| \chapter{Worked calibration architectures and failure certificates} |
| \label{ch:examples} |
|
|
| This chapter follows the exact examples distributed with the release. Every |
| reported certificate value is taken from the executed rational checks in |
| \path{results/robust_results.json} or from an accompanying planner |
| certificate. The derivations explain why those values occur, so that the |
| examples can also be checked without executing the software. The final |
| conditioning example is an analytic construction, explicitly separate from |
| the computational test counts. |
|
|
| Vertices and channel indices in the JSON files start at zero. A channel |
| index is distinct from the coordinate it measures: channel zero may measure |
| vertex one, for example. We keep this indexing when describing particular |
| failure sets. The unknown $Q$ is the inverse-gauge parameter obtained from |
| the aligned discrepancy $DB_0-C=CQ$. All the examples concern that linear |
| contract. The theorem connecting a sufficiently small $Q$ to a strict |
| causal alternative is separate from the finite decoding tests. |
|
|
| \section{A complete four-vertex graph: six parameters and three noiseless channels} |
|
|
| The first graph has vertices $0,1,2,3$ and every edge $i\to j$ with $i<j$. |
| There are six edges. A parent $j$ surrounds a later vertex $i$ because every |
| child of $i$ is also a child of $j$. The ambiguity supports are consequently |
| \[ |
| I_0=\{1,2,3\},\qquad I_1=\{2,3\},\qquad |
| I_2=\{3\},\qquad I_3=\varnothing. |
| \] |
| There are no installed trusted rows in this example. The residual spaces |
| are therefore the full coordinate spaces on these supports, and their |
| dimensions are $(3,2,1,0)$. A general parameter matrix is |
| \[ |
| Q=\begin{pmatrix} |
| 0&0&0&0\\ |
| a&0&0&0\\ |
| b&d&0&0\\ |
| c&e&f&0 |
| \end{pmatrix}. |
| \] |
| The six symbols are independent algebraic parameters. Nevertheless the |
| noiseless dense-channel minimum is three, because a common three-row |
| measurement matrix can identify each of its labeled columns. Six unknown |
| scalar entries do not require six independently designed channels when the |
| same channel supplies the full labeled operator row. |
|
|
| For one known erasure and one arbitrary corrupted channel, the relevant |
| fault burden is $1+2=3$. The dense frontier therefore requires |
| $3+1+2=6$ channels. The test constructs the positive moment rows |
| \begin{equation}\label{examples:moment} |
| C_6=\begin{pmatrix} |
| 1&1&1&1\\ |
| 1&2&4&8\\ |
| 1&3&9&27\\ |
| 1&4&16&64\\ |
| 1&5&25&125\\ |
| 1&6&36&216 |
| \end{pmatrix}. |
| \end{equation} |
| The first column of $C_6$ does not affect these residual spaces, since the |
| root coordinate is zero in every residual vector. It remains part of the |
| known physical readout map and is included in the certificate. |
|
|
| A nonzero vector in $E_0$ has the form $(0,a,b,c)$. Its channel value at |
| parameter $t$ is $at+bt^2+ct^3=t(a+bt+ct^2)$. At the six distinct positive |
| parameters, it can vanish at no more than two positions. Hence its measured |
| support has at least four entries. A vector in $E_1$ gives |
| $t^2(d+et)$ and has at most one zero, while a nonzero vector in $E_2$ gives |
| $ft^3$ and has none. The three distances are exactly $4,5,6$. |
|
|
| More generally, every surviving set of $h$ channels has restricted rank |
| $\min(h,d_j)$ on column $j$. The complete severity profiles are |
| \[ |
| (D_{0,1},D_{0,2},D_{0,3})=(4,5,6),\qquad |
| (D_{1,1},D_{1,2})=(5,6),\qquad D_{2,1}=6. |
| \] |
| The fourth column has an empty profile because it has no ambiguity. |
| Erasing three channels leaves every column identifiable. Erasing four |
| leaves one possible ambiguity dimension; erasing five leaves three in |
| total; erasing all six restores all six. The exact total dimension as a |
| function of the number $e$ of erased channels is thus |
| \[ |
| \begin{array}{c|rrrrrrr} |
| e&0&1&2&3&4&5&6\\\hline |
| \text{unresolved dimension}&0&0&0&0&1&3&6. |
| \end{array} |
| \] |
| Here the value depends only on the number erased, because every restricted |
