗**P* is the normal to *P*, then $\\vec{N}\\_P \\cdot (\\vec{0},1) = 0$. We let P denote the collection of all *t*-independent planes. If *P* ∈ P we let *P**x* ⊂ R*n* be the (*n* − 1)-dimensional plane that defines *P* = *P**x* × R.\nGiven a function $\\psi:\\mathbb R^{n-1}\\times\\mathbb R\\to \\mathbb R$ we let *D*1/2*t**ψ*(*x*, *t*) denote the 1/2 derivative in *t* of *ψ*(*x*, ⋅ ), *x* fixed. This half derivative in time can be defined by way of the Fourier transform or by\n \n$$\\begin{aligned}\n \\label{1.8}\n D\\_{1/2}^t \\psi (x, t) \\equiv \\hat c \\int\\_{ \\mathbb R }\n\\, \\frac{ \\psi ( x, s ) - \\psi ( x, t ) }{ | s - t |^{3/2} } \\, {\\, \\mathrm{d}}s,\\end{aligned}$$ \nfor properly chosen $ \\hat c$. We let ∥ ⋅ ∥\\* denote the norm in parabolic $BMO(\\mathbb R^{n})$ (replace standard cubes by parabolic cubes in the definition of *B**M**O*).\n[goodgraph.def] A function $\\psi:\\mathbb R^{n-1}\\times\\mathbb R\\to \\mathbb R$ is called Lip(1,1/2) with constant *b*1, if\n \n$$\\begin{aligned}\n\\label{1.1}\n|\\psi(x,t)-\\psi(y,s)|\\leq b\\_1(|x-y|+|t-s|^{1/2})\\\\end{aligned}$$ \nwhenever $(x,t)\\in\\mathbb R^{n}$, $(y,s)\\in\\mathbb R^{n}$. If Σ = {(*x*, *ψ*(*x*, *t*), *t*) : (*x*, *t*) ∈ R*n* − 1 × R} in the coordinates *P**x* × (*P**x*)⊥ × R, for some *t*-independent plane *P* ∈ P and Lip(1,1/2) function *ψ*, then we say that Σ is a Lip(1,1/2) graph. We say that $ \\psi = \\psi ( x, t ) : \\mathbb R^{n-1}\\times\\mathbb R\\to \\mathbb R$ is a *regular* Lip(1,1/2) function with parameters *b*1 and *b*2, if *ψ* satisfies and if\n \n$$\\begin{aligned}\n \\label{1.7}\n D\\_{1/2}^t\\psi\\in BMO(\\mathbb R^n), \\ \\ \\|D\\_{1/2}^t\\psi\\|\\_\\*\\leq b\\_2<\\infty.\\end{aligned}$$ \nIf Σ = {(*x*, *ψ*(*x*, *t*), *t*) : (*x*, *t*) ∈ R*n* − 1 × R} in the coordinates *P**x* × (*P**x*)⊥ × R, for some *t*-independent plane *P* ∈ P and *regular* Lip(1,1/2) function *ψ*, then we say that Σ is a *regular* Lip(1,1/2) graph.\n[def1.ADR] Let $\\Sigma \\subset \\mathbb R^{n+1}$ be a closed set. We say that a measure m, defined on Σ, is *parabolic Ahlfors-David regular*, parabolic ADR for short (or simply p-ADR, or just ADR) with constant *M* ≥ 1, if\n \n*M*− 1 *r**d* ≤ m(Δ(*X*, *t*, *r*)) ≤ *M* *r**d*, \nwhenever 0 < *r* < diamΣ, (*X*, *t*) ∈ Σ, *T*0 < *t* < *T*1 and where diamΣ is the (parabolic) diameter of Σ (which may be infinite). If\n \n*M*− 1 *r**d* ≤ *σ*(Δ(*X*, *t*, *r*)) ≤ *M* *r**d*, \nthen we simply say that Σ is parabolic ADR (p-ADR, or just ADR).\nAs noted above (see Remark [r-measures] (i)), if holds for a measure m on Σ, then it holds for *σ* as in, i.e. holds for a possibly different but still universal choice of *M*.\nFollowing, we next introduce parabolic *β*2 numbers, which we will denote by *γ*. These are defined in a way analogous to the usual *β* numbers in except that they incorporate parabolic scalings, parabolic surface measure and they only allow for approximation by *t*-independent planes. Indeed, assume that $\\Sigma \\subset \\mathbb R^{n+1}$ is parabolic ADR in the sense of Definition [def1.ADR]. We then let\n \n$$\\begin{aligned}\n \\gamma( Z, \\tau, r ):=\\gamma\\_\\Sigma( Z, \\tau, r ):= \\inf\\_{P \\in \\mathcal{P}} \\biggl ( \\, \\bariint\\_{ \\Delta ( Z, \\tau,r) } \\, \\biggl (\\frac {d\\_p ( Y,s, P )}{r}\\biggr )^2 {\\, \\mathrm{d}}\\sigma (Y, s )\\biggr )^{1/2}, \\end{aligned}$$ \nwhenever (*Z*, *τ*) ∈ Σ, *r* > 0, and where P is the set of *n*-dimensional hyperplanes *P* containing a line parallel to the *t* axis. We also introduce\n \n$$\\begin{aligned}\n\\label{me}\\quad {\\, \\mathrm{d}}\\nu( Z, \\tau,\nr ) :={\\, \\mathrm{d}}\\nu\\_\\Sigma ( Z, \\tau,\nr ) \\, = \\, ({\\gamma}\\_\\Sigma ( Z, \\tau, r))^2 \\, {\\, \\mathrm{d}}{\\sigma}( Z, \\tau) \\, r^{ - 1 }\n{\\, \\mathrm{d}}r. \\end{aligned}$$ \nRecall that *ν* is defined to be a Carleson measure on Δ(*Y*, *s*, *R*) × (0, *R*), if there exists Γ < ∞ such that\n \n$$\\begin{aligned}\n\\label{1.9}\n \\nu ( \\Delta( X, t,\\rho ) \\times ( 0, \\rho) ) \\, \\leq \\,\n\\Gamma \\, \\rho^{d},\\end{aligned}$$ \nwhenever (*X*, *t*) ∈ Σ and *C**ρ*(*X*, *t*) ⊂ *C**R*(*Y*, *s*). The least such Γ in is called the Carleson norm of *ν* on Δ(*Y*, *s*, *R*) × (0, *R*).\nWe are now ready to give the definition of parabolic uniform rectifiability.\n[def1.UR] Assume that $\\Sigma \\subset \\mathbb R^{n+1}$ is parabolic ADR in the sense of Definition [def1.ADR] with constant *M*. Let *ν* = *ν*Σ be defined as in. Then Σ is parabolic uniformly rectifiable, parabolic UR for short, with UR constants (*M*, Γ) if\n \n$$\\begin{aligned}\n\\label{eq1.sf}\n\\| \\nu \\|:=\\sup\\_{(X,t)\\in\\Sigma,\\ \\rho>0} \\rho^{ -d }\\nu ( \\Delta(X,t,\\rho) \\times ( 0, \\rho) ) \\, \\leq \\,\n\\Gamma.\\end{aligned}$$ \nWe will also need a parabolic dyadic decomposition of Σ. This is a special case of a more general result in.\n[cubes] Assume that $\\Sigma \\subset \\mathbb R^{n+1}$ is parabolic ADR in the sense of Definition [def1.ADR] with constant *M*. Then Σ admits a parabolic dyadic decomposition in the sense that there exist constants *α* > 0, *β* > 0 and *c*\\* < ∞, such that for each *k* ∈ Z there exists a collection of Borel sets, D*k*, which we will call (dyadic) cubes, such that \nD*k* : = {*Q**j**k* ⊂ Σ : *j* ∈ I*k*}, \n where I*k* denotes some index set depending on *k*, satisfying\n \n$$\\begin{aligned}\n(i)&&\\mbox{$\\Sigma=\\cup\\_{j}Q\\_{j}^k\\,\\,$ for each\n$k\\in{\\mathbb Z}$.}\\notag\\\\\n(ii)&&\\mbox{If $m\\geq k$ then either $Q\\_{i}^{m}\\subset Q\\_{j}^{k}$ or\n$Q\\_{i}^{m}\\cap Q\\_{j}^{k}=\\emptyset$.}\\notag\\\\\n(iii)&&\\mbox{For each $(j,k)$ and each $m<k$, there is a unique\n$i$ such that $Q\\_{j}^k\\subset Q\\_{i}^m$.}\\notag\\\\\n(iv)&&\\mbox{$\\operatorname{diam}\\big(Q\\_{j}^k\\big)\\leq c\\_\\* 2^{-k}$.}\\notag\\\\\n(v)&&\\mbox{Each $Q\\_{j}^k$ contains $\\Sigma\\cap C\\_{\\alpha2^{-k}}(Z^k\\_{j},t^k\\_{j})$ for some $(Z^k\\_{j},t^k\\_j)\\in\\Sigma$.}\\notag\\\\\n(vi)&&\\mbox{$\\sigma(\\{(Z,t)\\in Q^k\\_j: d\\_p(Z,t,\\Sigma\\setminus Q^k\\_j)\\leq \\varrho \\,2^{-k}\\big\\})\\leq\nc\\_\\*\\,\\varrho^\\beta\\,\\sigma(Q^k\\_j),$}\\notag\\\\\n&&\\mbox{for all $k,j$ and for all $\\varrho\\in (0,\\alpha)$.}\\end{aligned}$$ \nWe denote by D = D(Σ) the collection of all *Q**j**k*, i.e., \nD : = ∪ *k*D*k*. \n For a dyadic cube *Q* ∈ D*k*, we let ℓ(*Q*) : = 2− *k*, and we will refer to this quantity as the size of *Q*. Evidently, ℓ(*Q*) ∼ diam(*Q*) with constants of comparison depending at most on *n* and *M*. Note that (*i**v*) and (*v*) of Lemma [cubes] imply, for each cube *Q* ∈ D*k*, that there is a point (*X**Q*, *t**Q*) = (*X**Q*, *t**Q*) ∈ Σ, and a cube *C**r*(*X**Q*, *t**Q*), such that\n \n*r* ≈ 2− *k* ≈ diam(*Q*) , and Σ ∩ *C**r*(*X**Q*, *t**Q*) ⊂ *Q* ⊂ Σ ∩ *C**c**r*(*X**Q*, *t**Q*), \nfor some uniform constant *c*. We will denote the associated surface ball by\n \nΔ*Q*(*r*) : = Σ ∩ *C**r*(*X**Q*, *t**Q*), \nand we shall refer to the point (*X**Q*, *t**Q*) as the center of *Q*. Given a dyadic cube *Q* ⊂ Σ and *λ* > 1, we define\n \n*λ**Q* : = Δ*Q*(*λ*diam(*Q*)). \n[d3.11] Let ${{\\bf S}}\\subset {\\mathbb{D}}(\\Sigma)$. We say that ${{\\bf S}}$ is coherent if the following conditions hold.\n* ${{\\bf S}}$ contains a unique maximal element $Q({{\\bf S}})$ which contains all other elements of ${{\\bf S}}$ as subsets.\n* If *Q* belongs to ${{\\bf S}}$, and if $Q\\subset \\widetilde{Q}\\subset Q({{\\bf S}})$, then $\\widetilde{Q}\\in {\\bf S}$.\n* Given a cube $Q\\in {{\\bf S}}$, either all of its children belong to ${{\\bf S}}$, or none of them do.\nWe say that ${{\\bf S}}$ is semi-coherent if only conditions (*a*) and (*b*) hold.\nStatement of the main results\n=============================\nAs stated in the introduction, the purpose of this work is to prove (a) $\\implies$ (b) and (a) $\\implies$ (c) in Theorem [thrm.char]. To prove (a) $\\implies$ (b) we show the following.\n[main] Suppose that $\\Sigma\\subset\\mathbb R^{n+1}$ is parabolic uniformly rectifiable with constants (*M*, Γ). Let *δ* ≪ 1 and *κ* ≫ 1 be two given positive constants. Then there exists a disjoint decomposition D(Σ) = B ∪ G, satisfying the following properties.\n1. The collection B satisfies the Carleson packing condition: \n∑*Q*ʹ ⊆ *Q*, *Q*ʹ ∈ B*σ*(*Q*ʹ) ≤ *c*(*n*, *M*, Γ, *δ*, *κ*) *σ*(*Q*) , ∀*Q* ∈ D(Σ) .\n2. The collection G is further subdivided into disjoint stopping time regimes $\\{{{\\bf S}}\\}\\_{{{\\bf S}}\\in \\mathcal{F}}$, such that each such regime **S** is coherent.\n3. The maximal cubes $Q({{\\bf S}})$ for the stopping time regimes satisfy the Carleson packing condition: \n$$\\sum\\_{{{\\bf S}}: Q({{\\bf S}})\\subseteq Q}\\sigma\\big(Q({{\\bf S}})\\big)\\,\\leq\\, c(n,M,\\Gamma,\\delta,\\kappa)\\, \\sigma(Q)\\,,\n\\quad \\forall Q\\in {\\mathbb{D}}(\\Sigma)\\,.$$\n4. For each ${{\\bf S}}$, there exists a coordinate system[8](#fn8) and a regular Lip(1,1/2) function $ \\psi\\_{{{\\bf S}}}:=\\psi = \\psi ( x, t ) : \\mathbb R^{n-1}\\times\\mathbb R\\to \\mathbb R$ with parameters *b*1 = *c*(*n*, *M*) ⋅ *δ* and *b*2 = *b*2(*n*, *M*, Γ), such if we define $\\Sigma\\_{{{\\bf S}}}:=\\Sigma\\_\\psi:=\\{(x,\\psi(x,t),t):\\ (x,t)\\in \\mathbb R^{n-1}\\times\\mathbb R\\}$, then\n \n$$\\label{eq2.2a-RE}\n\\sup\\_{(X,t)\\in \\kappa Q} d\\_p(X,t,\\Sigma\\_{{{\\bf S}}} )\\,\n\\leq\\, \\delta\\,\\operatorname{diam}(Q),$$ \nfor all $Q\\in {{\\bf S}}$.\nThe second theorem we prove is a strengthening of Theorem [main] in the sense that it gives instead a bilateral approximation in (4) and proves (a) $\\implies$ (c) in Theorem [thrm.char].\n[main.bil.thrm] Suppose that $\\Sigma\\subset\\mathbb R^{n+1}$ is parabolic uniformly rectifiable with constants (*M*, Γ). Let *δ* ≪ 1 and *κ* ≫ 1 be two given positive constants. Then there exists a disjoint decomposition D(Σ) = B\\* ∪ G\\*, satisfying the following properties.\n1. The collection B\\* satisfies the Carleson packing condition: \n∑*Q*ʹ ⊆ *Q*, *Q*ʹ ∈ B\\**σ*(*Q*ʹ) ≤ *c*(*n*, *M*, Γ, *δ*, *κ*) *σ*(*Q*) , ∀*Q* ∈ D(Σ) .\n2. The collection G\\* is further subdivided into disjoint stopping time regimes $\\{{{\\bf S}}^\\*\\}\\_{{{\\bf S}}^\\* \\in \\mathcal{F}^\\*}$, such that each such regime ${{\\bf S}}^\\*$ is coherent.\n3. The maximal cubes $Q({{\\bf S}}^\\*)$ for the stopping time regimes satisfy the Carleson packing condition: \n$$\\sum\\_{{{\\bf S}}^\\*: Q({{\\bf S}}^\\*)\\subseteq Q}\\sigma\\big(Q({{\\bf S}}^\\*)\\big)\\,\\leq\\, c(n,M,\\Gamma,\\delta,\\kappa)\\, \\sigma(Q)\\,,\n\\quad \\forall Q\\in {\\mathbb{D}}(\\Sigma)\\,.$$\n4. For each ${{\\bf S}}^\\*$, there exists a coordinate system and a regular Lip(1,1/2) function $ \\psi\\_{{{\\bf S}}^\\*}:=\\psi = \\psi ( x, t ) : \\mathbb R^{n-1}\\times\\mathbb R\\to \\mathbb R$ with parameters *b*1 = *c*(*n*, *M*) ⋅ *δ* and *b*2 = *b*2(*n*, *M*, Γ), such if we define $\\Sigma\\_{{{\\bf S}}^\\*}:=\\Sigma\\_\\psi:=\\{(x,\\psi(x,t),t):\\ (x,t)\\in \\mathbb R^{n-1}\\times\\mathbb R\\}$, then\n \n$$\\begin{aligned}\n\\sup\\_{(X,t)\\in \\kappa Q} d\\_p(X,t,\\Sigma\\_{{{\\bf S}}^\\*} )\\, + \\, \\sup\\_{(Y,s)\\in C\\_{\\kappa \\operatorname{diam}(Q)}(X\\_Q,t\\_Q) \\cap \\Sigma\\_{{{\\bf S}}^\\*}} d\\_p(Y,s, \\Sigma) \\,\n\\leq\\, \\delta\\,\\operatorname{diam}(Q),\\end{aligned}$$ \nfor all $Q\\in {{\\bf S}}^\\*$.\nThe proof of Theorem [main]\n---------------------------\nWe here briefly discuss the outline of the proof of Theorem [main] to be presented in subsequent sections.\nIn the sequel, $\\Sigma\\subset\\mathbb R^{n+1}$ is parabolic uniformly rectifiable with constants (*M*, Γ). Throughout the proof, four parameters will consistently appear: *K*, *ε*, *δ* and *κ*. The parameters *δ* and *κ* are exactly as in the statement of Theorem [main]. *K* and *ε* are auxiliary parameters. Both *ε* and *δ* will be small, *ε* ≪ *δ* ≪ 1, and *ε* is chosen/defined in. This implies that *ε* = *ε*(*n*, *M*, Γ, *κ*, *K*, *δ*). *K* and *κ* will be large, *K* ≫ *κ* ≫ 1, and to achieve that the constants in Theorem [main] only depend on *n*, *M*, Γ, *δ* and *κ*, we will enforce the relation *K*/*κ* = *c*(*n*, *M*) ≫ 1.\nFor *K* ≫ *κ* ≫ 1 fixed, we define, for each *Q* ∈ D(Σ), where\n \n*β*∞(*Q*, *K*) : = inf*P* ∈ Pdiam(*Q*)− 1sup{(*Y*, *s*) ∈ 8*K**Q*}*d**p*(*Y*, *s*, *P*) , \nand\n \n$$\\label{4}\n\\quad\\quad\\quad\\beta\\_2(Q,K) := \\inf\\_{P \\in\\mathcal{P}} \\left(\\operatorname{diam}(Q)^{-d} {\\int\\_{}\\kern-.34em \\int}\\_{8KQ} \\biggl (\\frac{ d\\_p(Y,s,P)}{\\operatorname{diam}(Q)}\\biggr )^2 \\,d\\sigma(Y,s)\\right)^{1/2}.$$ \nIn the following we drop *K* to ease the notation. Note that\n \n$$\\label{5}\n\\beta\\_\\infty(Q)^{n+3}\\lesssim \\beta\\_2^{2}(Q^\\*),$$ \nwhere *Q*\\* is an appropriately chosen ancestor to *Q* such that \n$$100c\\operatorname{diam}(Q) \\le \\operatorname{diam}(Q^\\*) \\lesssim \\operatorname{diam}(Q)$$ \n for *c* = *c*(*n*, *M*) ≫ 2 to be chosen. Indeed, let $ {\\widetilde}P $ be a plane realizing the infimum in the definition of *β*2(*Q*\\*). Clearly there exists (*Z*1, *τ*1) ∈ 8*K**Q* with\n \n$$\\rho := \\operatorname{diam}(Q) \\beta\\_\\infty(Q)\\leq d\\_p( Z\\_1, \\tau\\_1, {\\widetilde}P ).$$ \nWe have $ {\\sigma}( {\\Delta}( Z\\_1, \\tau\\_1, \\rho/8) )\\gtrsim \\rho^{d}$ as Σ is parabolic ADR. Moreover every point in Δ(*Z*1, *τ*1, *ρ*/8) is contained in 8*c**K**Q*, for some *c* = *c*(*n*, *M*) ≫ 2, and lies at least *ρ*/2 from $ {\\widetilde}P. $ Thus, is valid.\nUsing that *ν* is a Carleson measure, a standard argument (see e.g. ) shows that\n \n$$\\label{beta2cmest.eq}\n\\sum\\_{Q \\subset Q^\\*} \\beta\\_2^2(Q) \\sigma(Q) \\lesssim \\sigma(Q^\\*), \\quad Q^\\* \\in \\mathbb{D}(\\Sigma),$$ \nwith implicit constant depending on *K*, *n*, *M* and Γ.\nTo this end, we define\n \n$$\\begin{aligned}\n\\mathcal{B} &:= \\mathcal{B}(\\epsilon,K) := \\{Q \\in \\mathbb{D}(\\Sigma): \\beta\\_\\infty(Q) \\ge \\epsilon\\},\\\\\n\\mathcal{G} &:= \\mathcal{G}(\\epsilon, K) = \\mathbb{D}(\\Sigma) \\setminus \\mathcal{B}(\\epsilon,K).\\end{aligned}$$ \nThen and imply that\n \n$$\\label{bad-pack}\n\\sum\\_{\\substack{Q \\subseteq {\\widetilde{Q}}\\\\ Q \\in \\mathcal{B}}} \\sigma(Q)\n\\lesssim \\sigma( \\widetilde{Q}), \\quad \\widetilde{Q} \\in \\mathbb{D}(\\Sigma),$$ \nwhere the implicit constant now depends on *ε*, *K*, *n*, *M* and Γ.\nFor *Q* ∈ D(Σ) we let *P**Q* ∈ P be a *t*-independent plane which achieves the infimum in the definition of *β*∞(*Q*). Then, by the definition of G,\n \n*d**p*(*Y*, *s*, *P**Q*) ≤ *ε*diam(*Q*)for all (*Y*, *s*) ∈ 8*K**Q*, \nand for all *Q* ∈ G.\nFollowing (or equivalently, ) we may construct an augmented collection Bʹ ⊃ B, and a refined collection Gʹ ⊂ G, for which we have a disjoint decomposition of the dyadic cubes D(Σ) = Bʹ ∪ Gʹ. Moreover, Bʹ still satisfies the packing condition, and Gʹ may be partitioned into a collection of coherent stopping time regimes $\\{{{\\bf S}}\\}\\_{{{\\bf S}}\\in \\mathcal{F}}$, with maximal cubes $\\{Q({{\\bf S}})\\}\\_{{{\\bf S}}\\in \\mathcal{F}}$, with the following properties:\n \n$$\\begin{aligned}\n\\label{dico}\n&\\mbox{(A) If $Q \\in {{\\bf S}}$, then ${\\operatorname{Angle}}(P\\_Q, P\\_{Q({{\\bf S}})}) \\le \\delta$.}\\notag\\\\\n&\\mbox{(B) If $Q$ is a minimal cube of ${{\\bf S}}$, then either a child of $Q$ is in $\\mathcal{B}'$}\\notag\\\\\n&\\quad\\quad\\mbox{or ${\\operatorname{Angle}}(P\\_Q,P\\_{Q({{\\bf S}})}) \\ge \\delta/2$.}\\end{aligned}$$ \nGiven a stopping time regime ${{\\bf S}}$, we define $m({{\\bf S}})$ to be the collection of minimal cubes in ${{\\bf S}}$.\nFollowing, we will construct a Lip(1,1/2) graph associated to each stopping time regime. After doing so, proving Theorem [main] is a matter of demonstrating that the graph is, in fact, a *regular* Lip(1,1/2) graph and that the maximal cubes $\\{Q({{\\bf S}})\\}$ pack. These objectives will be achieved in a somewhat simultaneous manner in a way similar to. The insight (and courage) of David and Semmes in providing such a proof is quite remarkable.\nThe proof of Theorem [main] is divided up into three sections: in Section [subsgraphconst], we construct a Lip(1,1/2) graph with small constant for each stopping time regime. In Section [pushsqfn], we push the geometric square function estimate (the Carleson measure estimate for *ν*) to the graph. The control of the geometric square function on the graph will verify, by, that the graph is indeed regular. In Section [packsttime], we use the fact that we pushed the geometric square function in a sharp manner to pack the stopping time regimes.\nOur argument will also rely on two simple lemmas concerning the approximation of surface cubes by planes, these lemmas can be found in Appendix [subsplanes]. The construction of the graph relies on condition (A) and that the geometric lemma holds for Σ. It is also worth making the remark that we only use condition (B) above when packing the stopping time regimes in Section [packsttime].\nConstruction of Lip(1,1/2) graphs for stopping time regimes\n===========================================================\nIn this section we shall follow very closely the arguments in to construct, for ${{\\bf S}}\\in \\mathcal{F}$ fixed, a Lip(1,1/2) graph which approximates $Q \\in {{\\bf S}}$ as in Theorem [main](4). We can without loss of generality assume that \n$$P\\_{Q({{\\bf S}})} := \\{(x,x\\_n, t) \\in \\mathbb{R}^{n-1} \\times \\mathbb{R} \\times \\mathbb{R}: x\\_n = 0\\}.$$ \n We will often identify $P\\_{Q({{\\bf S}})}$ with R*n* and employ the notation *C*ʹ*r*(*x*, *t*) introduced in for a *n*-dimensional parabolic cube in $P\\_{Q({{\\bf S}})}$. We let *π* denote the projection onto $P\\_{Q({{\\bf S}})}$, that is, *π*(*x*, *x**n*, *t*) = (*x*, *t*) ∈ R*n*. We let *π*⊥ denote the projection onto the normal to $P\\_{Q({{\\bf S}})}$, that is, *π*⊥(*x*, *x**n*, *t*) = *x**n*. Given *Q* ∈ D(Σ) we let (*x**Q*, *t**Q*) : = *π*(*X**Q*, *t**Q*) denote the projected center of *Q*.\nGiven ${{\\bf S}}\\in\\mathcal{F}$, and following, we introduce distance functions adapted to the stopping regime. Indeed, if $(X,t)\\in\\mathbb R^{n+1}$ then we define \n$$d(X,t)=d\\_{{{\\bf S}}}(X,t):=\\inf\\_{Q\\in{{\\bf S}}}[d\\_p(X,t,Q)+\\mbox{diam}(Q)],$$ \n and we let $D=D\\_{{{\\bf S}}} : \\mathbb R^n\\to [0,\\infty)$ be defined as\n \n$$\\begin{aligned}\n\\label{6-}D(x,t):=\\inf\\_{(X,t)\\in \\pi^{-1}(x,t)}d(X,t)=\\inf\\_{Q\\in{{\\bf S}}}[d\\_p(x,t,\\pi(Q))+\\mbox{diam}(Q)].\\end{aligned}$$ \nIn the sequel, we set \n$$F:=\\big\\{(X,t)\\in \\Sigma: d(X,t)=0\\big\\}\\,.$$ \n[lem1] Let ${{\\bf S}}\\in\\mathcal{F}$ and let $P\\_{ Q({{\\bf S}})}=\\mathbb R^n$. Then *π* is one-to-one on *F* and we can define a function $\\hat\\psi$ on $\\pi(F)\\subset \\mathbb R^n$ as follows. Given (*X*, *t*) = (*x*, *x**n*, *t*) ∈ *F*, we set $\\hat\\psi(x,t):=\\pi^\\perp(X,t)=x\\_n$. Then $\\hat \\psi$ is a well-defined Lip(1,1/2) function with constant bounded by 2*δ* on *π*(*F*), i.e.,\n \n$$\\begin{aligned}\n\\label{hatlip}\n|\\hat \\psi(x,t)-\\hat \\psi(y,s)|\\leq 2\\delta d\\_p(x,t,y,s),\\end{aligned}$$ \nwhenever (*x*, *t*), (*y*, *s*) ∈ *π*(*F*). More generally, suppose that $(X,t),\\ (Y,s)\\in 16 \\kappa Q({{\\bf S}})$, and that for some *N* ≥ 1, we have min{*d*(*X*, *t*), *d*(*Y*, *s*)} ≤ *N**d**p*(*X*, *t*, *Y*, *s*); then\n \n$$\\begin{aligned}\n\\label{6}\n|\\pi^\\perp(X,t)-\\pi^\\perp(Y,s)|\\leq 2\\delta d\\_p\\big(\\pi(X,t),\\pi(Y,s)\\big)\\,,\\end{aligned}$$ \nprovided that *ε* ≪ *δ* is small enough, depending on *N*, and *K* ≫ 1 large enough, depending on *N* and *κ*. (Remark: in the sequel, we apply the lemma with *N* equal to an absolute constant.)\nClearly, it suffices to prove. The proof follows that of. Let $(X,t),\\ (Y,s)\\in 16 \\kappa Q({{\\bf S}})$ and without loss of generality, suppose that *d*(*X*, *t*) ≤ *N**d**p*(*X*, *t*, *Y*, *s*). Let $Q\\in {{\\bf S}}$ be such that\n \n$$\\begin{aligned}\n\\label{6a}d\\_p(X,t,Q)+\\mbox{diam}(Q)\\leq 2N d\\_p(X,t,Y,s).\\end{aligned}$$ \nWe can, if necessary, replace *Q* by one of its ancestors (which is still in ${{\\bf S}}$) to obtain $\\kappa^{-1} d\\_p(X,t,Y,s)\\lesssim \\operatorname{diam}Q\\lesssim N d\\_p(X,t,Y,s)$. By construction, *Q* ∈ G and hence *β*∞(*Q*) ≤ *ε*, so for *K* = *K*(*N*, *κ*) large enough, there is a plane *P**Q* such that \n$$d\\_p(X,t,P\\_Q)+ d\\_p(Y,s,P\\_Q)\\lesssim \\epsilon \\operatorname{diam}Q \\lesssim\n \\epsilon N d\\_p(X,t,Y,s)\\ll \\delta d\\_p(X,t,Y,s),$$ \n as *ε* ≪ *δ*. The conclusion in now follows since, by construction of the stopping time, the angle between $P\\_{Q({{\\bf S}})}$ and *P**Q* is at most *δ*. In particular, the reader can verify this claim using elementary geometry and the fact that sin2(*δ*) ≈ *δ*2 for *δ* small.\nWith Lemma [lem1] as the starting point we next construct a function *ψ* on $\\mathbb R^n=P\\_{ Q({{\\bf S}})}$ which coincides with $\\hat\\psi$ on *π*(*F*). We will extend ${\\hat\\psi}$ off *π*(*F*) using an appropriate Whitney type extension following the corresponding construction in. For $(x,t)\\in \\mathbb R^n$ and (*x*, *t*) not on the boundary of any dyadic cube, let *I*(*x*, *t*) be the largest (closed) dyadic cube in $\\mathbb R^n$ containing (*x*, *t*) and satisfying \n$$\\operatorname{diam}(I\\_{(x,t)})\\leq \\frac 1 {20}\\inf\\_{(z,\\tau)\\in I\\_{(x,t)}}D(z,\\tau).$$ \n Let {*I**i*} be a labelling of the set of all these cubes *I*(*x*, *t*) without repetition. By construction the collection {*I**i*} consists of pairwise non-overlapping closed dyadic cubes in $\\mathbb R^n$. These cubes cover $ \\mathbb R^{n} {\\setminus}\\pi( F) $ and they do not intersect *π*(*F*). For the construction we need the following, whose proof follows that of, essentially verbatim.\n[10-60lemma] Given *I**i* we have\n \n$$\\begin{aligned}\n\\label{10-60.eq}\n10 \\operatorname{diam}I\\_i \\leq D(y,s) \\leq 60 \\operatorname{diam}I\\_i \\indent \\forall (y,s) \\in 10I\\_i.\\end{aligned}$$ \nIn particular, if 10*I**i* ∩ 10*I**j* ≠ ∅, then\n \n$$\\begin{aligned}\n\\label{eq1}\n\\operatorname{diam}I\\_i \\approx \\operatorname{diam}I\\_j.\\end{aligned}$$ \nFix *i* and let (*y*, *s*) ∈ *I**i*. As *D*( ⋅ , ⋅ ) is Lip(1, 1/2) with norm 1, we have \n*D*(*y*, *s*) ≥ inf(*x*, *t*) ∈ *I**i**D*(*x*, *t*) − 10diam*I**i* ≥ 10diam*I**i*, \n where we used that diam*I**i* ≤ 20− 1inf(*x*, *t*) ∈ *I**i**D*(*x*, *t*). On the other hand, if *I* is the dyadic parent of *I**i*, then there exists (*z*, *τ*) ∈ *I* such that *D*(*z*, *τ*) < 20diam*I* = 40diam*I**i*. Thus, \n*D*(*y*, *s*) ≤ *D*(*z*, *τ*) + 20diam*I**i* ≤ 60diam*I**i*. \n This proves, and hence also.\nWe continue to follow the constructions in. Set\n \n$$\\begin{aligned}\n\\label{nota2}R=R\\_{{{\\bf S}}}:=\\operatorname{diam}( Q({{\\bf S}})).\\end{aligned}$$ \nGiven the center $(X\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})})$ of $ Q({{\\bf S}})$, we recall that \n$$(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})})=\\pi(X\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})}),$$ \n and we introduce the index set\n \n$$\\label{la-def}\n {\\Lambda}\\, = \\{ i :\n I\\_i \\cap C'\\_{2\\kappa R}(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})}) \\not = \\emptyset \\}.$$ \nWe claim, for each *i* ∈ Λ, that there exists $Q(i) \\in {{\\bf S}}$ such that\n \n$$\\begin{aligned}\n\\label{eq1+}\n\\kappa^{-1} \\operatorname{diam}I\\_i \\lesssim \\operatorname{diam}Q(i) \\leq 120 \\operatorname{diam}I\\_i, \\quad d\\_p(\\pi(Q(i)),I\\_i) \\leq 120 \\operatorname{diam}I\\_i.\\end{aligned}$$ \nTo see this, note that given *I**i* and (*x*, *t*) ∈ *I**i*, there exists a cube $Q\\in {{\\bf S}}$ such that *d**p*(*x*, *t*, *π*(*Q*)) + diam(*Q*) ≤ 2*D*(*x*, *t*) ∼ diam(*I**i*). Moreover, $D(x,t) \\le c \\kappa \\operatorname{diam}(Q({{\\bf S}}))$, for a constant *c* ≥ 1 depending only on *n*, *M*, and \n*d**p*(*x*, *t*, *π*(*Q*\\*)) ≤ 2*D*(*x*, *t*) ≤ 120diam*I**i* , \n for all $Q \\subseteq Q^\\* \\subseteq Q({{\\bf S}})$. Therefore we can choose *Q*(*i*) so that $Q \\subseteq Q(i) \\subseteq Q({{\\bf S}})$ and\n \n(*c**κ*)− 1*D*(*x*, *t*) ≤ diam(*Q*(*i*)) ≤ 2*D*(*x*, *t*) ≤ 120diam*I**i* , \nprovided *κ* is chosen large enough.\nIn the sequel, we will, at instances, use the notation\n \n$$\\begin{aligned}\n\\label{nota1}r\\_i:=\\operatorname{diam}I\\_i,\\,\\, \\text{ so that } \\, r\\_i/\\kappa \\lesssim\n\\operatorname{diam}Q(i) \\lesssim r\\_i,\\end{aligned}$$ \nby.\nLet $ \\{ \\tilde \\nu\\_i \\} $ be a class of infinitely differentiable functions on $\\mathbb R^{n}$ such that\n \n$$\\begin{aligned}\n (i)&\\ \\tilde \\nu\\_i \\equiv 1 \\mbox{ on } 2I\\_i\n \\mbox{ and } \\tilde \\nu\\_i \\equiv 0 \\mbox{ in } \\mathbb R^{n} {\\setminus}3I\\_i \\mbox{ for all } i,\\notag\\\\\n (ii)&\\ r\\_i^{ l} \\, \\left| \\frac{\\partial^l}{\\partial x^l } \\tilde \\nu\\_i \\right|\n\\, + \\, r\\_i^{ 2 l} \\left| \\frac{\\partial^l}{\\partial t^l } \\tilde \\nu\\_i \\right| \\, \\leq\n\\, c ( l, n ) \\mbox{ for } l = 1, 2, \\dots.\\end{aligned}$$ \nIn (*i**i*), $\n\\frac{{\\partial}^l}{{\\partial}x^l } $ denotes an arbitrary partial derivative with respect to the space variable *x* and of order *l*. We also introduce {*ν**i*} defined as a partition of unity adapted to {2*I**i*}*i*, i.e., \n$$\\nu\\_i(x,t)=\\tilde \\nu\\_i(x,t)\\big (\\sum\\_{j} \\tilde \\nu\\_j(x,t)\\big)^{-1}.$$ \n Then {*ν**i*} is also a class of infinitely differentiable functions on $\\mathbb R^{n}$ and\n \n$$\\begin{aligned}\n\\label{2.16+}\nr\\_i^{ l} \\, \\left| \\frac{\\partial^l}{\\partial x^l } \\nu\\_i \\right|\n\\, + \\, r\\_i^{ 2 l} \\left| \\frac{\\partial^l}{\\partial t^l } \\nu\\_i \\right| \\, \\leq\n\\, c ( l, n ) \\mbox{ for } l = 1, 2, \\dots.\\end{aligned}$$ \nWe now ready to construct the extension of $ \\hat \\psi $ off *π*(*F*). If *i* ∈ Λ, then for *I**i* we have an associated dyadic cube $Q(i)\\in {{\\bf S}}$ as in, we first express the associated hyperplane *P**Q*(*i*) as\n \n$$\\begin{aligned}\n\\label{2.19-} P\\_{Q(i)}=\\{ (x, B\\_i ( x,t),t ) : ( x,t) \\in \\mathbb R^n \\},\\end{aligned}$$ \ni.e., $B\\_i:\\mathbb R^n\\to \\mathbb R$ is the affine function whose graph is *P**Q*(*i*). If *i* ∉ Λ, set *B**i* = 0 and we note, in this case, *B**i* is the affine function whose graph is $P\\_{Q({{\\bf S}})}$. Using this notation we let\n \n$$\\begin{aligned}\n\\label{2.19}\n\\psi ( x,t)&=\\hat \\psi ( x,t)\\mbox{ when } ( x,t) \\in \\pi (\nF),\\notag\\\\\n\\psi ( x,t) &=\n { {\\displaystyle}\\sum\\_{ i } } \\, B\\_i ( x,t)\n\\, \\nu\\_i ( x,t ) \\mbox{ when } ( x,t ) \\in \\mathbb R^n {\\setminus}\\pi (\nF). \\end{aligned}$$ \nThis defines *ψ* on $\\mathbb R^n$, with $\\psi=\\hat\\psi$ on *π*(*F*) and by construction\n \n$$\\label{supportpsi}\n\\psi\\equiv 0 \\text{ on } \\mathbb R^n\\setminus C'\\_{4\\kappa R}(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})}).$$ \nBased on the construction we can prove the following lemma.\n[lemma2.21] For $ ( y,s ),\\ (z, \\tau) \\in \\mathbb R^{n}$ we have\n \n$$\\begin{aligned}\n(i)&\\ | B\\_i ( y, s ) - B\\_i ( z, \\tau ) | \\, \\lesssim \\delta d\\_p(y,s,z,\\tau) \\mbox{ for all } i, \\notag\\\\\n(ii)&\\ | B\\_i ( y, s ) - B\\_j ( y, s ) |\\lesssim \\epsilon \\min ( r\\_i, r\\_j ), \\mbox{ if }\n ( y, s ) \\in 100 I\\_i \\cup 100 I\\_j,\\ 10 I\\_i \\cap 10 I\\_j \\not =\\emptyset.\n \\end{aligned}$$ \nBy construction of the stopping regime, the Lip(1,1/2) norm of *B**i* is bounded by 2*δ*, so (*i*) is trivial. To prove (*i**i*), we follow the proof of. Let *I**i* and *I**j* be two cubes such that $10 I\\_i \\cap 10 I\\_j \\not =\n\\emptyset$, thus diam*I**i* = *r**i* ≈ *r**j* = diam*I**j* by Lemma [10-60lemma]. We claim that if *i* ∈ Λ or *j* ∈ Λ, then\n \n$$\\begin{aligned}\n\\label{6++-}d\\_p(Q(i),Q(j))\\lesssim r\\_i\\approx r\\_j.\\end{aligned}$$ \nLet us prove the claim. Let *i* ∈ Λ. Let (*X*, *t*) ∈ *Q*(*i*) and (*Y*, *s*) ∈ *Q*(*j*), We may assume that *d**p*(*X*, *t*, *Y*, *s*) ≥ diam*Q*(*i*), the other case being trivial. Since *d*(*X*, *t*) ≤ diam*Q*(*i*), by we see that\n \n$$\\begin{aligned}\n\\label{6++}\n|\\pi^\\perp(X,t)-\\pi^\\perp(Y,s)|\\leq d\\_p(\\pi(X,t),\\pi(Y,s)).\\end{aligned}$$ \nBy, $d\\_p(\\pi(X,t),\\pi(Y,s))\\lesssim r\\_i\\approx r\\_j$, so the claim holds.\nSuppose that *i* ∈ Λ. Then by and, there is a ``surface ball\" Δ*r*, centered in *Q*(*i*), such that *r* ≈ *r**i* ≈ *r**j*, and \n*Q*(*i*) ∪ *Q*(*j*) ⊂ Δ*r* ⊂ Δ2*r* ⊂ *K**Q*(*i*) ∩ *K**Q*(*j*) , \n for *K* large enough. Using the second part of Lemma [aux2] (with Δ = Δ*r*, *P*1 = *P**Q*(*i*), *P*2 = *P**Q*(*j*)), we see that if (*y*, *s*) ∈ 100*I**i* ∪ 100*I**j*, with $10 I\\_i \\cap 10 I\\_j \\not =\n\\emptyset$, then\n \n$$\\begin{aligned}\n\\label{2.21+}\n| B\\_i ( y, s ) - B\\_j ( y, s ) |\\lesssim \\epsilon r.\\end{aligned}$$ \nThe case *j* ∈ Λ is exactly the same, so this completes the proof in the case that at least one of *i* or *j* belongs to Λ.\nIf, on the other hand, $i \\notin \\Lambda$ and $j \\notin \\Lambda$, then $Q(i)=Q(j)=Q({{\\bf S}})$, and hence *B**i* ≡ *B**j*, so the estimate holds trivially. This completes the proof of Lemma [lemma2.21].\nBefore proving that *ψ* is Lip(1,1/2), we present one more preliminary lemma, which is the parabolic version of.\n[d=D] Let\n \n$$\\begin{aligned}\n\\label{2.21+++b}\n (x,t)\\in C'\\_{4\\kappa R}(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})})\\mbox{ and $r>0$ with }D(x,t)\\leq r<2\\kappa R,\\end{aligned}$$ \nand let $Q\\in{{\\bf S}}$ be such that *d**p*(*x*, *t*, *π*(*Q*)) ≤ *λ**r* and *λ*− 1*r* ≤ diam*Q* ≤ *λ**r* for some *λ* ≥ 1. Then\n \n$$\\begin{aligned}\n\\label{2.21+++c}\\mbox{$\\pi^{-1}\\left(C'\\_r(x,t)\\right)\\cap 4\\kappa Q({{\\bf S}})$ \n is contained in $\\tilde\\lambda Q$},\\end{aligned}$$ \nwhere $\\tilde\\lambda$ depends on *λ* and *κ*.\nFurthermore, for some $\\hat\\lambda$ depending on *κ*, we have\n \n$$\\begin{aligned}\n\\label{2.21+++d}\\mbox{$\\hat\\lambda^{-1}d(X,t)\\leq D(\\pi(X,t))\\leq d(X,t)$ \n for all $(X,t)\\in 2\\kappa Q({{\\bf S}})$}.\\end{aligned}$$ \nThe proof follows that of essentially verbatim. To prove, we consider (*Y*, *s*) ∈ *Q*, $(Z,\\tau) \\in \\pi^{-1}\\left(C'\\_r(x,t)\\right)\\cap 4\\kappa Q({{\\bf S}})$. First, if \n$$d\\_p\\big((Y,s), (Z,\\tau)\\big) \\leq \\operatorname{diam}Q,$$ \n then $(Z,\\tau) \\in \\tilde\\lambda Q$, for, say, $\\tilde\\lambda \\geq 5$. Second, if \n$$d\\_p\\big((Y,s), (Z,\\tau)\\big) >\\operatorname{diam}Q,$$ \n then \n$$d(Y,s) \\leq \\operatorname{diam}Q \\leq d\\_p\\big((Y,s), (Z,\\tau)\\big).$$ \n Hence, by Lemma [lem1] \n$$| \\pi^{\\perp}(Y,s) - \\pi^{\\perp}(Z,\\tau)| \\leq d\\_p\\big(\\pi(Y,s), \\pi(Z,\\tau)\\big) \\lesssim\\_\\lambda \\operatorname{diam}Q.$$ \n (The last inequality follows from our assumptions that $d\\_p(x,t,\\pi(Q))\\lesssim r\\approx \\operatorname{diam}Q$, and that (*Y*, *s*) ∈ *Q*, and *π*(*Z*, *τ*) ∈ *C*ʹ*r*(*x*, *t*).) This proves.\nTo deduce, we note that the second inequality is trivial by definition. To prove the first inequality, we consider first the case that *π*(*X*, *t*) ∈ *π*(*F*), thus, *π*(*X*, *t*) = *π*(*X*ʹ, *t*) for some (*X*ʹ, *t*) ∈ *F*, so that *d*(*X*ʹ, *t*) = 0 by definition. But then by Lemma [lem1], (*X*, *t*) = (*X*ʹ, *t*), so *d*(*X*, *t*) = 0, and holds in this case. Otherwise, if $\\pi(X,t) \\notin \\pi(F)$, we then set (*x*, *t*) : = *π*(*X*, *t*), and *r* : = *D*(*x*, *t*) > 0, and note that by definition of *D*(*x*, *t*), there is a $Q\\in{{\\bf S}}$ such that $d\\_p(x,t,\\pi(Q))\\lesssim r\\approx \\operatorname{diam}Q$ (here, there can be possible implicit dependence on *κ* in the case that $R\\ll r\\lesssim \\kappa R$). Hence, we may apply to deduce that $(X,t) \\in \\tilde\\lambda Q$, and therefore, \n$$d(X,t) \\leq d\\_p(X,t,Q) +\\operatorname{diam}Q \\lesssim \\operatorname{diam}Q \\approx r=D\\big(\\pi(X,t)\\big).$$ \n[graphlip.lem]Let $\\psi:\\mathbb R^n\\to\\mathbb R$ be defined as in. Then $\\psi: \\mathbb R^n\\to\\mathbb R$ is Lip(1,1/2) with constant on the order of *δ*, i.e.\n \n$$\\label{lemma4.26est}\n|\\psi(y,s)- \\psi(z,\\tau)| \\lesssim \\delta d\\_p(y,s,z,\\tau)\\,, \\quad\n\\forall \\, (y,s),(z,\\tau) \\in \\mathbb{R}^n \\,.$$ \nAgain, we shall follow the corresponding arguments in. However, we point out that the proof in is not fully explicated, and in particular, does not address the case that *F* = ∅ (which, in contrast to the case of the classical Whitney extension theorem, can actually happen in the present setting). Fortunately, the methods of require only a modest refinement to address this situation.\nRecall that we use the notation *r**i* : = diam*I**i*. There are four cases to consider:\n1. (*y*, *s*), (*z*, *τ*) ∈ *π*(*F*).\n2. (*y*, *s*) ∈ R*n* \\ *π*(*F*), with (*y*, *s*) ∈ 2*I**j*, and (*z*, *τ*) ∈ *π*(*F*).\n3. (*y*, *s*), (*z*, *τ*) ∈ R*n* \\ *π*(*F*), with (*y*, *s*) ∈ 2*I**j*, (*z*, *τ*) ∈ 2*I**k*, and \n(WLOG) *r**k* ≤ *r**j* , (*z*, *τ*) ∈ 10*I**j* .\n4. (*y*, *s*), (*z*, *τ*) ∈ R*n* \\ *π*(*F*), with (*y*, *s*) ∈ 2*I**j*, (*z*, *τ*) ∈ 2*I**k*, and \n$$\\text{ (WLOG) } \\, r\\_k \\leq r\\_j\\,,\\quad (z,\\tau)\\notin 10I\\_j\\,.$$\nWe make one more preliminary observation: by, we may suppose that (*y*, *s*) and (*z*, *τ*) both lie in the closure of $C\\_\\*':=C'\\_{4\\kappa R}(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})})$. Indeed, if both points lie outside of this closed cube, then is trivial, and if one (say (*y*, *s*)) lies outside, and the other inside, then we may replace (*y*, *s*) by another point (*y*ʹ, *s*ʹ) ∈ ∂*C*ʹ\\* such that *d**p*(*z*, *τ*, *y*ʹ, *s*ʹ) ≤ *d**p*(*z*, *τ*, *y*, *s*), since for such points (*y*ʹ, *s*ʹ) and (*y*, *s*), we have *ψ*(*y*, *s*) = 0 = *ψ*(*y*ʹ, *s*ʹ).\nLet us now discuss the various cases (1)-(4).\nCase (1) follows immediately from Lemma [lem1], and the definition of *ψ* (see ).\nConsider next case (3). In this case, (*y*, *s*), (*z*, *τ*) ∈ 10*I**j*. By Lemma [10-60lemma], and Lemma [lemma2.21], and the fact that ∑*i*(*ν**i*(*y*, *s*) − *ν**i*(*z*, *τ*)) = 0,\n \n$$\\begin{gathered}\n\\label{2.21++}\n| \\psi( y, s ) - \\psi ( z,\\tau) | \\,\\leq \\, \\big|\\sum\\_{i} (B\\_i(y,s) - B\\_i(z,\\tau))\\nu\\_i(y,s)\\big|\n\\\\[4pt]\n +\\, \\big|\\sum\\_{i} (B\\_i(z,\\tau)-B\\_j(z,\\tau))(\\nu\\_i(y,s) - \\nu\\_i(z,\\tau))\\big| \\\\[4pt]\n\\lesssim \\,\\delta d\\_p(y,s,z,\\tau)\\,+\\,r\\_j^{-1}\\epsilon d\\_p(y,s,z,\\tau)r\\_j \n\\, \\lesssim \\,\\delta d\\_p(y,s,z,\\tau),\\end{gathered}$$ \nsince each *B**i* has Lip(1,1/2) norm at most 2*δ*, and *ε* ≪ *δ*.\nIt remains to treat cases (2) and (4), which we shall do more or less simultaneously. We remark that if *F* is non-empty, one can reduce case (4) to case (2), as in the proof of the classical Whitney extension theorem. However, in the present setting, it may be that *F* is empty, so we shall treat case (4) directly. For the sake of specificity, let us do this first, as the proof in case (2) will be similar, but a bit simpler.\nIn case (4), we decompose \n∣*ψ*(*y*, *s*) − *ψ*(*z*, *τ*)∣ ≤ *a*1 + *a*2 + *a*3 + *a*4 + *a*5 + *a*6 + *a*7, \n where for some (*U*, *t*) ∈ *Q*(*j*), (*U*ʹ, *t*ʹ) ∈ *Q*(*k*) to be specified more precisely below, setting (*u*, *t*) : = *π*(*U*, *t*), and (*u*ʹ, *t*ʹ) : = *π*(*U*ʹ, *t*ʹ), we define\n \n$$\\begin{aligned}\n&a\\_1 = |\\psi(y,s) - B\\_j(y,s)|\\,, \\quad a\\_2 = |B\\_j(y,s) - B\\_j(u,t)|\\,, \\\\[4pt]\n&a\\_3 = |B\\_j(u,t) - \\pi^{\\perp}(U,t)|\\,, \\quad a\\_4 = |\\pi^{\\perp}(U,t) -\\pi^\\perp(U',t')|\\,, \\\\[4pt]\n&a\\_5= |\\pi^\\perp(U',t') -B\\_k(u',t')|\\,, \\quad a\\_6 =|B\\_k(u',t') - B\\_k(z,\\tau)|\\,, \\\\[4pt]\n&a\\_7 = |B\\_k(z,\\tau) - \\psi(z,\\tau)|\\,.\\end{aligned}$$ \nWe estimate these terms as follows. For future reference, we observe that since $(z,\\tau)\\notin 10I\\_j$, and (*y*, *s*) ∈ 2*I**j*, therefore, for a purely dimensional implicit constant,\n \n$$\\label{4.29}\nr\\_k\\leq r\\_j \\lesssim d\\_p(y,s,z,\\tau)\\,.$$ \nConsider for now any (*U*, *t*) ∈ *Q*(*j*), and any (*U*ʹ, *t*ʹ) ∈ *Q*(*k*). By definition of *ψ*, since ∑*i**ν**i*(*y*, *s*) = 1, we have \n$$a\\_1 = \\big| \\sum\\_i \\nu\\_i(y,s)\\, \\big(B\\_i(y,s) -B\\_j(y,s)\\big)\\big| \\,\\lesssim \\,\\epsilon r\\_j \\,\\lesssim \\,\n\\epsilon d\\_p(y,s,z,\\tau)\\,,$$ \n by Lemma [lemma2.21] (ii) and. Similarly, \n$$a\\_7\\,\\lesssim\\, \\epsilon d\\_p(y,s,z,\\tau)\\,.$$ \nSince each *B**i* has Lip(1,1/2) norm at most 2*δ*, by the second inequality in,\n \n$$\\label{a2bound}\na\\_2 \\lesssim \\delta d\\_p(y,s,u,t) \\lesssim \\delta r\\_j \\lesssim \\delta d\\_p(y,s,z,\\tau)\\,,$$ \nagain using. Similarly, \n$$a\\_6\\,\\lesssim\\, \\delta d\\_p(y,s,z,\\tau)\\,.$$ \nSince *P**Q*(*j*) is the graph of *B**j*, which has small slope, and since *β*∞(*Q*(*j*)) ≤ *ε*, we have\n \n$$\\label{a3bound}\na\\_3\\approx {\\operatorname{dist}}\\big((U,t), P\\_{Q(j)}\\big) \\leq \\epsilon \\operatorname{diam}Q(j) \\leq 120 \\epsilon r\\_j \\lesssim\n\\epsilon d\\_p(y,s,z,\\tau)\\,,$$ \nwhere we have used and then in the last two steps. Similarly, \n$$a\\_5 \\lesssim \\epsilon d\\_p(y,s,z,\\tau)\\,.$$ \nIt remains to deal with *a*4. Since (*U*, *t*) ∈ *Q*(*j*) and (*y*, *s*) ∈ 2*I**j*, we have\n \n*d*(*U*, *t*) ≤ diam*Q*(*j*) ≤ 120*r**j* , *d**p*(*u*, *t*, *y*, *s*) ≤ 300*r**j* , \nby. Similarly, since (*U*ʹ, *t*ʹ) ∈ *Q*(*k*), and (*z*, *τ*) ∈ 2*I**k*,\n \n*d*(*U*ʹ, *t*ʹ) ≤ diam*Q*(*k*) ≤ 120*r**k* ≤ 120*r**j* , *d**p*(*u*ʹ, *t*ʹ, *z*, *τ*) ≤ 300*r**k* ≤ 300*r**j* . \nConsequently, by and,\n \n$$\\label{4.30} d(U,t) \\leq 120 r\\_j \\lesssim d\\_p(y,s,z,\\tau) \\,.$$ \nTo treat *a*4, we now consider two sub-cases:\n* (Sub-case A) *d**p*(*y*, *s*, *z*, *τ*) ≥ 1000*r**j*.\n* (Sub-case B) *d**p*(*y*, *s*, *z*, *τ*) < 1000*r**j*.\nWe first consider the former. In this case, since *d**p*(*y*, *s*, *z*, *τ*) ≥ 1000*r**j*, the right hand inequalities in and imply that \n*d**p*(*y*, *s*, *z*, *τ*) ≈ *d**p*(*u*, *t*, *u*ʹ, *t*ʹ) ≤ *d**p*(*U*, *t*, *U*ʹ, *t*ʹ) , \n Combining the latter estimate with, we obtain \n$$d(U,t)\\, \\lesssim \\,d\\_p(U,t,U',t')\\,.$$ \n Thus, we may apply Lemma [lem1], with *N* equal to an absolute constant, and with *ε* ≪ *δ*, to obtain\n \n*a*4 ≤ 2*δ**d**p*(*u*, *t*, *u*ʹ, *t*ʹ) ≈ *δ**d**p*(*y*, *s*, *z*, *τ*) . \nOn the other hand, consider now sub-case B, i.e., *d**p*(*y*, *s*, *z*, *τ*) < 1000*r**j*. Thus\n \n*d**p*(*y*, *s*, *z*, *τ*) ≈ *r**j* , \nby. In the current scenario, we consider two further sub-cases.\n(Sub-case B1): *Q*(*j*) meets *Q*(*k*). In this case we simply let (*U*, *t*) = (*U*ʹ, *t*ʹ) be any point in *Q*(*j*) ∩ *Q*(*k*), so that, trivially, *a*4 = 0.\n(Sub-case B2): *Q*(*j*) and *Q*(*k*) are disjoint. In this case, we let (*U*ʹ, *t*ʹ) be any point in *Q*(*k*), and we select (*U*, *t*) as follows. Let *C*(*j*) : = *C**r*(*X**Q*(*j*), *t**Q*(*j*)), with *r* ≈ diam*Q*(*j*), be the cube defined in, with respect to *Q* = *Q*(*j*). Since Σ ∩ *C*(*j*) ⊂ *Q*(*j*) by construction (see ), it follows that $(U',t') \\notin C(j)$. We then set (*U*, *t*) : = (*X**Q*(*j*), *t**Q*(*j*)) (the center of *C*(*j*)), and observe that in this case\n \n$$\\label{4.34}\nd\\_p(U,t,U',t') \\gtrsim \\operatorname{diam}Q(j)\\,,$$ \nwhere the implicit constants here depend on those in.