| submatrix has the strongest rank compatible with its size. This dependence |
| on cardinality alone is a property of this design, not a general property |
| of constrained architectures. |
|
|
| \section{Reading a concrete certificate rather than a genericity claim} |
|
|
| The moment construction examines the residual spaces left by subsets of |
| previously selected rows and avoids a row that annihilates any active one. |
| Its executed trace is |
| \[ |
| \begin{array}{c|rrrrrr} |
| \text{step}&1&2&3&4&5&6\\\hline |
| \text{active families}&3&5&8&12&17&23\\ |
| \text{search bound}&10&16&25&37&52&70\\ |
| \text{chosen parameter}&1&2&3&4&5&6. |
| \end{array} |
| \] |
| The bound is three times the number of active families plus one, reflecting |
| the maximum polynomial degree in four ambient coordinates. The fact that |
| small parameters succeed here does not replace the finite search argument |
| for arbitrary residual spaces. |
|
|
| The planner independently searches the same six-channel architecture. Its |
| certificate contains a different positive integer matrix: |
| \[ |
| C_{\mathrm{plan}}=\begin{pmatrix} |
| 5&1&5&2\\ |
| 1&1&5&1\\ |
| 3&2&3&3\\ |
| 1&2&1&3\\ |
| 5&2&5&4\\ |
| 3&3&4&2 |
| \end{pmatrix}. |
| \] |
| This matrix passed after sixteen search attempts. The certificate checks |
| $3\cdot2^6=192$ restricted ranks, one for each subset and each nonzero |
| column space. Its profiles agree with those of~\eqref{examples:moment}. |
| The two files therefore demonstrate two valid designs, rather than two |
| different descriptions of the same numerical matrix. |
|
|
| For example, the planner supplies $v=(0,-5,1,0)\in E_0$ as a first-failure |
| witness. Direct multiplication gives |
| \[ |
| C_{\mathrm{plan}}v=(0,0,-7,-9,-5,-11)^T. |
| \] |
| The first two surviving rows do not see it, and deleting rows $2,3,4,5$ |
| makes it invisible. This proves that the distance cannot exceed four. |
| The all-subset rank certificate proves the matching lower bound. A |
| witness alone establishes an upper bound; a list of successful subsets |
| alone needs complete coverage to establish a universal lower bound. |
|
|
| The finite decoding experiment uses three exact vectors, |
| \[ |
| x_0=(0,1/3,-2,5/7),\qquad |
| x_1=(0,0,2,-1/2),\qquad x_2=(0,0,0,-3/5). |
| \] |
| It checks no erasure or one erasure, no error or one error on a retained |
| channel, and base error amplitudes $-7/3,1,5/2$. The amplitude is multiplied |
| by $j+1$ for column $j$. There are $19$ scenarios without erasure and |
| $6\cdot16=96$ with one erasure, giving $115$ scenarios and $345$ successful |
| column decodes. These are finite implementation checks using rational |
| arithmetic. They are not empirical accuracy estimates and do not sample |
| the full continuum of possible errors; the theorem supplies that universal |
| claim. |
|
|
| \section{Five channels: an explicit pair of indistinguishable observations} |
|
|
| Delete the final row of $C_6$, giving five moment rows. Its root-column |
| distance is three. That is sufficient for one unknown corrupted channel |
| alone, but insufficient for one corruption together with one erasure. |
| The difference is visible in an exact two-model certificate. |
|
|
| Take |
| \[ |
| v=(0,2,-3,1),\qquad C_5v=(0,0,6,24,60)^T. |
| \] |
| The formula follows from $2t-3t^2+t^3=t(t-1)(t-2)$. Erase channel $2$, |
| whose parameter is $t=3$, and supply the retained observations |
| \[ |
| y=(0,0,\text{erased},24,0)^T. |
| \] |
| They have two explanations under exactly the requested budget: |
| \begin{center} |
| \begin{tabular}{@{}lll@{}} |
| \toprule |
| Parameter & Clean encoded values & Single corrupted channel\\ |
| \midrule |
| $0$ & $(0,0,0,0,0)$ & channel $3$, error $24$\\ |
| $v$ & $(0,0,6,24,60)$ & channel $4$, error $-60$\\ |
| \bottomrule |
| \end{tabular} |
| \end{center} |
| The erased channel is common to both explanations. A decoder that returns |
| one of these parameters without reporting the ambiguity makes an unjustified |
| choice. The supplied exact decoder rejects this input because it does not |
| have exactly one identifiable candidate. |
|
|
| The example also supplies a local causal impossibility witness. Put $v$ in |
| column zero of a matrix $\Delta Q$, scale it by a sufficiently small |
| nonzero $\lambda$, and use $U=(I+\lambda\Delta Q)^{-1}$. The scaled pair |
| remains a pair of strict admissible models under the finite-context |
| hypotheses, and the same support pattern of erased and corrupted channels |
| still makes them indistinguishable. Large error magnitudes in the displayed |
| integer example are therefore not responsible for the impossibility. |
|
|
| The separate five-channel planner certificate reaches the same architectural |
| distance three with a different numerical matrix. It reports that the |
| requested tolerance is impossible even though its chosen design is |
| certified optimal for that architecture. An optimal design can still be |
| insufficient for a requested fault budget. |
|
|
| \section{A five-vertex star whose distance is three} |
|
|
| Take a root $0$ with four leaves and no other edges. Its sole nonzero |
| ambiguity column is the root column. Install the two trusted rows |
| $[0\ A]$, where |
| \[ |
| A=\begin{pmatrix}1&0&1&1\\0&1&1&-1\end{pmatrix}. |
| \] |
| The residual leaf vector is in $\ker A$ and has the parametrization |
| \[ |
| x=(-u-v,-u+v,u,v). |
| \] |
| The basis recorded by the certificate is |
| $(-1,-1,1,0)$ and $(-1,1,0,1)$, with an initial zero added for the root |
| coordinate. The residual dimension is two although the leaf support has |
| four coordinates. |
|
|
| Every pair of columns of $A$ is independent: their $2\times2$ determinants |
| are nonzero. Some three columns are dependent, since $A$ has rank two. |
| Thus the smallest support of a nonzero vector in its kernel is three. |
| The vector $(-1,-1,1,0)$ is an explicit witness. Dedicated leaf sensors |
| therefore have distance three and profile $(3,4)$. |
|
|
| The planner chooses gains $(1,2,1,2)$ on the four dedicated channels. |
| Their readings are |
| \[ |
| (-u-v,-2u+2v,u,2v). |
| \] |
| Nonzero gains cannot change which channel values are zero, so they preserve |
| the distance. In the certificate, deleting channels $1,2,3$ leaves channel |
| zero alone and allows the vector $(0,2,-1,1)$ on the leaves. Deleting all |
| four channels releases both dimensions. Conversely, every two retained |
| channels identify the two-dimensional kernel, which is why one arbitrary |
| corrupted channel is correctable. |
|
|
| This is a small instance of the star realization used in the complexity |
| theorem. Its graph is simple, but the trusted calibration matrix carries |
| nontrivial linear dependence information. The simplicity of the causal |
| graph does not make that information disappear. |
|
|
| \section{Doubling support: distance six with two residual parameters} |
|
|
| Replace $A$ by |
| \[ |
| A'=\begin{pmatrix}A&0\\I_4&-I_4\end{pmatrix}. |
| \] |
| The associated star now has eight leaves and nine vertices. The lower four |
| equations force its leaf vector to have the form $(x,x)$; the upper two |
| equations require $Ax=0$. Hence |
| \[ |
| \ker A'=\{(x,x):x\in\ker A\}. |
| \] |
| Its dimension stays two. Every nonzero support is duplicated, so the first |
| distance doubles from three to six. The union of the supports of the full |
| two-dimensional kernel is all eight coordinates, giving profile $(6,8)$. |
|
|
| An explicit minimal-support witness is |
| \[ |
| (-1,-1,1,0,-1,-1,1,0). |
| \] |
| The numerical certificate also uses the vector |
| $(0,2,-1,1,0,2,-1,1)$, which vanishes on channels $0$ and $4$ and is |
| nonzero on the other six. These two surviving channels read the same |
| underlying coordinate from the two copies. They impose only one |
| independent condition on the two-dimensional residual space. |
|
|
| The architecture therefore corrects two arbitrary corrupted channels and |
| tolerates one known erasure: its distance six exceeds $1+2\cdot2=5$. |
| It cannot achieve distance seven merely by changing its dedicated sensor |
| gains. The two-copy restriction fixes a six-channel support obstruction. |
| With seven freely designed dense channels, the same residual dimension |
| would suffice for this fault budget. This one-channel difference measures |