\nNote that and continue to hold, since we still have (*U*, *t*) ∈ *Q*(*j*), (*y*, *s*) ∈ 2*I**j*, (*U*ʹ, *t*ʹ) ∈ *Q*(*k*), and (*z*, *τ*) ∈ 2*I**k*. In particular,\n \n$$\\label{4.35}\nd(U,t) \\leq \\operatorname{diam}Q(j)\\,,\\quad d\\_p(u,t,y,s) + d\\_p(u',t',z,\\tau) \\lesssim\\, r\\_j\\,.$$ \nBy and the second inequality in, we see that\n \n$$\\label{4.36}\nd\\_p(u,t,u',t') \\lesssim r\\_j \\approx d\\_p(y,s,z,\\tau)\\,.$$ \nCombining and the first inequality in, we find that \n$$d(U,t) \\leq \\operatorname{diam}Q(j) \\lesssim d\\_p(U,t,U',t')\\,.$$ \n Consequently, we may as above apply Lemma [lem1] to obtain\n \n$$a\\_4 \\leq 2 \\delta d\\_p(u,t,u',t') \\lesssim \\delta d\\_p(y,s,z,\\tau)\\,,$$ \nwhere in the last step we have used.\nSumming our estimates for the terms *a*1 through *a*7, and using that *ε* ≪ *δ*, we obtain the desired bound in case (4), in general.\nIn case (2), we proceed in a similar fashion to case (4), but now matters are a bit simpler: we decompose \n∣*ψ*(*y*, *s*) − *ψ*(*z*, *τ*)∣ ≤ *a*1 + *a*2 + *a*3 + *a*4, \n with *a*1, *a*2 and *a*3 exactly as above (and enjoying the very same bounds), and with \n*a*4 = ∣*π*⊥(*U*, *t*) − *ψ*(*z*, *t*)∣ , \n where (*U*, *t*) is an arbitrary point in *Q*(*j*). Thus, again, holds, and combining the latter with, we see as before that\n \n$$\\label{4.41}\nd\\_p(u,t,y,s) \\lesssim d\\_p(y,s,z,\\tau)\\,.$$ \nRecall that in case (2), we assume (*z*, *τ*) ∈ *π*(*F*). Consequently, by (the first part of) Lemma [lem1], and the fact that *ψ* = *ψ̂* on *π*(*F*), we have *d*(*z*, *τ*, *ψ*(*z*, *τ*)) = 0, and *π*⊥(*z*, *τ*, *ψ*(*z*, *τ*)) = *ψ*(*z*, *τ*). Hence, by the second part of Lemma [lem1], and, we find that \n$$a\\_4 \\leq 2\\delta d\\_p(u,t,z,\\tau) \\lesssim \\delta d\\_p(y,s,z,\\tau)$$ \n We now sum the bounds for terms *a*1 through *a*4 to complete the proof of the lemma.\n[closegraph.lem] Let $(X,t)\\in 2\\kappa Q({{\\bf S}})$. Then\n \n$$\\begin{aligned}\n\\label{2.21+++a} d\\_p(X,t,(x,\\psi(x,t),t))\\lesssim \\epsilon d(X,t),\\quad (x,t):=\\pi(X,t).\\end{aligned}$$ \nIn particular, holds provided *ε* ≪ *δ*/*κ*.\nNote that if $Q \\in {{\\bf S}}$, then $d(X,t) \\lesssim \\kappa \\operatorname{diam}(Q)$ for all (*X*, *t*) ∈ *κ**Q*. Using this observation, we see that implies provided *ε* ≪ *δ*/*κ*.\nWe turn now to, following the proof of. If (*X*, *t*) ∈ *F*, there is nothing to prove, so we consider $(X,t)\\in 2\\kappa Q({{\\bf S}})\\setminus F$. In particular, *d*(*X*, *t*) > 0. Set (*x*, *t*) : = *π*(*X*, *t*). Estimate implies that *D*(*x*, *t*) > 0 and hence (*x*, *t*) ∈ *I**i* for some *i*. Using with *Q* : = *Q*(*i*) and *r* : = *D*(*x*, *t*) ≈ diam*I**i*, we see that $(X,t)\\in\\tilde \\lambda Q(i)$, where by, the constant *λ* in Lemma [d=D] depends on *κ*, and therefore $\\tilde \\lambda$ depends on *κ*. Since we assume that *K* is large, depending on *κ*, we may in particular take *K* much larger than $\\tilde\\lambda$, so that \n$$|\\pi^\\perp(X,t)-B\\_i(x,t)|\\leq 2\\epsilon\\operatorname{diam}(Q(i)) \\lesssim \\epsilon \\operatorname{diam}I\\_i \\approx \\epsilon D(x,t) \\lesssim \\epsilon d(X,t)\\,,$$ \n where we have used with *Q* = *Q*(*i*), along with the definition of *B**i*. Estimate now follows, by Lemma [lemma2.21] (ii) and the definition of *ψ*.\n \n$$\\begin{aligned}\n\\label{2.21++bo}\n|\\nabla\\_x^2\\psi(y,s)|+|\\partial\\_t\\psi(y,s)|\\lesssim \\epsilon r\\_j^{-1}\\mbox{ if } (y,s)\\in 2I\\_j.\\end{aligned}$$ \nWe first prove the bound for ∣∇*x*2*ψ*(*y*, *s*)∣ by following. By the definition of Λ (see ), and the adapted partition of unity {*ν**i*}, we have\n \n$$\\begin{aligned}\n\\label{diff}\n\\sum\\_{i \\in \\Lambda} \\nu\\_i(y,s) =1, \\text{ and } \\,\n\\sum\\_{i \\in \\Lambda} \\nabla\\_x\\nu\\_i(y,s) =\n\\nabla\\_x\\Big(\\sum\\_{i \\in \\Lambda} \\nu\\_i(y,s) \\Big) = 0\\,,\\end{aligned}$$ \nwhenever (*y*, *s*) ∈ 2*I**j*. Let *α*, *β* be two spatial indices and let ∂*α*, ∂*β*, and ∂*α**β* denote the corresponding partial differential operators of order one and two. Using we see that if (*y*, *s*) ∈ 2*I**j*, then\n \n$$\\begin{aligned}\n&\\partial\\_{\\alpha\\beta} \\psi(y,s) = \\partial\\_{\\alpha\\beta} \\left(\\sum\\_{i} \\nu\\_i(y,s) B\\_i(y,s) \\right) \\\\\n&= \\sum\\_{i }(\\partial\\_{\\alpha\\beta} \\nu\\_i(y,s))B\\_i(y,s) \n+ \\sum\\_{i} (\\partial\\_{\\alpha} \\nu\\_i(y,s))(\\partial\\_\\beta\n B\\_i(y,s)) + \\sum\\_{i} (\\partial\\_{\\beta} \\nu\\_i(y,s))(\\partial\\_\\alpha B\\_i(y,s))\n\\\\\n&= \\sum\\_{i}(\\partial\\_{\\alpha\\beta}\\nu\\_i(y,s))(B\\_i(y,s)-B\\_j(y,s)) + \\sum\\_{i} (\\partial\\_{\\alpha} \\nu\\_i(y,s))\n(\\partial\\_{\\beta} B\\_i(y,s) -\\partial\\_\\beta B\\_j(y,s)) \\\\\n&+ \\sum\\_{i} (\\partial\\_{\\beta} \\nu\\_i(y,s)) (\\partial\\_{\\alpha} B\\_i(y,s) -\\partial\\_\\alpha B\\_j(y,s))\\, =:\\, I+II+III.\\end{aligned}$$ \nThe desired bound for term *I* follows immediately from Lemma [lemma2.21] (ii) and, and the fact that the cubes 3*I**i* (which contain the support of *ν**i*) have bounded overlaps, by virtue of Lemma [10-60lemma] (specifically ). A similar argument yields the desired bound for terms *I**I* and *I**I**I*, once we observe that, if 10*I**i* ∩ 10*I**j* ≠ ∅, then $|\\nabla\\_x B\\_i -\\nabla\\_x B\\_j| \\lesssim \\epsilon$. To see the latter, we translate and (spatially) rotate coordinates (only for purposes of the present argument) so that *P**Q*(*j*) is the hyperplane {(*x*, *x**n*, *t*) : *x**n* = 0}, hence in these new coordinates *B**j* ≡ 0. By Lemma [aux2], the angle between *P**Q*(*i*) and *P**Q*(*j*) is at most *C**ε*, i.e., in the new coordinates *B**i* is affine with ∣∇*x**B**i* − ∇*x**B**j*∣ = ∣∇*x**B**i*∣ ≤ *C**ε*.\nThe bound for ∣∂*t**ψ*(*y*, *s*)∣ can be produced similarly. In this case we first note that ∂*t**B**i* = 0 (since *P**Q*(*i*) ∈ P) and that \n0 = ∂*t*(∑*i**ν**i*(*y*, *s*)) = ∑*i*∂*t**ν**i*(*y*, *s*) \n for (*y*, *s*) ∈ 2*I**j*. Using this, Lemma [lemma2.21] (ii), and, we deduce, for (*y*, *s*) ∈ 2*I**j*,\n \n$$\\begin{aligned}\n|\\partial\\_t \\psi(y,s)| &= \\bigl|\\partial\\_t \\sum\\_i \\nu\\_i(y,s) B\\_i(y,s)\\bigr| = \\bigl| \\sum\\_i \\partial\\_t\\nu\\_i(y,s) B\\_i(y,s)\\bigr|\n\\\\ &= \\bigl| \\sum\\_i \\partial\\_t\\nu\\_i(y,s) (B\\_i(y,s)-B\\_j(y,s)) \\bigr|\\\\\n& \\lesssim \\sum\\_i |\\partial\\_t \\nu\\_i | \\epsilon r\\_j \\lesssim \\epsilon r\\_j^{-1}.\\end{aligned}$$ \nHere we have also used diam(*I**i*) ∼ diam(*I**j*) whenever 10*I**i* ∩ 10*I**j* ≠ ∅.\nPushing the geometric square function to the graph\n==================================================\nIn the previous section we proved that there exists, for arbitrary but fixed ${{\\bf S}}\\in \\mathcal{F}$, a coordinate system and a Lip(1,1/2) function $\\psi\\_{{{\\bf S}}}:=\\psi=\\psi ( x, t ): \\mathbb R^{n-1}\\times\\mathbb R\\to \\mathbb R$ with parameter $b\\_1\\lesssim \\delta$, such that if we define \n$$\\Sigma\\_{{{\\bf S}}}:=\\Sigma\\_\\psi:=\\{(x,\\psi(x,t),t):\\ (x,t)\\in \\mathbb R^{n-1}\\times\\mathbb R\\},$$ \n then\n \n$$\\label{eq2.2a}\n\\sup\\_{(X,t)\\in 2\\kappa Q} d\\_p(X,t,\\Sigma\\_{{{\\bf S}}} )\\,\n\\lesssim\\epsilon\\ell(Q)\\,,\\qquad \\forall\\,Q\\in {{\\bf S}}.$$ \n(The latter bound is proved in Lemma [closegraph.lem].) Recall that we have already fixed some of the relations amongst the parameters *K*, *ε*, *δ* and *κ*: e.g., we have specified that *ε* ≪ *δ*, depending on *κ* (see Lemma [closegraph.lem]), and that *K* is very large, also depending on *κ* (see Lemma [lem1] and the proof of Lemma [closegraph.lem]). A little more precisely, we have 0 < *C*(*κ*) *ε* ≪ *δ* ≪ 1, for some large enough constant *C*(*κ*), and *K* = *c*(*κ*, *n*, *M*) ≫ 1. In addition to these existing relations, in the final part of the argument, see, *ε* will be chosen to depend further upon *n*, *K*, the ADR constant *M*, and the parabolic UR constant Γ (see Definition [def1.UR]), as well as on *κ* and *δ*.\nTo complete the proof of Theorem [main] we have to prove two things. First, we have to prove that the constructed function $ \\psi\\_{{{\\bf S}}}:=\\psi = \\psi ( x, t ) : \\mathbb R^{n-1}\\times\\mathbb R\\to \\mathbb R$ in fact is a *regular* parabolic Lip(1,1/2) function with parameters $b\\_1\\lesssim\\delta$ and *b*2 = *b*2(*n*, *M*, Γ). Second, we have to prove that the maximal cubes $Q({{\\bf S}})$ for the stopping time regimes satisfy the Carleson packing condition\n \n$$\\begin{aligned}\n\\label{pack}\\sum\\_{{{\\bf S}}: Q({{\\bf S}})\\subset Q}\\sigma\\big(Q({{\\bf S}})\\big)\\,\\leq\\, c(n,M,\\Gamma,\\delta,\\kappa)\\, \\sigma(Q)\\,,\n\\quad \\forall Q\\in {\\mathbb{D}}(\\Sigma).\\end{aligned}$$ \nThe key to both of these arguments is to make use of the geometric square function which is part of the definition of parabolic uniform rectifiability. In this section we prove how this information can be pushed to the constructed graph.\nRecall that Σ is equipped with the geometric square function *γ* and the Carleson measure *ν*. We let ${\\widetilde}\\Sigma$ be the graph of *ψ* and we denote the corresponding geometric square function and Carleson measure by $\\tilde\\gamma$ and $\\tilde \\nu$, respectively. In addition, for $ r > 0, ( z, \\tau ) \\in\n\\mathbb R^{n} $ we introduce\n \n$$\\begin{aligned}\n\\hat\\gamma ( z, \\tau, r ) \\, := \\, \\biggl ( \\inf\\_{ L } \\, \\bariint\\_{ C'\\_r ( z, \\tau ) } \\, \\biggl (\\frac {| \\psi ( y, s ) - L ( y ) |}{r}\\biggr )^2 \\, {\\, \\mathrm{d}}y{\\, \\mathrm{d}}s \\biggr )^{1/2},\\end{aligned}$$ \nwhere the infimum is with respect to all linear functions of *y* (only). Using the regularity of the *ψ* constructed, we see that[9](#fn9)\n \n$$\\begin{aligned}\n\\label{measure}\n\\ d {\\sigma}\\_{{\\widetilde}\\Sigma} \\approx \\ d(\\mu|\\_{{\\widetilde}\\Sigma})\n( Y, s ) = \\sqrt{ 1 + | \\nabla\\_y \\psi ( y, s ) |^2 } \\, {\\, \\mathrm{d}}y{\\, \\mathrm{d}}s\\sim {\\, \\mathrm{d}}H^{n-1 } ( y){\\, \\mathrm{d}}s,\\end{aligned}$$ \nfor all $( Y, s ) \\in {\\widetilde}\\Sigma$, and we recall that *μ* is the slice-wise measure (see ). The reader can easily verify that the slice-wise measure on the graph is parabolic ADR, and hence after an application of Proposition [muhpsim.prop] that the implicit constants in the first comparability in display [measure] above depend only on constants which depend on dimension and *δ*. Combining with and it follows that\n \n$$\\begin{aligned}\n\\label{2.30}\\hat\\gamma ( z, \\tau, r\n)\\sim \\tilde {\\gamma}( Z, \\tau, r )\\mbox{ for all }(Z,\\tau)=(z,\\psi(z,\\tau),\\tau)\\in{\\widetilde}\\Sigma,\\ r>0.\\end{aligned}$$ \nThe essence of the section is to prove that we can control integrals of $\\hat\\gamma ( z, \\tau, r\n)$, and hence integrals of *γ̃*(*Z*, *τ*, *r*), with corresponding quantities involving *γ*, with an error controlled by *ε*2. In particular, the most important lemma in this section is the following.\n[count++] Fix ${{\\bf S}}\\in {\\mathcal{F}}$. Consider $(\\hat z,\\hat\\tau)\\in\\mathbb R^n$, *ρ* > 0, and let\n \n$$\\begin{aligned}\n\\label{pl11n}\n \\hat\\nu(\\hat z,\\hat\\tau,\\rho):=\n \\int\\_0^{\\rho}{\\int\\_{}\\kern-.34em \\int}\\_{C'\\_{\\rho}(\\hat z,\\hat\\tau)}(\\hat\\gamma ( z, \\tau, r ))^2\\frac {{\\, \\mathrm{d}}z{\\, \\mathrm{d}}\\tau{\\, \\mathrm{d}}r}r.\\end{aligned}$$ \nAssume further that\n \n$$\\begin{aligned}\n \\label{local}\nC'\\_{2\\rho}(\\hat z,\\hat\\tau)\\subseteq C'\\_{2\\kappa R}(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})}),\\quad\n\\rho\\leq\\kappa R/10.\\end{aligned}$$ \nThen\n \n$$\\begin{aligned}\n\\label{pl11n+}\n \\hat\\nu(\\hat z,\\hat\\tau,\\rho)\\lesssim \\epsilon^2\\rho^{d}+{{\\int\\_{}\\kern-.34em \\int}\\_{}\\kern-.34em \\int}\\_{E(\\hat z,\\hat\\tau,\\rho)} ({\\gamma}( Z, \\tau, Kr))^2 \\, \\frac{{\\, \\mathrm{d}}{\\sigma}( Z, \\tau){\\, \\mathrm{d}}r} {r}\\end{aligned}$$ \nwhere $E(\\hat z,\\hat\\tau,\\rho)$ is the set of all $(Z,\\tau,r)\\in \\big[\\Sigma\\cap\\pi^{-1}(C'\\_{K\\rho}(\\hat z,\\hat\\tau))\\big]\\times(0,\\infty)$ such that *K*− 1*d*(*Z*, *τ*) ≤ *r* ≤ *K**ρ*.\nFrom the previous lemma we shall be able to deduce the following lemma which will yield the desired regularity of the graph (see Lemma [pushlemma2]).\n[pushlemma1] Let $(\\hat z,\\hat\\tau)\\in\\mathbb R^n$, *ρ* > 0, and let $\\hat\\nu(\\hat z,\\hat\\tau,\\rho)$ be defined as in Lemma [count++]. Then\n \n$$\\begin{aligned}\n\\label{pl11}\n \\hat\\nu(\\hat z,\\hat\\tau,\\rho)&\\lesssim (\\epsilon^2+||\\nu||)\\rho^{d}.\\end{aligned}$$ \nThe rest of this section is devoted to the proofs of Lemma [count++] and Lemma [pushlemma1]. To achieve this we will prove a number of auxiliary lemmas. Recall that $R = \\operatorname{diam}(Q({{\\bf S}}))$.\nAuxiliary lemmas\n----------------\n[count] Let $(z,\\tau)\\in C'\\_{2\\kappa R}(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})})$ and *D*(*z*, *τ*)/60 ≤ *r* ≤ *κ**R*/10. Let *I*(*z*, *τ*, *r*) be the set of all *i* ∈ Λ such that *C**r*(*z*, *τ*) ∩ *I**i* ≠ ∅. Let, for *i* ∈ Λ, *J*(*i*) be the subset of *I*(*z*, *τ*, *r*) which consists of those *j* which satisfy diam(*Q*(*j*)) ≤ diam(*Q*(*i*)) and 2*Q*(*j*) ∩ 2*Q*(*i*) ≠ ∅. Let \n*N**i*(*Y*, *s*) : = ∑*j* ∈ *J*(*i*)*χ*2*Q*(*j*)(*Y*, *s*). \n Then\n \n$$\\label{Niavgbnd.eq}\n{\\int\\_{}\\kern-.34em \\int}N\\_i(Y,s)\\, {\\, \\mathrm{d}}\\sigma (Y,s)\\lesssim \\sigma(Q(i)).$$ \nFurthermore, there exists, for any *β* > 1, a constant *c* = *c*(*n*, *M*, *β*) > 1, such that\n \n∑*i* ∈ *I*(*z*, *τ*, *r*)(*N**i*(*Y*, *s*))− *β**χ*2*Q*(*i*)(*Y*, *s*) ≤ *c*, \nfor all (*Y*, *s*) ∈ ∪ *i* ∈ *I*(*z*, *τ*, *r*)(2*Q*(*i*)). Here we interpret[10](#fn10) (*N**i*(*Y*, *s*))− *β* = 0 if *N**i*(*Y*, *s*) = ∞.\nTo prove of the lemma it suffices to note that\n \n$$\\begin{aligned}\n{\\int\\_{}\\kern-.34em \\int}N\\_i(Y,s)\\, {\\, \\mathrm{d}}\\sigma (Y,s)\\lesssim \\sum\\_{j\\in J(i)}\\sigma(Q(j))\\lesssim \\sum\\_{j\\in J(i)}\\sigma(I\\_j)\\lesssim\n\\sigma(Q(i))\\end{aligned}$$ \nsince {*I**j*} is a disjoint collection, with $d\\_p\\big(I\\_j,\\pi(Q(i))\\big)\\lesssim r\\_i$ for each *j* ∈ *J*(*i*).\nTo prove, we consider (*Y*, *s*) ∈ ⋃*i* ∈ *I*(*z*, *τ*, *r*)2*Q*(*i*) fixed. Note that in the sum in we only consider indices *i* in *I*(*z*, *τ*, *r*) such that (*Y*, *s*) ∈ 2*Q*(*i*) and in the following we, for simplicity, denote this collection $\\widetilde{I}$. An important observation is that there exists *a*ʹ = *a*ʹ(*n*, *M*) ∈ N, such that if $i, i' \\in \\widetilde{I}$, *Q*(*i*) ∈ D*k* and *Q*(*i*ʹ) ∈ D*k*ʹ, for some *k* and *k*ʹ such that *k* ≤ *k*ʹ − *a*ʹ, then *i*ʹ ∈ *J*(*i*). This is a consequence of the facts that diam(*Q*) ≈ 2− *k* for all *Q* ∈ D*k*, and that (*Y*, *s*) ∈ 2*Q*(*i*) ∩ 2*Q*(*i*ʹ).\nWe divide the proof of into two cases. First, assume that $\\# \\widetilde{I} = \\infty$. In this case, since each $Q(i) \\subseteq Q({\\bf S})$, there must exist a sequence of cubes {*Q*(*i**m*)}*m* with $i\\_m \\in \\widetilde{I}$, such that diam(*Q*(*i**m*)) → 0 as *m* → ∞. Using the observation made above, this implies that *N**i*(*Y*, *s*) = ∞ for all $i \\in \\widetilde{I}$. In particular, \n$$\\sum\\_{i \\in I(z,\\tau, r)} (N\\_i(Y,s))^{-\\beta} \\mathbbm{1}\\_{2Q(i)}(Y,s) = 0.$$ \nSecond, assume that $\\# \\widetilde{I} < \\infty$. In this case, the sets {*Q*(*i*)}, where $i \\in \\widetilde{I}$, belong to a finite number of generations of D. With this in mind, we let {*k**m*}*m* = 1*m*0, *k*1 > *k*2 > … > *k**m*0, be all the integers for which there exists *Q*(*i*) ∈ D*k**m* with $i \\in \\widetilde{I}$. For *m* ∈ {1, 2, …, *m*0} we introduce \n$$\\mathcal{G}\\_m := \\{Q(i) : i \\in \\widetilde{I}, Q(i) \\in \\mathbb{D}\\_{k\\_m}\\},$$ \n and we note that G*m* is non-empty. Again by the observation made above, we can conclude that if *Q*(*i*) ∈ G*m* and *Q*(*i*ʹ) ∈ G*m*ʹ, with *m* ≥ *m*ʹ + *a*ʹ, then *i*ʹ ∈ *J*(*i*). Thus, \n*N**i*(*Y*, *s*) ≥ max{1, *m* − *a*ʹ}, ∀*Q*(*i*) ∈ G*m*. \n For each dyadic generation *k*, every *Q* ∈ D*k* such that (*Y*, *s*) ∈ 2*Q*, must lie within a distance $\\lesssim \\operatorname{diam}Q \\approx 2^{-k}$ of (*Y*, *s*), hence, by the dyadic cube construction, there are at most a uniformly bounded number of such cubes; in particular, #G*m* ≤ *c*(*n*, *M*). Consequently,\n \n$$\\begin{aligned}\n\\sum\\_{i \\in I(z,\\tau, r)} (N\\_i(Y,s))^{-\\beta} \\mathbbm{1}\\_{2Q(i)}(Y,s) &= \\sum\\_{m = 1}^{m\\_0} \\sum\\_{Q(i) \\in \\mathcal{G}\\_m} (N\\_i(Y,s))^{-\\beta} \\mathbbm{1}\\_{2Q(i)}(Y,s)\n\\\\& \\le \\sum\\_{m = 1}^{m\\_0} \\#\\mathcal{G}\\_m (\\max\\{1, m-a'\\})^{-\\beta}\\\\\n& \\le c'(n,M,\\beta),\\end{aligned}$$ \nwhere we have used that *a*ʹ = *a*ʹ(*n*, *M*).\nThe following lemma will be an important ingredient in the proof of Lemma [count++].\n[count+] Let $(z,\\tau)\\in C'\\_{2\\kappa R}(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})})$ and *D*(*z*, *τ*)/60 ≤ *r* ≤ *κ**R*/10. Let *I*(*z*, *τ*, *r*) be the set of all *i* ∈ Λ such that *C*ʹ*r*(*z*, *τ*) ∩ *I**i* ≠ ∅. Let (*X*, *t*) ∈ Σ, (*X*, *t*) = (*X*(*z*, *τ*, *r*), *t*(*z*, *τ*, *r*)), be such that $(X,t)\\in Q({{\\bf S}})$ and $d\\_p(z,\\tau,\\pi(X,t))\\lesssim r$. Then\n \n$$\\begin{gathered}\n\\label{5.17}\n(\\hat\\gamma (z,\\tau, r ))^2\n\\\\[4pt]\n\\lesssim r^{-d}{\\int\\_{}\\kern-.34em \\int}\\_{\\Sigma\\cap C\\_r(X,t)}(\\gamma(Y,s,Kr/10))^2\\, {\\, \\mathrm{d}}\\sigma(Y,s) \\,\n+\\,\\sum\\_{i\\in I( z, \\tau, r )} \\epsilon^2(r\\_i/r)^{d+2},\\end{gathered}$$ \nLet $(z,\\tau)\\in C'\\_{2\\kappa R}(x\\_{ Q({{\\bf S}})},t\\_{ Q({{\\bf S}})})$, *I*(*z*, *τ*, *r*) and \n(*X*, *t*) = (*X*(*z*, *τ*, *r*), *t*(*z*, *τ*, *r*)) ∈ Σ, \n be as stated in the lemma. We emphasize that by construction the point (*X*, *t*) depends on (*z*, *τ*) and *r*. Let *P* : = *P*(*X*, *t*, *r*) ∈ P be the time independent plane for which the infimum in the definition of *γ*(*X*, *t*, *K**r*/10) is realized. Recall the notation \n$${\\widetilde}\\Sigma=\\{(y,\\psi(y,s),s):\\ (y,s)\\in\\mathbb R^n\\}.$$ \n To denote points on the approximating graph, we will in the following also use the notation ${\\widetilde}\\Sigma(y,s):=(y,\\psi(y,s),s)$.\nTo start the proof we first note that\n \n$$\\begin{aligned}\n\\label{aeq1}\n(\\hat\\gamma ( z,\\tau, r ))^2&\\lesssim r^{-d}{\\int\\_{}\\kern-.34em \\int}\\_{C'\\_r( z,\\tau)}\\biggl (\\frac{d\\_p(\\tilde\\Sigma(y,s),P)}r\\biggr )^2\\, {\\, \\mathrm{d}}y{\\, \\mathrm{d}}s,\\end{aligned}$$ \nand we introduce\n \n$$\\begin{aligned}\n\\label{aeq2}\nT&:= r^{-d}{\\int\\_{}\\kern-.34em \\int}\\_{C'\\_r( z,\\tau)\\cap \\pi(F) }\\biggl (\\frac{d\\_p(\\tilde\\Sigma(y,s),P)}r\\biggr )^2\\, {\\, \\mathrm{d}}y{\\, \\mathrm{d}}s,\\notag\\\\\nT\\_i&:=r^{-d}{\\int\\_{}\\kern-.34em \\int}\\_{C'\\_r( z,\\tau)\\cap I\\_i }\\biggl (\\frac{d\\_p(\\tilde\\Sigma(y,s),P)}r\\biggr )^2\\, {\\, \\mathrm{d}}y{\\, \\mathrm{d}}s.\\end{aligned}$$ \nUsing this notation we can continue the estimate in and write\n \n$$\\begin{aligned}\n\\label{aeq3}\n(\\hat\\gamma ( z,\\tau, r ))^2&\\lesssim T+\\sum\\_{i\\in I( z,\\tau, r )}T\\_i.\\end{aligned}$$ \nIt is straightforward to lift the integral in the definition of *T* to the graph, and to produce the estimate\n \n$$\\begin{aligned}\n\\label{aeq4}\nT&\\lesssim r^{-d}{\\int\\_{}\\kern-.34em \\int}\\_{\\pi^{-1}(C'\\_r( z,\\tau))\\cap F }\\biggl (\\frac{d\\_p((Y,s),P)}r\\biggr )^2\\, {\\, \\mathrm{d}}\\sigma(Y,s)\\lesssim (\\gamma(X,t,Kr/10))^2.\\end{aligned}$$ \nNote that in this estimate we have also used.\nFix *i* ∈ *I*(*z*, *τ*, *r*). Then \n$$T\\_i\\lesssim \\hat T\\_i+{\\widetilde}T\\_i,$$ \n where\n \n$$\\begin{aligned}\n\\hat T\\_i&:=r^{-d}{\\int\\_{}\\kern-.34em \\int}\\_{C'\\_r( z,\\tau)\\cap I\\_i }\\biggl (\\frac{d\\_p(\\tilde\\Sigma(y,s),P\\_{Q(i)})}r\\biggr )^2\\, {\\, \\mathrm{d}}y{\\, \\mathrm{d}}s,\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n{\\widetilde}T\\_i&:=r^{-d}r\\_i^{d}\\sup\\biggl\\{\\biggl (\\frac {d\\_p(Y,s,P)}r\\biggr )^2:\\ (Y,s)\\in P\\_{Q(i)},\\ d\\_p(Y,s,Q(i))\\leq cr\\_i\\biggr\\}.\\end{aligned}$$ \nRecall and. Using that *P**Q*(*i*) is the graph of *B**i*,\n \n$$\\begin{aligned}\n\\label{aeq5}\n\\hat T\\_i&=r^{-d}{\\int\\_{}\\kern-.34em \\int}\\_{C'\\_r( z,\\tau)\\cap I\\_i }\\biggl (\\frac{|\\psi(y,s)-B\\_i(y,s)|}r\\biggr )^2\\, {\\, \\mathrm{d}}y{\\, \\mathrm{d}}s\\lesssim \\epsilon^2r^{-d}r\\_i^{d}(r\\_i/r)^2,\\end{aligned}$$ \nwhere we have used that $|\\psi(y,s)-B\\_i(y,s)|\\lesssim \\epsilon r\\_i$ on *I**i*, see Lemma [lemma2.21].\nTo estimate ${\\widetilde}T\\_i$ we use Lemma [aux1] applied to *Q* = *Q*(*i*). Indeed, let (*Z**j*, *τ**j*) ∈ *Q*(*i*) for *j* = 0, 1, ..., *n*, be as in the statement of the lemma and let *L**Q*(*i*) = *L**n* − 1 be the spatial (*n* − 1)-dimensional plane which passes through *Z*0, *Z*1, ..., *Z**n* − 1. Then\n \n$$\\begin{aligned}\n\\label{planeapprox}\n{\\widetilde}T\\_i\\lesssim r^{-d}r\\_i^{d}\\sup\\_{j\\in\\{0,...,n-1\\}}\\biggl \\{\\biggl (\\frac {d\\_p(Z\\_j,\\tau\\_j,P)}r\\biggr )^2\n+\\biggl (\\frac {d\\_p(Z\\_j,\\tau\\_j,P\\_{Q(i)})}r\\biggr )^2\\biggr \\}.\\end{aligned}$$ \nTo see this, we note that by Lemma [aux1] we have that the points {(*Z**j*, *τ**j*)} stay at an ample distance from each other, and the point (*Y**j*, *s**j*), realizing the infimum \ninf(*Y*, *s*) ∈ *P**Q*(*i*)*d**p*((*Y*, *s*), (*Z**j*, *τ**j*)), \n is within *ε*diam*Q*(*i*) of (*Z**j*, *τ**j*). Hence, for sufficiently small *ε*, we can ensure that \n$$\\|(Y\\_j,s\\_j)-(Y\\_{j'},s\\_{j'})\\| \\gtrsim \\operatorname{diam}Q(i)\\mbox{ whenever }{j'}\\neq j.$$ \n In particular, the points {(*Y**j*, *s**j*)} generate the plane *P**Q*(*i*), and for each (*Y*, *s*) ∈ *P**Q*(*i*) such that *d**p*(*Y*, *s*, *Q*(*i*)) ≤ *c**r**i*, we have \n$$d\\_p(Y,s,L\\_{n-1}) \\lesssim \\max\\_{j \\in \\{0,...,n-1\\}} d\\_p(Y\\_j,s\\_j,Z\\_j,\\tau\\_j) = \\max\\_{j \\in \\{0,...,n-1\\}} d\\_p(Z\\_j,\\tau\\_j,P\\_{Q(i)}).$$ \n An even simpler argument shows that *d**p*(*L**n* − 1, *P*) is controlled by \nmax*j* ∈ {0, ..., *n* − 1}*d**p*(*Z**j*, *τ**j*, *P*). \n In this case we do not need to find a generating set for *P*. Put together we can conclude that holds.\nUsing that $Q(i)\\in {{\\bf S}}$ we see that\n \n$$\\begin{aligned}\n\\sup\\_{j\\in\\{0,...,n\\}}\\biggl \\{\\biggl (\\frac {d\\_p(Z\\_j,\\tau\\_j,P\\_{Q(i)})}r\\biggr )^2\\biggr \\}\\lesssim\\epsilon^2(r\\_i/r)^2,\\end{aligned}$$ \nand hence, based on,\n \n$$\\begin{aligned}\n\\label{aeq6}\n{\\widetilde}T\\_i\\lesssim r^{-d}r\\_i^{d}\\sup\\_{j\\in\\{0,...,n\\}}\\biggl \\{\\biggl (\\frac {d\\_p(Z\\_j,\\tau\\_j,P)}r\\biggr )^2\\biggr \\}\n+ r^{-d}r\\_i^{d}\\epsilon^2(r\\_i/r)^2.\\end{aligned}$$ \nWe now note that the statement in remains true with (*Z**j*, *τ**j*) replaced by any $(\\tilde Z\\_j,\\tilde \\tau\\_j)\\in \\Sigma$ satisfying $d\\_p(Z\\_j,\\tau\\_j,\\tilde Z\\_j,\\tilde \\tau\\_j)\\leq \\eta r\\_i$ for *η* = *η*(*n*, *M*) sufficiently small. This can be seen from the proof of Lemma [aux1] because we are free, at each iteration of the argument, to replace the points (*Z**j*, *τ**j*) with sufficiently close points $(\\tilde Z\\_j,\\tilde \\tau\\_j)$, as this will preserve the desired properties of the (new) approximating planes and collection of points. Combining this observation,,,, and averaging over such (*Z̃**j*, *τ̃**j*), we conclude that\n \n$$\\begin{gathered}\n\\label{aeq7}\n(\\hat\\gamma ( z,\\tau, r ))^2 \\\\[4pt]\n\\lesssim \\, (\\gamma(X,t,Kr/10))^2\\,+\\sum\\_{i\\in I( z,\\tau, r )} \\epsilon^2r^{-d}r\\_i^{d}(r\\_i/r)^2 \\,\n+\\sum\\_{i\\in I( z,\\tau, r )} r^{-d}r\\_i^{d}\\beta\\_{Q(i)}\\,,\\end{gathered}$$ \nwhere\n \n$$\\begin{aligned}\n\\label{aeq7}\n\\beta\\_{Q(i)}:=\\biggl(\\bariint\\_{2Q(i)} \\biggl (\\frac {d\\_p(Y,s,P)}r\\biggr )^{2/3}{\\, \\mathrm{d}}\\sigma (Y,s)\\biggr )^{3}\\,.\\end{aligned}$$ \nWe now let *J*(*i*) be the subset of *I*(*z*, *τ*, *r*) which consists of those *j* which are such that diam(*Q*(*j*)) ≤ diam(*Q*(*i*)) and 2*Q*(*j*) ∩ 2*Q*(*i*) ≠ ∅. We let\n \n$$\\begin{aligned}\nN\\_i(Y,s):=\\sum\\_{j\\in J(i)}\\chi\\_{2Q(j)}(Y,s).\\end{aligned}$$ \nThen using Lemma [count] we have, for *β* > 1 given, that there exists a constant *c* = *c*(*n*, *M*, *β*), 1 ≤ *c* < ∞, such that\n \n∑*i*(*N**i*(*Y*, *s*))− *β**χ*2*Q*(*i*)(*Y*, *s*) ≤ *c*, \nfor all (*Y*, *s*) and\n \n$$\\label{aeq9}\n\\bariint\\_{2Q(i)}N\\_i(Y,s)\\, {\\, \\mathrm{d}}\\sigma (Y,s)\\lesssim 1.$$ \nTo estimate *β**Q*(*i*), we apply Hölder’s inequality and use to deduce that\n \n$$\\begin{gathered}\n\\beta\\_{Q(i)} =\n\\biggl(\\bariint\\_{2Q(i)} \\biggl (\\frac {d\\_p((Y,s),P)}r\\biggr )^{2/3}\n(N\\_i(Y,s))^{-2/3}(N\\_i(Y,s))^{2/3}{\\, \\mathrm{d}}\\sigma (Y,s)\\biggr )^{3} \\\\[4pt]\n\\lesssim \\,\\biggl(\\bariint\\_{2Q(i)} \\biggl (\\frac {d\\_p((Y,s),P)}r\\biggr )^{2}(N\\_i(Y,s))^{-2}{\\, \\mathrm{d}}\\sigma (Y,s)\\biggr ).\\end{gathered}$$ \nIn particular we can rewrite as\n \n$$\\begin{gathered}\n(\\hat\\gamma ( z,\\tau, r ))^2 \\,\n\\lesssim \\,(\\gamma(X,t,Kr/10))^2\\,+\\sum\\_{i\\in I( z,\\tau, r )} \\epsilon^2r^{-d}r\\_i^{d}(r\\_i/r)^2\\\\[4pt]\n+ \\,r^{-d}\\sum\\_{i\\in I( z,\\tau, r )}\n{\\int\\_{}\\kern-.34em \\int}\\_{2Q(i)} \\biggl (\\frac {d\\_p(Y,s,P)}r\\biggr )^{2}(N\\_i(Y,s))^{-2}{\\, \\mathrm{d}}\\sigma (Y,s).\\end{gathered}$$ \nThen using with *β* = 2, we conclude that\n \n$$\\begin{gathered}\n(\\hat\\gamma ( z,\\tau, r ))^2 \n\\,\\lesssim \\,(\\gamma(X,t,Kr/10))^2 \\,\n+\\sum\\_{i\\in I( z,\\tau, r )} \\epsilon^2r^{-d}r\\_i^{d}(r\\_i/r)^2 \\\\[4pt]\n+ \\,r^{-d} {\\int\\_{}\\kern-.34em \\int}\\_{\\cup\\_{i\\in I( z,\\tau, r )}2Q(i)} \\biggl (\\frac {d\\_p(Y,s,P)}r\\biggr )^{2}{\\, \\mathrm{d}}\\sigma ("},"data_id":{"kind":"string","value":"arxiv_0183122"}}},{"rowIdx":45798,"cells":{"content":{"kind":"string","value":"Perturbative reduction of derivative order in EFT\n=================================================\nHigher derivative corrections are ubiquitous in effective field theories, which seemingly introduces new degrees of freedom at successive order. This is actually an artefact of the implicit local derivative expansion defining effective field theories. We argue that higher derivative corrections that introduce additional degrees of freedom should be removed and their effects captured either by lower derivative corrections, or special combinations of higher derivative corrections not propagating extra derees of freedom. Three methods adapted for this task are examined and field redefinitions are found to be most appropriate. First order higher derivative corrections in a scalar tensor theory are removed by field redefinition and it is found that their effects are captured by a subset of Horndeski theories. A case is made for restricting the effective field theory expansions in principle to only terms not introducing additional degrees of freedom.\nIntroduction\n============\nEffective theories are a physical framework based on an observation that not all scales of nature are relevant for describing of a particular given system . Usually it pertains to descriptions of some low energy phenomenon in terms of the degrees of freedom and interactions relevant for that system, with typical energy scales well below some higher scales typical for the interactions and degrees of freedom which we neglect. It is not to say that we have to neglect the effect of physics on higher scales entirely. We may study the small corrections that it supplies to our low energy system, but we do not need to know exactly what that physics is. Its effects can be captured in terms of the degrees of freedom of the low energy system by introducing additional effective interactions suppressed by the ratio of scales typical for the system at hand and the high scale of new physics. In field theories such effects are encoded by all the possible corrections to the leading order action organized as an expansion in the inverse powers of the high energy (or mass) scale. At each order all that is demanded from the correction terms is to satisfy the assumed symmetries and to have the right dimensionality.\nHigher derivative corrections in effective field theory (EFT) are ubiquitous, and are very hard to avoid. Such corrections, taken at face value, tend to exhibit unphysical behaviour. The issues regarding them were most recently discussed by Burgess and Williams , who also argued we typically need not worry about the said issues. Here we make a case for the issues actually not being there if one treats higher derivatives in EFTs correctly in the same perturbative spirit in which they arise in the first place[1](#fn1)\nIt is known that higher derivative theories in general exhibit unphysical behavior. The most well known issue encountered is that of Ostrogradsky instabilities . Higher derivative theories generally contain more degrees of freedom per dynamical variable as compared to the lower derivative theories, and such theories we refer to as *genuine* higher derivative theories in this paper. Usually these extra degrees of freedom are ghost-like, and their excitation can lower the energy of the system without bounds. This is always true for theories without constraints, but can in certain cases be avoided in constrained theories (e.g. *f*(*R*) gravity). Another issue often exhibited by higher derivative theories is the existence of runaway solutions. It is important to emphasize that there are theories with higher derivatives in the action that are not genuine in the sense defined above, since they contain the same number of degrees of freedom as their lower order counterparts with the same number of dynamical variables. Examples of such non-genuine higher derivative theories are the Horndeski theory , generalized Galileons , and more generally the beyond Horndeski theories , and all the the DHOST theories , which are defined as the most general scalar-tensor theories with three propagating degrees of freedom. These are studied very much today in the context of the effective field theory of dark energy For convenience, we sometimes refer to non-genuine higher derivative theories as lower derivative theories when it is obvious we are emphasizing their property of not carrying extra degrees of freedom.\nEffective field theories in actuality are not genuine higher derivative theories. This becomes clear when we examine cases where we can actually derive them from a more fundamental theory by integrating out some heavy degrees of freedom. In principle, we do not need to assume much about the degrees of freedom being integrated out. This procedure produces an effective action for the system which contains nonlocal terms which do not carry extra degrees of freedom. By construction, the procedure of integrating out degrees of freedom can only lower their total number in the resulting theory. Only when we decide that the degrees of freedom integrated out are heavy with respect to some energy scale we are interested in can the nonlocal terms in the action be considered small, and expanded in inverse powers of the heavy mass scale. This expansion corresponds to the local derivative expansion of the nonlocal terms, and this is how higher derivative terms appear in the low energy EFT. The modern approach to field theories is to consider them as EFTs up to some higher energy scale. Then naturally all field theories can be considered to have corrections suppressed by this high energy (mass) scale. But here we maintain that these corrections ought to descend from some in principle non-local corrections.\nAt successively higher orders in EFTs terms with higher and higher derivatives appear. It makes no physical sense that each order of perturbation theory introduces new degrees of freedom. Rather, this is an artifact of our approximation scheme. The observation that the higher derivative terms appear perturbatively, and that their effects should be taken into account perturbatively as well, resolves the issue of extra degrees of freedom. They never appear in perturbatively expandable solutions.\nThis problem was first encountered and studied in the context of Abraham-Lorentz theory of the electron , where the third time derivative of the electron’s position appears in the equation of motion. This third derivative is there to take into account the electron’s reaction to its own radiation. The space of solutions of this theory contains some clearly unphysical runaway solutions – one can set initial conditions for the electron to be at rest, and it can still accelerate to velocities arbitrarily close to the speed of light. This is an example of the runaway solution due to equations of motion requiring the specification of the initial acceleration in addition to the initial position and velocity. It is an unphysical behavior introduced by treating the higher derivative corrections on the same footing as the leading order equation. One must discard all the solutions of the equations of motion that are not analytic in the expansion parameter as unphysical.\nThe aim of this paper is to investigate three different methods of correctly accounting for the higher derivative terms in the EFT expansion. They all aim to reduce the derivative order of the higher derivative corrections, representing to them in terms on lower derivative ones, and hence we refer them as *derivative reduction* methods. In particular we consider (i) derivative reduction on the level of the equations of motion, (ii) derivative reduction by method of perturbative constraints (described in Sec. [sec: Formalism]), and (iii) derivative reduction via field redefinition. Their advantages and disadvantages, utility and drawbacks are analyzed in detail in Sec. [sec: Scalar] on a series of examples. They all serve to motivate the overall claim we wish to make about the nature of the EFT expansion in general: *Genuine higher derivative corrections appearing in EFTs are completely captured by the most general corrections not introducing extra degrees of freedom*.\nIt is common lore, mostly due to the work of Simon and Parker , that the higher derivative terms can always be removed from the equations of motion by perturbatively applying lower order equations of motion. This is certainly true for any single particle systems, such as the minisuperspace model they considered, where the only dynamical variable is the scale factor (see also ). Even though the study of this single particle models and insightful, multiple issues and subtleties of this derivative reduction method are missed. One of the things that we examine in this paper is the analogous derivative reduction procedure at the equation of motion level in the field theory system possessing Lorentz symmetries, and whether this derivative reduction procedure is consistent with maintaining Lorentz symmetry. Our findings are that in general it is not, without allowing for field redefinitions. The same conclusion ought to be valid when considering local symmetries, such as in the EFT of gravity.\nReduction of the derivative order at the level of the equations of motion (if needed supplemented by field redefinition) is sufficient if one is interested in obtaining solutions as power series in the expansion parameter truncated to a given order (for some applications of this strategy to EFT of gravity see ). Sometimes such solutions are not sufficient. This is often the case in nonequilibrium systems , where spurious secular phenomena might be introduced by considering perturbative corrections to the leading order solutions. In such cases a resummation scheme is called for, which, in broad strokes, corresponds to solving exactly the truncated EFT equation with reduced derivatives. For instance, such circumstances arise when studying cosmological solutions in general relativity with EFT corrections. The problem that arises here is that the derivative reduction at the level of the equations of motion is not unique, in the sense that multiple equations accomplish the same task of capturing perturbative solutions to a given order. Attempting the resummed solution for different equations might lead to rather different results, and even exhibit effects such as energy non-conservation in conservative systems[2](#fn2) The correct way to avoid such spurious effects is to base the resummed solutions on the equations of motion that derive from an action principle, which guarantees properties such as energy conservation to all orders. The consideration of nonequilibrium systems motivates us to develop a formalism for reducing the derivative order of EFT corrections directly on the level of the action. Most of this work is devoted to the investigation of such methods.\n+0.5cm\nHigher derivative EFT corrections are particularly awkward if one wishes to consider them in the canonical (Hamiltonian) formalism. The initial motivation for this work came from the problem of constructing and quantizing higher order gauge-invariant cosmological perturbations[3](#fn3) in a single scalar inflationary model , with the idea of addressing certain issues regarding the loop correction to the primordial power spectrum (see and references therein). The formalism most suited for finding the gauge-invariant perturbations seems to be the canonical one, since it makes the constraint structure of the theory explicit . The quantization of these higher order perturbations necessarily requires the introduction of higher derivative terms to the action in order to guarantee one-loop renormalizability . But these higher derivative terms alter the structure of cosmological perturbations beyond linear order, and should be considered right from the start. Hence we need to consider these higher derivative terms in the canonical formalism.\nIn the canonical formalism every degree of freedom is promoted to an independent field, including the degrees of freedom associated to higher derivative terms in the EFT expansion . In the EFTs this is very cumbersome, since the resulting Hamiltonian is no longer perturbative in the inverse powers of the heavy mass scale assumed to be the control parameter in the EFT expansion, but rather has term containing positive powers of this mass parameter. This is clearly a signal of doing something wrong. In fact, it means precisely what we have stated, considering higher derivative terms to introduce extra degrees of freedom is inconsistent with the suppositions of EFT.\nThe very derivation of the canonical formalism in the presence of higher derivative EFT corrections has to be modified to accommodate for the requirement of perturbativity in the expansion parameter. This was noted and clarified by Jaen, Llosa, and Molina, , and by Eliezer and Woodard . The perturbativity requirement effectively introduces perturbative second-class constraints into the theory , which when solved for reduce the total number of degrees of freedom to the correct one, resulting in the lower order phase space action. Section [sec: Formalism] is devoted to recasting this method in a somewhat different form, and to the discussion of its possible drawbacks. Such a method would be preferable as a way to reconcile higher derivative corrections with the physical assumptions EFT expansion makes, since it would automatically mean that perturbativity requirement maps higher derivative EFT corrections onto lower derivative EFT ones. Larger part of Sec. [sec: Scalar] is devoted to applying this method to a scalar field system, where it is shown that even though the method always accomplishes what it is designed to do – produces a lower derivative phase space action – it in general does not provide a configuration space action (the Lagrangian) which makes it rather impractical for applications.