| the specified architectural restriction; it does not compare acquisition |
| costs or physical reliability of the two systems. |
|
|
| The jump from six to eight in the profile is also informative. Six erased |
| channels can hide one direction, but there is no seven-channel threshold |
| for releasing the second: at least one surviving dedicated coordinate |
| always imposes a nonzero functional on this kernel. All eight must be |
| erased to leave both parameters completely free. |
|
|
| \section{The same zero pattern can have different fault tolerance} |
|
|
| The next pair uses the same four-leaf star and the same dedicated sensors. |
| Only the trusted coefficients change: |
| \[ |
| A_{\mathrm g}=\begin{pmatrix}1&1&1&1\\1&2&3&4\end{pmatrix}, |
| \qquad |
| A_{\mathrm e}=\begin{pmatrix}1&1&1&1\\1&1&3&4\end{pmatrix}. |
| \] |
| Both matrices have rank two, both have no zero entries, and both leave a |
| two-dimensional residual space. Their support diagrams are identical. |
| For $A_{\mathrm g}$, any two columns have distinct lower entries and thus |
| a nonzero determinant. Its spark and calibration distance are three, with |
| profile $(3,4)$. Its witness $(1,-2,1,0)$ is the first basis vector in the |
| certificate. |
|
|
| For $A_{\mathrm e}$, the first two columns coincide. The vector |
| $(-1,1,0,0)$ is now in the kernel, so the distance falls to two and the |
| profile becomes $(2,4)$. The certificate's architectural cut is the pair of |
| channels $T=\{2,3\}$. Their restricted row spaces have total dimension one; |
| the cut calculation gives $4-2+1-2+1=2$. |
|
|
| Both planner designs use the same sensor gains $(1,2,1,2)$. The exceptional |
| witness then has readings $(-1,2,0,0)$. The observation $(-1,0,0,0)$ is |
| explained either by the zero parameter with an error in channel zero, or |
| by this witness with an error in channel one. Thus the exceptional case |
| cannot correct every single-channel corruption. The generic-coefficient |
| case can. The distinction is a numerical calibration dependency, and a |
| graph-only support test cannot certify it away. |
|
|
| \section{A direction that several dense-looking sensors cannot see} |
|
|
| The exhaustive architecture test contains the three row spaces |
| \[ |
| H_0=H_1=\operatorname{span}\{(1,1)\},\qquad |
| H_2=\operatorname{span}\{(1,-1)\}. |
| \] |
| Its common certified design is |
| \[ |
| C=\begin{pmatrix}2&2\\1&1\\2&-2\end{pmatrix}. |
| \] |
| Every row has both coordinates nonzero. Nevertheless the first two channels |
| measure the same direction. On $E=\mathbb R^2$ the readings of $(a,b)$ are |
| $(2(a+b),a+b,2(a-b))$. The vector $(1,-1)$ is seen only by channel two, |
| with readings $(0,0,4)$, so the architecture has distance one and profile |
| $(1,3)$. Losing that single channel leaves a one-dimensional ambiguity. |
|
|
| On the residual line $\operatorname{span}\{(1,1)\}$, the same matrix has |
| distance two; on $\operatorname{span}\{(1,-1)\}$ it has distance one. A |
| sensor can therefore be informative for one residual column and invisible |
| for another. The data needed for a common certificate are the restricted |
| row spaces $H_a|_{E_j}$, not merely the ambient support of $H_a$. |
|
|
| For comparison, the test with three unrestricted row spaces supplies |
| \[ |
| C_{\mathrm{free}}=\begin{pmatrix}1&4\\3&1\\4&3\end{pmatrix}. |
| \] |
| Its three pairwise determinants are $-11,-13,5$, so every pair is |
| independent on $\mathbb R^2$. Its full-space profile is $(2,3)$. It also |
| has nonzero values on all three channels for each of the four lines spanned |
| by $(1,0),(0,1),(1,1),(1,-1)$. It is consequently optimal for all five |
| spaces simultaneously. A full-space certificate by itself would not have |
| checked these four additional restrictions. |
|
|
| \section{What the finite architecture enumeration establishes} |
|
|
| The six permitted row-space types in the exhaustive test are zero, the two |
| coordinate lines, the sum line, the difference line, and the full plane. |
| Three indexed channels give $6^3=216$ architectures. The prescribed finite |
| row choices yield $729$ total candidate assignments. Each is checked on all |