\n+0.5cm\nThe third method of derivative reduction that we consider in this work is the field redefinition method. It ought be true that the physics (whatever is measured in the experiments) should not be altered by one’s choice of field variables describing the system. Hence, it is in principle permissible to perform transformations from one set of field variables to another (field redefinitions). There is a physical requirement on these redefinitions that they should not change the number of degrees of freedom system. Considering derivatives of the field, it means they should not contain any. In other words, we are not allowed to make a higher derivative theory out of the lower derivative one.\nConsider now field redefinitions is the EFTs. Assume that we know the exact form of the nonlocal correction to the action, before it is approximated by a local derivative expansion. Now we can imagine making a nonlocal field redefinition in such an action, resulting in another nonlocal action. We can take this field redefinition to be a special one – by requiring that the resulting nonlocal action has a local derivative expansion that contains no genuine higher derivative terms. Such special nonlocal transformation thus eliminates spurious degrees of freedom from the low energy EFT. Now consider taking a short-cut of this construction, directly on the low energy EFT. It is equivalent to making a local field redefinition containing derivatives, that removes all the genuine higher derivative terms from the EFT expansion (to a given order). It is due to these considerations that field redefinitions containing derivatives are allowed in EFTs, in the sense that field redefinitions themselves are treated as an EFT local derivative expansion. It is found in this paper that the field redefinition method of the derivative reduction is in general superior to the remaining two, especially when considering symmetries of the field systems.\n+0.5cm\nThe three mentioned methods for reducing the derivative order of EFT corrections are studied in this paper on several examples of a single scalar field in flat space. Attention is paid to the compatibility of these methods with Lorentz symmetry, and the possibility of obtaining a configuration space (Lagrangian) formulation of the reduced system. Field redefinitions are found to be the preferable method. The field redefinition method is then applied to the first order EFT corrections of a scalar-tensor theory, where it is found that the corrections are captured by the subset of Horndeski actions, which are the most general scalar-theories possessing second order equations of motion.\n+0.5cm\nThe introductory section that is drawing to a close serves to define and motivate the study of higher derivatives in the context of effective field theories. Section [sec: Formalism] is devoted to the presentation of the method of derivative reduction via perturbative constraints. In Section [sec: Scalar] we apply the three derivative reduction methods to four cases of a single scalar field with higher derivative corrections. The lessons learned from these examples are summarized at the end of Sec. [sec: Scalar]. In Section [sec: ST] the field redefinition method is employed to eliminate the first order higher derivative corrections from the EFT expansion of a scalar-tensor theory. The last section contains the discussion of the results and the concluding remarks.\nMethod of perturbative constraints\n==================================\nThis section presents the method of perturbative derivative reduction by implementing perturbative constraints[4](#fn4). The method has been developed for systems without constraints by Jaén, Molina and Llosa , and by Eliezer and Woodard . Here we present a modified version of it, which has a few advantages: (i) it applies to systems with arbitrary leading order actions, not only to the case of a free leading order system, and (ii) it applies to explicitly time dependent systems, and ones where Noether current is cumbersome to define, (iii) it is easily generalized to systems with constraints.\nEven though in this work we aim to investigate field systems, the method here is presented for a multi-particle systems. The reason is that the notation is simpler, and we do not have to explicitly write gradients, which just obscures the expressions. Nevertheless, the generalization to field systems is obvious and straightforward.\nThroughout this section, and later when applying the method we use Dirac’s notation for equalities , where we distinguish between strong equality (off-shell equality) “ = ” denotes the equality which is true on the level of the action, without applying the equations of motions, and a weak equality (on-shell equality) “ ≈ ” designating equalities valid only after the imposition of equations of motion.\nSay that we are interested in studying a system of particles described by variables *x**i*, that at leading order is specified by the *leading order action*,\n \n$$S^{(0)}[x\\_i] = \\int\\! dt \\, \\mathscr{L}^{(0)}(x\\_i, \\dot{x}\\_i) \\,,$$ \nwhich is assumed not to be singular, *i.e.* its Hessian is supposed invertible,\n \n$${\\rm det} \\Bigl( \\frac{\\partial^2 \\mathscr{L}^{(0)}}{\\partial \\dot{x}\\_i \\partial\\dot{x}\\_j} \n \\Bigr) \\neq 0 \\,.$$ \nand there are no constraints. The generalization of this method to systems with constraints is beyond the scope of this paper. We consider the first *N* corrections to this action in the form of an EFT expansion in terms (operators) of successively higher dimension, with *ε* being the expansion parameter, given by the configuration space action,\n \n*S*[*x**i*] = *S*(0)[*x**i*] + *ε**S*(1)[*x**i*] + *ε*2*S*(2)[*x**i*] + … + *ε**N**S*(*N*)[*x**i*] = ∑*n* = 0*N**ε**n**S*(*n*)[*x**i*] . \nThe higher order corrections are also assumed to contain irreducible higher time derivatives[5](#fn5), with each order introducing one more derivative,\n \n$$S^{(n)}[x\\_i] = \\int\\! dt \\,\n \\mathscr{L}^{(n)} \\bigl( x\\_i, \\dot{x}\\_i, \\dots, x\\_i^{[n]}, x\\_i^{[n+1]} \\bigr) \\,,\n\\qquad\nx\\_i^{[m]} \\equiv \\frac{d^m}{dt^m} x\\_i \\,,$$ \nso that each new order introduces one derivative more. In what follows we construct a method to isolate a subspace of solutions of this theory that are regular in the expansion parameter *ε*, *i.e.* that are perturbatively expandable in *ε*.\nThe starting point of the method is to define the *extended action* , which can be seen as the intermediate step between the Lagrangian and the Hamiltonian formalism, and is particularly suitable for theories with constraints. The extended action is defined by promoting all the time derivatives to independent velocity variables, *x**i*[*n*] → *V**i**n*, and introducing Lagrange multipliers *π**i*, *π**i*1,…,*π**i**N* − 1 to ensure the on-shell equivalence of the two formulations,\n \n$$\\begin{aligned}\n\\MoveEqLeft[4]\n\\mathcal{S}[x\\_i, V^1\\_i, V^2\\_i, \\dots, V^{N+1}\\_i, \\pi\\_i,\\pi\\_i^1,\\dots,\\pi\\_i^N] \n\\nonumber \\\\\n={}& \\int\\! dt \\, \\biggl\\{\n \\mathscr{L}^{(0)}(x\\_i, V^1\\_i)\n + \\epsilon \\mathscr{L}^{(1)}(x\\_i, V^i\\_i, V^2\\_i)\n + \\dots \n + \\epsilon^N \\mathscr{L}^{(N)}(x\\_i, V^1\\_i, \\dots, V^{N+1}\\_i)\n\\nonumber \\\\\n& \\hspace{2.5cm}\n + \\pi\\_i (\\dot{x}\\_i - V^1\\_i) + \\pi\\_i^1 (\\dot{V}\\_1-V^2\\_i) + \\dots\n + \\pi\\_i^{N} (\\dot{V}^{N}\\_i - V^{N+1}\\_i)\n \\biggr\\}\n\\label{EA}\\end{aligned}$$ \nThe canonical formalism, namely the phase space action S is now one step away. One needs to solve for the highest velocities *V**i**N* + 1 from the equation descending from the variation of the extended action with respect to *V**i**N* + 1’s,\n \n$$\\frac{\\delta \\mathcal{S}}{\\delta V^{N+1}\\_i}\n = \\epsilon^N \\frac{\\partial \\mathscr{L}^{(N)}}{\\partial V^{N+1}\\_i}\n - \\pi\\_i^{N} \\approx 0 \\,.\n\\label{V^N equation}$$ \nwhich is an algebraic relation involving no time derivatives and plug it back into S. The solutions for *V**i**N*’s,\n \n$$V^{N+1}\\_i \\approx \\overline{V}^{N+1}\\_i(x\\_i, V^1\\_i,\\dots V^{N}\\_i, \n \\pi\\_i,\\pi^1\\_i, \\dots \\pi^{N}\\_i ) \\,,$$ \nare then plugged back into S as strong equalities, which results in the phase space action,\n \n$$\\begin{aligned}\n\\MoveEqLeft[4]\n\\mathscr{S}[x\\_i,\\pi\\_i, V^1\\_i, \\pi^1\\_i, \\dots, V^{N}\\_i, \\pi^{N}\\_i]\n\\nonumber \\\\\n={}& \\int\\! dt \\, \\biggl\\{ \\pi\\_i \\dot{x}\\_i + \\pi^1\\_i \\dot{V}^1\\_i\n + \\dots + \\pi^N\\_i \\dot{V}^N\\_i \n - H\\bigl(x\\_i,\\pi\\_i, V^1\\_i, \\pi^1\\_i, \\dots V^{N}\\_i, \\pi^{N}\\_i \\bigr) \\biggr\\} \\,,\\end{aligned}$$ \nwhere the Hamiltonian is\n \n$$\\begin{aligned}\nH ={}& \\pi\\_i V^1\\_i + \\pi^1\\_i V^2\\_i + \\dots \n + \\pi^{N-1}\\_i V\\_i^N \n + \\pi^{N} \\overline{V}\\_i^{N+1}\n\\nonumber \\\\\n& - \\mathscr{L}^{(0)}(x\\_i, V^1\\_i) - \\epsilon \\mathscr{L}^{(1)}(x\\_i, V^1\\_i, V^2\\_i)\n - \\dots - \\epsilon^N \\mathscr{L}^{(N)}(x\\_i, V^1\\_i, V^2\\_i,\\dots, \\overline{V}^{N+1}\\_i) \\,.\\end{aligned}$$ \nThis phase space action is on-shell equivalent to the extended action ([EA]), and hence to the starting configuration space action ([CSA]) as well. This construction is equivalent to the one introduced by Ostrogradsky . There is nothing about this construction specific to higher derivative theories. It is perfectly applicable to standard lower derivative theories, and in fact allows for a much more pedagogical introduction of the canonical (Hamiltonian) formalism.\nThe obstruction to the construction of the phase space action in the case at hand is that equation ([VN equation]) does not admit solutions for *V**i**N* + 1’s analytic in the perturbative expansion parameter *ε*, and is therefore not admissible by our assumptions[6](#fn6). Given such assumptions, the equation ([VN equation]) actually represents a constraint, and should be treated as such. This observation allows us to isolate from the full theory only the subset of solutions perturbatively expandable in *ε*.\nIn the language of Dirac’s canonical formalism equation ([VN equation]) is a *primary* constraint. The requirement that this constraint is consistent, *i.e.* that its time derivative vanishes on-shell produces a *secondary* constraint. Again requiring consistency, and repeating this process generates a set of constraints that actually end up being *second-class*, meaning that in the end the variables *V**i**N* + 1’s are uniquely determined (by the equations of motion). Even though we can cast the whole method in the form usually employed in canonical formalism, by defining Poisson brackets, and working with the Hamiltonian, it is actually not the most practical way to proceed here. It is more convenient to work with the extended action ([EA]), and the equations of motion descending from varying it with respect to independent variables.\nThe systematic way to proceed is to write down all the equations descending from varying the extended action,\n \n$$\\begin{aligned}\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta V\\_i^{N+1}} \\approx 0\n\\qquad &\\Rightarrow \\qquad \n\\pi\\_i^{N} \\approx \\epsilon^N \\frac{\\partial \\mathscr{L}^{(N)}}{\\partial V\\_i^{N+1}} \\,,\n\\label{first eq}\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta V\\_i^{N}} \\approx 0\n\\qquad & \\Rightarrow \\qquad \n\\dot{\\pi}\\_i^{N} \\approx - \\pi\\_i^{N-1} \n + \\epsilon^N \\frac{\\partial \\mathscr{L}^{(N)}}{\\partial V\\_i^{N}}\n + \\epsilon^{N-1} \\frac{\\partial \\mathscr{L}^{(N-1)}}{\\partial V\\_i^{N}} \\,,\n\\\\\n&\\displaystyle \\vdots \\qquad& \\nonumber \n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta V\\_i^2} \\approx 0\n\\qquad & \\Rightarrow \\qquad \n\\dot{\\pi}\\_i^{2} \\approx - \\pi\\_i^1\n + \\epsilon^N \\frac{\\partial \\mathscr{L}^{(N)}}{\\partial V\\_i^{2}}\n + \\dots + \\epsilon^{1} \\frac{\\partial \\mathscr{L}^{(1)}}{\\partial V\\_i^{2}} \\,,\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta V\\_i^1} \\approx 0\n\\qquad & \\Rightarrow \\qquad \n\\dot{\\pi}\\_i^1 \\approx - \\pi\\_i\n + \\epsilon^N \\frac{\\partial \\mathscr{L}^{(N)}}{\\partial V\\_i^{1}}\n + \\dots \n + \\epsilon^{0} \\frac{\\partial \\mathscr{L}^{(0)}}{\\partial V\\_i^{1}} \\,, \\qquad\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta x\\_i} \\approx 0\n\\qquad & \\Rightarrow \\qquad \n\\dot{\\pi}\\_i \\approx \\epsilon^N \\frac{\\partial \\mathscr{L}^{(N)}}{\\partial x\\_i}\n + \\dots \n + \\epsilon^{0} \\frac{\\partial \\mathscr{L}^{(0)}}{\\partial x\\_i} \\,, \\qquad\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta \\pi\\_i} \\approx 0\n\\qquad &\\Rightarrow \\qquad \n\\dot{x}\\_i \\approx V\\_i^1 \\,,\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta \\pi\\_i^1} \\approx 0\n\\qquad &\\Rightarrow \\qquad \n\\dot{V}\\_i^1 \\approx V\\_i^2 \\,,\n\\\\\n&\\displaystyle \\vdots \\qquad& \\nonumber\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta \\pi\\_i^{N}} \\approx 0\n\\qquad &\\Rightarrow \\qquad \n\\dot{V}\\_i^{N} \\approx V\\_i^{N+1} \\,.\n\\label{last eq}\\end{aligned}$$ \nThese equations need to be satisfied at each order in *ε*. Therefore, we may solve them order by order. Multiplying all of the equations by *ε**N* yields the leading order equations,\n \n$$\\begin{aligned}\n\\epsilon^N \\pi\\_i^N \\approx{}& 0 \\,,\n\\label{first}\n\\\\\n\\epsilon^N \\dot{\\pi}\\_i^N \\approx{}& - \\epsilon^N \\,,\n\\\\\n\\vdots \\ \\ \\nonumber \n\\\\\n\\epsilon^N \\dot{\\pi}\\_i^1 \\approx{}& - \\epsilon^N \\pi\\_i \\,,\n\\\\\n\\epsilon^N \\dot{\\pi}\\_i \\approx{}& \n \\epsilon^N \\frac{\\partial \\mathscr{L}^{(0)}}{\\partial x\\_i} \\,,\n\\\\\n\\epsilon^N \\dot{x}\\_i \\approx{}& \\epsilon^N V\\_i^N \\,,\n\\\\\n\\vdots \\ \\ \\nonumber \n\\\\\n\\epsilon^N \\dot{V}\\_i^N \\approx{}& \\epsilon^N V\\_i^{N+1} \\,.\\end{aligned}$$ \nThe first equation ([first]) is a constraint. Taking derivative of it generates another constraint. Taking derivative of that one generates yet another one, etc. The result of this process are leading order solutions of the constraints,\n \n$$\\epsilon^N \\pi\\_i^n \\approx \\epsilon^N \\overline{\\pi}\\_i^{n}(x\\_i, \\pi\\_i) \\,,\n\\qquad\n\\epsilon^N V\\_i^n \\approx \\epsilon^N \\overline{V}\\_i^{n}(x\\_i,\\pi\\_i) \\,,$$ \nwhere we have determined just the leading order term in the *ε* expansion of $\\overline{\\pi}\\_i^{n}(x\\_i, \\pi\\_i)$ and $\\overline{V}\\_i^{n}(x\\_i,\\pi\\_i)$.\nNext step is to multiply equations ([first eq]-[last eq]) by *ε**N* − 1 and repeat the same process to obtain\n \n$$\\epsilon^{N-1} \\pi\\_i^n \\approx \\epsilon^{N-1} \\overline{\\pi}\\_i^{n}(x\\_i, \\pi\\_i) \\,,\n\\qquad\n\\epsilon^{N-1} V\\_i^n \\approx \\epsilon^{N-1} \\overline{V}\\_i^{n}(x\\_i,\\pi\\_i) \\,,$$ \nmeaning that we now have first two terms in the *ε* expansion of $\\overline{\\pi}\\_i^{n}(x\\_i, \\pi\\_i)$ and $\\overline{V}\\_i^{n}(x\\_i,\\pi\\_i)$.\nThis is repeated until we have *solved for all the constraints* to obtain\n \n$$\\pi\\_i^n \\approx \\overline{\\pi}\\_i^{n}(x\\_i, \\pi\\_i) \\,,\n\\qquad\nV\\_i^n \\approx \\overline{V}\\_i^{n}(x\\_i,\\pi\\_i) \\,,$$ \nto order *ε**N*. The final step involves plugging these of-shell relations as strong equalities into the extended action ([EA]). This produces the reduced phase space action,\n \n$$\\mathscr{S}\\_{\\rm red}[x\\_i,\\pi\\_i] = \\int\\! d^{D\\!}x \\, \\biggl\\{\n \\pi\\_i \\dot{x}\\_1 + \\overline{\\pi}\\_i^1 \\dot{\\overline{V}}\\_i^1 + \\dots\n + \\overline{\\pi}\\_1^N \\dot{\\overline{V}}\\_i^N - H\\_{\\rm red}(x\\_i,\\pi\\_i)\n \\biggr\\} \\,,\n\\label{reduced phase action}$$ \nwhere the reduced Hamiltonian is\n \n$$\\begin{aligned}\nH\\_{\\rm red}(x\\_i,\\pi\\_i) ={}& \\overline{\\pi}\\_i \\overline{V}\\_i^1\n + \\overline{\\pi}\\_i^1 \\overline{V}\\_i^1 + \\dots \n + \\overline{\\pi}\\_i^N \\overline{V}\\_i^{N+1}\n - \\mathscr{L}^{(0)}(x\\_i, \\overline{V}\\_i^1) \n\\nonumber \\\\\n& \n - \\epsilon \\mathscr{L}^{(1)}(x\\_i, \\overline{V}\\_i^1, \\overline{V}\\_i^2)\n - \\dots\n - \\epsilon^N \\mathscr{L}^{(N)}\n (x\\_i, \\overline{V}\\_i^1, \\overline{V}\\_i^2,\\dots, V\\_i^N) \\,.\\end{aligned}$$ \nThis phase space action now depends explicitly only on the degrees of freedom found in the leading order action.\nThe nonstandard feature of the phase space action ([reduced phase action]) is the non-canonical kinetic term (the part containing the time derivatives). The reader familiar with Dirac’s canonical formalism will recognize this actually defining the Dirac brackets between variables. We are not interested in the brackets here so we will not work them out (for examples of this see ). We are interested though whether there is a configutation space equivalent of the reduced phase space action ([reduced phase action]). The symplectic part plays a crucial role in the answer to this question.\nSince $\\overline{V}\\_i^n$’s and $\\overline{\\pi}\\_i^n$’s are functions of *x**i* and *π**i* only, the symplectic part of the action can be written as\n \n$$\\int\\! d^{D\\!}x \\, \\Bigl\\{\n \\bigl[ \\pi\\_i + \\epsilon\\Pi\\_i( x\\_j, \\pi\\_j) \\bigr] \\dot{x}\\_i \n + \\epsilon X\\_i (x\\_j, \\pi\\_j) \\dot{\\pi}\\_i\n \\Bigr\\} \\,,$$ \nwhere Π*i* and *X**i* have some expansion in *ε*. In general no amount of partial integrations can remove the time derivative from *π**i* completely. In such a case one cannot follow the standard procedure to obtain the configuration space action. Usually what one does when there is no $\\dot{\\pi}\\_i$ is solve for the conjugate momenta from the Hamilton’s equations as functions of *x**i* and $\\dot{x}\\_i$,\n \n$$\\pi\\_i \\approx \\overline{\\pi}\\_i (x\\_i,\\dot{x}\\_i) \\,,$$ \nand then plugs this into the phase space action as a strong equality. This then results in the configuration space action, and the Lagrangian formulation. The reason why it works is that it can easily be shown that two formulations are on-shell equivalent. But when $\\dot{\\pi}\\_i$ is present in the symplectic part of the action in an irreducible way this procedure does not result in the on-shell equivalent configuration space action, meaning there is no Lagrangian formulation of the reduced system only in terms of *x**i*’s. Note that this never happens for systems with a single degree of freedom where $\\dot{\\pi}$ can always be removed[7](#fn7).\nOne can remove the irreducible $\\dot{\\pi}\\_i$’s from the symplectic part by a redefinition of the variables *x**i*’s. Unfortunately this procedure is not unique, and more likely than not one will make a redefinition which for example breaks manifest Lorentz covariance. Even though there is nothing fundamentally wrong with that, this is reason why we shy away from making variable redefinitions in the phase space action, since it is very likely to complicate things. That being said, if one is interested only in the phase space formulation, action ([reduced phase action]) can always be brought to the canonical form by redefining the variables and their conjugate momenta perturbatively in *ε*.\nThe method presented in this section is applied to a few cases involving a single scalar field in the next section. If the reader find the presentation of the method in this section too dense, two examples in the next section are worked out in detail.\nDerivative reduction in single scalar EFT\n=========================================\nThis section serves to test, examine, and compare the three methods of perturbative derivative reduction on simple examples of real scalar field systems. Note that this is not a one-particle system, but a system with one *local* degree of freedom, interacting with the neighboring ones via gradients. We work in *D* spacetime dimensions, with a positive signature Minkowski metric; the D’Alembertian is ▫ = ∂*μ*∂*μ*, and the Laplacian ∂*i*∂*i*, and a dot denotes a time derivative, while a prime denote a derivative with respect to the field.\nWe consider an EFT expansion in the system of a single scalar field, up to cubic order in the expansion parameter *ε* = *M*− 2, where *M* is some heavy mass scale,\n \n*S*[*ϕ*] = *S*(0)[*ϕ*] + *ε**S*(1)[*ϕ*] + *ε*2*S*(2)[*ϕ*] + *ε*3*S*(3)[*ϕ*] + O(*ε*4) , \nwhere the leading order action is taken to be a scalar with a canonical kinetic term, and a potential term,\n \n$$S^{(0)} = \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - U(\\phi) \\biggr\\} \\,.\n\\label{leading order action}$$ \nThe corrections to this action are assumed to be all possible Lorentz invariant terms of certain mass dimension, and the expansion is ordered in the number of these mass dimensions. The first order correction,\n \n*ε**S*(1) = *ε*∑*n* = 13*α**n*(1)*S**n*(1) , \nis composed out of all possible independent dimension 6 terms (operators),\n \n$$\\begin{aligned}\nS^{(1)}\\_1 ={}& \\int\\! d^{D\\!}x \\, \\phi^6 \\,,\n\\label{Ss(1)1}\n\\\\\nS^{(1)}\\_2 ={}& \\int\\! d^{D\\!}x \\, \\phi^2 (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) \\,,\n\\label{Ss(1)2}\n\\\\\nS^{(1)}\\_3 ={}& \\int\\! d^{D\\!}x \\, (\\square\\phi) (\\square\\phi) \\,,\n\\label{Ss(1)3}\\end{aligned}$$ \nthe second order correction,\n \n*ε*2*S*(2) = *ε*2∑*n* = 16*α**n*(2)*S**n*(2) , \nout of all possible independent dimension 8 terms,\n \n$$\\begin{aligned}\nS^{(2)}\\_1 ={}& \\int\\! d^{D\\!}x \\, \\phi^8 \\,,\n\\label{Ss(2)1}\n\\\\\nS^{(2)}\\_2 ={}& \\int\\! d^{D\\!}x \\, \\phi^4 (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) \\,,\n\\label{Ss(2)2}\n\\\\\nS^{(2)}\\_3 ={}& \\int\\! d^{D\\!}x \\, (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) \n (\\partial\\_\\nu\\phi) (\\partial^\\nu\\phi) \\,,\n\\label{Ss(2)3}\n\\\\\nS^{(2)}\\_4 ={}& \\int\\! d^{D\\!}x \\, \n \\phi (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) (\\square\\phi) \\,,\n\\label{Ss(2)4}\n\\\\\nS^{(2)}\\_5 ={}& \\int\\! d^{D\\!}x \\, \\phi^2 (\\square\\phi) (\\square\\phi) \\,,\n\\label{Ss(2)5}\n\\\\\nS^{(2)}\\_6 ={}& \\int\\! d^{D\\!}x \\, \\phi^2 (\\partial\\_\\mu \\square \\phi) \n (\\partial^\\mu \\square \\phi) \\,,\n\\label{Ss(2)6}\\end{aligned}$$ \nand the third order correction,\n \n*ε*3*S*(3) = *ε*3∑*n* = 113*α**n*(3)*S**n*(3) , \nout of all possible independent dimension 10 terms,\n \n$$\\begin{aligned}\nS^{(3)}\\_1 ={}& \\int\\! d^{D\\!}x \\, \\phi^{10} \\,,\n\\label{Ss(3)1}\n\\\\\nS^{(3)}\\_2 ={}& \\int\\! d^{D\\!}x \\, \\phi^{6} (\\partial\\_\\mu \\phi) (\\partial^\\mu \\phi) \\,,\n\\label{Ss(3)2}\n\\\\\nS^{(3)}\\_3 ={}& \\int\\! d^{D\\!}x \\, \\phi^{2} (\\partial\\_\\mu \\phi) (\\partial^\\mu \\phi)\n (\\partial\\_\\nu \\phi) (\\partial^\\nu \\phi) \\,,\n\\label{Ss(3)3}\n\\\\\nS^{(3)}\\_4 ={}& \\int\\! d^{D\\!}x \\, \\phi^3 (\\partial\\_\\mu \\phi) (\\partial^\\mu \\phi) \n (\\square\\phi) \\,,\n\\label{Ss(3)4}\n\\\\\nS^{(3)}\\_5 ={}& \\int\\! d^{D\\!}x \\, \\phi^4 (\\square \\phi) (\\square \\phi) \\,,\n\\label{Ss(3)5}\n\\\\\nS^{(3)}\\_6 ={}& \\int\\! d^{D\\!}x \\, (\\partial\\_\\mu \\phi) (\\partial^\\mu \\phi)\n (\\square \\phi) (\\square \\phi) \\,,\n\\label{Ss(3)6}\n\\\\\nS^{(3)}\\_{7} ={}& \\int\\! d^{D\\!}x \\, (\\partial\\_\\mu \\phi) (\\partial^\\mu \\phi)\n (\\partial\\_\\nu \\partial\\_\\rho \\phi) (\\partial^\\nu \\partial^\\rho \\phi) \\,,\n\\label{Ss(3)7}\n\\\\\nS^{(3)}\\_{8} ={}& \\int\\! d^{D\\!}x \\, (\\partial\\_\\mu\\phi) (\\partial\\_\\nu \\phi)\n (\\partial^\\mu \\partial^\\nu \\phi) (\\square\\phi) \\,,\n\\label{Ss(3)8}\n\\\\\nS^{(3)}\\_{9} ={}& \\int\\! d^{D\\!}x \\, \\phi (\\square \\phi) (\\square \\phi) (\\square\\phi) \\,,\n\\label{Ss(3)9}\n\\\\\nS^{(3)}\\_{10} ={}& \\int\\! d^{D\\!}x \\, \\phi (\\partial\\_\\mu\\partial\\_\\nu \\phi)\n (\\partial^\\mu\\partial^\\nu \\phi) (\\square\\phi) \\,,\n\\label{Ss(3)10}\n\\\\\nS^{(3)}\\_{11} ={}& \\int\\! d^{D\\!}x \\, \\phi (\\partial\\_\\mu\\partial\\_\\nu \\phi)\n (\\partial^\\nu\\partial^\\rho \\phi) (\\partial\\_\\rho \\partial^\\mu \\phi) \\,,\n\\label{Ss(3)11}\n\\\\\nS^{(3)}\\_{12} ={}& \\int\\! d^{D\\!}x \\, \\phi^2 (\\partial\\_\\mu \\square \\phi)\n (\\partial^\\mu \\square \\phi) \\,,\n\\label{Ss(3)12}\n\\\\\nS^{(3)}\\_{13} ={}& \\int\\! d^{D\\!}x \\, (\\square \\square \\phi) (\\square \\square \\phi) \\,.\n\\label{Ss(3)13}\\end{aligned}$$ \nThe EFT expansion introduces one higher derivative term ([Ss(1)3]) at linear order, three higher derivative terms ([Ss(2)4]-[Ss(2)6]) at second order, and ten ([Ss(3)4]-[Ss(3)13]) at third order. This is a rather generic feature of the EFT expansion, and it is hard to come up with a symmetry principle that would exclude them. These higher derivative terms do not mean that at each order in *ε* new degrees of freewhatdom appear. It is rather the assumed approximation scheme that represents corrections as local terms that is to blame for the seeming introduction of extra degrees of freedom. These degrees are spurious, and ought to be treated in the same spirit in which they arise – perturbatively, which actually eliminates them.\nThe following subsections analyze the reduction of derivative order at successive orders in the expansion parameter *ε* using three methods – reduction at the equation of motion level, reduction via perturbative constraints method of section [sec: Formalism], and reduction via field redefinitions. Considering all the possible corrections ([Ss(1)1]-[Ss(3)13]) in is not necessary for the points that we want to make, and would only obscure them with the sheer number of formulas without introducing anything essentially new to the discussion. Rather we will only consider the highest derivative term at each order ([Ss(1)3]), ([Ss(2)6]), and ([Ss(3)13]), and in addition ([Ss(3)7]). The emphasis in this section is not on the exact form of the reduced theories, but rather on the lessons that can be learned about methods from considering simpler examples.\nIt is worth noting that in the Lorentz invariant systems of multiple interacting scalar fields with actions containing irreducible second derivatives were analysed, and classified according whether they can be brought to explicit lower order form, and whether this preserves manifest Lorentz covariance.\nDerivative reduction at first order\n-----------------------------------\nLet us consider the the action ([leading order action]) with the first order correction ([Ss(1)3]),\n \n$$S = \\int\\! d^{D\\!}x \\, \\biggl\\{ - \\frac{1}{2} (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi)\n - U + \\epsilon \\alpha\\_1 (\\square\\phi) (\\square\\phi) \\biggr\\} \\,,\n\\label{first corrected action}$$ \nwhile neglecting all the contributions of order O(*ε*2). We will be looking to reduce the derivative order of this system, and represent it with an equivalent lower derivative system that captures correctly the solutions to order *ε*. Three methods are considered in the following: reducing the derivative order at the level of the equation of motion, reducing the derivative order by implementing perturbative second class constraints of Section [sec: Formalism], and reducing the order via field redefinition.\n### Reduction at the equation of motion level\nThe equation of motion descending from the action ([first corrected action]) is\n \n▫*ϕ* − *U*ʹ + 2*ε**α*1▫▫*ϕ* = 0 . \nApplying the leading order equation of motion to the first order term is straightforward and yields\n \n$$\\square \\phi - U' + 2\\epsilon\\alpha\\_1 \\bigl[ U'' \\square\\phi \n + U''' (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) \\bigr] = 0 \\,,$$ \nwhich correctly reproduces the perturbative solutions of the original equation ([first order unreduced equation]) to linear order in *ε*. But this is not the only lower (time) derivative equation that does it. In fact, since we can always use ▫*ϕ* = *U*ʹ at linear order, there is a one-parameter family of equations that accomplish this task,\n \n$$\\square\\phi - U' + 2\\epsilon \\alpha\\_1 \\bigl[ (1\\!+\\!b) U'' \\square\\phi\n + U'''(\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - b \\, U'' U' \\bigr] = 0 \\,,\n\\label{first order b equation}$$ \nwhere *b* is the free parameter. There is one distinguished choice for this parameter *b* = 1, for which there is an action formulation of the equation of motion, in the sense that the equation following from the action is exactly the one for *b* = 1 with no further manipulations required. We will derive this action in the following subsection via the method of perturbative second-class constraints. As noted in the Introduction, an action formulation allows for resummation that conserves exactly the charges associated to global symmetries.\n### Reduction via perturbative constraints\nHere we apply the method described in Sec. [sec: Formalism] to reduce the derivative order of the action ([first corrected action]). The starting point is the definition of the extended action, which is obtained from the configuration space one by promoting time derivatives to independent fields,\n \n$$\\dot{\\phi}\\rightarrow V\\_1, \n\\qquad\n- \\square\\phi = \\ddot{\\phi} - \\nabla^2\\phi = \\dot{V}\\_1 - \\nabla^2\\phi \\rightarrow V\\_2 \\,,$$ \nand by introducing appropriate Lagrange multipliers *π* and *π*1 to ensure the on-shell equivalence,\n \n$$\\begin{aligned}\n\\mathcal{S} \n={}& \\int\\! d^{D\\!}x \\, \\biggl\\{ \\frac{1}{2} V\\_1^2 - \\frac{1}{2}(\\nabla\\phi)^2 - U\n + \\epsilon \\alpha\\_1 V\\_2^2\n + \\pi (\\dot{\\phi} - V\\_1) + \\pi\\_1 (\\dot{V}\\_1 - \\nabla^2\\phi -V\\_2) \\biggr\\}\n\\nonumber \\\\\n={}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n \\pi \\dot{\\phi} + \\pi\\_1 \\dot{V}\\_1 - \\biggl[\n \\pi V\\_1 + \\pi\\_1 ( V\\_2 + \\nabla^2\\phi) \n - \\frac{1}{2} V\\_1^2 + \\frac{1}{2}(\\nabla\\phi)^2 + U\n - \\epsilon \\alpha\\_1 V\\_2^2\n \\biggr]\n \\biggr\\} \\,.\n\\label{first order extended action}\\end{aligned}$$ \nProceeding with the recipe given in Sec. [sec: Formalism], the equations resulting from the action principle with respect to all the independent fields are\n \n$$\\begin{aligned}\n&\\displaystyle \\frac{\\delta \\mathcal{S}}{\\delta V\\_2} \\approx 0\n\\qquad \\Rightarrow \\qquad&\n\\pi\\_1 \\approx 2 \\epsilon \\alpha\\_1 V\\_2\n\\\\\n&\\displaystyle \\frac{\\delta \\mathcal{S}}{\\delta V\\_1} \\approx 0\n\\qquad \\Rightarrow \\qquad&\n\\dot{\\pi}\\_1 \\approx - \\pi + V\\_1 \\,,\n\\\\\n&\\displaystyle \\frac{\\delta \\mathcal{S}}{\\delta \\phi} \\approx 0\n\\qquad \\Rightarrow \\qquad&\n\\dot{\\pi} \\approx\\nabla^2\\phi - U' - \\nabla^2\\pi\\_1 \\,,\n\\\\\n&\\displaystyle \\frac{\\delta \\mathcal{S}}{\\delta \\pi} \\approx 0\n\\qquad \\Rightarrow \\qquad&\n\\dot{\\phi} \\approx V\\_1 \\,,\n\\\\\n&\\displaystyle \\frac{\\delta \\mathcal{S}}{\\delta \\pi\\_1} \\approx 0\n\\qquad \\Rightarrow \\qquad&\n\\dot{V}\\_1 \\approx V\\_2 + \\nabla^2\\phi \\,.\\end{aligned}$$ \nMultiplying these equations by *ε* and discarding the terms O(*ε*2) gives us the leading order equations,\n \n$$\\begin{aligned}\n\\epsilon \\pi\\_1 \\approx{}& 0 \\,,\n\\label{first order constraint}\n\\\\\n\\epsilon \\dot{\\pi}\\_1 \\approx{}& \\epsilon (- \\pi + V\\_1 ) \\,,\n\\\\\n\\epsilon \\dot{\\pi} \\approx{}& \\epsilon (\\nabla^2\\phi - U' - \\nabla^2\\pi\\_1 ) \\,,\n\\\\\n\\epsilon \\dot{\\phi} \\approx{}& \\epsilon V\\_1 \\,,\n\\\\\n\\epsilon \\dot{V}\\_1 \\approx{}& \\epsilon (V\\_2 + \\nabla^2\\phi ) \\,,\n\\label{first order dotV1}\\end{aligned}$$ \nwhere now it is clear that Eq. ([first order constraint]) takes the form of a primary constraint, and *V*2 is its associated Lagrange multiplier. Requiring the consistency of the primary constraint (vanishing of its time derivative) generates a secondary constraint\n \n$$\\epsilon \\dot{\\pi}\\_1 \\approx \\epsilon (- \\pi + V\\_1) \\approx 0 \\, \n\\qquad \\Rightarrow \\qquad \\epsilon V\\_1 \\approx \\epsilon \\pi \\,.$$ \nThe two constraint are second-class, and determine the Lagrange multiplier of the primary constraint,\n \n$$\\epsilon \\frac{d}{dt} (-\\pi + V\\_1) \\approx \\epsilon (U' + V\\_2) \\approx 0 \n\\qquad \\Rightarrow \\qquad\n\\epsilon V\\_2 \\approx - \\epsilon U' \\,.\n\\label{leading order V2}$$ \nThis determines *π*1, *V*1, *V*2 to leading order depending algebraically of *ϕ* and *π*.\nAfter solving for the leading order constraints, we turn to the first order equations ([first order constraint]-[first order dotV1]). The first of these equations again takes the form of the primary constraint, because upon using the leading order solution of the constraint equations ([leading order V2]) it reads\n \n*π*1 ≈ − 2*ε**α*1*U*ʹ . \nIn the same way as for the leading order, this primary constraint generates a secondary constraint, the two of them being second-class, and determining the Lagrange multiplier,\n \n$$\\begin{aligned}\nV\\_1 \\approx{}& \\pi - 2\\epsilon \\alpha\\_1 U'' \\pi \\,,\n\\\\\nV\\_2 \\approx{}& - U' - 2 \\epsilon\\alpha\\_1 \\Bigl\\{ U''' \\bigl[\\pi^2 - (\\nabla\\phi)^2 \\bigr]\n - U'' U' \\Bigr\\} \\,.\\end{aligned}$$ \nWe have solved for *π*1, *V*1, and *V*2 to first order as functions of *π* and *ϕ*. Since these relations are algebraic, in the sense that they do not involve any time derivatives, we can promote them to strong (off-shell) equalities, and plug them back into the extended action ([first order extended action]) to obtain the phase space reduced action, which after a couple of partial integrations reads\n \n$$\\mathscr{S}\\_{\\rm red} = \\int\\! d^{D\\!}x \\, \\biggl\\{\n \\bigl( 1 + 2\\epsilon \\alpha\\_1 U'' \\bigr) \\pi \\dot{\\phi} \n - \\biggl[ \\frac{1}{2} \\pi^2 + \\frac{1}{2} (\\nabla\\phi)^2 + U\n + \\epsilon \\alpha\\_1 \\bigl[ 2 U'' (\\nabla\\phi)^2 + (U')^2 \\bigr]\n \\biggr]\n \\biggr\\} \\,.\n\\label{phase action reduced}$$ \nIn fact, because of the structure of the extended action, we do not actually need to solve for velocities in this example beyond leading order since their higher orders cancel when plugged into the action.\nNote that the phase space action ([phase action reduced]) obtained via the imposition of perturbative second-class constraints is not in a canonical form, due to the symplectic part not being just $\\pi \\dot{\\phi}$, but is modified by the field dependent prefactor. This modified symplectic part means that the the Poissons bracket between *ϕ* and *π* is not canonical, but rather\n \n$$\\bigl\\{ \\phi (t,\\vec{x}), \\pi(t,\\vec{x}^{\\,\\prime}) \\bigr\\} \n = \\frac{\\delta^{D-1}(\\vec{x}\\!-\\!\\vec{x}^{\\,\\prime})}{1 + 2\\epsilon \\alpha\\_1 U''}\n = \\bigl[ 1 - 2\\epsilon \\alpha\\_1 U'' \\bigr] \\delta^{D-1}(\\vec{x}\\!-\\!\\vec{x}^{\\,\\prime}) \\,.$$ \nThis is not surprising, since such a bracket results from solving the second-class constraints, and is in fact nothing else then the Dirac bracket.\nEven though it is not necessary, the symplectic part of the phase space action can be canonicalized by a non-canonical transformation of the conjugate momentum field,\n \n$$\\pi \\rightarrow \\bigl( 1 - 2\\epsilon \\alpha\\_1 U'' \\bigr) \\pi \\,,\n\\label{momentum transformation}$$ \nafter which the phase space action takes the form\n \n$$\\mathscr{S}\\_{\\rm red} = \\int\\! d^{D\\!}x \\, \\biggl\\{\n \\pi \\dot{\\phi}\n - \\biggl[ \\frac{1}{2} \\pi^2 + \\frac{1}{2} (\\nabla\\phi)^2 + U\n + \\epsilon \\alpha\\_1 \\Bigl( 2 U'' [- \\pi^2 + (\\nabla\\phi)^2 ] + (U')^2 \\Bigr) \\biggr] \n \\biggr\\} \\,.\n\\label{first order canonical}$$ \nThe transformation of the conjugate momentum field ([momentum transformation]) is not the unique transformation that puts the phase space action into a canonical form. This can also be accomplished by transforming the scalar field itself,\n \n*ϕ* → *ϕ* − 2*ε**α*1*U*ʹ . \nWe are hesitant to make field transformations in the phase space action. Even though it is not the case in this specific example, it is very easy to lose manifest Lorentz covariance when making a field transformation that involves gradients of the field, or the conjugate momentum.\nThe reduced configuration space action is obtained from the phase space one ([first order canonical]) in the usual manner. First we solve the Hamilton’s equation for the conjugate momentum filed in terms of the scalar and its time derivative,\n \n$$\\frac{\\delta \\mathscr{S}\\_{\\rm red}}{\\delta \\pi} \\approx 0\n\\qquad \\Rightarrow \\qquad\n\\pi \\approx \\dot{\\phi} - 4 \\epsilon \\alpha\\_1 U'' \\dot{\\phi} \\,.$$ \nThis relation is then promoted to the on-shell equality and plugged into the phase space action to obtain the configuration space one, which can be written in a manifestly Lorentz invariant form\n \n$$S\\_{\\rm red} = \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} \\bigl[ 1 + 4\\epsilon \\alpha\\_1 U'' \\bigr] \n (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - U - \\epsilon \\alpha\\_1 (U')^2\n \\biggr\\} \\,.\n\\label{first order configuration action}$$ \nThe effect of the higher derivative correction when reduced is to renormalize the kinetic term (by a field-dependent factor), and to add another potential term. The equation of motion following from the reduced lower derivative action is\n \n$$\\square\\phi - U' + 2\\epsilon \\alpha\\_1 \\bigl[ 2 U'' (\\square\\phi)\n + U''' (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - U'' U' \\bigr] \\,,$$ \nwhich is precisely the *b* = 1 equation ([first order b equation]) we have obtained by reducing the derivative order at the level of the equation of motion.\nWe have derived the lower derivative action ([first order configuration action]) that captures completely to first order in *ε* the subspace of perturbatively expandable solutions of the higher derivative action ([first corrected action]).\n### Derivative reduction via field redefinition\nSince the higher derivative terms in the leading order action ([first corrected action]) are assumed to arise perturbatively, and are to be treated perturbatively, we may employ field redefinitions that include higher derivatives. It is simple to see that the following redefinition\n \n*ϕ* → *ϕ* − *ε**α*1▫*ϕ* , \nremoves all the higher derivatives to order O(*ε*2),\n \n$$S \\rightarrow \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} \\bigl[ 1 + 2\\alpha\\_1\\epsilon U'' \\bigr] \n (\\partial\\_\\mu\\phi) (\\partial^\\mu \\phi) - U\n \\biggr\\} \\,,$$ \naccomplishing the desired task. This is not the same action obtained by the method of perturbative constraints ([first order configuration action]). The discrepancy is resolved by noting that here we are still left with the freedom of field redefinitions not including derivatives, which do not change the derivative order of the action. In particular, an additional redefinition,\n \n*ϕ* → *ϕ* + *ε**α*1*U*ʹ , \ntransforms the action exactly into ([first order configuration action]),\n \n$$S \\rightarrow \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} \\bigl[ 1 + 4\\alpha\\_1\\epsilon U'' \\bigr] \n (\\partial\\_\\mu\\phi) (\\partial^\\mu \\phi) - U - \\epsilon\\alpha\\_1 (U')^2\n \\biggr\\} \\,.