| eight surviving subsets by an independent two-dimensional determinant |
| oracle, giving $5{,}832$ subset-rank comparisons. Of those assignments, |
| $449$ attain every predicted subset rank for their architecture. |
|
|
| The additional common-design test uses the full plane and the four lines |
| just listed. It finds a shared design for all $216$ architectures and checks |
| $216\cdot5\cdot8=8{,}640$ restricted subset ranks. The largest observed |
| search-attempt count is eleven. These counts describe the complete bounded |
| test family, not all possible real row spaces or all matrix sizes. In |
| particular, the attempt count is not a performance guarantee for the |
| planner on larger inputs. |
|
|
| The empty cases also have different meanings. If every residual column |
| space is zero, no calibration parameter remains, and no new channel is |
| required. Its distance minimum is over an empty family; the certificates |
| use an empty profile and a null distance entry for such a column. This is |
| why the zero-dimensional fourth column of the complete graph does not |
| reduce its overall distance to zero. |
|
|
| In contrast, if $E=\mathbb R^2$ remains unresolved but all three allowed |
| row spaces are zero, every measurement is zero. The architecture test |
| correctly returns profile $(0,0)$. Both ambiguity dimensions are already |
| invisible without any channel failure. An empty parameter space and an |
| uninformative measurement system must not be merged into the same |
| software case. |
|
|
| \section{A decoder must reject a feasible family of alternatives} |
|
|
| One regression check takes $E=\mathbb R^2$, $C=I_2$, observation $y=0$, |
| and allowance for one corrupted channel. The zero parameter is a valid |
| explanation, but it is not unique: every vector $(a,0)$ can be hidden by |
| an error in the first channel, and every vector $(0,b)$ by an error in |
| the second. A decoder that examines only full-rank candidate clean sets |
| can accidentally retain the zero explanation and skip these |
| rank-deficient feasible families. |
|
|
| The implementation instead tests consistency first. If a proposed clean |
| set is consistent but rank deficient, it reports a non-identifiable |
| feasible family. That behavior is exercised by this regression case. The |
| remaining input checks reject a negative error budget, a wrong observation |
| length, and an out-of-range erasure index, while ordinary zero-error |
| decoding of $(3/2,-7/4)$ is also checked. These are concrete guards against |
| false recovery claims, rather than additional mathematical assumptions. |
|
|
| \section{Returning to the mixed calibration in the supplied package} |
|
|
| The original package contains a complete five-vertex graph with three |
| different trusted resources. Its known readout is $(1,2,3,4,5)$, its known |
| baseline row is the incoming mechanism at target two, and its calibrated |
| nonunit gain acts at target three. Their combined left-constraint matrix is |
| \[ |
| M=\begin{pmatrix} |
| 1&2&3&4&5\\ |
| -1&-1&1&0&0\\ |
| 0&0&0&1&0 |
| \end{pmatrix}. |
| \] |
| For the root-column support $I_0=\{1,2,3,4\}$, write the residual vector |
| as $(0,x_1,x_2,x_3,x_4)$. The baseline equation gives $x_2=x_1$; the |
| gain equation gives $x_3=0$; and the readout equation then gives |
| $5x_1+5x_4=0$. Therefore |
| \[ |
| E_0=\operatorname{span}\{(0,1,1,0,-1)\}. |
| \] |
| For the next column, coordinate one is unavailable. The same baseline |
| equation forces coordinate two to zero, the gain removes coordinate |
| three, and the readout removes coordinate four. Thus all later residual |
| spaces vanish, yielding dimensions $(1,0,0,0,0)$ as in the input |
| certificate. Its recorded kernel vector is the negative of the displayed |
| basis vector, which describes the same line. |
|
|
| The sum row $c=(1,1,1,1,1)$ evaluates to one on the displayed basis. It |
| identifies the scalar residual parameter $\lambda$ without faults. |
| Three separately indexed copies return $(\lambda,\lambda,\lambda)$ and |
| correct one arbitrary corrupted channel; the median is an exact decoder |