$$ \nThis demonstrates the equivalence of field redefinitions to the other two derivative reduction methods.\nDerivative reduction at second order\n------------------------------------\nAt the second order in *ε* we consider only the two higher derivative corrections to the leading order action ([leading order action]), the first one ([Ss(1)3]) arising at linear level, and the second one ([Ss(2)6]) at the quadratic,\n \n$$S = \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} (\\partial\\_\\mu \\phi) (\\partial^\\mu \\phi) - U\n + \\epsilon \\alpha\\_1 (\\square\\phi) (\\square\\phi)\n + \\epsilon^2 \\alpha\\_2 (\\partial\\_\\mu \\square\\phi) (\\partial^\\mu\\square\\phi)\n \\biggr\\} \\,.\n\\label{2nd action}$$ \n### Reduction at the equation of motion level\nThe equation of motion following from the starting higher derivative action is\n \n▫*ϕ* − *U*ʹ + 2*ε**α*1▫▫*ϕ* − 2*ε*2*α*2▫▫▫*ϕ* = 0 . \nBy successive applications of the lower order equations in the higher order terms this equation can be reduced to the second order one in derivatives,\n \n$$\\begin{aligned}\n\\MoveEqLeft[3]\n\\square\\phi - U'\n + 2\\epsilon\\alpha\\_1 \\bigl[ U'' U' + U'''(\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) \\bigr]\n - 2\\epsilon^2(\\alpha\\_2 + 2\\alpha\\_1^2) \\Bigl\\{\n [U''U']' U' \n\\nonumber \\\\\n& +\\bigl[ 2 U^{(4)} U' + 5 U''' U'' \\bigr] (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi)\n + U^{(5)} (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) (\\partial\\_\\nu\\phi) (\\partial^\\nu\\phi)\n\\nonumber \\\\\n& + 4 U^{(4)} (\\partial^\\mu\\phi) (\\partial^\\nu\\phi) (\\partial\\_\\mu\\partial\\_\\nu\\phi)\n + 2 U''' (\\partial\\_\\mu\\partial\\_\\nu\\phi) (\\partial^\\mu\\partial^\\nu \\phi)\n \\Bigr\\} = 0 \\,.\\end{aligned}$$ \nThere is no obstacle to accomplishing this task. Just as in the first order case, this is not a unique equation capturing solutions to the second order in *ε*, and only particular forms of the reduced equation have action formulations.\n### Reduction via perturbative constraints\nThe starting point of the method of derivative reduction via perturbative constraints is the definition of the extended action. It is obtained by promoting time derivatives to independent fields,\n \n$$\\dot{\\phi}\\rightarrow V\\_1, \n\\qquad\n-\\square\\phi = \\dot{V}\\_1 - \\nabla^2\\phi \\rightarrow V\\_2\\,,\n\\qquad\n-\\partial\\_0 \\square\\phi = \\dot{V}\\_2 \\rightarrow V\\_3,$$ \nand introducing Lagrange multipliers *π*, *π*1, *π*2 to ensure on-shell equivalence,\n \n$$\\begin{aligned}\n\\mathcal{S} ={}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n \\frac{1}{2} V\\_1^2 - \\frac{1}{2} (\\nabla\\phi)^2 - U\n + \\epsilon\\alpha\\_1 V\\_2^2\n - \\epsilon^2 \\alpha\\_2 V\\_3^2 + \\epsilon^2 \\alpha\\_2 (\\nabla^2V\\_1)^2\n\\nonumber \\\\\n& \\hspace{4cm}\n + \\pi(\\dot{\\phi} - V\\_1) + \\pi\\_1 (\\dot{V}\\_1 - \\nabla^2\\phi - V\\_2)\n + \\pi\\_2 (\\dot{V}\\_2 - V\\_3)\n \\biggr\\}\n\\nonumber \\\\\n={}& \\int\\! d^{D\\!}x \\, \\biggl\\{ \n \\pi\\dot{\\phi} + \\pi\\_1 \\dot{V}\\_1 + \\pi\\_2 \\dot{V}\\_2\n - \\biggl[\n \\pi V\\_1 + \\pi\\_1(\\nabla^2\\phi+V\\_2) + \\pi\\_2 V\\_3\n\\nonumber \\\\\n& \\hspace{3cm}\n - \\frac{1}{2}V\\_1^2 + \\frac{1}{2} (\\nabla\\phi)^2 + U\n - \\epsilon\\alpha\\_1 V\\_2^2\n + \\epsilon^2 \\alpha\\_2 [V\\_3^2 - (\\nabla V\\_2)^2]\n \\biggr]\n \\biggr\\} \\,.\n\\label{2nd extended}\\end{aligned}$$ \nThe procedure here mirrors the one for the first order case. We provide all the steps in order to illustrate the method better. The equations resulting from the action principle with respect to all the independent fields are\n \n$$\\begin{aligned}\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta V\\_3} \\approx 0\n\\qquad \\Rightarrow \\qquad &\n\\pi\\_2 \\approx - 2\\epsilon^2\\alpha\\_2 V\\_3 \\,,\n\\label{fe}\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta V\\_2} \\approx 0\n\\qquad \\Rightarrow \\qquad &\n\\dot{\\pi}\\_2 \\approx - \\pi\\_1 + 2\\epsilon\\alpha\\_1 V\\_2\n - 2\\epsilon^2 \\alpha\\_2 \\nabla^2 V\\_2 \\,,\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta V\\_1} \\approx 0\n\\qquad \\Rightarrow \\qquad &\n\\dot{\\pi}\\_1 \\approx - \\pi V\\_1 \\,,\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta \\phi} \\approx 0\n\\qquad \\Rightarrow \\qquad &\n\\dot{\\pi} \\approx - \\nabla^2 \\pi\\_1 + \\nabla^2\\phi - U' \\,,\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta \\pi} \\approx 0\n\\qquad \\Rightarrow \\qquad &\n\\dot{\\phi} \\approx V\\_1 \\,,\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta \\pi\\_1} \\approx 0\n\\qquad \\Rightarrow \\qquad &\n\\dot{V}\\_1 \\approx \\nabla^2\\phi + V\\_2 \\,,\n\\\\\n&\\displaystyle \\frac{\\delta\\mathcal{S}}{\\delta \\pi\\_2} \\approx 0\n\\qquad \\Rightarrow \\qquad &\n\\dot{V}\\_2 \\approx V\\_3 \\,.\n\\label{le}\\end{aligned}$$ \nMultiplying these equations by *ε*2 gives the leading order equations,\n \n$$\\begin{aligned}\n\\epsilon^2 \\pi\\_2 \\approx{}& 0 \\,,\n\\\\\n\\epsilon^2 \\dot{\\pi}\\_2 \\approx{}& - \\epsilon^2 \\pi\\_1 \\,,\n\\\\\n\\epsilon^2 \\dot{\\pi}\\_1 \\approx{}& \\epsilon^2 (-\\pi + V\\_1) \\,,\n\\\\\n\\epsilon^2 \\dot{\\pi} \\approx{}& \\epsilon^2 (-\\nabla^2\\pi\\_1 + \\nabla^2\\phi - U') \\,,\n\\\\\n\\epsilon^2 \\dot{\\phi} \\approx{}& \\epsilon^2 V\\_1 \\,,\n\\\\\n\\epsilon^2 \\dot{V}\\_1 \\approx{}& \\epsilon^2 (\\nabla^2\\phi+V\\_2) \\,,\n\\\\\n\\epsilon^2 \\dot{V}\\_2 \\approx{}& \\epsilon^2 V\\_3 \\,.\\end{aligned}$$ \nThe first of these is a constraint equation. Requiring its time derivative to vanish on-shell produces another constraint,\n \n$$\\epsilon^2 \\dot{\\pi}\\_2 \\approx - \\epsilon^2 \\pi\\_1 \\approx 0 \\,.$$ \nConsistency of this constraint produces the next one,\n \n$$\\epsilon^2 \\dot{\\pi}\\_2 \\approx \\epsilon^2 (- \\pi + V\\_1) \\approx 0\n\\qquad \\Rightarrow \\qquad \n\\epsilon^2 V\\_1 \\approx \\epsilon^2 \\pi \\,,$$ \nAgain requiring consistency produces a constraint,\n \n$$\\epsilon^2 \\frac{d}{dt} (- \\pi + V\\_1) \\approx\n \\epsilon^2 (V\\_2 + U') \\approx 0\n\\qquad \\Rightarrow \\qquad \n\\epsilon^2 V\\_2 \\approx - \\epsilon^2 U' \\,.$$ \nFinally, the consistency of this constraint determines the Lagrange multiplier,\n \n$$\\epsilon^2 \\frac{d}{dt} (V\\_2 + U') \\approx \\epsilon^2 (V\\_3 + U'' \\pi) \\approx 0\n\\qquad \\Rightarrow \\qquad \n\\epsilon^2 V\\_3 \\approx - \\epsilon^2 U'' \\pi \\,.$$ \nAfter solving the leading order constraints we proceed to the fist order ones. Multiplying equations ([fe]-[le]) by *ε*, and making use of the leading order constraints gives the equations at first order\n \n$$\\begin{aligned}\n\\epsilon \\pi\\_2 \\approx{}& 0 \\,,\n\\\\\n\\epsilon \\dot{\\pi}\\_2 \\approx{}& - \\epsilon\\pi\\_1 - 2\\epsilon^2\\alpha\\_1U' \\,,\n\\\\\n\\epsilon \\dot{\\pi}\\_1 \\approx{}& \\epsilon (-\\pi + V\\_1) \\,,\n\\\\\n\\epsilon \\dot{\\pi} \\approx{}& \\epsilon (-\\nabla^2\\pi\\_1 + \\nabla^2\\phi - U') \\,,\n\\\\\n\\epsilon \\dot{\\phi} \\approx{}& \\epsilon V\\_1 \\,,\n\\\\\n\\epsilon \\dot{V}\\_1 \\approx{}& \\epsilon (\\nabla^2\\phi+V\\_2) \\,,\n\\\\\n\\epsilon \\dot{V}\\_2 \\approx{}& \\epsilon V\\_3 \\,.\\end{aligned}$$ \nThe first of these equations is a constraint. Its consistency generates another constraint,\n \n$$\\epsilon \\dot{\\pi}\\_2 \\approx - \\epsilon \\pi\\_1 -2\\epsilon^2 \\alpha\\_1 U' \\approx 0\n\\qquad \\Rightarrow \\qquad \n\\epsilon \\pi\\_1 \\approx - 2\\epsilon^2 U' \\,.$$ \nIts consistency generates the next constraint,\n \n$$\\epsilon \\frac{d}{dt} (\\pi\\_1 + 2\\epsilon\\alpha\\_1 U') \\approx 0\n\\qquad \\Rightarrow \\qquad \n\\epsilon V\\_1 \\approx \\epsilon \\pi - 2\\epsilon^2\\alpha\\_1 U'' \\pi \\,,$$ \nwhich by consistency requirements generates yet another one\n \n$$\\epsilon \\frac{d}{dt} (V\\_1 - \\pi + 2\\epsilon\\alpha\\_1 U'' \\pi) \\approx 0\n\\qquad \\Rightarrow \\qquad \n\\epsilon V\\_2 \\approx - \\epsilon U' \n +2\\epsilon^2\\alpha\\_1 \\bigl( U'''[-\\pi^2+(\\nabla\\phi)^2] + U''U' \\bigr) \\,.$$ \nFinally consistency of this constraint determines *V*3 at first order, but we do not require this to construct the reduced action.\nIn the last stage of solving the constraints we turn to the second order equations ([fe]-[le]). By successively requiring consistency of constraints solve the perturbative constraints fully. Here we give just the terms that do not cancel when plugged into the action,\n \n$$\\begin{aligned}\n\\pi\\_2 \\approx{}& 2 \\epsilon^2 \\alpha\\_2 U'' \\pi \\,,\n\\\\\n\\pi\\_1 \\approx{}& - 2\\epsilon \\alpha\\_1 U' \n + 2\\epsilon^2(\\alpha\\_2+2\\alpha\\_1^2)\n \\bigl( U'''[-\\pi^2+(\\nabla\\phi)^2] + U''U' \\bigr) \\,,\n\\\\\nV\\_1 \\approx{}& \\pi - 2\\epsilon\\alpha\\_1 U'' \\pi + \\mathcal{O}(\\epsilon^2) \\,,\n\\\\\nV\\_2 \\approx{}& - U' + \\mathcal{O}(\\epsilon) \\,,\n\\\\\nV\\_3 \\approx{}& - U'' \\pi + \\mathcal{O}(\\epsilon) \\,.\\end{aligned}$$ \nThese are now plugged into the extended action ([2nd extended]) as strong equalities, resulting in the reduced phase space action, which after some partial integrations can be written as\n \n$$\\begin{aligned}\n\\mathscr{S}\\_{\\rm red} ={}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n \\biggl[ \\pi + 2\\epsilon\\alpha\\_1 U'' \\pi\n + 2\\epsilon^2 (\\alpha\\_2 + 2\\alpha\\_1^2) \\Bigl( \\frac{1}{3} U^{(4)} \\pi^3\n - U^{(4)} (\\nabla\\phi)^2 \\pi \n\\nonumber \\\\\n& \\hspace{6cm}\n + 2 \\nabla(U'''\\pi\\nabla\\phi) - [U'''U' + 2(U'')^2] \\pi \\Bigr) \\biggr] \\dot{\\phi}\n\\nonumber \\\\\n& - \\biggl[ \\frac{1}{2} \\pi^2 + \\frac{1}{2}(\\nabla\\phi)^2 + U\n + \\epsilon \\alpha\\_1 \\bigl( 2 U''(\\nabla\\phi)^2 + (U')^2 \\bigr) \n - \\alpha\\_2 \\epsilon^2 (U'')^2 (\\nabla\\phi)^2\n\\nonumber \\\\\n& + 2\\epsilon^2 (\\alpha\\_2 + 2\\alpha\\_1^2) \n \\bigl( U'''[-\\pi^2 + (\\nabla\\phi)^2] + U''U' \\bigr) (\\nabla^2\\phi - U')\n -(\\alpha\\_2 + 2\\alpha\\_1^2) \\epsilon^2 (U'')^2 \\pi^2 \\biggr]\n \\biggr\\} \\,.\n\\label{2nd phase action}\\end{aligned}$$ \nThe reduced phase space action at second order comes with the noncanonical symplectic part, but such that it can be canonicalized by redefining the conjugate momentum field, just as we had at the first order reduction. We do not bother with canonicalizing the phase space action, but proceed to construct the reduced configuration space action. First we solve for the conjugate momentum field\n \n$$\\frac{\\delta\\mathscr{S}\\_{\\rm red}}{\\delta \\pi} \\approx 0\n\\qquad \\Rightarrow \\qquad\n\\pi \\approx \\dot{\\phi} + 2\\epsilon \\alpha\\_1 U'' \\dot{\\phi} + \\mathcal{O}(\\epsilon^2) \\,,$$ \nand then plug it into the phase space action ([2nd phase action]). Thus we obtain the reduced configuration space action, which after some number of partial integrations and groupings of the terms can be written in a manifestly Lorentz invariant form,\n \n$$\\begin{aligned}\nS\\_{\\rm red} ={}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} \\Bigl[ 1 + 4\\epsilon \\alpha\\_1 U''\n - 8(\\alpha\\_2 + 2\\alpha\\_1^2) U''' U'\n - 2(3\\alpha\\_2 + 4\\alpha\\_1^2) (U'')^2 \\Bigr] \n (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi)\n\\nonumber \\\\\n& \\hspace{0.3cm}\n - U - \\epsilon \\alpha\\_1 (U')^2\n + 2(\\alpha\\_2 + 2\\alpha\\_1^2) U'' (U')^2\n - 2(\\alpha\\_2 + 2\\alpha\\_1^2) U''' (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) (\\square \\phi)\n \\biggr\\} \\,.\n\\label{II reduced}\\end{aligned}$$ \nApart from renormalizing the kinetic term, and modifying the potential already encountered at first order, the effect of reduction the considered higher derivative is also to introduce a new term – the last one in the action above. This term technically is a higher derivative one due to the presence of d’Alembertian operator, but it is not a genuine higher derivative terms, since the higher time derivative in it can be removed by partial integration, and are only introduced for the sake of manifest Lorentz invariance. Moreover, the very method used to derive it guarantees no higher derivatives in the final result. It is worth noting that derivative reduction introduces new types of terms containing four derivatives into the reduced action, but the kind that are not genuine higher derivative ones.\n### Derivative reduction via field redefinition\nWith a bit of work one can see that the field transformation,\n \n$$\\phi \\rightarrow \\phi - \\epsilon \\alpha\\_1 \\square\\phi\n + \\frac{3}{2} \\epsilon^2 \\alpha\\_1^2 \\square \\square \\phi\n + 2\\epsilon^2 \\alpha\\_1^2 U'' \\square\\phi\n + 2 \\epsilon^2 \\alpha\\_2 \\square \\bigl[ \\square\\phi + U' \\bigr] \\,,$$ \nremoves all the genuine higher derivative terms from the configuration space action ([2nd action]), and transforms it into\n \n$$\\begin{aligned}\nS \\rightarrow{}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} \\Bigl[ 1 + 2\\epsilon\\alpha\\_1 U''\n - 2\\epsilon^2(\\alpha\\_2 + 2\\alpha\\_1^2)(U'')^\n - 4\\epsilon^2 \\alpha\\_1^2 U''' U' \\Bigr] \n (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) \n\\nonumber \\\\\n& \\hspace{2cm}\n - U - \\frac{3}{2} \\epsilon^2 \\alpha\\_1^2 U''' U' \n (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) (\\square\\phi)\n \\biggr\\} \\,.\\end{aligned}$$ \nThis action is not in the form of the reduced one ([II reduced]) obtained via perturbative constraints method. But again, this is not an inconsistency, since an additional field redefinition\n \n$$\\phi \\rightarrow \\phi \n + \\epsilon \\alpha\\_1 U' \n - \\frac{5}{2} \\epsilon^2 \\alpha\\_1^2 U''' (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi)\n - \\frac{9}{2} \\epsilon^2 \\alpha\\_1^2 U'' U'\n - 2\\epsilon^2\\alpha\\_2 U''' (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi)\n - 2\\epsilon^2 \\alpha\\_2 U''U' \\,,$$ \nbring the above action precisely to the form of ([II reduced]). The field redefinition method is consistent with the perturbative constraints method.\nDerivative reduction at third order – Part I\n--------------------------------------------\nIn order to illustrate the points made at this section we consider the leading order action ([leading order action]) with the only nonvanishing higher order correction ([Ss(3)13]) coming in at the third order EFT expansion,\n \n$$S = \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - U\n + \\epsilon^3 \\alpha\\_3 (\\square\\square\\phi) (\\square\\square\\phi) \n \\biggr\\} \\,.\n\\label{third order HD action}$$ \nAt this order we make the point that field redefinitions are necessary in order to define an equivalent lower derivative, manifestly Lorentz invariant, configuration space action formulation of the higher derivative system.\n### Reduction at the equation of motion level\nThe equation of motion descending from the action ([third order HD action]) is\n \n▫*ϕ* − *U*ʹ + 2*ε*3*α*3▫▫▫▫*ϕ* = 0 . \nJust as in the cases of the first and the second order reduction, we make use of the lower order equation to reduce the derivative order of the higher order part of the equation. The resulting equation takes the form\n \n$$\\begin{aligned}\n\\MoveEqLeft[4]\n\\square\\phi - U' + 2\\epsilon^3 \\alpha\\_3 \\biggl\\{\n A(\\phi) + B(\\phi) \\times (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi)\n + C(\\phi) \\times (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) \n (\\partial\\_\\nu\\phi) (\\partial^\\nu\\phi)\n\\nonumber \\\\\n& + D(\\phi) \\times (\\partial\\_\\mu \\partial\\_\\nu \\phi) (\\partial^\\mu\\phi) (\\partial^\\nu\\phi)\n + E(\\phi) \\times (\\partial\\_\\mu \\partial\\_\\nu\\phi) (\\partial^\\mu\\partial^\\nu\\phi)\n\\nonumber \\\\\n& + 8 U^{(5)} (\\partial\\_\\mu\\partial\\_\\nu\\partial\\_\\rho \\phi) (\\partial^\\mu\\phi) \n (\\partial^\\nu\\phi) (\\partial^\\rho\\phi)\n + 24 U^{(4)} (\\partial\\_\\mu\\partial\\_\\nu\\partial\\_\\rho\\phi) \n (\\partial^\\mu\\partial^\\nu\\phi) (\\partial^\\rho\\phi)\n\\nonumber \\\\\n& + 4 U''' (\\partial\\_\\mu\\partial\\_\\nu\\partial\\_\\rho\\phi) \n (\\partial^\\mu\\partial^\\nu\\partial^\\rho\\phi) \n \\biggr\\} = 0 \\,.\\end{aligned}$$ \nThe functions *A*-*E* are composed out of various derivatives of the potential function *U*, and are not important for this discussion. The important thing to notice is that the last three terms in the brackets are of third order in derivatives, and that there is no way of reducing their (time) derivative order while maintaining Lorentz covariance. In the first and second order derivative reductions whenever we needed to use the lower order equation in the higher order part we were always encountering second time derivatives in the combination with the Laplacian so that they form the d’Alembertian ▫. Here this is no longer the case, and if we want to reduce third derivatives above we need to pick out second time derivatives individually, which will break manifest covariance. This example teaches us that reducing the derivative order at the level of the equations of motion is not always compatible with the symmetries of the system.\nNote also that a field redefinition including higher derivatives,\n \n$$\\phi \\rightarrow \\phi - 4 \\epsilon^3\\alpha \\_3 U''' \\bigl[ \n (\\partial\\_\\mu\\partial\\_\\nu \\phi) (\\partial^\\mu \\partial^\\nu \\phi)\n +2 U^{(4)} (\\partial\\_\\mu\\partial\\_\\nu \\phi)\n (\\partial^\\mu\\phi) (\\partial^\\nu\\phi) \\bigr] \\,,$$ \nprecisely removes the third derivatives from the equation of motion (and modifies functions *A*-*E*), while preserving covariance. There seems to be no way around making field redefinitions when perturbatively reducing the derivative order while maintaining covariance.\n### Reduction via perturbative constraints\nSimilar maladies afflicting derivative reduction at the level of the equation of motion exhibited in the previous subsection affect the method of perturbative constraints. It will be shown here that field redefinitions are unavoidable.\nThe starting point is again the extended action, with time derivatives promoted to independent fields,\n \n$$\\dot{\\phi} \\rightarrow V\\_1\\,,\n\\quad\n- \\square\\phi = \\dot{V}\\_1 - \\nabla^2\\phi \\rightarrow V\\_2 \\,,\n\\quad\n-\\partial\\_0 \\square\\phi = \\dot{V}\\_2 \\rightarrow V\\_3 \\,,\n\\quad\n\\square\\square\\phi = \\dot{V}\\_3 - \\nabla^2V\\_2 \\rightarrow V\\_4 \\,,$$ \nand Lagrange multipliers *π*, *π*1, *π*2, and *π*3 introduced to ensure on-shell equivalence,\n \n$$\\begin{aligned}\n\\mathcal{S} ={}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n \\frac{1}{2} V\\_1^2 - \\frac{1}{2} (\\nabla\\phi)^2 - U + \\epsilon^3\\alpha\\_3 V\\_4^2\n + \\pi (\\dot{\\phi} - V\\_1) + \\pi\\_1 (\\dot{V}\\_1 - \\nabla^2\\phi - V\\_2)\n\\nonumber \\\\\n& \\hspace{3cm}\n + \\pi\\_2 (\\dot{V}\\_2 - V\\_3)\n + \\pi\\_3 (\\dot{V\\_3} - \\nabla^2 V\\_2 - V\\_4)\n \\biggr\\}\n\\nonumber \\\\\n={}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n \\pi\\dot{\\phi} + \\pi\\_1 \\dot{V}\\_1 + \\pi\\_2 \\dot{V}\\_2 + \\pi\\_3 \\dot{V}\\_3\n - \\biggl[\n \\pi V\\_1 + \\pi\\_2 (\\nabla^2\\phi + V\\_2)\n + \\pi\\_2 V\\_3 \n\\nonumber \\\\\n& \\hspace{3cm}\n + \\pi\\_3 (\\nabla^2 V\\_2 +V\\_4)\n - \\frac{1}{2} V\\_1^2 + \\frac{1}{2} (\\nabla\\phi)^2 + U\n - \\epsilon^3 \\alpha\\_3 V\\_4^2\n \\biggr]\n \\biggr\\} \\,.\n\\label{third order extended action}\\end{aligned}$$ \nWe do not bother here with presenting step-by-step how the perturbative constraints are imposed order by order, this should be clear from examples of the first and second order reductions, and the expressions at third order are rather lengthy while bringing noting new to the discussion. Instead we just cite the solutions of the perturbative second-class constraints needed to reduce the derivative order, and obtain the lower derivative phase space action,\n \n$$\\begin{aligned}\n\\pi\\_3 \\approx{}& - \\epsilon^3 \\alpha\\_3 \\bigl( U'' [\\pi^2 - (\\nabla\\phi)^2] - U''U'\\bigr) \\,,\n\\\\\n\\pi\\_2 \\approx{}& - 2 \\epsilon^3\\alpha\\_3 \\Bigl\\{ U^{(4)} \\pi [-\\pi^2 + (\\nabla\\phi)^2] \n - 2 U''' \\pi [\\nabla^2\\phi - U'] \n\\nonumber \\\\\n& \\hspace{5cm} \n + 2U''' (\\nabla\\phi) (\\nabla\\pi)\n + [U''U']' \\pi \\Bigr\\} \\,,\n\\\\\n\\pi\\_1 \\approx{}& 2\\epsilon^3 \\alpha\\_3 \\Bigl\\{\n 4 U''' (\\nabla\\pi)^2 + 8 U^{(4)} \\pi (\\nabla\\pi) (\\nabla\\phi)\n - U^{(5)} [\\pi^2-(\\nabla\\phi)^2]^2\n\\nonumber \\\\\n& \\hspace{2cm}\n - U^{(4)} \\bigl[ 4 \\pi^2 \\nabla^2\\phi + U' (\\nabla\\phi)^2\n + 2 (\\nabla\\phi) \\nabla (\\nabla\\phi)^2 \\bigr]\n\\nonumber \\\\\n& \\hspace{2cm}\n + U''' \\bigl[ - 2 (\\nabla^2\\phi - U')^2 + 2(\\nabla\\phi) \\nabla[\\nabla^2\\phi-U']\n - \\nabla^2 (\\nabla\\phi)^2 \\bigr]\n\\nonumber \\\\\n& \\hspace{2cm}\n + [U''U']''' [\\pi^2 - (\\nabla\\phi)^2]\n + [5 U^{(4)}U' + 2U'''U''] \\pi^2 - [U''U']'U'\n \\Bigr\\} \\,,\n\\\\\nV\\_4 \\approx{}& - U''' [\\pi^2 - (\\nabla\\phi)^2] - U'' U' + \\mathcal{O}(\\epsilon) \\,,\n\\\\\nV\\_3 \\approx{}& - \\epsilon^3 U' \\,,\n\\\\\nV\\_2 \\approx{}& - U' + \\mathcal{O}(\\epsilon) \\,,\n\\\\\nV\\_1 \\approx{}& \\pi + \\mathcal{O}(\\epsilon) \\,.\\end{aligned}$$ \nPromoting these to strong equalities and plugging them in the extended action ([third order extended action]) gives the phase space reduced action, which takes the form\n \n$$\\mathscr{S}\\_{\\rm red} = \\int\\! d^{D\\!}x \\, \\biggl\\{\n 8 \\epsilon^3 \\alpha\\_3 \\bigl[ U''' (\\nabla\\pi)^2 \n + 2 U^{(4)} \\pi (\\nabla\\pi) (\\nabla\\phi) \\bigr] \\dot{\\pi} +\n \\bigl( \\pi + \\epsilon^3\\alpha\\_3 \\Pi[\\phi,\\pi] \\bigr) \\dot{\\phi} - H[\\phi,\\pi] \n \\biggr\\} \\,,\n\\label{third order phase space}$$ \nwhere Π is a function of *ϕ*, *π*, and their spatial derivatives, and so is the Hamiltonian *H*. The method of derivative reduction via perturbative second-class constraints accomplishes exactly what it is designed to do – produces a phase space formulation of a lower derivative system capturing completely the perturbative solutions of the original system to order O(*ε*4). But phase space actions are considerably less convenient to work with than the configuration space ones, and it is desirable to construct one here.\nThe specific form of the phase space action ([third order phase space]) is of interest for the discussion here, but rather just the symplectic part. The fact that the symplectic part appears appears in a non-canonical form is by now not surprising, since we have encountered similar situations at lower orders in derivative reduction. The novel feature here is that the symplectic part *cannot* be canonicalized by only redefining the conjugate momentum field. The part containing the function Π can easily be removed by\n \n*π* → *π* − *ε*3*α*3Π , \nbut the part of ([third order phase space]) containing $\\dot{\\pi}$, cannot be removed this way. There is no way to rewrite it so that the time derivative is only on the scalar field *ϕ*. This is the behaviour mentioned in Sec [sec: Formalism] that appears in systems with multiple degrees of freedom. A field has infinitely many *local* degrees of freedom, and their coupling is expressed through gradients. It is precisely gradients acting on the conjugate momentum field that are preventing the canonicalization of the action.\nIf we insist on not redefining the scalar field itself, this also means there is *no* configuration space formulation of the action ([third order phase space]) solely in terms of the scalar field *ϕ*. There perhaps is a Lagrangian description with additional constraints, but they are likely to require explicit perturbative treatment, which defeats the purpose of this construction.\nThe action ([third order phase space]) can, of course, be canonicalized by the scalar field redefinition,\n \n$$\\phi \\rightarrow \\phi + 8\\epsilon^3\\alpha\\_3 \n \\bigl[ U''' (\\nabla\\phi)^2 + 2 U^{(4)} \\pi (\\nabla\\pi)(\\nabla\\phi) \\bigr] \\,.$$ \nAfter performing this, we will surely be able to find a configuration space action. But as soon as we start making such fields redefinitions in the phase space action, we do not really know (unless we investigate it in detail) whether we are preserving Lorentz covariance or not, and things become rather complicated.\nThe conclusion here is that field redefinitions are unavoidable when looking for a lower derivative configuration space action formulation of the systems where higher derivatives enter perturbatively. This being the case, we might as well consider field redefinitions of the configuration space action ([third order HD action]) right from the beginning.\n### Reduction via field redefinition\nThe field redefinition method when employed at the first and second order yielded the equivalent results as reducing derivative order in the equation of motion, and as the perturbative constraints method. Moreover, it was somewhat easier and faster to implement. At the third order the difference in the practicality of the methods becomes huge. In any case we need to employ field redefinitions, but doing it directly on the configuration space action is now tremendously faster and more straightforward. With comparatively much less work one can see that the scalar field redefinition\n \n$$\\phi \\rightarrow \\phi - \\epsilon^3 \\alpha^3 \\square\\square \\bigl[ \\square\\phi - U' \\bigr]\n - \\epsilon^3 \\alpha\\_3 (U'')^2 (\\square\\phi) \\,,$$ \ntransforms the action ([third order HD action]) into\n \n$$\\begin{aligned}\nS \\rightarrow{}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} \\Bigl( 1 + 2\\epsilon^3 \\alpha\\_3 [(U'')^2U']' \\Bigr)\n (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - U\n\\nonumber \\\\\n& \\hspace{2cm}\n + \\epsilon^3\\alpha\\_3 (U'')^2 (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) \n (\\partial\\_\\nu\\phi) (\\partial^\\nu\\phi)\n + 2\\epsilon^3\\alpha\\_3 U''' U'' (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) (\\square\\phi)\n \\biggr\\} \\,,\\end{aligned}$$ \nremoving all the genuine higher derivative terms, leaving only terms propagating three degrees of freedom.\nDerivative reduction at third order – Part II\n---------------------------------------------\nAs a last example of this section we examine the leading order action ([leading order action]) with a single higher order correction ([Ss(3)7]) from the third order EFT expansion,\n \n$$S = \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - U\n + \\epsilon^3 \\alpha\\_3 (\\partial\\_\\mu\\phi) (\\partial^\\mu \\phi)\n (\\partial\\_\\nu \\partial\\_\\rho\\phi) (\\partial^{\\nu}\\partial^\\rho\\phi) \n \\biggr\\} \\,.\n\\label{third order HD action II}$$ \nThe conclusions of the preceding subsection were that field redefinitions are generally inevitable in derivative reduction methods, therefore we consider only field redefinition here.\nThe higher order correction term in ([third order HD action II]) is indeed a genuine higher derivative term. It is not difficult to recognize that no Lorentz covariant field redefinition will be able to remove it (though non-covariant redefinitions would allow for this ). At this point we draw the attention to the difference between genuine and non-genuine higher derivative terms, former carrying extra degrees of freedom, compared to the latter that do not but still contain higher (spatial) derivatives. One such non-genuine higher derivative term is\n \n$$\\int\\! d^{D\\!}x \\, (\\partial\\_\\mu \\phi)(\\partial^\\nu\\phi) \\bigl[\n (\\square\\phi) (\\square\\phi) \n - (\\partial\\_\\mu\\partial\\_\\nu \\phi) (\\partial^\\mu\\partial^\\nu \\phi)\n \\bigr] \\,,$$ \nwhich is a specific flat space specialization of the fourth Horndeski action . We can rewrite the action ([third order HD action II]) in terms of it,\n \n$$\\begin{aligned}\nS ={}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - U\n - \\epsilon^3 \\alpha\\_3 (\\partial\\_\\mu \\phi)(\\partial^\\nu\\phi) \\bigl[\n (\\square\\phi) (\\square\\phi) \n - (\\partial\\_\\mu\\partial\\_\\nu \\phi) (\\partial^\\mu\\partial^\\nu \\phi)\n \\bigr]\n\\nonumber \\\\\n& \\hspace{2cm}\n + \\epsilon^3 \\alpha\\_3 (\\partial\\_\\mu \\phi)(\\partial^\\nu\\phi) (\\square\\phi) (\\square\\phi) \n \\biggr\\} \\,.\\end{aligned}$$ \nNow the only genuine higher derivative term in the action above can be taken the term in the bottom row, and that one is easily removed by the following field redefinition,\n \n*ϕ* → *ϕ* − *ε*3*α*3(∂*μ**ϕ*)(∂*μ**ϕ*)(▫*ϕ*) , \ntransforming the action to\n \n$$\\begin{aligned}\nS\\rightarrow{}& \\int\\! d^{D\\!}x \\, \\biggl\\{\n - \\frac{1}{2} (\\partial\\_\\mu\\phi) (\\partial^\\mu\\phi) - U\n - \\epsilon^3 \\alpha\\_3 (\\partial\\_\\mu \\phi)(\\partial^\\nu\\phi) \\bigl[\n (\\square\\phi) (\\square\\phi) \n - (\\partial\\_\\mu\\partial\\_\\nu \\phi) (\\partial^\\mu\\partial^\\nu \\phi)\n \\bigr]\n\\nonumber \\\\\n& \\hspace{2cm}\n + \\epsilon^3 \\alpha\\_3 U' (\\partial\\_\\mu \\phi)(\\partial^\\nu\\phi) (\\square\\phi) \n \\biggr\\} \\,,\\end{aligned}$$ \nwhere now the term in the bottom row is not genuinely higher derivative, and the whole action propagates no extra degrees of freedom. This field transformation teaches us that sometimes seemingly we do not need to eliminate a term by field transformation, but produce it in order to eliminate the genuine higher derivative behavior.\nSummary of the single scalar case study\n---------------------------------------\nStudying different examples of of perturbative derivative reductions in single scalar field systems we have learned several lessons summarized by the following points:\n* Reducing derivative order at the level of equations of motion is ambiguous. Many different equations capture perturbatively expandable solutions to a given order, and only specific ones have an action formulation. Those are the ones that should be used when attempting to solve the truncated lower order equation exactly.\n* It is not always possible to reduce the derivative order at the equations of motion level completely, and at the same time retain Lorentz covariance. Field redefinitions are needed to complete the task in general. This point was proved in for the case of multiple interacting scalar fields, and second derivatives in the action.\n* Derivative reduction by perturbative second-class constraints always accomplishes what it is meant to do – remove spurious degrees of freedom and deliver a lower derivative (or non-genuine higher derivative) phase space action. In general there is no equivalent configuration space action corresponding to this phase space reduced one, unless we allow for additional non-covariant field redefinitions.\n* Field redefinition method successfully eliminates genuine higher derivative terms from the configuration space action to a desired order, and is consistent with the first two methods.\n* Genuine higher derivative corrections are mapped to lower order or non-genuine higher derivative corrections. These correctly capture the EFT corrections. Still, we need to know which higher derivative terms are not genuine, the reduction method does not tell us this.\n* Removal of genuine higher derivative EFT corrections by field redefinition is generally not unique (but then again neither is any action where field redefinitions are allowed).\nIn conclusion, the field redefinition method is the preferred method for reducing genuine higher order EFT corrections.\nDerivative reduction in scalar-tensor theory\n============================================\nIn this section we apply the lessons learned in the single scalar case study of the preceding section, and employ the field redefinition method to the higher order EFT corrections of a scalar-tensor theory. Scalar theories are particularly suited for our purposes, since many of the non-genuine higher derivative terms have been worked out and classified, first within Horndeski theory , and later beyond Horndeski , and lately within the more general DHOST program . This actually makes our job much easier, since we can concentrate only on removing the genuine higher derivative corrections.\nHere we also work in *D* spacetime dimensions, and use the following conventions: metric field is of positive signature, the Christoffel symbol is $\\Gamma^{\\alpha}\\_\\mu\\nu = \\frac{1}{2} g^{\\alpha\\beta} (\\partial\\_\\mu g\\_{\\nu\\beta}\n+ \\partial\\_\\nu g\\_{\\mu\\beta} - \\partial\\_\\beta g\\_{\\mu\\nu})$, the Riemann tensor *R**ρ**μ**σ**ν* = ∂*σ*Γ*μ**ν**ρ* − …, the Ricci tensor *R**μ**ν* = *R**ρ**μ**ρ**ν*, the Ricci scalar *R* = *R**μ**μ*, and ∇*μ* is the covariant derivative.\nThe leading order action in our scalar-tensor theory is assumed to be\n \n$$S^{(0)} = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, \\biggl\\{\n \\frac{1}{\\kappa^2} f(\\phi) R\n - \\frac{1}{2} z(\\phi) g^{\\mu\\nu} (\\nabla\\_\\mu\\phi) (\\nabla\\_\\nu\\phi)\n - u(\\phi)\n \\biggr\\} \\,,\n\\qquad\n\\kappa = \\sqrt{16\\pi G\\_N} \\,,$$ \nand it propagates one scalar and two tensor degrees of freedom in *D* = 4.\nThe EFT expansion of corrections to this action is somewhat awkward with the metric dimensionless and the scalar field of mass dimension one. It is far more convenient to define a dimensionless scalar,\n \n*φ* = *κ**ϕ* , \nso that the leading order action reads\n \n$$S^{(0)} = \\frac{1}{\\kappa^2} \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, \\biggl\\{\n F(\\varphi) R \n - \\frac{1}{2} Z(\\varphi) g^{\\mu\\nu} (\\nabla\\_\\nu \\varphi) (\\nabla\\_\\nu \\varphi)\n - U(\\varphi)\n \\biggr\\}\n\\label{leading action}$$ \nwith the identifications *F*(*φ*) = *f*(*ϕ*), *Z*(*φ*) = *z*(*ϕ*), *U*(*φ*) = *u*(*ϕ*). Now the EFT expansion is naturally organized as an expansion in total number of derivatives. The leading order action contains terms with zero and two derivatives. The first order correction terms then contain four derivatives, and are one *κ*2 order higher then the leading order action,\n \n*S*(1) = ∑*n* = 19*S**n*(1) , \nwhere all the independent terms containing four derivatives are\n \n$$\\begin{aligned}\nS^{(1)}\\_1 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_1(\\varphi) \\,\n (\\nabla\\_\\mu\\varphi) (\\nabla^\\mu\\varphi) (\\nabla\\_\\nu\\varphi) (\\nabla^\\nu\\varphi) \\,,\n\\label{S(1)1}\n\\\\\nS^{(1)}\\_2 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_2(\\varphi) \\, \n (\\nabla\\_\\mu \\varphi) (\\nabla^\\mu \\varphi) (\\square\\varphi) \\,,\n\\label{S(1)2}\n\\\\\nS^{(1)}\\_3 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_3(\\varphi) \\, \n (\\square\\varphi) (\\square\\varphi) \\,,\n\\label{S(1)3}\n\\\\\nS^{(1)}\\_4 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_4(\\varphi) \\, \n (\\nabla\\_\\mu\\varphi) (\\nabla^\\mu\\varphi) R \\,,\n\\label{S(1)4}\n\\\\\nS^{(1)}\\_5 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_5(\\varphi) \\, \n (\\nabla\\_\\mu\\varphi) (\\nabla\\_\\nu\\varphi) R^{\\mu\\nu} \\,,\n\\label{S(1)5}\n\\\\\nS^{(1)}\\_6 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_6(\\varphi) \\, \n (\\square \\varphi) R \\,,\n\\label{S(1)6}\n\\\\\nS^{(1)}\\_7 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_7(\\varphi) \\, R^2 \\,,\n\\label{S(1)7}\n\\\\\nS^{(1)}\\_8={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_8(\\varphi) \\, R\\_{\\mu\\nu} R^{\\mu\\nu} \\,,\n\\label{S(1)8}\n\\\\\nS^{(1)}\\_9={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, f\\_9(\\varphi) \\, \n R\\_{\\mu\\nu\\rho\\sigma} R^{\\mu\\nu\\rho\\sigma} \\,,\n\\label{S(1)9}\\end{aligned}$$ \nand *f*1-*f*9 are arbitrary functions of the scalar field. All the other scalar-tensor actions containing four derivatives that one can write down can be expressed in terms if the nine above my making use of partial integrations, commutators of covariant derivatives, and the Bianchi identity.\nNot all of the terms ([S(1)1]-[S(1)9]) are genuinely higher-derivative ones, in the sense that they do not introduce additional degrees of freedom compared to the three present in the leading order action. Luckily, the most general scalar-tensor action containing four derivatives and propagating only three degrees of freedom is known. Most of it is captured by the Horndeski action specialized to four derivatives[8](#fn8),\n \n$$\\begin{aligned}\nS^{\\rm H}\\_2 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, g\\_2(\\varphi) \\, \n (\\nabla\\_\\mu \\varphi) (\\nabla^\\mu \\varphi) \n (\\nabla\\_\\nu \\varphi) (\\nabla^\\mu \\varphi) \\,,\n\\label{SH2}\n\\\\\nS^{\\rm H}\\_3 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, g\\_3(\\varphi) \\,\n (\\nabla\\_\\mu \\varphi) (\\nabla^\\mu \\varphi) (\\square\\varphi) \\,,\n\\label{SH3}\n\\\\\nS^{\\rm H}\\_4 ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, g\\_4(\\varphi) \\,\n \\Bigl\\{ (\\nabla\\_\\mu\\varphi) (\\nabla^\\mu\\varphi) R \n - 2 \\bigl[ (\\square\\varphi) (\\square\\varphi) \n - (\\nabla\\_\\mu\\nabla\\_\\nu \\varphi) (\\nabla^\\mu \\nabla^\\nu \\varphi) \\bigr]\n \\Bigr\\} \\,.\n\\label{SH4}\\end{aligned}$$ \nThe last of these actions can be rearranged into a more convenient form\n \n$$\\begin{aligned}\nS^{\\rm H}\\_4 ={}&\n \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, \\biggl\\{\n - 2 g\\_4(\\varphi) \\, G^{\\mu\\nu} (\\nabla\\_\\mu \\varphi) (\\nabla\\_\\nu \\varphi)\n + 3 g\\_4'(\\varphi) \\, (\\nabla\\_\\mu\\varphi) \n (\\nabla^\\mu \\varphi) (\\square\\varphi)\n\\nonumber \\\\\n& \\hspace{4cm}\n + g\\_4''(\\varphi)\\, (\\nabla\\_\\mu\\varphi) (\\nabla^\\mu \\varphi)\n (\\nabla\\_\\nu\\varphi) (\\nabla^\\nu \\varphi)\n \\biggr\\} \\,,\n\\label{StH4}\\end{aligned}$$ \nso instead of it we will use a more convenient one\n \n$$\\widetilde{S}^{\\rm H}\\_4 = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\,\n \\widetilde{g}\\_4(\\varphi) \\, G^{\\mu\\nu} \n (\\nabla\\_\\mu\\varphi) (\\nabla\\_\\nu\\varphi) \\,.$$ \nThere is one more remaining action with four derivatives propagating only three degrees of freedom, and it is known from Lanczos-Lovelock gravity , and more generally from scalar-Gauss-Bonnet models ,\n \n$$S^{\\rm SGB} = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, \n b(\\varphi) \\bigl[ R^2 - 4R\\_{\\mu\\nu} R^{\\mu\\nu} \n + R\\_{\\mu\\nu\\rho\\sigma} R^{\\mu\\nu\\rho\\sigma} \\bigr]\n\\label{SGB}$$ \nThe actions ([SH2]), ([SH3]), ([StH4]), and ([SGB]) comprise the most general four-derivative action propagating one scalar and two tensor degrees of freedom.\nOut of nine independent four-derivative actions ([S(1)1]-[S(1)9]) four do not need to be considered as higher derivative since we can instead write them as combinations forming actions ([SH2]), ([SH3]), ([StH4]), and ([SGB]) which are not genuinely higher derivative. They can be added to the leading order scalar-tensor action ([leading action]) as genuine EFT corrections not introducing spurious degrees of freedom. These considerations reduce the number of genuinely higher derivative terms we need to worry about to only five, which we relabel for convenience,\n \n$$\\begin{aligned}\nS^{\\rm HD}\\_1 ={}& S\\_8^{(1)} = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\,\n h\\_1(\\varphi) \\, R\\_{\\mu\\nu} R^{\\mu\\nu} \\,,\n\\label{SHD1}\n\\\\\nS^{\\rm HD}\\_2 ={}& S\\_7^{(1)} = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\,\n h\\_2(\\varphi) \\, R^2 \\,,\n\\label{SHD2}\n\\\\\nS^{\\rm HD}\\_3 ={}& S\\_6^{(1)} = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\,\n h\\_3(\\varphi) \\, (\\square\\varphi) R \\,,\n\\label{SHD2}\n\\\\\nS^{\\rm HD}\\_4 ={}& S\\_5^{(1)} = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\,\n h\\_4(\\varphi) \\, (\\nabla\\_\\mu\\varphi) (\\nabla^\\mu \\varphi) R \\,,\n\\label{SHD4}\n\\\\\nS^{\\rm HD}\\_5 ={}& S\\_3^{(1)} = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\,\n h\\_5(\\varphi) \\, (\\square\\varphi) (\\square\\varphi) \\,.\n\\label{SHD5}\\end{aligned}$$ \nWe proceed to eliminate the five higher derivative corrections ([SHD1]-[SHD5]) to the action by field redefinitions of the dimensionless scalar and the metric,\n \n*φ* → *φ* + *κ*2*δ**φ* , *g**μ**ν* → *g**μ**ν* + *κ*2*δ**g**μ**ν* . \nThe leading order action ([leading action]) transforms under these field redefinitions as\n \n*S*(0) → *S*(0) + *κ*2[*δ**φ**S*(0)] + *κ*2[*δ**g**S*(0)] , \nwhere the first subleading term on the right hand side descends form the scalar field redefinition,\n \n$$\\kappa^2 [\\delta\\_\\varphi S^{(0)}] = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, \\biggl\\{\n F' R + Z(\\square\\varphi) \n - \\frac{1}{2} Z' (\\nabla\\_\\mu\\varphi) (\\nabla^\\mu\\varphi) - U' \n \\biggr\\} \\delta\\varphi \\,,\n\\label{action phi transformation}$$ \nand the second one descends from the metric field redefinition,\n \n$$\\begin{aligned}\n\\MoveEqLeft[3]\n\\kappa^2 [\\delta\\_g S^{(0)}] = \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, \\biggl\\{\n - F G\\_{\\mu\\nu} \n + F' \\bigl[ (\\nabla\\_\\mu\\nabla\\_\\nu \\varphi) - (\\square\\varphi) g\\_{\\mu\\nu} \\bigr]\n\\nonumber \\\\\n&\n + \\frac{1}{2} [Z+2F''] (\\nabla\\_\\mu \\varphi) (\\nabla\\_\\nu \\varphi)\n - \\frac{1}{4} [Z + 4 F''] (\\nabla\\_\\rho \\varphi) (\\nabla^\\rho \\varphi) g\\_{\\mu\\nu}\n - \\frac{1}{2}U g\\_{\\mu\\nu}\n \\biggr\\} \\delta g^{\\mu\\nu} \\,.\n\\label{action g transformation}\\end{aligned}$$ \nIn order to accomplish the task of removing higher derivatives from the first order corrections ([SHD1]-[SHD5]) while preserving general covariance, quantities *δ**φ* and *δ**g**μ**ν* have to contain two derivatives and have to be covariant themselves. Their general form is thus\n \n$$\\begin{aligned}\n\\delta\\varphi ={}& C\\_1\\times (\\square\\varphi)\n + C\\_2 \\times (\\nabla\\_\\mu \\varphi) (\\nabla^\\mu \\varphi) \n + C\\_3 \\times R \\,\n\\label{scalar field redef}\n\\\\\n\\delta g\\_{\\mu\\nu} ={}& C\\_4 \\times g\\_{\\mu\\nu} R\n + C\\_5 \\times R\\_{\\mu\\nu}\n + C\\_6 \\times g\\_{\\mu\\nu} (\\square \\varphi)\n\\nonumber \\\\\n& \\hspace{1cm}\n + C\\_7 \\times (\\nabla\\_\\mu \\! \\nabla\\_\\nu \\varphi)\n + C\\_8 \\times g\\_{\\mu\\nu} (\\nabla\\_\\rho\\varphi) (\\nabla^\\rho\\varphi)\n + C\\_9 \\times (\\nabla\\_\\mu\\varphi) (\\nabla\\_\\nu\\varphi) \\,.\n\\label{metric field redef}\\end{aligned}$$ \nwhere the coefficients *C*1-*C*9 are functions of the scalar field, *C**i* = *C**i*(*φ*). We plug these into the expressions for the transformation of leading order action, ([action phi transformation]) and ([action g transformation]), and after performing some partial integrations and making use of the Bianchi identity, the transformed leading order action is put into the form\n \n$$S^{(0)} \\rightarrow S^{(0)} + \\int\\! d^{D\\!}x\\, \\sqrt{-g} \\, \\biggl\\{\n \\sum\\_A B\\_A(\\varphi) \\times \\mathcal{R}\\_A[g\\_{\\mu\\nu}, \\nabla\\_\\mu\\varphi]\n \\biggr\\} \\,,\n\\label{full action transform}$$ \nwhere all the terms R*A* dependent on the metric and the covariant derivatives of the scalar field, and the scalar dependent functions *B**A* are given in Table [transformation coefficients].\n[h!]\n|c||c| R*A* & *B**A*(*φ*) \n*R**μ**ν**R**μ**ν* & − *C*5*F* \n*R*2 & $C\\_3 F' + \\bigl( \\frac{D-2}{2} \\bigr) C\\_4 F + \\frac{1}{2} C\\_5 F$ \n*R* (▫*φ*) & $C\\_1 F' + C\\_3 Z - (D\\!-\\!1) C\\_4 F' + \\bigl( \\frac{D-2}{2} \\bigr) C\\_6 F$ \n*R* (∇*μ**φ*)(∇*μ**φ*) & $C\\_2 F' - \\frac{1}{2} C\\_3 Z' - \\frac{1}{4} C\\_4 \\bigl[ (D\\!-\\!2)Z + 4(D\\!-\\!1) F'' \\bigr]\n + \\frac{1}{2} C\\_5' F' $ \n& $- \\frac{1}{2} C\\_7 F' + \\bigl( \\frac{D-2}{2} \\bigr) C\\_8 F$ \n(▫*φ*)(▫*φ*) & *C*1*Z* − (*D* − 1)*C*6*F*ʹ \n*G**μ**ν*(∇*μ**φ*)(∇*ν**φ*) & $\\frac{1}{2} \\bigl[ C\\_5 Z - 2 C\\_5' F' \\bigr] + C\\_7' F\n - C\\_9 F$ \n(∇*μ**φ*)(∇*μ**φ*)(▫*φ*) & $- \\frac{1}{2} C\\_1 Z' + C\\_2 Z - \\frac{1}{4} C\\_6 \\bigl[ (D\\!-\\!2)Z + 4(D\\!-\\!1) F'' \\bigr]$ \n& $+ \\frac{1}{2} \\bigl[ 3 C\\_7' F' - C\\_7 Z \\bigr] - (D\\!-\\!1)C\\_8 F' - \\frac{3}{2} C\\_9 F'$ \n(∇*μ**φ*)(∇*μ**φ*)(∇*ν**φ*)(∇*ν**φ*) & $- \\frac{1}{2} C\\_2Z' + \\frac{1}{4} \\bigl\\{ 2 [C\\_7' F']' - [C\\_7 Z]' \\bigr\\}\n - \\frac{1}{4} \\bigl[ (D\\!-\\!2) Z + 4(D\\!-\\!1) F'' \\bigr]$ \n& $+ \\frac{1}{4} \\bigl\\{ C\\_9 Z - 2[C\\_9F']' \\bigr\\}$ \n*R* & $ -C\\_3 U' - \\frac{D}{2} C\\_4 U - \\frac{1}{2} C\\_5 U$ \n(∇*μ**φ*)(∇*μ**φ*) & $[C\\_1U']' - C\\_2 U' + \\frac{D}{2} [C\\_6U]' + \\frac{1}{2} [C\\_7U]'\n - \\frac{D}{2} C\\_8 U - \\frac{1}{2} C\\_9 U$ \nWe need to choose the nine coefficient functions *C**i*’s so as to absorb five distinct terms ([SHD1]-[SHD5]), and the way to do this is not unique. In particular, the first five rows of Table [transformation coefficients] need to absorb the genuine higher derivative terms. For the argument that we intend to make here it suffices to demonstrate that one particular solution exists,\n \n$$\\begin{aligned}\n& C\\_2 = C\\_6 = C\\_7 = C\\_9 = 0 \\,,\n\\qquad\nC\\_1 = - \\frac{h\\_5}{Z} \\,,\n\\qquad\nC\\_5 = \\frac{h\\_1}{F} \\,,\n\\\\\n& C\\_4 = \\frac{- Z h\\_1 + 2 Z h\\_2 - 2 F' h\\_3 + \\frac{2(F')^2}{Z} h\\_5}\n {(D\\!-\\!2)ZF + 2(D\\!-\\!1) (F')^2} \\,,\n\\qquad\n C\\_3 = \\frac{1}{Z} \\biggl\\{\n - h\\_3 + \\frac{F'}{Z} h\\_5 + (D\\!-\\!1) F' C\\_4\n \\biggr\\} \\,,\n\\\\\n& C\\_8 = \\frac{-2}{(D\\!-\\!2)F} \\biggl\\{ h\\_4 + \\frac{F'}{F}h\\_1' - \\frac{(F')^2}{2F^2} h\\_1\n - \\frac{Z'}{2} C\\_3 - \\frac{1}{4} \\bigl[ (D\\!-\\!2)Z + 4(D\\!-\\!1)F'' \\bigr] C\\_4 \\biggr\\} \\,.\\end{aligned}$$ \nTherefore, the first order correction in the EFT expansion of the scalar-tensor theory is captured correctly by the most general scalar-tensor theory with at most four derivatives that contains only three (in *D* = 4) degrees of freedom,\n \n$$\\begin{aligned}\nS ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\Biggl\\{ \n \\frac{1}{\\kappa^2} \\biggl[ F\\_1 \\, R\n - \\frac{1}{2} F\\_2\n (\\nabla\\_\\mu \\varphi) (\\nabla^\\mu \\varphi)\n - F\\_3 \\biggr]\n + F\\_4 \\, (\\nabla\\_\\mu\\varphi) (\\nabla^\\mu\\varphi) \n (\\nabla\\_\\nu\\varphi) (\\nabla^\\nu\\varphi)\n\\nonumber \\\\\n&\n + F\\_5 \\, (\\nabla\\_\\mu \\varphi) (\\nabla^\\mu \\varphi) (\\square \\varphi)\n + F\\_6 \\, G^{\\mu\\nu} (\\nabla\\_\\mu \\varphi) (\\nabla\\_\\nu \\varphi)\n + F\\_7 \\, \\bigl( R^2 - R\\_{\\mu\\nu} R^{\\mu\\nu}\n - R\\_{\\mu\\nu\\rho\\sigma} R^{\\mu\\nu\\rho\\sigma} \\bigr)\n \\Biggr\\} \\,,\\end{aligned}$$ \nwhere *F*1-*F*7 are some functions of the scalar field, *F**i* = *F**i*(*φ*) that in principle have an expansion in *κ*2. This action is a subset of Horndeski actions.\nConclusions\n===========\nThe aim of this paper was to examine the nature of higher derivative corrections in the effective field theory expansions. The point of view that we take on them is that they arise as a peculiarity of the approximation scheme of local derivative expansion employed in the EFT expansion. The higher derivative corrections cannot be treated on the same footing as the lower derivative leading order action describing the system. Rather, their effect should be taken into account in the same spirit in which they arise – perturbatively. That implies capturing their effects with terms that do not introduce extra degrees of freedom. Three methods of derivative reduction of such higher derivative terms were examined and compared on several examples of single scalar field systems, at different order in perturbation parameter.\nThe examined methods are (i) derivative reduction at the equations of motion level, (ii) derivative reduction via perturbative constraints, and (iii) derivative reduction via field redefinitions. The lessons learned are summarized in the bullet points at the end of Sec. [sec: Scalar]. The main conclusion is that the field redefinition method is the most practical one, compatible with global (Lorentz) symmetries, yielding as a result a configuration space action with no genuine higher derivative corrections to a given order.\nFew words are in order on why field redefinition is a legitimate method to use. When employed to eliminate higher derivative terms to certain order of interest, field redefinition necessarily involves derivatives itself which is usually not admissible. It is often pragmatically argued that in EFTs this is admissible due to pushing the problem of higher derivatives beyond the perturbative order of interest. Even though this rationale is strictly not wrong, a stronger, but more subtle argument can be made. In general, nonlocal field redefinitions are admissible if they do not change the number of degrees of freedom in the action. We can imagine such nonlocal field redefinitions that have a local derivative expansion analogous to the derivative expansion of the nonlocal correction to the action defining the EFT expansion. In particular, we can imagine a special nonlocal field redefinition whose local expansion is responsible for cancelling the genuine higher derivative terms from the EFT expansion of the action. In this sense the presence of higher derivatives in the field redefinition is not seen as fundamental, but just as a consequence of the approximation scheme. This is the reasoning behind employing field redefinitions to remove spurious degrees of freedom from the EFT expansion.\nGenuine higher derivatives in EFTs are thus seen as artefacts of the approximation scheme, not as essential features. Hence, there is no reason to look for a UV principle that would forbid their appearance in the low energy EFTs, since the. Rather, we ought to write EFT expansions with an additional requirement – no terms in the expansion should be genuine higher derivative terms, *i.e.* we should not write the terms introducing new degrees of freedom. This would remedy the error we are making by using implicitly the local derivative expansion as the definition of the EFT. We can make a claim that the *EFTs should consist of all the possible corrections respecting assumed symmetries, and not introducing new degrees of freedom*. This work does not provide a general proof of this statement, but it certainly seems compelling in the light of examples presented here.\nObservation about EFTs has made here bear relevance for the scalar-tensor theories often studied today. The most general scalar-tensor theories with no genuine higher derivative terms go under the acronym DHOST (Degenerate-Higher-Order-Scalar-Tensor). Here we make a claim that, together with the most general such theories involving curvature tensors as well, they should capture completely all the possible EFT corrections, and can in fact be seen as arising as EFTs.\nHere we have shown explicitly that the first order correction in the EFT expansion of the scalar-tensor theory is captured correctly by the most general scalar-tensor theory with at most four derivatives that does not contain extra degrees of freedom,\n \n$$\\begin{aligned}\nS ={}& \\int\\! d^{D\\!}x \\, \\sqrt{-g} \\, \\Biggl\\{ \n \\frac{1}{\\kappa^2} \\biggl[ F\\_1 \\, R\n - \\frac{1}{2} F\\_2\n (\\nabla\\_\\mu \\varphi) (\\nabla^\\mu \\varphi)\n - F\\_3 \\biggr]\n + F\\_4 \\, (\\nabla\\_\\mu\\varphi) (\\nabla^\\mu\\varphi) \n (\\nabla\\_\\nu\\varphi) (\\nabla^\\nu\\varphi)\n\\nonumber \\\\\n&\n + F\\_5 \\, (\\nabla\\_\\mu \\varphi) (\\nabla^\\mu \\varphi) (\\square \\varphi)\n + F\\_6 \\, G^{\\mu\\nu} (\\nabla\\_\\mu \\varphi) (\\nabla\\_\\nu \\varphi)\n + F\\_7 \\, \\bigl( R^2 - R\\_{\\mu\\nu} R^{\\mu\\nu}\n - R\\_{\\mu\\nu\\rho\\sigma} R^{\\mu\\nu\\rho\\sigma} \\bigr)\n \\Biggr\\} \\,,\n\\label{final action}\\end{aligned}$$ \nwhere *F*1-*F*7 are some functions of the scalar field, *F**i* = *F**i*(*φ*) that in principle have an expansion in *κ*2. Even though higher derivatives appear in this action they are not genuine higher derivatives. In *D* = 4 spacetime dimensions this action propagates only one scalar and two tensor degrees of freedom.\nIt is worth noting that just recently Solomon and Trodden have made many of the points presented in this paper, and have performed similar analyses.\nIt is now possible to address the initial motivation for this study – the construction and quantization of gauge-invariant cosmological perturbations to cubic order in small diffeomorphism transformations. This should to be possible with using the action ([final action]). The perturbations of the fields are introduced as $g\\_{\\mu\\nu} = \\overline{g}\\_{\\mu\\nu} + \\kappa \\delta g\\_{\\mu\\nu}$ and $\\varphi = \\overline{\\varphi} + \\kappa \\delta \\varphi$, and the background fields are assumed to satisfy the leading order equation of motion. The higher order terms in the action will modify the structure of the gauge-invariant perturbations at quadratic and cubic order. Functions *F*1-*F*7 should accommodate the one-loop renormalization of the two-point functions of gauge-invariant perturbations, by absorbing the divergences into their parameters. This is consistent with a well known result regarding the absence of one-loop divergences in pure gravity . Namely, the ones proportional to *R*2 and *R**μ**ν**R**μ**ν* can be absorbed by the metric field redefinition, in the same way as we have considered it in Sec. [sec: ST], and the remaining one can only be a pure Gauss-Bonnet term, which is a total derivative in *D* = 4.\nGeneral proofs of the statements made here about the absence of genuine higher derivative corrections in EFTs would be very very powerful, as the authors of point out. 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Eliezer and"},"data_id":{"kind":"string","value":"arxiv_0183124"}}},{"rowIdx":45799,"cells":{"content":{"kind":"string","value":"Microlocal Morse theory of wrapped Fukaya categories\n====================================================\n### August 27, 2023\nThe Nadler–Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspondence to incorporate the infinite-dimensional spaces of morphisms ‘at infinity’, given on the Floer side by Reeb trajectories (also known as “wrapping”) and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact. When combined with existing sheaf theoretic computations, our results confirm many new instances of homological mirror symmetry.\nMore precisely, given a real analytic manifold *M* and a subanalytic isotropic subset Λ of its co-sphere bundle *S*\\**M*, we show that the partially wrapped Fukaya category of *T*\\**M* stopped at Λ is equivalent to the category of compact objects in the unbounded derived category of sheaves on *M* with microsupport inside Λ. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.\nIntroduction\n============\nThe calculation of Fukaya categories of symplectic manifolds has emerged as a question of central interest in geometry. Within symplectic geometry itself, many questions may be phrased in terms of intersections of Lagrangian submanifolds; as the Fukaya category is built from these, it is a natural tool for their study. In low-dimensional topology, a number of invariants of smooth manifolds and smooth knots can be extracted from Fukaya categories of associated symplectic manifolds. Homological mirror symmetry, a largely conjectural correspondence arising from non-rigorous reasoning in mathematical physics, further predicts that many categories of interest in algebraic geometry and representation theory also arise as Fukaya categories.\nBeyond the intrinsic interest in confirming or explaining these predictions, knowing that a category of interest arises as a Fukaya category suggests the existence of additional structures. For one example, morphism spaces in the Fukaya category are Floer homology chain complexes, and as such come with a natural basis; this often ‘explains’ the existence of previously known ‘canonical’ bases of these Hom spaces. For another, the relative ease of constructing symplectomorphisms (which act on the relevant Fukaya categories) gives a natural source of automorphisms of these categories that are far less apparent from other points of view. The difficulty in calculating Fukaya categories, which is present in all of the aforementioned settings, stems from the global and analytic nature of the pseudo-holomorphic disks appearing in the definition.\n*In this paper, we obtain a combinatorial description of the partially wrapped Fukaya categories of all stably polarized Weinstein manifolds (more generally, sectors), by showing that they are isomorphic to certain corresponding categories of microlocal sheaves.*\nWeinstein manifolds and partially wrapped Fukaya categories\n-----------------------------------------------------------\nA vector field *Z* on a symplectic manifold (*X*, *ω*) is said to be *Liouville* when *L**Z**ω* = *ω*. Recall that in this case *λ* : = *ω*(*Z*, ⋅ ) is a primitive for *ω*, and that such symplectic manifolds are necessarily non-compact. Such a triple (*X*, *ω*, *Z*) (equivalently (*X*, *λ*)) is called a *Liouville manifold* if, in addition, the non-compact ends of *X* are identified (necessarily uniquely) by *Z* with the positive end of the symplectization of a contact manifold. The *core* of a Liouville manifold is the locus of points ${\\mathfrak c}\\_X$ which do not escape to infinity under the Liouville flow. The inclusion ${\\mathfrak c}\\_X\\subseteq X$ is a homotopy equivalence, and in some sense ${\\mathfrak c}\\_X$ carries all of the symplectic topology of *X* as well.\nA Liouville manifold (*X*, *ω*, *Z*) is said to be *Weinstein* if *Z* is gradient-like. The key feature of such manifolds is that the core ${\\mathfrak c}\\_X$ is a union of isotropic submanifolds, and moreover admits transverse Lagrangian disks (‘cocores’) at its smooth Lagrangian points. Prototypical examples include cotangent bundles, affine algebraic varieties, and more generally (finite type) Stein complex manifolds. Many examples of interest in geometric representation theory, such as conical symplectic resolutions, quiver varieties, moduli of Higgs bundles, and cluster varieties are in this class. Moreover, any compact symplectic manifold whose symplectic form has rational periods can be presented as a compactification of a Weinstein manifold, and hence, through a strategy introduced in, understanding the Fukaya categories of Weinstein manifolds serves as a stepping stone to studying the Fukaya categories of closed symplectic manifolds.\nWhile various analytic difficulties in Floer theory are simplified in the Liouville setting (due to strong topological and geometric control on pseudo-holomorphic disks), there is a significant new layer of complexity possible, thanks to the non-compactness of the target space. Namely, as has been understood for some time, it is desirable in this context to enlarge the Fukaya category of compact Lagrangians by adding certain non-compact (properly embedded, conical at infinity) Lagrangians as well. These larger Fukaya categories often have better formal properties due to there being an ample supply of non-compact Lagrangians, and they are also required by mirror symmetry, where non-compact Lagrangians in non-compact targets arise as mirror objects to sheaves on non-compact or non-Calabi–Yau manifolds, whose Ext groups could be of infinite rank or fail to satisfy Poincaré duality. Fukaya categories of non-compact Lagrangians also have bearing on questions about the Reeb dynamics at infinity. There are many different ways to add non-compact Lagrangians to the Fukaya category, with substantially different results; the basic parameters are (1) in which directions at infinity to allow non-compact Lagrangians and (2) in what direction and by how much to perturb (‘wrap’) Lagrangians at infinity when computing Floer homology.\nThe framework of *partially wrapped Fukaya categories* has emerged as a way to describe and relate different prescriptions for asymptotics and wrapping. One specifies a subset at infinity which Lagrangians cannot limit to or wrap past, called the *stop*. In the resulting category, Lagrangians which are isotopic in the complement of the stop induce isomorphic objects, and symplectomorphisms preserving the stop induce autoequivalences. The resulting category is also invariant under isotopies of the complement of the stop. A *stopped Liouville manifold* (*X*, Λ) consists of a Liouville manifold *X* and a stop Λ. The *relative core* ${\\mathfrak c}\\_{X,\\Lambda}$ of (*X*, Λ) is the set of those which do not escape to the complement of Λ at infinity under the positive Liouville flow.\nThe partially wrapped setting includes variants of many previous constructions:\n* When the stop is empty one obtains the (fully) wrapped Fukaya category of Abouzaid–Seidel.\n* Given a smooth Legendrian Λ, there is a naive ‘infinitesimally wrapped’ Fukaya category with asymptotics along Λ given equivalently by either: (1) take the stop to be the complement of a small regular neighborhood of Λ, or (2) take the stop to be Λ and consider just the full subcategory of Lagrangians which admit wrappings converging to Λ (see Section [NZ] for further discussion, including a comparison with ).[1](#fn1) More generally, we can take Λ to be the core of a Liouville hypersurface.\n* The Fukaya category of a *Landau–Ginzburg model* $\\mathrm{w}: Y \\to {\\mathbb C}$, also known as the *Fukaya–Seidel category* when w is a Lefschetz fibration, can be modeled as the partially wrapped Fukaya category of *Y* stopped at (the core of) the Weinstein hypersurface w− 1( − ∞).[2](#fn2)\nOf course, partially wrapped Fukaya categories form a significantly broader class than infinitesimal or Fukaya–Seidel categories (e.g. they can have infinite-dimensional morphism spaces). An illustrative example: categories of coherent complexes on arbitrary (not necessarily compact or smooth) *n*-dimensional toric stacks can be shown equivalent to partially wrapped Fukaya categories of $({\\mathbb C}^\\*)^n = T^\\* T^n$, by combining the sheaf theoretic work of with the main theorem of the present article.\nOf particular importance are (possibly singular) isotropic stops. Isotropicity of the stop plays the same role as the Weinstein condition on the symplectic manifold itself; for instance, the core of a *Weinstein hypersurface at infinity* is a typical singular isotropic stop of interest to mirror symmetry.\nTopological and sheaf theoretic interpretations\n-----------------------------------------------\nDespite their analytic origins, Fukaya categories have often been found to admit topological interpretations. The first prototype is the fact that the Lagrangian Floer homology of an exact Lagrangian is nothing other than its ordinary cohomology. The work of Nadler and Zaslow provides a sweeping generalization of this, identifying infinitesimally wrapped Floer homologies of exact Lagrangians in cotangent bundles with morphisms of sheaves on the base. In a seemingly different direction, it was observed that *wrapped* Floer homology also has a topological interpretation. Indeed, work of Abbondondalo and Schwarz (see also Cieliebak and Latschev ) found many instances where wrapped Floer homology is isomorphic to the homology of spaces of paths and loops. Building on these, Abouzaid showed that in fact the wrapped Fukaya category of a cotangent bundle is naturally identified with perfect modules over chains on the based loop space of the base (see also ). This last result may be restated (by the ∞ version of the van Kampen theorem): the wrapped Fukaya category of a cotangent bundle is the global sections of the constant cosheaf of categories on the zero section with costalk ${\\operatorname{Perf}}{\\mathbb Z}$. This formulation exhibits an instance of a more general conjecture of Kontsevich : the wrapped Fukaya category of any Weinstein manifold *X* should be the global sections of a cosheaf of categories on its core ${\\mathfrak c}\\_X$.\nNadler’s work unified these points of view, by proposing that while infinitesimally wrapped Fukaya categories are modeled by (micro)sheaves with perfect (micro)stalks (as in ), the partially wrapped category should be modeled by compact objects in the category of all (micro)sheaves with appropriate microsupport conditions. For essentially formal reasons, these categories of compact objects may be organized into a cosheaf of categories, so Nadler’s proposal is a strengthening of Kontsevich’s conjecture. At the time of Nadler’s original proposal, microsheaves were only defined for subsets of cotangent bundles, but the high codimension embedding trick from has now defined a category of microsheaves on the core of any Liouville manifold.\nSince microlocal sheaf categories are entirely combinatorial/topological in nature, Nadler’s proposal is a (conjectural) computation of (partially) wrapped Fukaya categories. An illustrative example calculation is given in, where it was shown that the relevant category of microlocal sheaves on the skeleton of the (higher dimensional) symplectic pairs of pants matched the mirror-symmetric prediction for the wrapped Fukaya category (which was more recently verified directly ).\nRemarkably, while wrapped and infinitesimal Floer homologies are rather different creatures, the same does not appear to be the case on the sheaf side of Nadler’s proposal. Notably, there is nothing like ‘wrapping’ on the sheaf side; instead, purely categorical operations. As it is difficult to appreciate the depth of the distinction at first, let us return to the example of the cotangent bundle, with empty stop. The core is just the zero section, and sheaves microsupported in the zero section are (almost by definition) nothing other than locally constant sheaves. If we require these to have finite stalks, then we are studying finite rank local systems (which coincides with the infinitesimally wrapped Fukaya category, i.e. in this case the Fukaya category of compact Lagrangians). On the other hand, a typical compact object in this category is the ‘tautological’ local system, with fiber given by chains on the based loop space. Usually of infinite rank, this object is best understood in terms of the functor it co-represents: taking the stalk at a point. Note that no ‘wrapping’ appears in this purely categorical procedure proposed by Nadler, yet it does in fact correctly recover the wrapped Fukaya category.\nMain results\n------------\nWe now fix notation and state our main results more precisely. Theorem [sheaffukayaequivalence] concerns the special case of cotangent bundles, and its proof comprises the bulk of the paper. Theorem [microsheaffukayaequivalence] is derived from Theorem [sheaffukayaequivalence] and concerns more general stably polarized Liouville manifolds.\nFor a Liouville symplectic manifold *X* and closed subset Λ ⊆ ∂∞*X*, we write ${\\mathcal W}(X, \\Lambda)$ for the (partially) wrapped Fukaya category of *X*, stopped at Λ. Its objects are Lagrangians *L* ⊆ *X* which are eventually conical and disjoint from Λ at infinity, and its morphism complexes are Floer cochains after wrapping Lagrangians in the complement of Λ. It is an *A*∞ category defined in (see also ); we review its definition at the beginning of Section [sec:fukaya]. Particularly important objects of ${\\mathcal W}(X,\\Lambda)$ include: the Lagrangian linking disks to the smooth Legendrian points of Λ, the Lagrangian cocore disks when *X* is Weinstein, and the cotangent fibers when *X* = *T*\\**M* (which may be viewed as a special case of cocore disks). We write ${\\operatorname{Perf}}{\\mathcal W}(X,\\Lambda)$ for the idempotent-completed pre-triangulated closure of ${\\mathcal W}(X,\\Lambda)$. Whenever Λ ⊆ Λʹ, there is a tautological functor ${\\mathcal W}(X, \\Lambda') \\to {\\mathcal W}(X, \\Lambda)$.\nFor a smooth manifold *M*, we write Sh(*M*) for the dg category of sheaves of dg ${\\mathbb Z}$-modules on *M*. The microsupport of a sheaf ${\\mathcal F}$ is a closed conical locus ${\\operatorname{ss}}({\\mathcal F}) \\subseteq T^\\*M$ whose role is to encode, infinitesimally, which restriction maps are quasi-isomorphisms. We write ShΛ(*M*) for the full subcategory of Sh(*M*) spanned by those sheaves whose microsupport at infinity is contained in Λ. Particularly nice functors ${\\operatorname{Sh}}\\_\\Lambda(M)\\to{\\operatorname{Mod}}{\\mathbb Z}$ include the stalk functors at points of *M* and the microstalk functors at smooth Legendrian points of Λ. These definitions are reviewed in Section [sec:sheaves]. We denote by ShΛ(*M*)*c* the category of compact objects in ShΛ(*M*). The reader is cautioned that the compact objects of ShΛ(*M*) do not necessarily have perfect stalks or bounded homological degree; that is, they need not be constructible sheaves in the usual sense. The relevance of such objects on the sheaf side was pointed out in, where Theorem [sheaffukayaequivalence] was implicitly conjectured by the terminology ‘wrapped sheaves’ for compact objects of ShΛ(*M*) and ‘wrapped (microlocal) skyscrapers’ for co-representatives of (micro)stalk functors.\n[sheaffukayaequivalence] Let *M* be a real analytic manifold, and let Λ ⊆ *S*\\**M* be a subanalytic closed isotropic subset. There there is a canonical equivalence of categories\n \n$$\\label{sheaffukayaequivalenceeqn}\n{\\operatorname{Perf}}{\\mathcal W}(T^\\*M,\\Lambda)^{\\mathrm{op}}={\\operatorname{Sh}}\\_\\Lambda(M)^c$$ \nwhich carries the linking disk at any smooth Legendrian point *p* ∈ Λ to a co-representative of the microstalk functor at *p* ∈ Λ and carries the cotangent fiber at a point *p* ∈ *M* not in the image of Λ to a co-representative of the stalk functor.\nRather than passing to the opposite category of the Fukaya category, we could equivalently negate either Λ or the Liouville form on *T*\\**M*.\nTo prove Theorem [sheaffukayaequivalence], we do not calculate a single ${\\mathcal W}(T^\\*M,\\Lambda)$ on its own. Instead, we calculate the functor $\\Lambda\\mapsto{\\operatorname{Perf}}{\\mathcal W}(T^\\*M,\\Lambda)^{\\mathrm{op}}$ (functoriality is with respect to inclusions Λ ⊆ Λʹ) which is a much more rigid object. In fact, we formulate a pair of axioms which uniquely charaterize such a system of categories $\\Lambda\\mapsto{\\mathcal C}(\\Lambda)$ (Section [sec:axioms]), so the proof of Theorem [sheaffukayaequivalence] then reduces to verifying these axioms for both $\\Lambda\\mapsto{\\operatorname{Perf}}{\\mathcal W}(T^\\*M,\\Lambda)^{\\mathrm{op}}$ (Section [sec:fukaya]) and Λ ↦ ShΛ(*M*)*c* (Section [sec:sheaves]).\nThe underlying reason this strategy can succeed is that there are special stops Λ (specifically $\\Lambda=N^\\*\\_\\infty{\\mathcal S}$, the union of conormals to the strata of a Whitney triangulation ${\\mathcal S}$) for which the Reeb dynamics in the complement of Λ are simple, thus making it tractable to show by direct calculation. Since every Λ is a subset of some $N^\\*\\_\\infty{\\mathcal S}$, it then suffices to show that both sides of transform in the same way when Λ gets smaller. On the Fukaya side, the functor ${\\mathcal W}(T^\\*M,\\Lambda')\\to{\\mathcal W}(T^\\*M,\\Lambda)$ for Λʹ ⊇ Λ is the quotient by the linking disks to Λʹ \\ Λ; this was established recently in. On the sheaf side, one quotients by co-representatives of microstalks; this is ultimately a consequence of co-isotropicity of the microsupport. The identification of linking disks with microstalks matches the wrapping exact triangle of with the microlocal Morse description of sheaf cohomology from. The conclusion is then that choosing a Whitney triangulation ${\\mathcal S}$ of *M* whose conormal $N^\\*\\_\\infty{\\mathcal S}$ contains Λ yields a description of both categories ${\\operatorname{Perf}}{\\mathcal W}(T^\\*M,\\Lambda)^{\\mathrm{op}}$ and ShΛ(*M*)*c* as the same localization of the category ${\\operatorname{Perf}}{\\mathcal W}(T^\\*M,N^\\*\\_\\infty{\\mathcal S})^{\\mathrm{op}}={\\operatorname{Perf}}{\\mathcal S}={\\operatorname{Sh}}\\_{N^\\*\\_\\infty{\\mathcal S}}(M)^c$.\nThis approach to the proof of Theorem [sheaffukayaequivalence] is rather different from the previous computations of Fukaya categories of cotangent bundles. In particular, we rely on no results from these articles. In fact, our ability to add geometry to simplify the situation is such that the only Floer cohomology calculations which need to be made in this entire article are between Lagrangians which intersect in at most one point, obviating, in particular, the need to ever compute a holomorphic disc. We therefore expect that the proofs of the results in this paper would apply to the case of more general (e.g. sphere spectrum) coefficients, provided one has access to the definitions of the sheaf and Fukaya categories in these settings.\nThe equivalence of Theorem [sheaffukayaequivalence] is also functorial under open inclusions:\n[sfecompatibility] Let $M'\\hookrightarrow M$ be an analytic open inclusion of analytic manifolds. For subanalytic isotropics Λʹ ⊆ *S*\\**M*ʹ and Λ ⊆ *S*\\**M* with Λʹ ⊇ Λ ∩ *S*\\**M*ʹ, the following diagram commutes\n \n$$\\begin{tikzcd}\n{\\operatorname{Perf}}{\\mathcal W}(M',\\Lambda')^{\\mathrm{op}}\\ar[equal]{d}{Thm\\ \\ref{sheaffukayaequivalence}}\\ar{r}&{\\operatorname{Perf}}{\\mathcal W}(M,\\Lambda)^{\\mathrm{op}}\\ar[equal]{d}{Thm\\ \\ref{sheaffukayaequivalence}}\\\\\n{\\operatorname{Sh}}\\_{\\Lambda'}(M')^c\\ar{r}&{\\operatorname{Sh}}\\_\\Lambda(M)^c\n\\end{tikzcd}$$ \nwhere the bottom horizontal arrow is (the restriction to compact objects of) the left adjoint of restriction.\nIn Section [sec:corollaries], we detail a number of applications and corollaries of Theorem [sheaffukayaequivalence]. These include a version of the original Nadler–Zaslow correspondence, translations of the microsheaf theoretic work on mirror symmetry for toric varieties and toric boundaries, and a sheaf theoretic description of the augmentation category of the partially wrapped Floer cochains of the linking disk to a Legendrian (expected to be equivalent to the Legendrian DGA), valid for the jet bundle of a manifold of any dimension. After the present work, a host of sheaf theoretic calculations can now be understood as computations of wrapped Fukaya categories.\nFinally, in Section [microsection], we turn from cotangent bundles to the general setting of stably polarized Weinstein manifolds. We proceed by combining the ‘doubling trick’ of and the ‘antimicrolocalization’ of to reduce to the cotangent bundle case. We arrive the following sheaf theoretic description of partially wrapped Fukaya categories:\n[microsheaffukayaequivalence] Let *X* be a real analytic Liouville manifold, and let Λ ⊆ ∂∞*X* be a stop whose relative core ${\\mathfrak c}\\_{X,\\Lambda}:={\\mathfrak c}\\_X\\cup(\\Lambda\\times{\\mathbb R})\\subseteq X$ is subanalytic singular isotropic. For any stable polarization[3](#fn3) of *X*, there is a fully faithful functor\n \n$$\\label{microsheaffukayaequivalenceeqn}\n{\\operatorname{Perf}}{\\mathcal W}(X,\\Lambda)^{\\mathrm{op}}\\hookrightarrow{\\operatorname{\\mu sh}}\\_{{\\mathfrak c}\\_{X,\\Lambda}}({\\mathfrak c}\\_{X,\\Lambda})^c,$$ \nwhere ${\\operatorname{\\mu sh}}\\_{{\\mathfrak c}\\_{X,\\Lambda}}({\\mathfrak c}\\_{X,\\Lambda})$ denotes the category of microlocal sheaves on ${\\mathfrak c}\\_{X,\\Lambda}$. This functor sends a homological cocore at a smooth Lagrangian point *p* of ${\\mathfrak c}\\_{X,\\Lambda}$ to a co-representative of the microstalk at *p*.\nIn particular, if *X* is Weinstein, or more generally admits homological cocores, then is an equivalence.\nLet us comment on the hypotheses of Theorem [microsheaffukayaequivalence] and, in particular, argue that it applies to all stably polarized Weinstein sectors. The analyticity assumptions hold in most concrete cases of interest, and abstractly speaking, any Weinstein manifold (more generally, sector) may be perturbed so as to be real analytic and to have subanalytic relative core (see Corollary [analyticrelativeskeleton]). A homological cocore at a smooth Lagrangian point $p\\in{\\mathfrak c}\\_{X,\\Lambda}$ is an object of ${\\operatorname{Perf}}{\\mathcal W}(X,\\Lambda)$ whose image in ${\\operatorname{Perf}}{\\mathcal W}((X,\\Lambda)\\times({\\mathbb C}\\_{{\\operatorname{Re}}\\geq 0},\\infty))$ is the linking disk at *p* × ∞. Admitting homological cocores means that every smooth Lagrangian point of ${\\mathfrak c}\\_{X,\\Lambda}$ has a homological cocore; this condition turns out to depend only on *X* up to deformation, and holds whenever *X* is Weinstein. For (*X*, Λ) as in Theorem [microsheaffukayaequivalence], the stabilization $X\\times{\\mathbb C}$ always admits homological cocores, so there is always an equivalence ${\\operatorname{Perf}}{\\mathcal W}((X,\\Lambda)\\times({\\mathbb C},\\pm\\infty))^{\\mathrm{op}}={\\operatorname{\\mu sh}}\\_{{\\mathfrak c}\\_{X,\\Lambda}\\times{\\mathbb R}}({\\mathfrak c}\\_{X,\\Lambda}\\times{\\mathbb R})^c={\\operatorname{\\mu sh}}\\_{{\\mathfrak c}\\_{X,\\Lambda}}({\\mathfrak c}\\_{X,\\Lambda})^c$.\nThe embedding of Theorem [microsheaffukayaequivalence] depends *a priori* on a choice of analytic Liouville hypersurface embedding of $X\\times{\\mathbb C}^k$ into *S*\\**M* for some auxiliary analytic manifold *M*, compatible with stable polarizations (part of the proof is to show such data always exists). We expect our methods could be extended to show that the embedding of Theorem [microsheaffukayaequivalence] is independent of this choice.