| in that noiseless one-error case. Four copies have distance four and |
| also tolerate the combined budget of one erasure and one corruption. |
| This is the dense frontier with $d_*=1$, whereas the complete four-vertex |
| example without the installed calibrations has $d_*=3$ and requires six |
| channels for the same combined budget. |
|
|
| The comparison accounts only for the new channels. The trusted mechanism, |
| gain, and readout used to reach the one-dimensional space are additional |
| resources and have not been assigned zero physical cost. It illustrates |
| how the residual spaces summarize their mathematical effect, allowing the |
| robust channel calculation to begin after those resources have been |
| specified. |
|
|
| This example also makes the inverse-coordinate step especially transparent. |
| If $Q=\lambda w e_0^T$ with $w=(0,1,1,0,-1)^T$, then $Q^2=0$ because |
| $e_0^Tw=0$. Therefore $(I+Q)^{-1}=I-Q$ exactly. Recovering $\lambda$ from |
| the calibrated readings determines the forward transformation without an |
| iterative matrix inversion. The finite strict-support conditions still |
| restrict how large $\lambda$ may be for a particular reference model; |
| the nilpotent identity itself is valid for every real $\lambda$. |
|
|
| \section{Positive coefficients do not supply a uniform noise margin} |
|
|
| Consider the analytic family |
| \[ |
| C_\epsilon=\begin{pmatrix} |
| 1&1\\1&1+\epsilon\\1&1+2\epsilon |
| \end{pmatrix},\qquad\epsilon>0, |
| \qquad E=\mathbb R^2. |
| \] |
| All coefficients are positive. Every pair of rows is independent, so each |
| matrix has distance two and tolerates one arbitrary erasure. These exact |
| integer-valued properties do not deteriorate as $\epsilon$ approaches zero. |
| The rows remain bounded and their norms approach $\sqrt2$, so no vanishing |
| global gain is being used to manufacture the example. |
|
|
| However, for the unit vector $w=(1,-1)/\sqrt2$, |
| \[ |
| C_\epsilon w=(0,-\epsilon/\sqrt2,-2\epsilon/\sqrt2)^T. |
| \] |
| The full-map minimum gain is at most $\epsilon\sqrt{5/2}$. After erasing |
| the last channel, the gain on this direction is only |
| $\epsilon/\sqrt2$. Thus the one-erasure robust singular-value margin is |
| at most that quantity and tends to zero. Small observation error can |
| produce arbitrarily large relative parameter uncertainty even though |
| every matrix passes the exact distance certificate. |
|
|
| This final example separates three reportable statements. A positive row |
| construction establishes a sign constraint. An all-subset rational rank |
| certificate establishes exact fault tolerance. A numerical lower bound on |
| the relevant restricted singular values establishes a quantitative noise |
| margin. Each answers a different question, and none of the first two |
| should be presented as evidence for the third. |
|
|
| \section{Figures derived from the executed examples} |
| \begin{figure}[htbp] |
| \centering |
| \includegraphics[width=0.95\textwidth]{../figures/complete_four_failure_profiles.pdf} |
| \caption{Exact deletion thresholds for the three nonzero residual columns of the complete four-node example. The plotted data are the rationally verified integer profiles. The curves describe individual columns; a common deletion can affect several columns simultaneously.} |
| \end{figure} |
| \begin{figure}[htbp] |
| \centering |
| \includegraphics[width=0.95\textwidth]{../figures/same_support_coefficient_dependence.pdf} |
| \caption{The trusted calibration matrices have identical nonzero-entry patterns and equal rank, but different dedicated-sensor distances. The difference comes from an exact numerical column dependency.} |
| \end{figure} |
| \begin{figure}[htbp] |
| \centering |
| \includegraphics[width=0.95\textwidth]{../figures/normalized_conditioning_margin.pdf} |
| \caption{Floating-point illustration of a proved conditioning obstruction: five strictly positive convex-average rows have exact distance four for every positive separation parameter, while the worst retained two-row singular-value margin tends to zero. Numerical values in this plot are illustrative rather than exact rank certificates.} |
| \end{figure} |
|
|