\nThe embedding of Theorem [microsheaffukayaequivalence] associated to a given analytic Liouville hypersurface embedding $X\\hookrightarrow S^\\*M$ is, by construction, compatible with the equivalence of Theorem [sheaffukayaequivalence] in the sense that the following diagram commutes:\n \n$$\\label{theoremscommute}\n\\begin{tikzcd}\n{\\operatorname{Perf}}{\\mathcal W}(X,\\Lambda)^{\\mathrm{op}}\\ar{r}\\ar[hook]{d}[swap]{\\text{Thm \\ref{microsheaffukayaequivalence}}}&{\\operatorname{Perf}}{\\mathcal W}(T^\\*M,\\overline{{\\mathfrak c}\\_{X,\\Lambda}})^{\\mathrm{op}}\\ar[equal]{d}{\\text{Thm \\ref{sheaffukayaequivalence}}}\\\\\n{\\operatorname{\\mu sh}}\\_{{\\mathfrak c}\\_{X,\\Lambda}}({\\mathfrak c}\\_{X,\\Lambda})^c\\ar{r}{\\mu^\\*}&{\\operatorname{Sh}}\\_{\\overline{{\\mathfrak c}\\_{X,\\Lambda}}}(M)^c\n\\end{tikzcd}$$ \n(see Proposition [sheafdescmicrolocalwithremoval]), where *μ*\\* denotes the left adjoint to microlocalization. Using this compatibility, it is proven in that for a Fano toric stack *Y* with toric divisor *D*, there is a commutative diagram\n \n$$\\label{fsmirror}\n\\begin{tikzcd}\n{\\operatorname{Perf}}{\\mathcal W}(W^{-1}(-\\infty)) \\ar{r} \\ar[equal]{d} & {\\operatorname{Perf}}{\\mathcal W}(X,W) \\ar{r} \\ar[equal]{d} & {\\operatorname{Perf}}{\\mathcal W}(X) \\ar{r} \\ar[equal]{d} & 0 \\\\\n{\\operatorname{Coh}}(D) \\ar{r} & {\\operatorname{Coh}}(Y) \\ar{r} & {\\operatorname{Coh}}(Y \\setminus D) \\ar{r} & 0\n\\end{tikzcd}$$ \nwhere $W:X\\to{\\mathbb C}$ is the mirror Landau–Ginzburg model (see Example [veryaffine]).\n**Convention.** Throughout this document, we work in the setting of dg and, equivalently, *A*∞ categories over ${\\mathbb Z}$ (or more generally any commutative ring). We only ever consider “derived” functors, we only ever mean “homotopy” limits or colimits, and we systematically omit the word “quasi”. By modules, we mean dg or *A*∞ modules, e.g. by ${\\mathbb Z}$-modules we mean the category of chain complexes of abelian ${\\mathbb Z}$-modules, localized at quasi-isomorphisms, except when, as in this sentence, we qualify it with the word ‘abelian’. In Section [categoricalsection] we detail our assumptions about these categories and collect relevant categorical notions which will appear throughout the paper.\nAcknowledgements\n----------------\nWe thank Mohammed Abouzaid, Roger Casals, Alexander Efimov, Tobias Ekholm, Benjamin Gammage, David Nadler, Amnon Neeman, and Lenhard Ng for helpful discussions, some of which took place during visits to the American Institute of Mathematics.\nS.G. was partially supported by NSF grant DMS–1907635, and would also like to thank the Simons Laufer Mathematical Sciences Institute (previously known as MSRI) for its hospitality during visits in Spring 2018 (supported by NSF grant DMS–1440140) and Fall 2022 (supported by NSF grant DMS–1928930) during which some of this work was completed. This research was conducted during the period J.P. served as a Clay Research Fellow and was partially supported by a Packard Fellowship and by the National Science Foundation under the Alan T. Waterman Award, Grant No. 1747553. V.S. was supported by the NSF CAREER grant DMS–1654545.\nStratifications\n===============\nGeneralities\n------------\nLet *X* be a topological space. By a stratification ${\\mathcal S}$ of *X*, we mean a locally finite decomposition into disjoint locally closed subsets $\\{X\\_\\alpha\\}\\_{\\alpha\\in{\\mathcal S}}$, called strata, such that each boundary $\\overline{X\\_\\alpha}\\setminus X\\_\\alpha$ is a union of other strata *X**β*. The collection of strata ${\\mathcal S}$ is naturally a poset, in which there is a map *β* → *α* iff $X\\_\\alpha \\subseteq \\overline{X\\_\\beta}$.\n[stratvsposet] The poset ${\\mathcal S}$ does not generally capture the homotopy type of the space *X*. Conditions under which it does (contractibility of various strata/stars) are well known and recalled below.\nWe will say a subset *Y* ⊆ *X* is ${\\mathcal S}$-constructible when it is a union of strata of the stratification ${\\mathcal S}$ of *X*. We say that a stratification ${\\mathcal T}$ refines a stratification ${\\mathcal S}$ when the strata of ${\\mathcal S}$ are ${\\mathcal T}$-constructible.\nWe recall that an abstract simplicial complex on a vertex set ${\\mathcal V}$ is a collection Σ of nonempty finite subsets of ${\\mathcal V}$, containing all singletons and all subsets of elements of Σ. By a simplicial complex, we mean the geometric realization ∣Σ∣ of an abstract simplicial complex Σ; it comes with a stratification by the ‘open simplices’ (which, of course, are locally closed, not necessarily open, subsets of ∣Σ∣). We say a stratification ${\\mathcal S}$ on *X* is a triangulation when there exists a homeomorphism $\\left|\\Sigma\\right|\\xrightarrow\\sim X$ identifying stratifications. We never impose any sort of regularity condition (differentiability, smoothness, analyticity, etc.) on this homeomorphism, even in the context of stratifications of a given regularity class. Note that the following are *not* triangulations: a stratification of a circle into single point and its complement, or into two points and their complement; the stratification into three points and their complement is a triangulation.\nThe open star of a stratum is the union of strata whose closures contain it. Taking stars reverses the inclusion: we have $X\\_\\alpha \\subseteq \\overline{X\\_\\beta} \\iff {\\operatorname{star}}(X\\_\\beta) \\subseteq {\\operatorname{star}}(X\\_\\alpha)$. Note that star(*X**α*) ∩ star(*X**β*) = ⋃*α* ← *γ* → *β*star(*X**γ*). For triangulations, we can do better: star(*X**α*) ∩ star(*X**β*) = star(*X**γ*) where *γ* is the simplex spanned by the vertices of *α* union the vertices of *β* (if this simplex is present), and otherwise star(*X**α*) ∩ star(*X**β*) = ∅.\nFor a *C**p* manifold *M*, we say a stratification ${\\mathcal S}$ is *C**p* if each stratum *M**α* is a (locally closed) *C**p* submanifold.\nA *C*1 stratification ${\\mathcal S}$ of a *C*1 manifold *M* is called a *Whitney stratification* iff it satisfies Whitney’s conditions (a) and (b). These are usually stated as the following conditions on pairs of strata *X* and *Y* of ${\\mathcal S}$:\n* For any sequence *x**i* ∈ *X* converging to *y* ∈ *Y* such that *T**x**i**X* converges to a subspace *V* ⊆ *T**y**M*, we have *T**y**Y* ⊆ *V*.\n* For any pair of sequences *x**i* ∈ *X* and *y**i* ∈ *Y* both converging to *y* ∈ *Y* such that *T**x**i**X* converges to a subspace *V* ⊆ *T**y**M* and the secant directions from *y**i* to *x**i* converge to a line *L* ⊆ *T**y**M*, we have *L* ⊆ *V*.\nBy compactness of flag varieties, we may pass to convergent subsequences, and hence conditions (a) and (b) may be reformulated as follows:\n* For strata $Y\\subseteq\\overline X$, as *X* ∋ *x* → *y* ∈ *Y*, the tangent spaces *T**x**X* become arbitrarily close to containing *T**y**Y*, uniformly over compact subsets of *Y*.\n* For strata $Y\\subseteq\\overline X$, as *X* ∋ *x* → *y* ∈ *Y*, the secant lines between *x* and *y* become arbitrarily close to being contained in *T**x**X*, uniformly over compact subsets of *Y*.\nWhitney’s condition (a) is equivalent to the assertion that the union of conormals $N^\\*{\\mathcal S}:=\\bigcup\\_\\alpha N^\\*M\\_\\alpha\\subseteq T^\\*M$ is closed. In fact, it is not hard to see that Whitney’s condition (b) implies Whitney’s condition (a). A Whitney stratification is, by definition, at least *C*1; it makes sense to consider *C**p* Whitney stratifications for any *p* ≥ 1, including *p* = ∞.\nIn order to guarantee the existence of Whitney stratifications, we will ultimately restrict to the setting of (real) analytic manifolds and subanalytic stratifications. We recall that a set is defined to be subanalytic when locally (i.e. in a neighborhood of every point of its closure) it is the analytic image of a relatively compact semianalytic set (i.e. locally defined by finitely many analytic inequalities). The canonical modern reference for subanalytic geometry is. By a subanalytic stratification, we mean a stratification in which all strata are subanalytic. Every subanalytic stratification admits a subanalytic refinement in which all strata are locally closed analytic submanifolds. It is a fundamental result that for any locally finite collection of subanalytic subsets of an analytic manifold, there exists a subanalytic stratification with respect to which all the subsets are constructible. For proofs of these results, see. We also require the result that every subanalytic stratification admits a refinement to a subanalytic Whitney triangulation.\nWherever we have written ‘subanalytic’, one could substitute ‘defineable’ with respect to any fixed analytic-geometric category. Every defineable stratification has a defineable refinement to a *C**p* Whitney triangulation for any given *p* < ∞. The fact that this is not known to hold for *p* = ∞ does not create any difficulties.\n[whitneyintersect] Let *M* be a manifold with Whitney stratification ${\\mathcal S}$. If *N* ⊆ *M* is a locally closed submanifold transverse to every stratum of ${\\mathcal S}$, then the intersected stratification ${\\mathcal S}\\cap N$ is a Whitney stratification of *N*.\nIt suffices to verify that ${\\mathcal S}\\cap N$ satisfies Whitney (b). Thus consider a pair of strata *X* ∩ *N* and *Y* ∩ *N* with $Y\\subseteq\\overline X$. Whitney (b) for the stratification ${\\mathcal S}$ guarantees that the secant line from *x* ∈ *X* ∩ *N* to *y* ∈ *Y* ∩ *N* becomes arbitrarily close to being contained in *T**x**X* as *x* → *y*, uniformly over compact subsets of *Y* ∩ *N*. On the other hand, Whitney (b) for the stratification for ${\\mathcal S}\\cap N$ requires this secant line to become arbitrarily close to being contained in *T**x**X* ∩ *T**x**N*, a stronger condition. Since *N* is a submanifold, the secant line certainly becomes arbitrarily close to being contained in *T**x**N*. Our task is thus to pass from being close to *T**x**X* and *T**x**N* to being close to their intersection *T**x**X* ∩ *T**x**N*. It thus suffices to show that *T**x**X* and *T**x**N* are uniformly transverse as *x* → *y*. Let us see how this follows from Whitney (a) for ${\\mathcal S}$. Since *N* is transverse to *Y*, we have *T**y**Y* + *T**y**N* = *T**y**M*. Whitney (a) for ${\\mathcal S}$ means that *T**x**X* is arbitrarily close to containing *T**y**Y* as *x* → *y*. Thus *T**x**X* + *T**x**N* = *T**x**M* uniformly as *x* → *y*, as desired.\n[addsubmanifoldremark] Here is a typical application of Lemma [whitneyintersect]. Let *M* be a manifold with a Whitney stratification ${\\mathcal S}$, and let *N* ⊆ *M* be a closed (as a subset) submanifold transverse to every stratum of ${\\mathcal S}$. Then the stratification ${\\mathcal S}\\_N$ of *M* with strata *M**α* ∩ *N* and *M**α* \\ *N* for $\\alpha\\in{\\mathcal S}$ (the poset of strata ${\\mathcal S}\\_N$ is thus ${\\mathcal S}\\times\\{(M\\setminus N)>N\\}$) is Whitney. Indeed, Whitney (b) for ${\\mathcal S}\\_N$ is a special case of Whitney (b) for ${\\mathcal S}$ and ${\\mathcal S}\\cap N$. A slight modification of this example will also come up later. Let *B* ⊆ *M* be a closed ball whose boundary is transverse to a Whitney stratification ${\\mathcal S}$. The stratification of *M* by *M**α* ∩ ∂*B*, *M**α* ∩ *B*∘, and *M**α* \\ *B*, for strata *M**α* ⊆ *M*, is now Whitney by the same argument using Lemma [whitneyintersect].\nMicrolocal approximation of constructible sets\n----------------------------------------------\nA constructible set *X* with respect to a Whitney stratification is in general quite singular. Our goal in this section is to show how such sets can be *microlocally* approximated by manifolds-with-corners *X**η* parameterized by small *η* > 0, in the sense that *X**η* → *X* and the conormal to *X**η* converges to (being contained in) the conormal of *X* as *η* → 0. This result will be used in the proof of Proposition [better841] and in Section [fukayacalculationssec]. We fix an integer *p* ≥ 1, possibly *p* = ∞, and work throughout with stratifications of class *C**p*.\nA *C**p* *radius function* for a locally closed *C**p* submanifold *Y* ⊆ *M* is a pair (*U*, *ρ*) where *U* ⊆ *M* is an open set containing *Y* and $\\rho:U\\to{\\mathbb R}\\_{\\geq 0}$ is of class *C**p* on *U* \\ *Y* and satisfies the following three conditions:\n* *ρ*− 1(0) = *Y*;\n* *ρ* is Lipschitz on a neighborhood of any compact subset of *Y*;\n* the lim inf of the evaluation of *d**ρ*(*x*) on the secant direction from *y* ∈ *Y* to *x* ∈ *M* is bounded below by some *ε* > 0 as *x* → *y*, provided *y* is constrained to a compact subset of *Y* and the ratio $\\frac{d(x,y)}{d(x,Y)}$ is bounded by some fixed *N* < ∞ (this condition is well defined since *ρ* is assumed Lipschitz; it implies ∣*d**ρ*(*x*)∣ is bounded away from zero over neighborhoods of compact subsets of *Y*).\nThe standard radius function for ${\\mathbb R}^n\\times 0\\subseteq{\\mathbb R}^n\\times{\\mathbb R}^m$ is of course (*a*1, …, *a**n*, *b*1, …, *b**m*) ↦ (*b*12 + ⋯ + *b**m*2)1/2. Every locally closed *C**p* submanifold admits a *C**p* radius function, as can be seen by choosing a collection of local coordinate patches and summing together standard radius functions via a partition of unity (a convex combination of radius functions is a radius function).\nThe following (trivial) restriction property for radius functions will be important: if (*U*, *ρ*) is a radius function for *Y* ⊆ *M*, and *N* ⊆ *M* is transverse to *Y*, then (*U* ∩ *N*, *ρ*∣*U* ∩ *N*) is a radius function for *Y* ∩ *N* ⊆ *N*.\nWhen *Y* ⊆ *M* is relatively compact, a radius function (*U*, *ρ*) for *Y* will be called *proper* when for every open set *V* containing $\\overline Y$, there exists *ε* > 0 such that *ρ*− 1([0, *ε*]) ⊆ *V*. It is easy to produce proper radius functions: for any radius function (*U*, *ρ*), there exists an open set *U*ʹ ⊆ *U* containing *Y* such that (*U*ʹ, *ρ*∣*U*ʹ) is proper.\nThe purpose of a radius function is to define tubular neighborhoods *ρ*− 1([0, *ε*]). The conditions in the definition of a radius function are chosen so as to be able to prove the following two key assertions:\n[tubeconormal] Let (*U*, *ρ*) be a radius function for *Y* ⊆ *M*. We have *d**ρ*(*x*) → *N*\\**Y* as *x* → *Y*, uniformly over compact subsets of *Y* (equivalently, *N*\\**ρ*− 1(*r*) approaches being contained in *N*\\**Y* as *r* → 0, uniformly over compact subsets of *Y*).\nSuppose for sake of contradiction that there is a sequence *x**i* → *y* ∈ *Y* with *d**ρ*(*x**i*) → *ξ* ∈ *T**y*\\**M* \\ *N**y*\\**Y*. Since $\\xi\\notin N^\\*\\_yY$, there exist *y**i* ∈ *Y* with $\\frac{d(x\\_i,y\\_i)}{d(x\\_i,Y)}$ uniformly bounded and the secant direction from *y**i* to *x**i* converging to the kernel of *ξ*. This contradicts the final axiom of a radius function.\n[tubetransverse] Let *Y* ⊆ *M* be a stratum of a Whitney stratification ${\\mathcal S}$, and let (*U*, *ρ*) be a radius function for *Y*. There exists an open set *V* ⊆ *U* containing *Y* such that the submanifold *ρ*− 1(*r*) is transverse to ${\\mathcal S}$ over *V* for all *r* > 0.\nThe secant line to *x* ∈ *ρ*− 1(*r*) from nearby *y* ∈ *Y* pairs positively with *d**ρ*(*x*), whereas Whitney (b) requires that this secant line approach the tangent space to the stratum containing *x* in the limit *x* → *y*. Since *d**ρ* is bounded, this gives a positive lower bound on the restriction of *d**ρ* to any stratum of ${\\mathcal S}$ in a neighborhood of any compact subset of *Y*.\n[scale=/6cm] (0,0)to(0,1)to(1,1)to(1,0)to cycle; (0,0)to(1,1);\n[even odd rule] (1.15,-0.15)rectangle(3.15,1.15) (1.985,-0.015)to(1.985,1.015)to(2.015,1.015)to(2.015,-0.015)to cycle (2.985,-0.015)to(2.985,1.015)to(3.015,1.015)to(3.015,-0.015)to cycle (1.985,-0.015)to(1.985,.015)to(3.015,.015)to(3.015,-0.015)to cycle (1.985,0.985)to(1.985,1.015)to(3.015,1.015)to(3.015,0.985)to cycle (2-.010606,.010606)to(3-.010606,1.010606)to(3+.010606,1-.010606)to(2+.010606,-.010606)to cycle; (2,0) circle (0.1); (3,0) circle (0.1); (2,1) circle (0.1); (3,1) circle (0.1);\n[even odd rule] (1.1,-0.1)rectangle(3.1,1.1) (2,0) circle (0.1) (3,0) circle (0.1) (2,1) circle (0.1) (3,1) circle (0.1); (1.985,-0.015)to(1.985,1.015)to(3.015,1.015)to(3.015,-0.015)to cycle; (2.015,0.015)to(2.015,0.985)to(2.985,0.985)to(2.985,0.015)to cycle; (2-.010606,.010606)to(3-.010606,1.010606); (2+.010606,-.010606)to(3+.010606,1-.010606);\n[figureblowup]\nWe now turn to the setting of a compact set *X* ⊆ *M* constructible with respect to a chosen Whitney stratification ${\\mathcal S}$. Given a proper radius function for each stratum of ${\\mathcal S}$ contained in *X*, we define the ‘outward cornering’ of *X* with respect to ${\\mathcal S}$ (see Figure [figureblowup]) to be\n \n$$\\label{outwardcornering}\nX^{\\underline\\epsilon}=\\bigcup\\_{M\\_\\alpha\\subseteq X}\\rho\\_\\alpha^{-1}([0,\\epsilon\\_\\alpha])$$ \nfor $\\underline\\epsilon=(\\epsilon\\_\\alpha)\\_{M\\_\\alpha\\subseteq X}$, where it is tacitly required that *ε**α* > 0 be sufficiently small as an unspecified function of $(\\epsilon\\_\\beta)\\_{M\\_\\beta\\subsetneqq\\overline{M\\_\\alpha}}$. Note that, no matter our notion of sufficiently small, we can always find a parameterization $\\underline\\epsilon(\\eta)$ where each *ε**α* > 0 is an increasing function of *η* > 0, limiting to zero as *η* → 0, such that *ε**α*(*η*) > 0 is sufficiently small in terms of $(\\epsilon\\_\\beta(\\eta))\\_{M\\_\\beta\\subsetneqq\\overline{M\\_\\alpha}}$ for all *η* > 0 (proof: by induction on strata).\nThe significance of properness of the given radius functions is that it (along with compactness of *X*) ensures that the part of $\\partial X^{\\underline\\epsilon}$ coming from *ρ**α*− 1(*ε**α*) is contained in a neighborhood of a *compact* subset of *M**α* ⊆ *X* (depending on $(\\epsilon\\_\\beta)\\_{M\\_\\beta\\subsetneqq\\overline{M\\_\\alpha}}$), hence falls within the scope of Lemma [tubetransverse].\nHere is the first key property of $X^{\\underline\\epsilon}$:\n[neighborhoodcorners] Fix a Whitney stratification ${\\mathcal S}$ of *M*. Let *X* ⊆ *M* be a compact constructible subset, with a choice of proper radius function for each of its strata. Then $M\\setminus X^{\\underline\\epsilon}$ is a manifold-with-corners, all of whose corner strata are transverse to ${\\mathcal S}$.\nWe proceed by induction on the number of strata of *X*. Thus let *X* = *X*0 ∪ *M**α*, and suppose the result is known for *X*0. This means, in particular, that $M\\setminus X\\_0^{\\underline\\epsilon}$ is a manifold-with-corners, all of whose corner strata are transverse to *M**α*. It follows that further removing the small regular neighborhood *ρ**α*− 1([0, *ε**α*]) produces a manifold-with-corners $M\\setminus X^{\\underline\\epsilon}$, provided *ε**α* > 0 is sufficiently small as a function of $(\\epsilon\\_\\beta)\\_{M\\_\\beta\\subsetneqq\\overline{M\\_\\alpha}}$.\nNow let us show that the corner strata of $M\\setminus X^{\\underline\\epsilon}$ are transverse to ${\\mathcal S}$. The boundary stratum of $M\\setminus X^{\\underline\\epsilon}$ coming from *ρ**α*− 1(*ε**α*) is transverse to ${\\mathcal S}$ by Lemma [tubetransverse]. A general corner stratum of $M\\setminus X^{\\underline\\epsilon}$ is the intersection *C* ∩ *ρ**α*− 1(*ε**α*) where *C* is a corner stratum of $M\\setminus X\\_0^{\\underline\\epsilon}$. To see that such an intersection is also transverse to ${\\mathcal S}$, apply Lemma [tubetransverse] to the stratum *C* ∩ *M**α* ⊆ *C* of the restriction to *C* of ${\\mathcal S}$ (which is Whitney by Lemma [whitneyintersect], locally uniformly in $\\underline\\epsilon\\setminus\\epsilon\\_\\alpha$) equipped with the restriction of *ρ**α* (which remains a radius function as noted earlier).\nNote that the conclusion of Corollary [neighborhoodcorners] (transversality of $\\partial X^{\\underline\\epsilon}$ and ${\\mathcal S}$) is equivalent to saying that $N^\\*{\\mathcal S}$ (the union of the conormals of all strata) and $N^\\*X^{\\underline\\epsilon}$ (the union of the conormals of all corner strata) are disjoint at infinity.\nGiven transversality of $\\partial X^{\\underline\\epsilon}$ and ${\\mathcal S}$, we can define the ‘big conormal’ $N^\\*({\\mathcal S}|(M\\setminus X^{\\underline\\epsilon}))$ to be the union of conormals of intersections of strata of ${\\mathcal S}$ and corner strata of $M\\setminus X^{\\underline\\epsilon}$. The second key property of $X^{\\underline\\epsilon}$ is the following convergence result:\n[cornerconormalconvergence] Fix a Whitney stratification ${\\mathcal S}$ of *M*. Let *X* ⊆ *M* be a compact constructible subset, with a choice of proper radius function for each of its strata. Then the big conormal $N^\\*({\\mathcal S}|(M\\setminus X^{\\underline\\epsilon}))$ lies in arbitrarily small neighborhoods of $N^\\*{\\mathcal S}$ as $\\underline\\epsilon\\to 0$.\nWe proceed by induction on the number of strata of *X*. Thus let *X* = *X*0 ∪ *M**α*, and suppose the result is known for *X*0. It thus suffices to show that $N^\\*({\\mathcal S}|(M\\setminus X^{\\underline\\epsilon}))$ lies in an arbitrarily small neighborhood of $N^\\*({\\mathcal S}|(M\\setminus X\\_0^{\\underline\\epsilon}))$ as *ε**α* → 0 (uniformly over compact subsets of $\\underline\\epsilon\\setminus\\epsilon\\_\\alpha$).\nThe conormal of the intersection of a stratum *M**γ* of ${\\mathcal S}$ with *ρ**α*− 1(*ε**α*) is the sum of the conormals of *M**γ* and *ρ**α*− 1(*ε**α*). These individually approach being contained in *N*\\**M**α* in the limit *ε**α* → 0 (the first by Whitney (a) and the second by Lemma [tubeconormal]), and they are quantitatively transverse by the axioms of a radius function (as was the main point of the proof of Lemma [tubetransverse]). It follows that their sum also approaches *N*\\**M**α* as *ε**α* → 0.\nThe general case is that of the conormal of *M**γ* ∩ *ρ**α*− 1(*ε**α*) ∩ *C* where *C* is a corner stratum of $M\\setminus X\\_0^{\\underline\\epsilon}$. It follows by applying the same argument to the intersection of the situation with *C*, as in the proof of Corollary [neighborhoodcorners].\nIn the context of Corollary [cornerconormalconvergence], note that any subanalytic family of Legendrians inside *S*\\**M*, whose projections converge to *X*, will themselves converge to a subset of the conormal of *X* *with respect to some refinement of ${\\mathcal S}$*. Corollary [cornerconormalconvergence] provides a stronger convergence result (we do not need to refine the stratification) for the particular family of outward cornerings defined in.\nCornering and conormals of constructible open sets\n--------------------------------------------------\nHere we develop some finer properties of the microlocal approximations constructed in the previous subsection. They will not be used until Section [fukayacalculationssec].\nLet ${\\mathcal S}$ be a Whitney stratification of *M* by locally closed smooth submanifolds. For any ${\\mathcal S}$-constructible relatively compact open set *U* ⊆ *M*, we define its *inward cornering*\n \n$$U^{-\\underline\\epsilon}:=U\\setminus\\overline{(\\partial U)^{\\underline\\epsilon}}.$$ \nwhere $(\\partial U)^{\\underline\\epsilon}$ denotes the outward cornering ∂*U* defined in. Thus $U^{-\\underline\\epsilon}$ is (the interior of) a codimension zero submanifold-with-corners (Corollary [neighborhoodcorners]), depending smoothly on $\\underline\\epsilon$, such that as $\\underline\\epsilon\\to 0$, its conormal $N^\\*U^{-\\underline\\epsilon}$ remains disjoint from $N^\\*{\\mathcal S}$ at infinity (Corollary [neighborhoodcorners]) yet limits inside it (Corollary [cornerconormalconvergence]). Strictly speaking, $U^{-\\underline\\epsilon}$ also depends on the choices of tubular neighborhoods of the strata comprising ∂*U*, however we will leave this choice out of the notation as it is never particularly relevant (note that it is a convex, hence contractible, choice). When even the choice of $\\underline\\epsilon$ is not relevant, we will simply write *U*−.\nTaking $\\underline\\epsilon\\to 0$, we learn that:\n[perturbationdiffeo] *U* and *U*− are diffeomorphic, and the diffeomorphism may be chosen to be the identity on any fixed compact subset of *U*.\nWhen ${\\mathcal S}$ is a triangulation, we may consider for any simplex $s\\in{\\mathcal S}$ its open star, star(*s*).\n[bdrywarmup] For ${\\mathcal S}$ a triangulation and any $s \\in {\\mathcal S}$, there is a homotopy equivalence ∂star(*s*)− ≃ ∂star(*s*).\nBoth star(*s*) and star(*s*)− are the interiors of compact (topological) manifolds-with-boundary (namely their closures). They are also diffeomorphic: star(*s*) ≅ star(*s*)− by Lemma [perturbationdiffeo]. It therefore suffices to recall the standard fact that the interior of a compact manifold-with-boundary remembers its boundary, up to homotopy equivalence.\nIndeed, let $\\overline M$ be a compact manifold-with-boundary and let $M=\\overline M\\setminus\\partial\\overline M$ denote its interior. The ‘end space’ *e*(*M*) is the space of proper maps ${\\mathbb R}\\_{\\geq 0}\\to M$ (this is a model for the homotopy inverse limit of *M* \\ *K* over compact subsets *K* ⊆ *M*). A choice of collar $\\partial\\overline M\\times[0,1)\\hookrightarrow\\overline M$ determines a homotopy equivalence\n \n$$e(M)\\xleftarrow\\sim e(\\partial\\overline M\\times(0,{\\textstyle\\frac 12}])=C({\\mathbb R}\\_{\\geq 0},\\partial\\overline M)\\times e((0,{\\textstyle\\frac 12}]),$$ \nand we have homotopy equivalences $C({\\mathbb R}\\_{\\geq 0},\\partial\\overline M)\\simeq\\partial\\overline M$ and $e((0,\\frac 12])\\simeq \\*$, so we have $e(M)\\simeq\\partial\\overline M$. (Compare.)\n[strangesimplexintersectionisstableball] For ${\\mathcal S}$ a triangulation and for simplices $s,t\\in{\\mathcal S}$ with star(*t*) ∩ star(*s*) ≠ ∅ and $t\\nrightarrow s$, the intersection ${\\operatorname{star}}(t)^{-\\underline\\epsilon}\\cap\\partial{\\operatorname{star}}(s)^{-\\underline\\delta}$ is contractible for $\\underline\\epsilon$ sufficiently small and $\\underline\\delta$ sufficiently small in terms of $\\underline\\epsilon$.\nAs in the proof of Lemma [bdrywarmup], the intersection ${{\\operatorname{star}}(t)^{-\\underline\\epsilon}}\\cap{\\partial{\\operatorname{star}}(s)^{-\\underline\\delta}}$ is the end space of $\\overline{{\\operatorname{star}}(t)^{-\\underline\\epsilon}}\\cap{\\operatorname{star}}(s)^{-\\underline\\delta}$. As in Lemma [perturbationdiffeo], this intersection $\\overline{{\\operatorname{star}}(t)^{-\\underline\\epsilon}}\\cap{\\operatorname{star}}(s)^{-\\underline\\delta}$ is diffeomorphic to $\\overline{{\\operatorname{star}}(t)^{-\\underline\\epsilon}}\\cap{\\operatorname{star}}(s)$, whose end space is in turn given by ${\\operatorname{star}}(t)^{-\\underline\\epsilon}\\cap\\partial{\\operatorname{star}}(s)$. Now we may take $\\underline\\epsilon\\to 0$ again mimicking the proof of Lemma [perturbationdiffeo] to see that this is homotopy equivalent to star(*t*) ∩ ∂star(*s*). The assumptions on *s* and *t* now imply that this is contractible.\n[smoothcornering] If *U* has smooth boundary, then there is a *C*0-small isotopy between *U*− and a small inward pushoff of *U*.\nThe definition of *U*− depends on choice of radius functions for the strata comprising ∂*U*. Fix coordinates $\\partial U\\times{\\mathbb R}\\subseteq M$ near ∂*U*, and choose radius functions whose inward derivative in the ${\\mathbb R}$-direction is positive inside *U*. Such radius functions exist locally, hence can be patched together using a partition of unity pulled back from ∂*U* (i.e. independent of the ${\\mathbb R}$-coordinate), which preserves the property of having positive inward derivative inside *U*. Now using these radius functions, each vertical line $p\\times{\\mathbb R}$ for *p* ∈ ∂*U* intersects ∂*U*− exactly once, transversally, which provides the desired isotopy.\nIt is *not* asserted that the isotopy in Lemma [smoothcornering] will ensure that the conormal remains disjoint from $N^\\*{\\mathcal S}$ at infinity. This will require us to exercise some care when applying it.\nFor the next result, consider a Whitney stratification ${\\mathcal S}$ and a point *q* ∈ *M* lying in a stratum *M**χ*. Let ${\\mathcal S}\\_q$ denote the Whitney stratification obtained from ${\\mathcal S}$ by replacing *M**χ* with *M**χ* \\ *q* and {*q*}. Given a compact ${\\mathcal S}$-constructible set *X* ⊆ *M* containing *q*, we can consider its outward cornerings $X^{\\underline\\epsilon}$ and $X^{\\underline\\epsilon,\\delta}$ with respect to ${\\mathcal S}$ and ${\\mathcal S}\\_q$, respectively, where $\\underline\\epsilon=(\\epsilon\\_\\alpha)\\_{M\\_\\alpha\\subseteq X}$ and *δ* > 0 is associated to *q*. Evidently\n \n$$X^{\\underline\\epsilon,\\delta}=X^{\\underline\\epsilon}\\cup B\\_\\delta(q),$$ \nand according to the definition of ‘outward cornering’ above, there is the implicit requirement that *ε**χ* > 0 be sufficiently small as a function of $(\\epsilon\\_\\beta)\\_{M\\_\\beta\\subsetneqq\\overline{M\\_\\alpha}}$ and *δ*. The next result concerns the behavior of $X^{\\underline\\epsilon,\\delta}$ when we *remove* the dependence of *ε**χ* on *δ*. The resulting neighborhoods are illustrated in Figure [figureXepsdel], which should be contrasted with Figure [figureblowup].\n[wrapnothitballnew] Let *X* ⊆ *M* be a compact ${\\mathcal S}$-constructible subset, and fix *q* ∈ *X* living in stratum *χ*. Let $\\tilde X^{\\underline\\epsilon,\\delta}=X^{\\underline\\epsilon}{\\mathbin{\\tilde\\cup}}B\\_\\delta(q)$ where $X^{\\underline\\epsilon}$ is the outward cornering and the notation $\\tilde\\cup$ indicates that the boundary of the union *ρ**χ*− 1([0, *ε**χ*]) ∪ *B**δ*(*q*) is smoothed along its potential corner locus *ρ**χ*− 1(*ε**χ*) ∪ ∂*B**δ*(*q*), and then the remaining tubes are added. Then for suitable choices of radius functions near *q*, these modified outward cornerings $X^{\\underline\\epsilon,\\delta}$ satisfy the conclusion of Corollary [neighborhoodcorners].\n[scale=/10cm] (1,0.2)–(-1,0.2); (0.55,0)arc(0:90:0.55); ((-0.55,0)arc(180:90:0.55); ; ; (-1,0)to(1,0); (a)–(1,0.2); (b)–(-1,0.2);\n(-1,0.2)–(1,0.2)–(1,1)–(-1,1)–cycle; (0,0)circle(0.55);\n(0,0)circle(0.03); at(0.1,-0.15)*q*;\n[scale=/10cm] (1,0.2)–(-1,0.2); (0.4,0)arc(0:90:0.4); ((-0.4,0)arc(180:90:0.4); ; ; (-1,0)to(1,0); (a)–(1,0.2); (b)–(-1,0.2);\n(-1,0.2)–(1,0.2)–(1,1)–(-1,1)–cycle; (0,0)circle(0.4);\n(0,0)circle(0.03); at(0.1,-0.15)*q*;\n[scale=/10cm] (1,0.2)–(-1,0.2); (0.25,0)arc(0:90:0.25); ((-0.25,0)arc(180:90:0.25); ; ; (-1,0)to(1,0); (a)–(1,0.2); (b)–(-1,0.2);\n(-1,0.2)–(1,0.2)–(1,1)–(-1,1)–cycle; (0,0)circle(0.25);\n(0,0)circle(0.03); at(0.1,-0.15)*q*;\n[scale=/10cm] (-1,0)to(1,0); (1,0.2)–(-1,0.2); (0,0)circle(0.03); at(0.1,-0.15)*q*;\n[figureXepsdel]\nChoose a local Euclidean chart near *q* in which *q* ∈ *M**χ* ⊆ *M* is locally modelled on $0\\in{\\mathbb R}^k\\subseteq{\\mathbb R}^n$, and let us choose the usual radius (i.e. distance) functions in this chart. Note that we thus have a completely explicit picture of how these tubes intersect near *q*; in particular, their union with smoothed boundary is well behaved. We regard the union of tubes (with smoothed boundary) $\\rho\\_\\chi^{-1}([0,\\epsilon\\_\\chi]){\\mathbin{\\tilde\\cup}}B\\_\\delta(q)$ as a single object associated to the stratum *χ*, albeit depending on two parameters *ε**χ* and *δ*. This single object satisfies the conclusion of Lemma [tubetransverse] by Whitney (b), which is all that is used in the inductive proof of Corollary [neighborhoodcorners].\nMicrolocal Morse categories\n===========================\nStrata poset categories and refinement functors\n-----------------------------------------------\nLet ${\\mathcal S}$ be a stratification. We fix the following notation for the Yoneda embedding:\n \n$$\\begin{aligned}\n{\\mathcal S}&\\to {\\operatorname{Fun}}({\\mathcal S}^{\\mathrm{op}},{\\mathrm{Set}}),\\\\\n\\alpha&\\mapsto{\\operatorname{Hom}}(\\cdot,\\alpha)=:1\\_{{\\operatorname{star}}(\\alpha)}.\\end{aligned}$$ \nNote that\n \n$${\\operatorname{Hom}}(1\\_{{\\operatorname{star}}(\\alpha)},1\\_{{\\operatorname{star}}(\\beta)}) = 1\\_{{\\operatorname{star}}(\\beta)}(\\alpha) = {\\operatorname{Hom}}(\\alpha, \\beta) = \\begin{cases} \\{1\\} & {\\operatorname{star}}(\\alpha) \\subseteq {\\operatorname{star}}(\\beta) \\\\ \\emptyset & \\textrm{otherwise.} \\end{cases}$$ \nFor any ${\\mathcal S}$-constructible open set *U*, we introduce the functor $1\\_U\\in{\\operatorname{Fun}}({\\mathcal S}^{\\mathrm{op}},{\\mathrm{Set}})$ defined by the analogous formula\n \n$$\\label{indicatorUdef}\n{\\operatorname{Hom}}(1\\_{{\\operatorname{star}}(\\alpha)}, 1\\_{U}) = 1\\_{U}(\\alpha) := \\begin{cases} \\{1\\} & {\\operatorname{star}}(\\alpha) \\subseteq U \\\\ \\emptyset & \\textrm{otherwise.} \\end{cases}$$ \n(The action of 1*U* on morphism sets is in fact uniquely determined by the above, since when *α* → *β*, the set HomSet(1*U*(*β*), 1*U*(*α*)) always consists of one element.) Note that star(*α*) ⊆ *U* iff *α* ⊆ *U*. More generally, for *U* ⊆ *V* there is a unique natural transformation\n \n1*U* → 1*V*. \nIt sends 1 ∈ 1*U*(*α*) to 1 ∈ 1*V*(*α*) for any star(*α*) ⊆ *U* ⊆ *V*.\nNow let ${\\mathcal S}'$ be a stratification refining ${\\mathcal S}$. There is a natural map $r:{\\mathcal S}'\\to{\\mathcal S}$, sending a stratum in ${\\mathcal S}'$ to the unique stratum in ${\\mathcal S}$ containing it. We write\n \n$$r^\\*:{\\operatorname{Fun}}({\\mathcal S}^{\\mathrm{op}},{\\mathrm{Set}})\\to{\\operatorname{Fun}}({\\mathcal S}^{\\prime{\\mathrm{op}}},{\\mathrm{Set}})$$ \nfor the pullback of functors along this map *r*. For $\\tau' \\in {\\mathcal S}'$ and an ${\\mathcal S}$-constructible open set *U*, we have\n \n$${\\operatorname{Hom}}(1\\_{{\\operatorname{star}}(\\tau')}, r^\\*1\\_U) = (r^\\*1\\_U)(\\tau') = 1\\_U(r(\\tau')) =\n\\begin{cases} \\{1\\} & {\\operatorname{star}}(r(\\tau')) \\subseteq U \\\\ \\emptyset & \\textrm{otherwise.} \\end{cases}$$ \nSince *U* is open and ${\\mathcal S}$-constructible, we have star(*r*(*τ*ʹ)) ⊆ *U* iff star(*τ*ʹ) ⊆ *U*, so we conclude that *r*\\*1*U* = 1*U*.\nWe now linearize. We write ${\\mathbb Z}[{\\mathcal S}]$ for the linearization of a poset ${\\mathcal S}$. We write ${\\operatorname{Mod}}{\\mathcal S}$ for the category of modules ${\\operatorname{Fun}}({\\mathcal S}^{\\mathrm{op}},{\\operatorname{Mod}}{\\mathbb Z}) = {\\operatorname{Fun}}({\\mathbb Z}[{\\mathcal S}]^{\\mathrm{op}}, {\\operatorname{Mod}}{\\mathbb Z})$, and we use $r^\\*:{\\operatorname{Mod}}{\\mathcal S}\\to{\\operatorname{Mod}}{\\mathcal S}'$ for pullback of modules as above. As with any pullback of modules, this functor has a left adjoint, typically termed extension of scalars or induction, which we write as $r\\_!:{\\operatorname{Mod}}{\\mathcal S}'\\to{\\operatorname{Mod}}{\\mathcal S}$, which fits into the commuting diagram\n \n$$\\begin{tikzcd}[column sep = large]\n{\\mathcal S}'\\ar{r}{s\\mapsto 1\\_{{\\operatorname{star}}(s)}}\\ar{d}[swap]{r}& {\\operatorname{Mod}}{\\mathcal S}'\\ar{d}{r\\_!}\\\\\n{\\mathcal S}\\ar{r}{s\\mapsto 1\\_{{\\operatorname{star}}(s)}}& {\\operatorname{Mod}}{\\mathcal S}.\n\\end{tikzcd}$$ \nRestriction of scalars *r*\\* is co-continuous, so its left adjoint *r*! extension of scalars preserves compact objects, giving a map $r\\_!:{\\operatorname{Perf}}{\\mathcal S}'\\to{\\operatorname{Perf}}{\\mathcal S}$ (which can also be viewed as the canonical extension of $r:{\\mathcal S}'\\to{\\mathcal S}$ to the idempotent-completed pre-triangulated hulls).\nA category for any\n------------------\nWe now wish to define a *microlocal Morse category* ${\\mathcal C}(\\Lambda)$ for any subanalytic (possibly) singular isotropic Λ ⊆ *S*\\**M*, together with functors ${\\mathcal C}(\\Lambda')\\to{\\mathcal C}(\\Lambda)$ for inclusions Λʹ ⊇ Λ. We define this system of categories $\\Lambda\\mapsto{\\mathcal C}(\\Lambda)$, the *microlocal Morse theatre*, by formulating axioms which characterize it uniquely. (Recall that $S^\\*M:=(T^\\*M\\setminus M)/{\\mathbb R}\\_{>0}$ denotes the co-sphere bundle of *M*, and a closed subanalytic set Λ ⊆ *S*\\**M* is called isotropic iff for some, or, equivalently, every, cover of Λ by locally closed *C*1 submanifolds, all of them are isotropic.)\nThe previous subsection defined categories ${\\operatorname{Perf}}{\\mathcal S}$ together with functors $r\\_!:{\\operatorname{Perf}}{\\mathcal S}'\\to{\\operatorname{Perf}}{\\mathcal S}$ whenever ${\\mathcal S}'$ is a refinement of ${\\mathcal S}$. For our current purpose, these categories do not have the correct significance for general stratifications ${\\mathcal S}$ (compare Remark [stratvsposet]). As such, we will consider these categories only for triangulations ${\\mathcal S}$.[4](#fn4) The microlocal Morse theatre is an extension of this functor ${\\mathcal S}\\mapsto{\\operatorname{Perf}}{\\mathcal S}$ on triangulations.\n[categorysystemextension] A *microlocal Morse pre-theatre* $\\Lambda\\mapsto{\\mathcal C}(\\Lambda)$ is a functor from the category of subanalytic singular isotropics inside *S*\\**M* to the category of dg categories over ${\\mathbb Z}$. A *normalized* microlocal Morse pre-theatre is one equipped with an isomorphism of functors $({\\mathcal S}\\mapsto{\\mathcal C}(N^\\*\\_\\infty{\\mathcal S}))=({\\mathcal S}\\mapsto{\\operatorname{Perf}}{\\mathcal S})$ on Whitney triangulations ${\\mathcal S}$.\nAny isomorphism of functors $({\\mathcal S}\\mapsto H^\\*{\\mathcal C}(N^\\*\\_\\infty{\\mathcal S}))=({\\mathcal S}\\mapsto H^\\*{\\operatorname{Perf}}{\\mathcal S})$ automatically lifts to an isomorphism $({\\mathcal S}\\mapsto{\\mathcal C}(N^\\*\\_\\infty{\\mathcal S}))=({\\mathcal S}\\mapsto{\\operatorname{Perf}}{\\mathcal S})$ by Proposition [ffhenceainf]. This will be crucial when discussing Fukaya categories (specifically Theorem [Frefinement]).\nWe will characterize the microlocal Morse theatre in terms of microlocal Morse theory.[5](#fn5) Let $f:M\\to{\\mathbb R}$ be a function and ${\\mathcal S}$ a stratification. An intersection of Γ*d**f* with $N^\\*{\\mathcal S}$ is called an ${\\mathcal S}$-critical point, which is said to be Morse if it is a transverse intersection at a smooth point of $N^\\*{\\mathcal S}$. The function *f* is said to be ${\\mathcal S}$-Morse when all its ${\\mathcal S}$-critical points are Morse. When ${\\mathcal S}$ is subanalytic, such functions are plentiful, and can be chosen analytic. (See for this assertion, which is collected there from various results in the literature.)\nMore generally, for any singular isotropic Λ ⊆ *S*\\**M*, a Λ-critical point of *f* is by definition an intersection of Γ*d**f* with the union of the zero section and ${\\mathbb R}\\_{>0}\\times\\Lambda$. That is,\n \n$${\\mathrm{crit}}\\_\\Lambda(f) = \\Gamma\\_{df}\\cap(0\\_M\\cup({\\mathbb R}\\_{>0}\\times\\Lambda))\\subseteq T^\\*M$$ \nSuch a Λ-critical point is said to be Morse if the intersection is transverse and occurs at a smooth point of $0\\_M\\cup({\\mathbb R}\\_{>0}\\times\\Lambda)$, and any *f* whose Λ-critical points are all Morse is called Λ-Morse.\n[morsecharacter] In any normalized microlocal Morse pre-theatre $\\Lambda\\mapsto{\\mathcal C}(\\Lambda)$, the *Morse characters* ${\\mathcal X}\\_{\\Lambda,p}(f,\\epsilon,{\\mathcal S})\\in{\\mathcal C}(\\Lambda)$ are defined as follows for smooth Legendrian points *p* ∈ Λ.\nLet $f:M\\to{\\mathbb R}$ be an analytic function with a Morse Λ-critical point at *p* (i.e. somewhere in ${\\mathbb R}\\_{>0}\\times\\{p\\}\\subseteq T^\\*M$) with critical value 0, no other Λ-critical points with critical values in the interval [ − *ε*, *ε*], and with relatively compact sublevel set *f*− 1( − ∞, *ε*). Let ${\\mathcal S}$ be a Whitney triangulation for which $\\Lambda\\subseteq N^\\*\\_\\infty{\\mathcal S}$ and for which both *f*− 1( − ∞, − *ε*) and *f*− 1( − ∞, *ε*) are ${\\mathcal S}$-constructible.\nThe Morse character ${\\mathcal X}\\_{\\Lambda,p}(f,\\epsilon,{\\mathcal S})$ is then defined as the image of\n \n$${\\operatorname{cone}}(1\\_{f^{-1}(-\\infty, -\\epsilon)} \\to 1\\_{f^{-1}(-\\infty, \\epsilon)}) \\in {\\operatorname{Perf}}{\\mathcal S}={\\mathcal C}(N^\\*\\_\\infty{\\mathcal S}).$$ \nunder the map ${\\mathcal C}(N^\\*\\_\\infty{\\mathcal S})\\to{\\mathcal C}(\\Lambda)$, where 1*f*− 1( − ∞, − *ε*) → 1*f*− 1( − ∞, *ε*) is (the linearization of) the unique map.\nThe Morse character ${\\mathcal X}\\_{\\Lambda,p}(f,\\epsilon,{\\mathcal S})\\in{\\mathcal C}(\\Lambda)$ depends *a priori* on the ‘casting directors’ $(f,\\epsilon,{\\mathcal S})$. Casting directors (*f*, *ε*) exist at any smooth Legendrian point *p* ∈ Λ by general position, and ${\\mathcal S}$ exists by the following argument. First, by every closed subanalytic singular isotropic Λ ⊆ *S*\\**M* is contained in $N^\\*\\_\\infty{\\mathcal S}$ for some subanalytic stratification ${\\mathcal S}$ of *M*. Next, by refining ${\\mathcal S}$ the subanalytic subsets *f*− 1( − ∞, ± *ε*) can be made constructible. Finally, ${\\mathcal S}$ can be made a subanalytic Whitney triangulation by.\n[microlocalmorsetheatre] A *microlocal Morse theatre* is a normalized microlocal Morse pre-theatre $\\Lambda\\mapsto{\\mathcal C}(\\Lambda)$ satisfying the *localization property*: for any inclusion Λ ⊆ Λʹ and any collection of Morse characters ${\\mathcal X}\\_{\\Lambda',p}(f,\\epsilon,{\\mathcal S})\\in{\\mathcal C}(\\Lambda')$ at smooth Legendrian points *p* ∈ Λʹ \\ Λ with at least one in every component of the smooth Legendrian locus of Λʹ \\ Λ, the functor ${\\mathcal C}(\\Lambda')\\to{\\mathcal C}(\\Lambda)$ is the idempotent-completed quotient by these Morse characters.\nThe definition of a microlocal Morse theatre allows one to readily compute any particular microlocal Morse category ${\\mathcal C}(\\Lambda)$: embed Λ into some $N^\\*\\_\\infty{\\mathcal S}$, cast Morse characters in ${\\mathcal C}(N^\\*\\_\\infty{\\mathcal S})={\\operatorname{Perf}}{\\mathcal S}$ for all Legendrian components of $N^\\*\\_\\infty{\\mathcal S}\\setminus\\Lambda$, and take the quotient of ${\\operatorname{Perf}}{\\mathcal S}$ by these characters and idempotent complete. It follows that:\n[microlocalmorseuniqueness] Any two microlocal Morse theatres $\\Lambda\\mapsto{\\mathcal C}(\\Lambda)$ are uniquely isomorphic.\nFor any normalized microlocal Morse pre-theatre ${\\mathcal C}$, let ${\\mathcal X}\\_{\\Lambda'\\setminus\\Lambda}\\subseteq{\\mathcal C}(\\Lambda')$ denote the collection of all Morse characters at all smooth Legendrian points of Λʹ \\ Λ. Now for any microlocal Morse theatre ${\\mathcal C}$, we have a canonical quasi-equivalence (functorial in Λ):\n \n$$\\varinjlim\\_{\\begin{smallmatrix} N^\\*{\\mathcal S}\\supseteq \\Lambda\\\\{\\mathcal S}\\textrm{ Whitney triangulation}\\end{smallmatrix}}({\\mathcal C}(N^\\*{\\mathcal S})/{\\mathcal X}\\_{N^\\*{\\mathcal S}\\setminus\\Lambda})^\\pi\\xrightarrow\\sim{\\mathcal C}(\\Lambda),$$ \nand the left hand side is independent of ${\\mathcal C}$ since ${\\mathcal C}$ is normalized.\nA *dramatic realization* is a particular construction of the microlocal Morse theatre $\\Lambda\\mapsto{\\mathcal C}(\\Lambda)$. We give two dramatic realizations, namely via sheaves and via Lagrangians in Sections [sec:sheaves] and [sec:fukaya], respectively. Both these dramatic realizations *cast* the Morse characters as certain familiar objects. They moreover show that the Morse characters in fact depend only on *p* (up to shifts) and are independent of the casting directors.\n[microlocalmorseexistence] The microlocal Morse theatre $\\Lambda\\mapsto{\\mathcal C}(\\Lambda)$ exists, and the Morse characters ${\\mathcal X}\\_{\\Lambda,p}\\in{\\mathcal C}(\\Lambda)$ are independent of the casting directors and form a local system over the smooth Legendrian locus of Λ.\nThis follows from either Theorem [sheafrealization] or Theorem [fukcasting].\n[Proof of Theorem [sheaffukayaequivalence]] Combine Theorem [sheafrealization] and Theorem [fukcasting] with Proposition [microlocalmorseuniqueness].\nIn fact, both dramatic realizations show that ${\\mathcal C}(\\Lambda)$ is invariant under contact isotopy of *S*\\**M*, something which is not apparent from the present combinatorial prescription. This is immediate on the Fukaya side, and on the sheaf side it is ‘sheaf quantization’. In fact, there are even stronger invariance statements: it is shown in that in fact ${\\mathcal C}(\\Lambda)$ is invariant under contact isotopy of *S*\\**M* \\ Λ inside *S*\\**M*; meanwhile, it is shown in that ${\\mathcal C}(\\Lambda)$ is invariant under ‘gapped’ deformations of Λ.\nThe construction of this subsection makes sense in any stable setting, e.g. over the sphere spectrum. To show existence of the microlocal Morse theatre in such a more general setting, one could set up either microlocal sheaf theory or the Fukaya category over the sphere spectrum. In principle, one could also show existence directly from the stratified Morse theory of, as it already establishes results about homotopy types of spaces (not just their cohomologies). A more interesting question is whether any symplectically invariant statement can be made beyond the stable setting.\nOpen inclusions\n---------------\nWe now discuss functoriality of the microlocal Morse theatre under open inclusions.\nGiven any analytic open inclusion of analytic manifolds $M'\\hookrightarrow M$, a microlocal Morse pre-theatre ${\\mathcal C}'$ on *M*ʹ determines a microlocal Morse pre-theatre $\\Lambda\\mapsto{\\mathcal C}'(\\Lambda\\cap S^\\*M')$ on *M*; let us call this the extension of ${\\mathcal C}'$ to *M*. For microlocal Morse pre-theatres ${\\mathcal C}'$ on *M*ʹ and ${\\mathcal C}$ on *M*, a morphism ${\\mathcal C}'\\to{\\mathcal C}$ means a morphism to ${\\mathcal C}$ from the extension of ${\\mathcal C}'$ to *M*. In other words, such a morphism consists of a coherent system of maps ${\\mathcal C}'(\\Lambda\\cap S^\\*M')\\to{\\mathcal C}(\\Lambda)$ for subanalytic singular isotropics Λ ⊆ *S*\\**M*. Equivalently, we may view such a morphism as a coherent system of maps\n \n$${\\mathcal C}'(\\Lambda')\\to{\\mathcal C}(\\Lambda)$$ \nfor subanalytic singular isotropics Λʹ ⊆ *S*\\**M*ʹ and Λ ⊆ *S*\\**M* with Λ ∩ *S*\\**M*ʹ ⊆ Λʹ. If ${\\mathcal C}$ and ${\\mathcal C}'$ are normalized, then restricting these maps to the case that Λ and Λʹ are both conormals of Whitney triangulations, we obtain a coherent collection of maps\n \n$$\\label{correctonperf}\n{\\operatorname{Perf}}{\\mathcal S}'={\\mathcal C}'(N^\\*\\_\\infty{\\mathcal S}')\\to{\\mathcal C}(N^\\*\\_\\infty{\\mathcal S})={\\operatorname{Perf}}{\\mathcal S}$$ \nfor every pair of Whitney triangulations ${\\mathcal S}$ of *M* and ${\\mathcal S}'$ of *M*ʹ such that ${\\mathcal S}'$ refines ${\\mathcal S}\\cap M'$. A normalized morphism ${\\mathcal C}'\\to{\\mathcal C}$ is one equipped with an isomorphism between and the extension to Perf of the tautological maps of posets ${\\mathcal S}'\\to{\\mathcal S}$ (sending a stratum of ${\\mathcal S}'$ to the unique stratum of ${\\mathcal S}$ containing it). A morphism of microlocal Morse theatres ${\\mathcal C}'\\to{\\mathcal C}$ is, by definition, a normalized morphism of underlying normalized microlocal Morse pre-theatres.\nWe now have the following refinement of Proposition [microlocalmorseuniqueness]:\n[morphismunique] For an analytic open inclusion of analytic manifolds $M'\\hookrightarrow M$ and microlocal Morse theatres ${\\mathcal C}'$ on *M*ʹ and ${\\mathcal C}$ on *M*, there exists a unique morphism of microlocal Morse theatres ${\\mathcal C}'\\to{\\mathcal C}$.\nIt follows from that the maps ${\\mathcal C}'(\\Lambda')\\to{\\mathcal C}(\\Lambda)$ send Morse characters to Morse characters (choose casting directors on *M* which are ‘supported inside *M*ʹ’, and appeal to the independence of Morse characters of the casting directors from Theorem [microlocalmorseexistence]). Now we have the following commutative diagram, functorial in Λ and Λʹ:\n \n$$\\begin{tikzcd}\n\\varinjlim{}({\\mathcal C}(N^\\*\\_\\infty{\\mathcal S}')/{\\mathcal X}\\_{N^\\*\\_\\infty{\\mathcal S}'\\setminus\\Lambda'})^\\pi\\ar{d}{\\sim}\\ar{r}&\\varinjlim{}({\\mathcal C}(N^\\*\\_\\infty{\\mathcal S})/{\\mathcal X}\\_{N^\\*\\_\\infty{\\mathcal S}\\setminus\\Lambda})^\\pi\\ar{d}{\\sim}\\\\\n{\\mathcal C}'(\\Lambda')\\ar{r}&{\\mathcal C}(\\Lambda)\n\\end{tikzcd}$$ \nwhere both direct limits take place over pairs of Whitney triangulations ${\\mathcal S}$ and ${\\mathcal S}'$ for which ${\\mathcal S}'$ refines $M'\\cap{\\mathcal S}$ and for which $\\Lambda\\subseteq N^\\*\\_\\infty{\\mathcal S}$ and $\\Lambda'\\subseteq N^\\*\\_\\infty{\\mathcal S}'$. Hence the maps ${\\mathcal C}'(\\Lambda')\\to{\\mathcal C}(\\Lambda)$ are determined uniquely.\n[Proof of Proposition [sfecompatibility]] Combine Propositions [sheaffunctorial] and [wfunctorial] with Proposition [morphismunique].\nSheaf categories\n================\nWe recall the general formalism of sheaves, and properties of stratifications. We then recall from the notion of microsupport, and the category ShΛ(*M*) of sheaves on *M* whose microsupport at infinity is contained in Λ. We show that the assignment Λ ↦ ShΛ(*M*) is a microlocal Morse theatre in the sense of Definition [microlocalmorsetheatre].\nCategories of sheaves and functors between them\n-----------------------------------------------\nHere we give a brief review of the general formalism of sheaves. Our presentation is somewhat modern in that we *never* discuss sheaves of abelian groups, rather we work at the dg level and with unbounded complexes from the beginning, but it is essentially the same as any standard account such as, complemented by in order to work with unbounded complexes, and in particular for the proper base change theorem in this setting. Some discussion about working in the unbounded setting can be found in.\nGiven a topological space *T*, we write Op(*T*) for the category whose objects are open sets and morphisms are inclusions. A (${\\mathbb Z}$-module valued) presheaf on *T* is by definition a functor ${\\operatorname{Op}}(T)^{\\mathrm{op}}\\to {\\operatorname{Mod}}{\\mathbb Z}$. In particular, a presheaf ${\\mathcal F}$ takes a value ${\\mathcal F}(U) \\in {\\operatorname{Mod}}{\\mathbb Z}$ on an open set *U* ⊆ *T*, termed its sections; given open sets *U* ⊆ *V* it gives a morphism ${\\mathcal F}(V) \\to {\\mathcal F}(U)$, termed the restriction, etcetera. Given any subset *X* ⊆ *T*, we write ${\\mathcal F}(X) = \\varinjlim\\_{X \\subseteq U} {\\mathcal F}(U)$; when *X* is a point, this is termed the stalk and is written ${\\mathcal F}\\_x$.\nThe category of sheaves is the full subcategory of presheaves on objects ${\\mathcal F}$ taking covers to limits:\n \n$${\\mathcal F}\\left(\\bigcup\\_{i \\in I} U\\_i\\right) \\xrightarrow{\\sim} \\lim\\_{\\emptyset\\ne J \\subseteq I} {\\mathcal F}\\left( \\bigcap\\_{j \\in J} U\\_j \\right)$$ \nThe inclusion of sheaves into presheaves has a left adjoint termed “sheafification”, giving, for any presheaf ${\\mathcal F}$, a sheaf ${\\mathcal F}^{\\mathrm{sh}}$ such that any map from ${\\mathcal F}$ to a sheaf factors uniquely through ${\\mathcal F}^{\\mathrm{sh}}$.\nWe write Sh(*T*) for the (dg) category of sheaves of (dg) ${\\mathbb Z}$-modules on *T*. It is complete and co-complete. Its homotopy category is what was classically called the unbounded derived category of sheaves on *T*.\nFor any continuous map *f* : *S* → *T*, there is an adjoint pair *f*\\* : Sh(*T*) ↔ Sh(*S*) : *f*\\*. The pushforward *f*\\* is given by the formula $(f\\_\\* {\\mathcal F})(U) = {\\mathcal F}(f^{-1}(U))$, while the pullback *f*\\* is the sheafification of the presheaf given by $(f^\\*{\\mathcal G})(V) = {\\mathcal G}(f(V))$.\nConsider *f* : *S* → pt, and the constant sheaf ${\\mathbb Z}:= f^\\* {\\mathbb Z}$. Note that in our conventions, ${\\mathbb Z}(U)$ is a chain complex computing the cohomology of *U*. This should illustrate where, in this account of sheaf theory, is hiding the usual homological algebra of resolutions: it is in the sheafification.\nBeing a left adjoint, *f*\\* is co-continuous (preserves colimits, in particular, sums). When *j* : *U* → *T* is the inclusion of an open set, *j*\\* is given by the simpler formula $(j^\\*{\\mathcal F})(V) = {\\mathcal F}(V)$, no sheafification required, and hence preserves limits as well. In particular, it must also be a right adjoint. The corresponding left adjoint *j*! is easy to describe: it is the sheafification of\n \n$$V\\mapsto\\begin{cases} {\\mathcal F}(V) & V \\subseteq U \\\\ 0 & \\textrm{otherwise.} \\end{cases}$$ \nThe sheaf $j\\_! {\\mathcal F}$ is termed the extension by zero, since its stalks in *U* are isomorphic to the corresponding stalks of ${\\mathcal F}$, and its stalks outside of *U* are zero. For a sheaf ${\\mathcal F}$ on *T*, we write ${\\mathcal F}\\_U := j\\_! j^\\* {\\mathcal F}$. By adjunction there is a canonical morphism ${\\mathcal F}\\_U \\to {\\mathcal F}$. The object ${\\mathbb Z}\\_U$ co-represents the functor of sections over *U*, i.e. ${\\operatorname{Hom}}({\\mathbb Z}\\_U,{\\mathcal F})={\\mathcal F}(U)$.\nBeing a right adjoint, *f*\\* is continuous. When *f* is proper, it is in addition co-continuous. More generally, for a morphism of locally compact spaces *f* : *S* → *T*, one defines[6](#fn6)\n \n$$\\begin{aligned}\nf\\_!: {\\operatorname{Sh}}(S) & \\to & {\\operatorname{Sh}}(T) \\\\\n{\\mathcal F}& \\mapsto & \\varinjlim\\_{U \\subset \\subset S} f\\_\\* {\\mathcal F}\\_U\\end{aligned}$$ \nHere the notation *U* ⊂ ⊂ *S* means that the closure of *U* is compact. When *S* is an open subset, this recovers the original definition. When *f* is proper, then *f*! = *f*\\*. When *f* is the map to a point, then $f\\_! f^\\* {\\mathbb Z}$ is the compactly supported cohomology.\nAs *f*! is built from colimits, left adjoints, and pushforwards from compact sets, it is co-continuous. As such it has a right adjoint, denoted *f*!. When *f* is the inclusion of an open subset, we already had the right adjoint *f*\\*, so in this case *f*\\* = *f*!.\nFor any locally closed subset *v* : *V* ⊆ *T*, we extend the notation ${\\mathcal F}\\_V := v\\_! v^\\* {\\mathcal F}$. This sheaf has the same stalks as ${\\mathcal F}$ at points in *V*, and has vanishing stalks outside.\nFor an open-closed decomposition $U \\xhookrightarrow{j} T \\xhookleftarrow{i} V$ (*j* open, *i* closed), the functors *j*\\*, *j*! and *i*\\*, *i*! are fully faithful, and there is an exact triangle\n \n$$\\label{triangle}\nj\\_! j^! \\to {\\operatorname{id}}\\to i\\_\\* i^\\* \\xrightarrow{[1]}$$ \nDenoting by Op(*M*) the poset of open sets, there are functors\n \n$$\\begin{aligned}\n\\label{generatingsheaves}\n{\\operatorname{Op}}(M)&\\xrightarrow{!}{\\operatorname{Sh}}(M)&{\\operatorname{Op}}(M)^{\\mathrm{op}}&\\xrightarrow{\\*}{\\operatorname{Sh}}(M)\\cr\nU&\\mapsto u\\_!{\\mathbb Z}&U&\\mapsto u\\_\\*{\\mathbb Z}\\end{aligned}$$ \nwhere *u* : *U* → *M* denotes the inclusion. We have the following criterion for when (pullbacks of) these functors are fully faithful:\n[lem:posetsheaf] Let Π be a poset with a map to Op(*M*), and let ${\\mathbb Z}[\\Pi]$ denote its dg linearization. The following are equivalent:\n* $H^\\*(U) \\cong {\\mathbb Z}$ for all *U* ∈ Π and $H^\\*(U) \\xrightarrow{\\sim} H^\\*(U \\setminus V)$ whenever *U* ⊈ *V*.\n* The composition ${\\mathbb Z}[\\Pi]\\to{\\mathbb Z}[{\\operatorname{Op}}(M)]\\xrightarrow{!}{\\operatorname{Sh}}(M)$ is fully faithful.\nWe have\n \n$$\\begin{aligned}\n{\\operatorname{Hom}}\\_M({\\mathbb Z}\\_U, {\\mathbb Z}\\_V) &= {\\operatorname{Hom}}\\_M(u\\_! {\\mathbb Z}, v\\_! {\\mathbb Z}) = {\\operatorname{Hom}}\\_U ({\\mathbb Z}, u^! v\\_! {\\mathbb Z}) = {\\operatorname{Hom}}\\_U({\\mathbb Z}, u^\\* v\\_! {\\mathbb Z})\\cr\n&=H^\\*(U, {\\mathbb Z}\\_{V \\cap U})={\\operatorname{cone}}(H^\\*(U) \\to H^\\*(U \\setminus V))[-1],\\end{aligned}$$ \nwhere we have used the exact triangle. The second condition asks that this be ${\\mathbb Z}$ when *U* ⊆ *V* and zero otherwise, which is exactly what is asserted in the first condition.\n[lem:sheafrefine] Let Π be a poset with a map to Op(*M*) satisfying the equivalent conditions in Lemma [lem:posetsheaf], and suppose that *W* ⊆ *M* is an open set such that $H^\\*(U) \\xrightarrow{\\sim} H^\\*(U \\setminus W)$ is an isomorphism whenever *U* ⊈ *W*. Then the pullback of the module ${\\operatorname{Hom}}(-, {\\mathbb Z}\\_W)$ along ${\\mathbb Z}[\\Pi] \\xrightarrow{!} {\\operatorname{Sh}}(M)$ is the indicator functor\n \n$$1\\_{W}: U \\mapsto \\begin{cases} {\\mathbb Z}& U \\subseteq W, \\\\ 0 & \\textrm{otherwise.} \\end{cases}$$ \nThis is true by the same calculation as above.\nConstructible sheaves\n---------------------\nLet *T* be a topological space and ${\\mathcal S}: T = \\coprod T\\_\\alpha$ a stratification. Write $T\\_{\\mathcal S}$ for the topological space with underlying set *T* and base given by the stars of strata in ${\\mathcal S}$ (note that the intersection of any two stars is expressible as a union of stars). Note the continuous map $T \\to T\\_{\\mathcal S}$.\n[weakremark] Let *π* : *T* → *T*ʹ be any continuous bijection. For any open set *U* of *T*ʹ, and any sheaf ${\\mathcal F}$ on *T*, one has by definition $\\pi\\_\\*{\\mathcal F}(U) = {\\mathcal F}(U)$. It follows that $\\pi^\\*{\\mathbb Z}\\_U = {\\mathbb Z}\\_U$, as this sheaf co-represents the functor of sections over *U*.\n[posetmeaning] Pulling back sheaves under ${\\mathcal S}\\xrightarrow{\\operatorname{star}}{\\operatorname{Op}}(T\\_{\\mathcal S})$ defines an equivalence\n \n$$\\begin{aligned}\n{\\operatorname{Sh}}(T\\_{\\mathcal S})&\\xrightarrow{\\sim}{\\operatorname{Fun}}({\\mathcal S}^{\\mathrm{op}},{\\operatorname{Mod}}{\\mathbb Z})={\\operatorname{Mod}}{\\mathcal S}\\\\\n{\\mathcal F}&\\mapsto(s\\mapsto{\\mathcal F}({\\operatorname{star}}(s)))={\\operatorname{Hom}}\\_{T\\_{\\mathcal S}}({\\mathbb Z}\\_{{\\operatorname{star}}(-)},{\\mathcal F})\\end{aligned}$$ \nwhich sends ${\\mathbb Z}\\_{{\\operatorname{star}}(s)}$ to ${\\operatorname{Hom}}\\_{{\\mathcal S}}(\\cdot,s)=1\\_{{\\operatorname{star}}(s)}$.\nThe functor in question is simply restricting a sheaf on $T\\_{\\mathcal S}$ to the base consisting of stars of strata. This functor is fully faithful because a map of sheaves is determined by its restriction to a base for the topology. It is essentially surjective because there are no nontrivial covers of stars of strata by stars of strata. The behavior on objects is as asserted because ${\\mathbb Z}\\_{{\\operatorname{star}}(s)}$ and *s* are the co-representatives of the functors of sections over *s* and the value of the module at *s*, respectively.\n[posetmeaningpushforward] If ${\\mathcal S}'$ refines ${\\mathcal S}$, then the following diagram commutes:\n \n$$\\begin{tikzcd}\n{\\operatorname{Sh}}(T\\_{{\\mathcal S}'})\\ar{r}&{\\operatorname{Mod}}{\\mathcal S}'\\\\\n{\\operatorname{Sh}}(T\\_{\\mathcal S})\\ar{r}\\ar{u}{\\pi^\\*}&{\\operatorname{Mod}}{\\mathcal S}\\ar{u}[swap]{r^\\*}\n\\end{tikzcd}$$ \nwhere *π*\\* denotes pullback of sheaves under the continuous map $\\pi:T\\_{{\\mathcal S}'}\\to T\\_{\\mathcal S}$ and $r^\\*:{\\operatorname{Mod}}{\\mathcal S}\\to{\\operatorname{Mod}}{\\mathcal S}'$ denotes the pullback along the natural map $r:{\\mathcal S}'\\to{\\mathcal S}$.\nBy Remark [weakremark] and the characterization of the horizontal functors as ${\\mathbb Z}\\_U\\mapsto 1\\_U$.\nA sheaf is said to be constant when it is isomorphic to the star pullback of a sheaf on a point, and locally constant when this is true after restriction to an open cover. For a stratification ${\\mathcal S}$ of *M*, we say a sheaf is ${\\mathcal S}$-constructible[7](#fn7) if it is locally constant when star restricted to each stratum of ${\\mathcal S}$. We write ${\\operatorname{Sh}}\\_{\\mathcal S}(T)$ for the full subcategory of Sh(*T*) on the ${\\mathcal S}$-constructible sheaves.\nNote that the image of the pullback map ${\\operatorname{Sh}}(T\\_{{\\mathcal S}}) \\to {\\operatorname{Sh}}(T)$ is contained in ${\\operatorname{Sh}}\\_{\\mathcal S}(T)$.\n[sheafposetequivalence] For a triangulation ${\\mathcal S}$, the map ${\\operatorname{Sh}}(T\\_{{\\mathcal S}}) \\to {\\operatorname{Sh}}\\_{{\\mathcal S}}(T)$ is an equivalence.\nTo show full faithfulness, in view of the equivalence of Lemma [posetmeaning] it is enough to check that ${\\operatorname{Hom}}\\_{T\\_{\\mathcal S}}({\\mathbb Z}\\_{{\\operatorname{star}}(s)},{\\mathbb Z}\\_{{\\operatorname{star}}(t)})={\\operatorname{Hom}}\\_T({\\mathbb Z}\\_{{\\operatorname{star}}(s)},{\\mathbb Z}\\_{{\\operatorname{star}}(t)})$. The former is the indicator of star(*s*) ⊆ star(*t*) again by Lemma [posetmeaning]. To show that ${\\operatorname{Hom}}\\_T({\\mathbb Z}\\_{{\\operatorname{star}}(s)},{\\mathbb Z}\\_{{\\operatorname{star}}(t)})$ is as well, by Lemma [lem:posetsheaf] it is enough to show that *H*\\*(star(*s*)) → *H*\\*(star(*s*) \\ star(*t*)) is an isomorphism for star(*s*) ⊈ star(*t*). If star(*s*) ⊈ star(*t*), then star(*s*) \\ star(*t*) is the join of something with *s*, and is hence contractible.\nRegarding essential surjectivity, note that the exact triangle of serves to decompose any sheaf into an iterated extension of (extensions by zero of) sheaves on the strata; hence any constructible sheaf into (extensions by zero of) locally constant sheaves on the strata. Since the strata are all contractible, these sheaves are in fact constant. This shows that the ${\\mathbb Z}\\_s$ generate. To conclude that the ${\\mathbb Z}\\_{{\\operatorname{star}}(s)}$ generate, use the exact triangle ${\\mathbb Z}\\_{{\\operatorname{star}}(s) \\setminus s} \\to {\\mathbb Z}\\_{{\\operatorname{star}}(s)} \\to {\\mathbb Z}\\_s \\xrightarrow{[1]}$ and induction on dimension of strata (noting that the first term is in the span of ${\\mathbb Z}\\_t$ for dim(*t*) < dim(*s*)).\nMicrosupport\n------------\nThe notion of microsupport is developed in.[8](#fn8) We recall some basic facts here.\nFor what follows, let *M* denote an analytic manifold. Given a sheaf ${\\mathcal F}$ and a smooth function $\\phi: M \\to {\\mathbb R}$, consider a point *m* in a level set *ϕ*− 1(*t*). We say that *m* ∈ *M* is a cohomological ${\\mathcal F}$-critical point of *ϕ* if, for inclusion of the superlevelset $i: \\phi^{-1}({\\mathbb R}\\_{\\ge t}) \\hookrightarrow M$, one has $(i^! {\\mathcal F})\\_m \\ne 0$.\nThe *microsupport* ${\\operatorname{ss}}({\\mathcal F}) \\subseteq T^\\*M$ is by definition the closure of the locus of differentials of functions at their cohomological ${\\mathcal F}$-critical points. It is conical.\nIf ${\\mathcal F}$ is locally constant, then a cohomological ${\\mathcal F}$-critical point can only occur where the function in question has zero derivative. Thus the microsupport of a locally constant sheaf is contained in the zero section (and is equal to it where the sheaf is not locally zero). If *U* ⊆ *M* is an open set and *m* is a point in the smooth locus of ∂*U*, then over *m*, the locus ${\\operatorname{ss}}({\\mathbb Z}\\_U)={\\operatorname{ss}}(u\\_!{\\mathbb Z})$ is the half-line of outward conormals to ∂*U*. The locus ${\\operatorname{ss}}(u\\_\\*{\\mathbb Z})$ is the inward conormal.\nFor a subset *X* ⊆ *T*\\**M*, we write Sh*X*(*M*) for the full subcategory of Sh(*M*) spanned by objects with microsupport contained in *X*. Similarly, for *X* ⊆ *S*\\**M*, we write Sh*X*(*M*) for the full subcategory of Sh(*M*) with microsupport at infinity contained in *X*. Evidently if 0*M* ⊆ *X*, then Sh*X*(*M*) = Sh∂∞*X*(*M*).\nThe following result is a strengthening of :\n[better841] For a Whitney stratification ${\\mathcal S}$ of a *C*1 manifold *M*, we have ${\\operatorname{Sh}}\\_{\\mathcal S}(M)={\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)$ (i.e. having microsupport contained in $N^\\*{\\mathcal S}$ is equivalent to being ${\\mathcal S}$-constructible).\nWe first show the inclusion ${\\operatorname{Sh}}\\_{\\mathcal S}(M)\\subseteq{\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)$. Let us first show that ${\\operatorname{ss}}({\\mathbb Z}\\_X)\\subseteq N^\\*{\\mathcal S}$. When *X* is relatively compact, express *X* as the ascending union of locally closed submanifolds-with-corners $X\\_i = X\\setminus(\\partial X)^{\\underline\\epsilon\\_i}$ where $(\\partial X)^{\\underline\\epsilon\\_i}$ denotes the outward cornering of $\\partial X=\\overline X\\setminus X$ in the sense of and $\\underline \\epsilon\\_i \\to 0$. Corollary [neighborhoodcorners] implies $X\\setminus(\\partial X)^{\\underline\\epsilon\\_i}$ are indeed locally closed submanifolds-with-corners, and Corollary [cornerconormalconvergence] implies that their conormals limit inside $N^\\*{\\mathcal S}$ as $\\underline\\epsilon\\_i\\to 0$, and hence that ${\\operatorname{ss}}({\\mathbb Z}\\_X)={\\operatorname{ss}}(\\varinjlim{\\mathbb Z}\\_{X\\_i})\\subseteq N^\\*{\\mathcal S}$. The case of general *X* may be reduced to the relatively compact case by refining the stratification as in Remark [addsubmanifoldremark] (the assertion ${\\operatorname{ss}}({\\mathbb Z}\\_X)\\subseteq N^\\*{\\mathcal S}$ is local). The same argument shows that for any locally constant sheaf on *X*, its lower shriek pushforward to *M* has microsupport contained in $N^\\*{\\mathcal S}$. Since any ${\\mathcal S}$-constructible sheaf is (locally) a finite iterated extension of such sheaves, we conclude that ${\\operatorname{Sh}}\\_{\\mathcal S}(M)\\subseteq{\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)$.\nWe now show that the inclusion ${\\operatorname{Sh}}\\_{\\mathcal S}(M)\\subseteq{\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)$ implies the reverse inclusion ${\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)\\subseteq{\\operatorname{Sh}}\\_{\\mathcal S}(M)$ by a straightforward dévissage argument. Suppose ${\\operatorname{ss}}({\\mathcal F})\\subseteq N^\\*{\\mathcal S}$, and let us show that ${\\mathcal F}$ is ${\\mathcal S}$-constructible. Let *X* be a maximal stratum over which ${\\mathcal F}$ is nonzero, and let *U* ⊆ *M* be an open set containing *X* so that *X* ⊆ *U* is the support of *F*∣*U* (hence, in particular, *X* ⊆ *U* is closed). Since ${\\operatorname{ss}}({\\mathcal F})\\subseteq N^\\*{\\mathcal S}$, there exists a (derived) local system on *X* whose lower shriek pushforward ${\\mathcal F}\\_0$ (which is ${\\mathcal S}$-constructible) agrees with ${\\mathcal F}$ over *U*. Since ${\\mathcal F}\\_0\\in{\\operatorname{Sh}}\\_{\\mathcal S}(M)\\subseteq{\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)$, it suffices to show that the cone of ${\\mathcal F}\\_0\\to{\\mathcal F}$ is ${\\mathcal S}$-constructible. We have thus reduced to a sheaf with smaller support. Iterating, we eventually reduce to the case of ${\\mathcal F}=0$ which is obviously ${\\mathcal S}$-constructible.\nMicrostalks\n-----------\nRecall that if ${\\mathcal F}$ is a sheaf and *ϕ* is a smooth function with *ϕ*(*x*) = *t* and *d**ϕ**x* = *ξ*, and we denote the inclusion $i: \\phi^{-1}({\\mathbb R}\\_{\\ge t}) \\to M$, then if (*i*!*F*)*x* ≠ 0, we have $(x, \\xi) \\in {\\operatorname{ss}}({\\mathcal F})$ (though not conversely). Given this, one wants to assign the complex $(i^! {\\mathcal F})\\_x$ itself as an invariant of ${\\mathcal F}$ at (*x*, *ξ*). This is not generally possible, but it can be done when *ξ* is a point in the smooth Lagrangian locus of ${\\operatorname{ss}}({\\mathcal F})$. Namely, at any smooth Lagrangian point (*x*, *ξ*) ∈ *X* ⊆ *T*\\**M*, there is a ‘microstalk’ functor\n \n*μ*(*x*, *ξ*) : Sh*X*(*M*) → Sh(pt). \nIt is given by a shift of ${\\mathcal F}\\mapsto (i^! {\\mathcal F})\\_x$ for any *ϕ* with *d**x**ϕ* = *ξ* with the graph of *d**ϕ* transverse to *X*. The shift can be fixed using the index of the three transverse Lagrangians $({\\operatorname{ss}}({\\mathcal F}), T^\\*\\_x M, \\Gamma\\_{d \\phi})$. When *ξ* = 0, the microstalk functor is simply the stalk functor.\n[microstalkcocontinuous] The microstalk functors are co-continuous.\nEvery stalk functor (*i*\\* for *i* the inclusion of a point) is a left adjoint, hence is co-continuous. To show co-continuity of the microstalk at a point (*x*, *ξ*) ∈ Λ with *ξ* ≠ 0, argue as follows. By applying a contact transformation, we may reduce to the case that Λ is (locally near (*x*, *ξ*)) the conormal of a smooth hypersurface *N* ⊆ *M*. Let *B* ⊆ *M* be an open ball with smooth boundary whose inward conormal at *x* ∈ ∂*B* is *ξ*. Moreover, choose *B* so that *N*\\*∂*B* and Λ = *N*\\**N* intersect cleanly at (*x*, *ξ*) (that is, ∂*B* and *N* are tangent at *x*, differing by a non-degenerate quadratic form). Define *B*− and *B*+ from *B* by pushing ∂*B* inward/outward near *x*. Now the cone of the map Γ*c*(*B*−, − ) → Γ*c*(*B*+, − ) is (up to a shift) the microstalk functor *μ*(*x*, *ξ*). The compactly supported sections functor Γ*c* is co-continuous, since it is the composition of the restriction and lower shriek pushforward functors, both of which are co-continuous.\n[removemicrosupport] Let *X* ⊆ *T*\\**M* be closed and conical, and let Λ ⊆ *T*\\**M* \\ *X* be closed, conical, and stratified by isotropic submanifolds. Then Sh*X*(*M*) ⊆ Sh*X* ∪ Λ(*M*) is the kernel of all microstalks at Lagrangian points of Λ.\nIf ${\\operatorname{ss}}({\\mathcal F})\\subseteq X$, then the microstalks of ${\\mathcal F}$ at Lagrangian points of Λ vanish by definition of microsupport. To prove the converse, suppose that ${\\operatorname{ss}}({\\mathcal F})\\subseteq X\\cup\\Lambda$ and that the microstalks of ${\\mathcal F}$ vanish at all Lagrangian points of Λ, and let us show that ${\\operatorname{ss}}({\\mathcal F})\\subseteq X$. By the fundamental result that the microsupport is co-isotropic, it is enough to show that $p\\notin{\\operatorname{ss}}({\\mathcal F})$ for every Lagrangian point *p* ∈ Λ. It is not quite immediate from the definitions that vanishing of the microstalk implies there is no microsupport, since the microsupport is defined in terms of arbitrary test functions, whereas microstalks are defined in terms of microlocally transverse test functions. To see it is true, and that moreover the microstalk is locally constant along Λ, one can apply a contact transformation so that Λ becomes locally the conormal to a smooth hypersurface; for details see.\nIt will be central to our discussion to find co-representatives of the microstalk functors. Here is a first step:\n[ or ][thm:sheafwex] Let *X* ⊆ *T*\\**M* be a closed conical subset, let $\\phi: M \\to {\\mathbb R}$ be a proper function, and assume that over *ϕ*− 1([*a*, *b*)), one has Γ*d**ϕ* ∩ *X* = (*x*, *ξ*), where (*x*, *ξ*) is a smooth Lagrangian point of *X*.\nLet *A* : *ϕ*− 1(( − ∞, *a*)) → *M*, *A*ʹ : *ϕ*− 1((*a*, ∞)) → *M*, *B* : *ϕ*− 1(( − ∞, *b*)) → *M*, and *B*ʹ : *ϕ*− 1((*b*, ∞)) → *M* be the inclusions. Then (up to a shift), the following functors Sh*X*(*M*) → Sh(pt) are isomorphic:\n* The microstalk functor *μ*(*x*, *ξ*).\n* ${\\operatorname{Hom}}({\\operatorname{cone}}(A\\_! {\\mathbb Z}\\to B\\_! {\\mathbb Z}), -)$.\n* ${\\operatorname{Hom}}({\\operatorname{cone}}(A'\\_\\* {\\mathbb Z}\\to B'\\_\\* {\\mathbb Z}), -)$.\nHere the maps are the canonical ones coming from restriction of sections.\nWe do not say that ${\\operatorname{cone}}(A\\_! {\\mathbb Z}\\to B\\_! {\\mathbb Z})$ co-represents the microstalk because it is not an element of Sh*X*(*M*). As observed in, such co-representatives do exist, for categorical reasons, as we now explain. First, we need to know that the categories in question are well generated in the sense of Neeman.\n[wellgeneration] The category Sh*X*(*M*) is well generated.\nThe category of all sheaves Sh(*M*) is the derived category of a Grothendieck abelian category, hence is well generated. A sheaf ${\\mathcal F}$ having singular support inside *X* is equivalent to the restriction maps ${\\mathcal F}(U\\_\\alpha)\\to{\\mathcal F}(V\\_\\alpha)$ being isomorphisms for some list of pairs (*U**α*, *V**α*)*α* depending on *X*. This condition is equivalent to ${\\mathcal F}$ being right-orthogonal to the cone of the map ${\\mathbb Z}\\_{V\\_\\alpha}\\to{\\mathbb Z}\\_{U\\_\\alpha}$. Now the right-orthogonal complement of a set of objects in a well generated category is well generated.\nNow note that the microsupport of a sum or product is contained in the closure of the union of the microsupports. Thus if *X* is closed, then the subcategory Sh*X*(*M*) ⊆ Sh(*M*) is closed under sums and products. In particular, Sh*X*(*M*) is complete and co-complete, and the inclusion Sh*X*(*M*) → Sh(*M*) is continuous and co-continuous. More generally, if *X* ⊆ *X*ʹ are closed, then the inclusion *ι* : Sh*X*(*M*) → Sh*X*ʹ(*M*) is continuous and co-continuous. It follows that:\n[adjoints] For closed *X* ⊆ *X*ʹ ⊆ *T*\\**M*, the inclusion *ι* : Sh*X*(*M*) → Sh*X*ʹ(*M*) has both adjoints: (*ι*\\*, *ι*, *ι*!).\nSince Sh*X*(*M*) is well generated and *ι* is co-continuous, it has a right adjoint *ι*! by Brown representability for well generated categories.\nFor the left adjoint *ι*\\*, it would be sufficient to know Brown representability for the opposite of Sh*X*(*M*). However according to Neeman, it is an open problem to establish Brown representability for the opposites of well generated categories. Instead, we may argue as follows. The categories in question are presentable (co-complete and accessible ; this is a version of well generation) and so a functor has a left adjoint iff it is continuous and accessible (preserves *κ*-filtered colimits ) by Lurie.\nAnother proof of the existence of the left adjoint *ι*\\* has been given by Efimov.\nFor example, if *V* ⊆ *M* is a closed subset, then taking *X* = *T*\\**M*∣*V* and *X*ʹ = *T*\\**M* recovers the adjoint triple for the pushforward along *V* → *M*, because Sh*T*\\**M*∣*V*(*M*) = Sh(*V*).\nUsing the left adjoint and Theorem [thm:sheafwex], we can obtain a co-representative for the microstalk as follows. Take any $X' \\supseteq {\\operatorname{ss}}({\\operatorname{cone}}(A\\_! {\\mathbb Z}\\to B\\_! {\\mathbb Z}))$, e.g. *X*ʹ = *T*\\**M*. Then $\\iota^\\*{\\operatorname{cone}}(A\\_! {\\mathbb Z}\\to B\\_! {\\mathbb Z}) \\in {\\operatorname{Sh}}\\_X(M)$ co-represents the microstalk.\nWe do not generally have a good understanding of (*ι*\\*, *ι*, *ι*!), but when *X*ʹ \\ *X* is isotropic we have the following (special cases of which have appeared in ):\n[thm:sheafstopremoval] Let *X* ⊆ *T*\\**M* be closed and conical, and let Λ ⊆ *T*\\**M* \\ *X* be closed, conical, and stratified by isotropic submanifolds. The left adjoint *ι*\\* to the inclusion *ι* : Sh*X*(*M*) → Sh*X* ∪ Λ(*M*) realizes the quotient\n \n$$\\label{sheafstopremovalqe}\n{\\operatorname{Sh}}\\_{X \\cup \\Lambda}(M)/{\\mathcal D}\\xrightarrow\\sim {\\operatorname{Sh}}\\_X(M),$$ \nwhere ${\\mathcal D}$ denotes co-representing objects for the microstalks at Lagrangian points of Λ.\nAccording to Proposition [removemicrosupport], the full subcategory Sh*X*(*M*) ⊆ Sh*X* ∪ Λ(*M*) is precisely the right-orthogonal to ${\\mathcal D}$. The left adjoint *ι*\\* to the inclusion *ι* is thus termed the quotient by ${\\mathcal D}$.\n[noneedwellgeneration] For our purposes in this paper, we do not need Lemma [adjoints] and Theorem [thm:sheafstopremoval] in their general formulations given above, rather only in the special case of subanalytic singular isotropic singular supports. In this setting, we give an elementary derivation (i.e. without appealing to general Brown representability type statements) in the next subsection.\nCompact objects\n---------------\nHere we elaborate upon some assertions of.\nWe write Sh*X*(*M*)*c* for the compact objects in the category Sh*X*(*M*). Be warned:\n[] When *M* is connected and non-compact, Sh(*M*)*c* = 0.\nThere are not many more compact objects in the compact case. However, for sheaves with prescribed isotropic microsupport, the situation is different:\n[isotropicimpliescompact] For Λ ⊆ *T*\\**M* a conic subset Whitney stratifiable by isotropics, the category ShΛ(*M*) is compactly generated by the (co-representatives of the) microstalk functors at the smooth Lagrangian points of Λ.\nIt was shown immediately after the proof of Lemma [adjoints] that the microstalk functors at smooth Lagrangian points of Λ are co-represented by objects of ShΛ(*M*). Since the microstalk functors are co-continuous (Lemma [microstalkcocontinuous]), these co-representatives are compact. Any sheaf right-orthogonal to these co-representatives has by definition vanishing microstalks at all smooth Lagrangian points of Λ, hence has microsupport contained in the complement of the smooth Lagrangian locus (see the proof of Proposition [removemicrosupport]). This complement, being stratified by subcritical isotropics, has no co-isotropic subset; hence by the involutivity of microsupports, the microsupport is in fact the empty set and the sheaf vanishes.\nAs promised in Remark [noneedwellgeneration], we now give arguments avoiding the use of representability theorems in non-compactly-generated categories. This comes at the cost of assuming subanalyticity in order to ensure the existence of triangulations, but for the main results we will anyway need this hypothesis.\n[constructibleimpliescompact] For ${\\mathcal S}$ a triangulation, the category ${\\operatorname{Sh}}\\_{\\mathcal S}(M)$ is compactly generated, and the objects of ${\\operatorname{Sh}}\\_{\\mathcal S}(M)^c$ are the sheaves with perfect stalks and compact support.\nUnder the identification (Lemma [sheafposetequivalence]) ${\\operatorname{Sh}}\\_{\\mathcal S}(M) = {\\operatorname{Mod}}{\\mathcal S}$, the ${\\mathbb Z}\\_{{\\operatorname{star}}(s)}$ go to compact generators. The dévissage in the proof of the same Lemma shows that ${\\mathbb Z}\\_s$ also generate, and can be expressed using finitely many ${\\mathbb Z}\\_{{\\operatorname{star}}(s)}$, hence are compact. The ${\\mathbb Z}\\_s$ evidently generate the sheaves with perfect stalks and compact support.\n[noncompacttriangulation] Note that while a non-compact manifold does not admit a finite triangulation, it can sometimes be a relatively compact constructible subset of a larger manifold.\nRecall that ${\\operatorname{Sh}}\\_{\\mathcal S}(M)={\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)$ for any Whitney stratification by Proposition [better841].\n[compactmicrostalks] For any subanalytic Whitney triangulation ${\\mathcal S}$, the category ${\\operatorname{Sh}}\\_{\\mathcal S}(M)$ is compactly generated by co-representatives of the microstalks at smooth points of $N^\\*{\\mathcal S}$.\nConsider the microstalk at some smooth point $(x, \\xi)\\in N^\\*{\\mathcal S}$. It is possible to choose real analytic *ϕ* as in Theorem [thm:sheafwex], see or. We keep the notation of Theorem [thm:sheafwex]. Refine ${\\mathcal S}$ to a subanalytic Whitney triangulation ${\\mathcal S}'$ for which $A\\_!{\\mathbb Z}$ and $B\\_!{\\mathbb Z}$ are constructible.\nBy Lemma [constructibleimpliescompact], ${\\operatorname{cone}}(A\\_! {\\mathbb Z}\\to B\\_! {\\mathbb Z})$ is a compact object in ${\\operatorname{Sh}}\\_{{\\mathcal S}'}(M)$. Lemmas [posetmeaning] and [sheafposetequivalence] give ${\\operatorname{Sh}}\\_{\\mathcal S}(M)={\\operatorname{Mod}}{\\mathcal S}$, and Lemma [posetmeaningpushforward] states that the inclusion $\\iota:{\\operatorname{Sh}}\\_{\\mathcal S}(M)\\to{\\operatorname{Sh}}\\_{{\\mathcal S}'}(M)$ corresponds to the map $r^\\*:{\\operatorname{Mod}}{\\mathcal S}\\to{\\operatorname{Mod}}{\\mathcal S}'$ from Section [posetcategories]. It was observed in Section [posetcategories] that *r*\\* has a left adjoint $r\\_!:{\\operatorname{Mod}}{\\mathcal S}'\\to{\\operatorname{Mod}}{\\mathcal S}$, thus giving us a left adjoint $\\iota^\\*:{\\operatorname{Sh}}\\_{{\\mathcal S}'}(M)\\to{\\operatorname{Sh}}\\_{\\mathcal S}(M)$. These left adjoints *r*!/*ι*\\* preserve compact objects since *r*\\*/*ι* are co-continuous. Thus the object $\\iota^\\*{\\operatorname{cone}}(A\\_! {\\mathbb Z}\\to B\\_! {\\mathbb Z}) \\in {\\operatorname{Sh}}\\_{\\mathcal S}(M)$, which co-represents the desired microstalk, is compact.\nCo-representatives of the microstalks at all smooth points of $N^\\*{\\mathcal S}$ generate ${\\operatorname{Sh}}\\_{\\mathcal S}(M)$ by Proposition [removemicrosupport].\n[compactstalks] A similar argument shows that the stalk at any point of *M* (not necessarily a smooth point of $N^\\*{\\mathcal S}$) is co-representable by a compact object of ${\\operatorname{Sh}}\\_{\\mathcal S}(M)$. Indeed, note that for any *x* ∈ *M*, the functor of taking stalks at *x*, which is by definition ${\\mathcal F}\\_x := \\varinjlim {\\mathcal F}(B\\_\\epsilon(x))$, is in fact computed by some fixed ${\\mathcal F}\\_x = {\\mathcal F}(B\\_{\\epsilon\\_x}(x))$. Indeed, further shrinking of the ball will be non-characteristic with respect to $N^\\*{\\mathcal S}$, as follows from Whitney’s condition (b) (or alternatively from microlocal Bertini–Sard ). We may now argue as above, choosing any analytic function with sublevelset *B**ε**x*(*x*).\n[microstalksgenerate] For any closed conical subanalytic isotropic Λ ⊆ *T*\\**M*, the category ShΛ(*M*) is compactly generated by co-representatives of the microstalks at smooth points of Λ.\nFix a subanalytic Whitney triangulation ${\\mathcal S}$ for which $\\Lambda\\subseteq N^\\*{\\mathcal S}$. Denote by ${\\mathcal D}\\subseteq{\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)^c$ the co-representatives of the microstalks at smooth points of $N^\\*{\\mathcal S}\\setminus\\Lambda$. By Proposition [removemicrosupport], $\\iota:{\\operatorname{Sh}}\\_\\Lambda(M)\\subseteq{\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)$ is precisely the inclusion of the right-orthogonal to ${\\mathcal D}$. Since the objects of ${\\mathcal D}$ are compact by Proposition [compactmicrostalks], Lemma [compactinquotient] applies to show that ${\\operatorname{Sh}}\\_\\Lambda(M)={\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)/{\\mathcal D}$ is compactly generated by ${\\operatorname{Sh}}\\_\\Lambda(M)^c=({\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)^c/{\\mathcal D})^\\pi$ and that the resulting functor $\\iota^\\*:{\\operatorname{Sh}}\\_{N^\\*{\\mathcal S}}(M)\\to{\\operatorname{Sh}}\\_\\Lambda(M)$ is left adjoint to *ι*.\n[compactsheafstopremoval] Let *X* ⊆ *T*\\**M* and Λ ⊆ *T*\\**M* \\ *X* be closed conical subanalytic isotropics. The inclusion *ι* : Sh*X*(*M*) → Sh*X* ∪ Λ(*M*) has a left adjoint *ι*\\* : Sh*X* ∪ Λ(*M*) → Sh*X*(*M*) whose restriction to compact objects defines an equivalence\n \n$$({\\operatorname{Sh}}\\_{X \\cup \\Lambda}(M)^c/{\\mathcal D})^\\pi\\xrightarrow\\sim {\\operatorname{Sh}}\\_X(M)^c,$$ \nwhere ${\\mathcal D}$ denotes co-representing objects for the microstalks at Lagrangian points of Λ.\nBy Corollary [microstalksgenerate], Sh*X* ∪ Λ(*M*) is compactly generated by the microstalks at smooth points of *X* ∪ Λ. Now argue as in the proof of Corollary [microstalksgenerate].\nThe following result was shown in using arborealization; here is a direct argument.\n[propersheaves] The Yoneda embedding induces an equivalence between the full subcategory of ShΛ(*M*) of objects with perfect stalks and the category PropShΛ(*M*)*c*.\nFrom the argument in Proposition [compactmicrostalks], we see that the microstalks are calculated by comparing sections over precompact sets; it follows that a sheaf microsupported in Λ (thus constructible) with perfect stalks has perfect microstalks. The microstalk functors split-generate ShΛ(*M*)*c* by Corollary [microstalksgenerate], so we see that a sheaf with perfect stalks defines a proper module over ShΛ(*M*)*c*.\nTo see the converse, recall from Remark [compactstalks] that the stalk functors can be expressed in terms of sections over open sets constructible with respect to some ${\\mathcal S}$ satisfying $N^\\*{\\mathcal S}\\supseteq\\Lambda$. The left adjoint to ${\\operatorname{Sh}}\\_\\Lambda(M)\\hookrightarrow{\\operatorname{Sh}}\\_{\\mathcal S}(M)$ preserves compact objects as observed previously, hence proper over ShΛ(*M*)*c* implies perfect stalks.\nFor compact *M*, we establish smoothness and/or properness for some of these categories.\n[sheafcompactsmoothproper] If *M* is compact and ${\\mathcal S}$ is a triangulation, then ${\\operatorname{Sh}}\\_{\\mathcal S}(M)^c$ is smooth and proper.\nThe ${\\mathbb Z}\\_{{\\operatorname{star}}(s)}$ give a finite generating exceptional collection which is proper, and this implies smoothness by Lemma [exceptionalpropersmooth].\nMore generally,\nIf *M* is compact and Λ is closed conical subanalytic singular isotropic, then ShΛ(*M*)*c* is smooth, and hence PropShΛ(*M*)*c* ⊆ PerfShΛ(*M*)*c* and PropSh"},"data_id":{"kind":"string","value":"arxiv_0183125"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":457,"numItemsPerPage":100,"numTotalItems":46503,"offset":45700,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjIwOTE1NCwianRpIjoiZDE3OGE5MGMtNTQyMC00NjUyLWJjYmEtOWQ4NjI3YTA1ODA3Iiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODYyMTI3NTQsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.7K6LsHrEC-d_I9DSfwL8U5D6E_lzUvyKrhBPZA_84YRq_NjX_GKzg3rLShq0OzmqhYmlrwol3KErSfyMyKrqDQ","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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A problem dependent analysis of SOCP algorithms in noisy compressed sensing
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**Abstract**
Under-determined systems of linear equations with sparse solutions have been the subject of an extensive research in last several years above all due to re... | arxiv_0182810 |
use it in our maps. So the final maps reduced with the 2 × 2 aperture results in 9 × 9 pixel maps, covering a slightly smaller region than the original maps. The coordinates assigned to each pixel of the resulting map are those of the center of the 2 × 2 aperture, and the pixel fluxes reported in Figures [fig:SH-maps1... | arxiv_0182815 |
The role of Alfvén wave heating in solar prominences
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### Received XXX; accepted XXX
INTRODUCTION
============
The energy balance in solar prominences, and the understanding of the processes involved with heating and cooling of the plasma, are difficult problems that ... | arxiv_0182818 |
x***γ* and **y***γ* are the Jacobi coordinates for three particles with unequal masses *m*1, *m*2 and *m*3 having positions **r**1, **r**2 and **r**3 defined as
$$\begin{aligned}
\mathbf{x}\_{i}& =\sqrt{\frac{m\_{j}m\_{k}}{m\_{j}+m\_{k}}}(\mathbf{r}\_{j}-\mathbf{r}\_{k}),\text{ \ } \notag \\
\mathbf{y}\_{i}& =\sqrt{... | arxiv_0182821 |
A Unified 2D/3D Large Scale Software Environment for Nonlinear Inverse Problems
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namedefver@subfigure.sty9999/12/31 namedefopt@subfigure.sty
Abstract
========
Large scale parameter estimation problems are among some of the most computationa... | arxiv_0182824 |
the weights of the respective dual faces, and compute the graph Forman-Ricci curvature and associated flow developed for Complex Networks in,,. Recall that Forman’s Ricci curvature for an edge is given by the following formula:
$$\label{FormanRicciEdge}
{ \rm Ric}\_{F,r}(e) = w\_e \left( \frac{w\_{v\_1}}{w\_e} + \f... | arxiv_0182825 |
Generation of smoothly-varying infill configurations from a continuous menu of cell patterns and the asymptotic analysis of its mechanical behaviour
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We here introduce a no... | arxiv_0182826 |
Distributed Quantum Computation Based-on Small Quantum Registers
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We describe and analyze an efficient register-based hybrid quantum computation scheme. Our scheme is based on probabilistic, heralded optical connection among local five-qubit quantum regist... | arxiv_0182830 |
Self-force correction to the deflection angle in black-hole scattering: a scalar charge toy model
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Using self-force methods, we consider the hyperbolic-type scattering of a pointlike particle carrying a scalar charge *Q* of... | arxiv_0182831 |
Learning Controllers from Data via Approximate Nonlinearity Cancellation
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We introduce a method to deal with the data-driven control design of nonlinear systems. We derive conditions to design controllers via (approximate) nonlinearity cancellation... | arxiv_0182832 |
Toric Eigenvalue Methods forSolving Sparse Polynomial Systems
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We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety *X*. Our starting point is a homogeneous ideal *I* in the... | arxiv_0182839 |
* by requiring that
− 2⟨*ω*\*, *v* ∧ *w*⟩ = *ω*(*v*, *w*)
for every *v*, *w* ∈ R4. In our case, we have
*ω*\* = *e*1 ∧ *e*3 + *e*2 ∧ *e*4,
and we notice that ⟨*ω*\*, *ω*\*⟩ = 1. The group Sp(4, R) acts naturally by isometries on (*V*, ⟨ ⋅ , ⋅ ⟩), and preserves *ω*\*. Therefore, it acts isometrically on (... | arxiv_0182844 |
III]*λ*5007/H*β* versus [OI]*λ*6300 diagram (from now on [OI]-BPT), which is insensitive to metallicity (e.g., ), and the use of X-ray observations (e.g., ; ; ; ; ; ), radio observations (e.g., ; ; ), optical and infrared variability (e.g., ; ; ; ), or coronal lines (e.g., ; ; ; ; ) to identify AGN. X-ray observations ... | arxiv_0182860 |
Drawing outerplanar graphs using thirteen edge lengths
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We show that every outerplanar graph *G* can be linearly embedded in the plane such that the number of distinct distances between pairs of adjacent vertices is at most thirteen and there is no intersection betw... | arxiv_0182865 |
Fiberwise convexity of Hill’s lunar problem
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In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transfo... | arxiv_0182868 |
Thermal Correlation Functions of KdV Charges in 2D CFT
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Introduction and summary of results
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Two dimensional conformal field theories (CFTs) have an infinite-dimensional symmetry algebra – the Virasoro algebra of local conformal tr... | arxiv_0182871 |
at (1,0) × ;
&
[scale=.8,baseline=(0,-.5)] (0,0).. controls +(0,-.5) and +(0,-.5).. +(.5,0); (1,0) – +(0,-.9);
([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt); ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt); at (.5,0) × ; at (1,0) × ;
+
[scale=.8,baseline=(0,-.5)] (.5,0).. controls +(0,... | arxiv_0182877 |
representative example, our N2LO predictions for selected Nd scattering observables at $E\_{\rm N} = 135$ MeV, along with the experimental data and calculations based on the CD Bonn NN potential with and without the TM99 3NF model.
[fig:135MeV]
As an important internal consistency check of the calculations, we have ve... | arxiv_0182891 |
Atmospheric Circulation of Hot Jupiters: Dayside-Nightside Temperature Differences
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The full-phase infrared light curves of low-eccentricity hot Jupiters show a trend of increasing dayside-to-nightside brightness temperature difference wi... | arxiv_0182892 |
Evolution of the habitable zone of low-mass stars
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Detailed stellar models and analytical relationships for different masses and chemical compositions [1](#fn1)
=============================================================================================================
... | arxiv_0182897 |
Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers
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We study multiplicative quiver varieties associated to specific extensions of cyclic quivers with *m* ≥ 2 vertices. Their global Poisson s... | arxiv_0182902 |
Balanced Meta-Softmaxfor Long-Tailed Visual Recognition
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Deep classifiers have achieved great success in visual recognition. However, real-world data is long-tailed by nature, leading to the mismatch between training and testing distributions. In this paper, we sho... | arxiv_0182905 |
be observed by looking for a peak in the *W**Z* invariant mass distribution, as well as final states containing multiple *W*, *Z*, and Higgs bosons. The models also generically predict large corrections (either enhancement or suppression) to the $h \to \ga\ga$ width.
Introduction
============
Supersymmetry (SUSY) rema... | arxiv_0182908 |
[arXiv:astro-ph/9511029 [astro-ph]].
V. F. Mukhanov and P. J. Steinhardt, “Density perturbations in multifield inflationary models,” Phys. Lett. B **422** (1998), 52-60 [arXiv:astro-ph/9710038 [astro-ph]].
X. Chen, “Primordial Non-Gaussianities from Inflation Models,” Adv. Astron. **2010** (2010), 638979 [arXiv:100... | arxiv_0182910 |
Ramond-Ramond fields and twisted differential K-theory
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We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computationa... | arxiv_0182912 |
+ D\_2 \Psi^{(2)} + D\_3 \Psi^{(3)}
+ \begin{pmatrix} {\frac{1}{{{\sqrt{2}}\,\kappa}}}e^{2\sigma} \frac{W\_2}{J W\_1} \\ 0 \end{pmatrix} \:.\end{aligned}$$
The auxiliary factors are
$$\begin{aligned}
C\_1 &= \int {\sqrt{2}}J \left( \frac{a\_1 W\_1 - a\_2 W\_2}{X} - \frac{WZ}{X^2} \right) dy \:,\\
C\_2 &= \int ... | arxiv_0182914 |
, y, \, z) \, \, \, \, \, \, \longrightarrow \, \quad \, \, \,
\Bigl( {{1} \over {x}}, \, \, {{1} \over {y}}, \, \,\, x^2 \, y^2 \, z\Bigr).
\end{aligned}$$
This involutive birational transformation transforms the rational function ([simplest]) into:
$$\begin{aligned}
\label{involutrationalmf}
\hspace{-0.96in... | arxiv_0182916 |
The Jones-Krushkal polynomial and minimal diagrams of surface links
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We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has min... | arxiv_0182917 |
The topology of any complex system is key to understanding its structure and function. Fundamentally, algebraic topology guarantees that any system represented by a network can be understood through its closed paths. The length of each path provides a notion of scale, which is vitally important in characterizing domina... | arxiv_0182918 |
≪ 1 .
From this and the fact that *a* is confined in the region
0 < *a* < 1/6
the separation of the instanton and cosmological sizes follows. Eq. ([DCondOnLambda]) imposes an upper bound on the absolute value of *λ*, which can always be satisfied provided that *ξ* ≠ 0.
It is worth to note that when the vacuum g... | arxiv_0182921 |
Clustering with Distributed Data
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We consider *K*-means clustering in networked environments (e.g., internet of things (IoT) and sensor networks) where data is inherently distributed across nodes and processing power at each node may be limited. We consider a clustering algorithm referre... | arxiv_0182926 |
4(1)
& Ψ3, –(0) &0 &6.4090875602(1) &0 &0 & − 3.2045437801(1) &0
& Ψ1, –(0) & − 4.5277999888(1) & − 3.3462569001(1) & − 4.5277999888(1) &2.2638999944(1) &1.6731284500(1) &2.2638999944(1)
& Ψ2, –(0) &4.5277999888(1) & − 3.0632694008(1) &4.5277999888(1) & − 2.2638999944(1) &1.5316347004(1) & − 2.2638999944(1)... | arxiv_0182934 |
Intelligent Reflecting Surface Enhanced Wireless Network via Joint Active and Passive Beamforming
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-0.1in
Intelligent reflecting surface (IRS) is a revolutionary and transformative technology for achieving spectrum and ener... | arxiv_0182936 |
On the stability of POD Basis Interpolation via Grassmann Manifolds for Parametric Model Order Reduction in Hyperelasticity
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This work considers the stability of Proper Orthogonal Decomposition (PO... | arxiv_0182940 |
On Preference Learning Based on Sequential Bayesian Optimization with Pairwise Comparison
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User preference learning is generally a hard problem. Individual preferences are typically unknown even to users themselves, while the space... | arxiv_0182955 |
Unconventional Bose-Einstein Condensations Beyond the “No-node” Theorem
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Feynman’s “no-node” theorem states that the conventional many-body ground-state wavefunctions of bosons in the coordinate representation is positive-definite. This implies that... | arxiv_0182956 |
Fractal properties of particle paths due to generalised uncertainty relations
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We determine the Hausdorff dimension of a particle path, $D\_{\rm H}$, in the recently proposed ‘smeared space’ model of quantum geometry. The model introduces addi... | arxiv_0182961 |
Horizon-Independent Optimal Prediction with Log-Loss in Exponential Families
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We study online learning under logarithmic loss with regular parametric models. showed that a Bayesian prediction strategy with Jeffreys prior and sequential norma... | arxiv_0182962 |
Hitchin Equation, Singularity, and *N* = 2 Superconformal Field Theories
========================================================================
Introduction
============
*D* = 6 is the maximal dimension in which we can formulate a superconformal field theory (SCFT). Six dimensional (0, 2) superconformal field theory ... | arxiv_0182968 |
in the universe. Relationalism thereby avoids the commitment to differences in the ontology that do not amount to a physical difference.
If there is no absolute space into which the matter points are embedded, there is no geometry fixed from the outset in the following sense: the spatial relations in any given initial... | arxiv_0182970 |
Orbital effects of a strong in-plane magnetic field on a gate-defined quantum dot
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We theoretically investigate the orbital effects of an in-plane magnetic field on the spectrum of a quantum dot embedded in a two-dimensional electron gas (... | arxiv_0182972 |
terms: Rʹ is independent, and for all *v* ∈ *V* there is a (*B*, *v*)-linkage where *B* is a base of N(*v*). It is easy to see that finding a branching packing {B*i*}*i* ∈ *S* for R is equivalent to finding a feasible extension Rʹ of R such that SRʹ(*v*) is a base of N(*v*) for all *v* ∈ *V*. Here *A*(B*i*) will consi... | arxiv_0182973 |
ries created by interactions between owners and renters in a sharing platform. To ground our modeling, we discuss the model in terms of ride-sharing and use data from Didi Chuxing to guide our modeling choices.
Model preliminaries
-------------------
We consider a marketplace consisting of a *sharing platform* and two... | arxiv_0182974 |
Pfaffian Schur processes and last passage percolation in a half-quadrant
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We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that... | arxiv_0182975 |
Relative Calabi–Yau structures andice quivers with potential
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### November 11, 2023
In 2015, Van den Bergh showed that complete 3-Calabi–Yau algebras over an algebraically closed field of characteristic 0 are equivalent to Ginzburg dg algebras associated with ... | arxiv_0182978 |
Stellar orbit evolution in close circumstellar disc encounters
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[firstpage]
The formation and early evolution of circumstellar discs often occurs within dense, newborn stellar clusters. For the first time, we apply the moving-mesh code AREPO, to circumstella... | arxiv_0182980 |
S. 2010, J. Korean Astron. Soc. 43, 191 Conroy, C. & Wechsler, R. H. 2009,, 696, 620 Doroshkevich, G. 1970, Astrophysika, 6, 320 Dubinski, J., Kim, J., Park, C., & Humble, R. 2004, New Astron, 9, 111 Eisenstein, D. J., et al. 2001,, 122, 2267 Eisenstein, D. J., Blanton, M., Zehavi, I., Bahcall, N., Brinkmann, J., Love... | arxiv_0182982 |
some consequences on the possible fundamental groups of the fillings. The driving condition is the following observation: let *p* : *Ŷ* → *Y* be a covering map between compact and connected 3-manifolds, and let $i:Y\hookrightarrow X$ be the inclusion of the boundary *Y* = ∂*X*. Then, by covering theory:
$$\begin{a... | arxiv_0182984 |
Optimal Adaptive Random Multiaccess in Energy Harvesting Wireless Sensor Networks
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Wireless sensors can integrate rechargeable batteries and energy-harvesting (EH) devices to enable long-term, autonomous operation, thus requiring intellige... | arxiv_0182985 |
On axiomatic aspects of *N* = 2 vertex superalgebras with odd formal variables, and deformations of *N* = 1 vertex superalgebras
================================================================================================================================
### October 30, 2007
The notion of ``*N* = 2 vertex superalgeb... | arxiv_0182986 |
, which are revealing in the sense that the best–reaction function of the worker and his optimal control can be computed straightforwardly;
2. prove that the restriction to revealing contracts is without loss of generality;
3. solve the supervisor’s problem, which boils down to a standard optimal control problem.
Consi... | arxiv_0182988 |
M* properties is no less than 1 − *δ*. From Lemma. [lemma:U-estVar], we choose the size for each set as $272/{\epsilon^2} \max\_{i} \mathcal{A}^{(2)}({\mathcal{O}\_i},{\bm{X}\_\text{t}})$. Consequently, the total number of copies consumed is $N\_\text{sample} = {544}/{\epsilon^2} \ln(2M/\delta)\max\_{i} \mathcal{A}^{(2... | arxiv_0182992 |
Highest weight categories arising from Khovanov’s diagram algebra II: Koszulity
===============================================================================
[1](#fn1) [2](#fn2) [3](#fn3)
This is the second of a series of four articles studying various generalisations of Khovanov’s diagram algebra. In this article we... | arxiv_0182993 |
2}\right]\notag\\
&\equiv&\frac{\hat{T}\_{ji}^{\pm,s}}{\mp 4s\pi Q\_{ji}^2\sqrt{1+4M^2/Q\_{ji}^2}}\.\label{Eq:tree}\end{aligned}$$
Also note that the momenta *K̃*(*i**j* ± )*s* + *Q**k**i* with *k* ≠ *i*, *j* instead, are not on-shell. Now, since *K̃*(*i**j* ± ) and *K̃*(*i**j* ± ) + *Q**j**i* are both on-shell, this... | arxiv_0182995 |
the C*q*, *t*-linear operators *σ*, $\varpi\in\operatorname{\operatorname{End}}\mathbb{C}\_{q,t}[X^{\pm1}]$ given by the formulas (*σ**f*)(*X*) = *f*(*X*− 1) and $(\varpi f)(X)=f(qX)$, and we define operators
$$\widehat{T}=t^{\frac{1}{2}}\sigma+\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{X^2-1}, \quad \widehat{X}=X, \q... | arxiv_0182997 |
Detection of Two-Way Outliers in Multivariate Data and Application to Cheating Detection in Educational Tests
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The paper proposes a new latent variable model for the simultaneous (two-way) detection of outlying ... | arxiv_0182998 |
*κ*) and of *f**c*. As noted in, a flat prior in the stellar structure parameters *M* and *R* does *not* correspond to a flat prior in the equation of state parameters, such as *P* and *ε*. If there is no other information available and the quantities of interest are *M* and *R*, then a flat prior in those variables m... | arxiv_0183005 |
Kernel vs. Kernel: Exploring How the Data Structure Affects Neural Collapse
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Recently, a vast amount of literature has focused on the “Neural Collapse” (NC) phenomenon, which emerges when training neural network (NN) classifiers beyond the zero ... | arxiv_0183006 |
The Gradient Convergence Bound of Federated Multi-Agent Reinforcement Learning with Efficient Communication
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The paper considers independent reinforcement learning (IRL) for multi-agent collaborative decision-maki... | arxiv_0183016 |
of observing an anomalous series is in the lower tail.
The literature discussing the detection of anomalous time series has involved introducing methods to measure deviations of latent shapes or features extracted from time series. That is, given a set of time series, the anomalous series are those having the most rel... | arxiv_0183018 |
*next, *σ*) *// Schedule *σ* for next update*
---
### Numerical Experiments
We first present a simple numerical example demonstrating the good energy behavior of our asynchronous integrator. The AVI was used to integrate in time over a 2-D rectangular cavity with perfectly electrically conducting (PEC) boundaries, so t... | arxiv_0183020 |
The Radial Dependence of Temperature and Iron Abundance
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Galaxy Clusters from z=0.14 to z=0.89
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### Received March 18, 2008, Accepted June 6, 2009
Introduction
============
Many dynamic processes occur in the high temperature, ... | arxiv_0183028 |
**Giant planet and brown dwarf formation**
==========================================
Understanding the dominant brown dwarf and giant planet formation processes, and finding out whether these processes rely on completely different mechanisms or share common channels represents one of the major challenges of astronomy ... | arxiv_0183032 |
The BluediskSurvey: molecular gas distribution and scaling relations in the context of galaxy evolution
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### Accepted 2016 August 18. Received 2016 August 18; in original form 2016 June 10
[firstpage]
One of the key... | arxiv_0183036 |
applied under the associated quasistatic assumptions, is ubiquitous in computational modeling of electrical and magnetic stimulation. The purpose of QSA is to reduce the computational complexity of calculating the electric fields (E-fields), potentials, and currents in the target tissue, e.g., brain, spinal cord, or p... | arxiv_0183039 |
Massive Spin-2 Scattering Amplitudes in Extra-Dimensional Theories
==================================================================
In this paper we describe in detail the computation of the scattering amplitudes of massive spin-2 Kaluza-Klein excitations in a gravitational theory with a single compact extra dimensio... | arxiv_0183040 |
LIGO Laboratory and Advanced LIGO are funded by the United States NSF as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600... | arxiv_0183041 |
Robust Relativistic Bit Commitment
==================================
Relativistic cryptography exploits the fact that no information can travel faster than the speed of light in order to obtain security guarantees that cannot be achieved from the laws of quantum mechanics alone. Recently, Lunghi *et al* [*Phys. Rev. L... | arxiv_0183042 |
, *x*2 ∈ C and assume there exists *ρ* > 0 such that
$$\label{A operator commutation condition}
\abs{\frac{1-x\_iy\_k}{1-qx\_iy\_k}\frac{q(1-x\_j/y\_k)}{1-qx\_j/y\_k}}\leq \rho <1,$$
for all *i* ≠ *j* and *k* ∈ N. Then the double-row operators from commute:
*A*(*x*1)*A*(*x*2) = *A*(*x*2)*A*(*x*1).
Let *μ*, *... | arxiv_0183043 |
and they are identified by the color bar. The (i, j) element of the matrix denotes the weight of the connection from jth to ith node.](upr10ANDmodel.jpg "fig:") [1]
In Tab. [tab1.3], we report the dGHD for the topological comparison of all the possible networks pair. The MDN sets results are confirmed by the topologic... | arxiv_0183047 |
in sections [secdiscussion] and [sectionconclusion], respectively.
Observational Properties of V2491 Cyg and RS Oph
================================================
[v2491cygrsoph]
Figure [v2491cygrsoph] shows the *V*-band and supersoft X-ray light curves of the classical nova V2491 Cyg (black symbols) as well as thos... | arxiv_0183048 |
Fourier transforms from strongly complementary observables
==========================================================
Ongoing work in quantum information emphasizes the need for a structural understanding of quantum speedups: in this work, we focus on the quantum Fourier transform and the structures in quantum theory t... | arxiv_0183049 |
A Wide-field Survey of Two Z ∼ 0.5 Galaxy Clusters: Identifying the Physical Processes Responsible for the Observed Transformation of Spirals into S0s
======================================================================================================================================================
We present new res... | arxiv_0183050 |
Resource theory of coherence based on positive-operator-valued measures
=======================================================================
Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoheren... | arxiv_0183051 |
:dash:04:e]. The orange part of *δ* has length *L*4 − ℓ, the blue part of *δ* has length $\frac{L\_2 + L\_3 - \ell}{2}$, and the green part of *δ* has length 2∣*τ*∣. Thus:
$$\ell' = 2|\tau| + L\_4 - \ell + \frac{L\_2 + L\_3 - \ell}{2}
=
L\_2 + L\_4 - \ell + \left[ 2|\tau| - \frac{L\_2 + \ell - L\_3}{2} \right]\_{+... | arxiv_0183057 |
The structure of allelic diversity in the presence of purifying selection
=========================================================================
In the absence of selection, the structure of equilibrium allelic diversity is described by the elegant sampling formula of Ewens. This formula has helped shape our expecta... | arxiv_0183058 |
thus intended to model reconnection. Emission of hard X-rays or microwaves from loop footpoints provide evidence of non-thermal electrons energizing the chromosphere. There are, however, flares in which substantial evaporation occurs in the absence of footpoint emission. In such cases the downward transport of reconne... | arxiv_0183059 |
-photon amplitude (where $a(\bm r,t)$ is the annihilation operator of a cavity-photon), $V\_{\bm k}=e^2/(2\epsilon|\bm k|)$ is the two-dimensional Coulomb interaction (*ε* is the dielectric constant), and *g* is a dipole coupling between carriers (electrons and holes) and photons. The coupling of the system to the bath... | arxiv_0183060 |
\_k(r \mathbf n (\varphi),\mu) =
\frac{Ab\_1b\_2}{2\pi\lambda^{2k}} \int\_{0}^\infty\int\_0^{2\pi}
\frac{\rho e^{ -\rho (\beta(\psi) - i r \cos (\psi-\varphi)) }} {(1+
\omega^2\rho^2)^k} \,d\rho d \psi.$$
Here one can evaluate the integral over *ρ*. For *k* = 1, 2, 3, we obtain
$$\begin{gathered}
\Theta\_1(r\mathb... | arxiv_0183061 |
A Search for Additional Planets in the NASA *EPOXI* Observations of the Exoplanet System GJ 436
===============================================================================================
We present time series photometry of the M dwarf transiting exoplanet system GJ 436 obtained with the EPOCh (Extrasolar Planet O... | arxiv_0183067 |
Giant Planets
=============
### To be published in *Treatise on Geophysics*, G. Shubert, T. Spohn Eds
The Giant Planets
Tristan Guillot
Observatoire de la Côte d’Azur
Daniel Gautier
Observatoire de Paris
We review the interior structure and evolution of Jupiter, Saturn, Uranus and Neptune, and extrasolar giant ... | arxiv_0183074 |
Distributed Constrained Recursive Nonlinear Least-Squares Estimation: Algorithms and Asymptotics
================================================================================================
This paper focuses on recursive nonlinear least squares parameter estimation in multi-agent networks, where the individual age... | arxiv_0183078 |
\\
&\{H\_{A}{\left|{\varphi}\right\rangle}, {\langle{H\_B}\rangle\_{{\left|{\varphi}\right\rangle}}}{\left|{\varphi}\right\rangle},
X\_{\frac{N}{2}-r-1} {\left|{\varphi}\right\rangle},\dots,X\_{\frac{N}{2}}
{\left|{\varphi}\right\rangle} \},
\end{split}$$
{∣*x̃**k*⟩}*k* equals
{*H**A*∣*φ*⟩, ∣*φ*⟩, ∑*k* = 2*r* ... | arxiv_0183079 |
The Algebraic Intersection Type Unification Problem
===================================================
The algebraic intersection type unification problem is an important component in proof search related to several natural decision problems in intersection type systems. It is unknown and remains open whether the alge... | arxiv_0183084 |
**Partial Recovery in the Graph Alignment Problem**
===================================================
In this paper, we consider the graph alignment problem, which is the problem of recovering, given two graphs, a one-to-one mapping between nodes that maximizes edge overlap. This problem can be viewed as a noisy vers... | arxiv_0183085 |
**Double-bootstrap methods that use a single double-bootstrap simulation**
==========================================================================
We show that, when the double bootstrap is used to improve performance of bootstrap methods for bias correction, techniques based on using a single double-bootstrap sampl... | arxiv_0183089 |
\_{\rm{MT}} < \widehat D\_1 = 38.7305$ m (see Table [TableIM1ball]). This confirms the findings in Sections [TrendsMinimumMaximum] and [TrendsSystemLevelOptimization]. Furthermore, it is interesting to note that, by adopting a realistic load model, the ASE monotonically increases with both $\lambda\_{\rm{MT}}$ and $\la... | arxiv_0183091 |
,0)(1,0)8.5 (0,0)(0,1)3 (.5,.5)(1,0)2 (0,.5)(1,0).5 (0,3)(1,0).5 (.5,.5)(0,1)2.5
(2.5,0)(0,1)3 (2.5,1.5)(1,0).5 (2.5,3)(1,0).5 (3,.5)(0,1)2.5 (3,.5)(1,0)2 (4,0)(0,1).5
(5,0)(0,1)2 (5,2)(1,0).5 (5.5,.5)(0,1)1.5 (5.5,.5)(1,0).5 (6,0)(0,1)3 (6,1.5)(1,0).5 (6,3)(1,0).5 (6.5,.5)(0,1)2.5 (6.5,.5)(1,0)2 (7.5,0)(0,1).5 (8.5,0)... | arxiv_0183094 |
The Mass Dependence of H Emission and Stellar Spindown for Fully Convective M Dwarfs
====================================================================================
=0
Fully convective M dwarfs typically remain rapidly rotating and magnetically active for billions of years, followed by an abrupt and mass-dependent... | arxiv_0183102 |
Filters in the Partition Lattice
================================
Given a filter Δ in the poset of compositions of *n*, we form the filter Π\*Δ in the partition lattice. We determine all the reduced homology groups of the order complex of Π\*Δ as ${{\mathfrak S}}\_{n-1}$-modules in terms of the reduced homology groups ... | arxiv_0183104 |
MIL setting by proposing a maximum likelihood estimation (MLE) framework, from which we derive a two-stage deep-learning-based algorithm. The algorithm identifies informative instances and assigns them local malignancy scores that are incorporated into a global malignancy prediction. We derive a lower bound of the MLE... | arxiv_0183108 |
opadhyay and D. Mukhopadhyay, J. Phys. B: At. Mol. Opt. Phys. **40**, 1787 (2007).
T.-C. Jagau and J. Gauss, J. Chem. Phys. **137**, 044116 (2012).
P. K. Samanta, D. Mukherjee, M. Hanauer, and A. Köhn, J. Chem. Phys. **140**, 134108 (2014).
D. Datta and M. Nooijen, J. Chem. Phys. **137**, 204107 (2012).
M. Nooijen, O. ... | arxiv_0183109 |
+ 3 &*P**k* + 1 = *H*(*N**k* + 1) & for *b**n* < *b* < *b**n* + 3:&
& &1em find **first** LV such that &
& &2em *H*(*P**k* + 1∣∣*L* ⊕ *P**k* + 1) = *V* &
& &**if found in block** *b̂*:&
& &1em**unless** ∃*b* ∈ (*b**n*, *b̂*), *L*′ ∈ *B**b* : &
& &2em*H*(*L*′ ⊕ *P**k* + 1) = *P**k*& Post
& &**else**& R... | arxiv_0183110 |
A Population of Radio-Loud Narrow Line Seyfert1 Galaxies with Blazar-like Properties?
=====================================================================================
Blazars with strong emission lines were found to be associated mostly with broad-line type 1 Active Galactic Nuclei (AGN). Hitherto, evidence for bl... | arxiv_0183115 |
The Secretary Recommendation Problem
====================================
In this paper we revisit the basic variant of the classical secretary problem. We propose a new approach in which we separate between an agent that evaluates the secretary performance and one that has to make the hiring decision. The evaluating a... | arxiv_0183117 |
High magnetic field theory for the local density of states in graphene with smooth arbitrary potential landscapes
=================================================================================================================
We study theoretically the energy and spatially resolved local density of states (LDoS) in g... | arxiv_0183118 |
. This is independent of the diagnostics used.
Finally, we discussed scenarios for the unusual explosion site of ASASSN-14jb. We conclude that, although the probablity of ejection from the disk is low from theory and the specific mechanism to initiate the star formation at this heights is uncertain, both scenarios req... | arxiv_0183121 |
Corona Decompositions for ParabolicUniformly Rectifiable Sets
=============================================================
[1](#fn1)
We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together with our previous work, this allows us to c... | arxiv_0183122 |
Perturbative reduction of derivative order in EFT
=================================================
Higher derivative corrections are ubiquitous in effective field theories, which seemingly introduces new degrees of freedom at successive order. This is actually an artefact of the implicit local derivative expansion def... | arxiv_0183124 |
Microlocal Morse theory of wrapped Fukaya categories
====================================================
### August 27, 2023
The Nadler–Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between correspo... | arxiv_0183125 |
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