{"id": "NYR-V-001", "title": "Budget-preserving scalar realization", "theorem": "Theorem 2.2", "statement": "Finite scalar division h=sum(a_j*g_j) has an isometric realization as the declared normalized higher-identity repair equation on a regular central foliation with compact torus leaves and a nonvanishing transverse higher anchor. Parameter regularity and pointwise Euclidean/L2 budgets are preserved.", "assumptions": ["Smooth or real-analytic scalar data as appropriate.", "Torus dimension n+2, n>=2, with binomial(n+2,2)>=k+1.", "Declared all-pairs multiderivation normalization."], "proof_pages": [3, 4], "verification_scripts": ["verify_coherent_repair.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-002", "title": "Attained optimal continuous Euclidean budget", "theorem": "Theorem 3.2", "statement": "For A(x)u(x)=h(x), continuous exact repair exists iff every stable affine Glaeser fiber is nonempty. When it exists, min_continuous_u max_x ||u(x)|| = max_x dist(0,H_x^infinity), and the minimum is attained.", "assumptions": ["Compact metric parameter space.", "Finite continuous Euclidean linear system.", "Classical finite affine Glaeser stabilization and Michael selection."], "proof_pages": [4, 5], "verification_scripts": ["verify_budget_resources.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-003", "title": "One-parameter analytic continuation within budget", "theorem": "Theorem 4.1", "statement": "A real-analytic one-parameter finite linear system admits an analytic solution iff its pointwise minimum repair norms are finite and locally bounded. A prescribed uniform norm bound is preserved by the analytic solution.", "assumptions": ["Real-analytic matrix and target on a connected open interval.", "Euclidean solution norm."], "proof_pages": [6, 7], "verification_scripts": ["verify_budget_resources.py", "verify_coherent_repair.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-004", "title": "Analytic-arc solvability without continuous coherence", "theorem": "Theorem 4.2", "statement": "The explicit two-parameter polynomial curvature example has bounded exact repairs at every point and analytic repairs along every real-analytic parameter arc, but no continuous neighborhood repair. The pointwise repair norm is at most 1/sqrt(2).", "assumptions": ["The polynomial central-foliation family of equation (9).", "Arcs restrict the repair equations; no arbitrary higher-anchor pullback is asserted."], "proof_pages": [7], "verification_scripts": ["verify_coherent_repair.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-005", "title": "Tangent minimax and exact quadratic certificate", "theorem": "Theorems 5.1 and 5.2", "statement": "The leading residual of a continuous bounded repair equals a finite tangent minimax over constant coefficient limits. Quadratic data yield an exact semidefinite epigraph. The explicit two-parameter example has sharp quadratic coherence floor 1/2.", "assumptions": ["Declared scalar realization and Euclidean norm budget.", "Homogeneous leading coefficients of common order d.", "Continuous parameter families of smooth leafwise cochains."], "proof_pages": [8], "verification_scripts": ["verify_coherent_repair.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-006", "title": "Arbitrary sharp finite differentiability loss", "theorem": "Theorem 6.1", "statement": "For x^q*g1+y^q*g2=x^(q-1)*y^(q-1), q>=3, a C^(q-3,1) exact repair exists but no C^(q-2) repair exists. For q=2 pointwise repairs are bounded but no continuous repair exists. Explicit projective blow-up formulas resolve the repairs analytically.", "assumptions": ["Integer q>=2.", "The norm-preserving scalar realization; radial costs refer to the s=0 slice."], "proof_pages": [9], "verification_scripts": ["verify_coherent_repair.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-007", "title": "Sharp regularity-horizon resource law", "theorem": "Theorem 7.2", "statement": "In the explicit graded geometric representation, ||G_N|| is comparable to N+r_g^2*N^3. For fixed correction norm and Holder exponent sigma, optimal residual e_(K,sigma) is comparable to K^(-2/sigma), and minimax normalized energy is comparable to 1+N^2*K^(-4/sigma). The required Lipschitz budget has order sqrt(N).", "assumptions": ["Disk radius 0<R<=1.", "Correction norm budget B>=1/sqrt(2).", "Holder exponent 0<sigma<=1 and sufficiently large seminorm budget K.", "Fixed finite weighted-degree cutoff L>=7."], "proof_pages": [10, 11, 12], "verification_scripts": ["verify_geometry_transport.py", "verify_budget_resources.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-008", "title": "Finite complete algebraic-horizon certificate with feedback", "theorem": "Theorem 8.2", "statement": "For M=D^2, baseline height h, moment trace orders v_k and characteristic coefficients a_j, b=min_j ord(a_j)/j and beta=min(b,min_(h<k<M) v_k/(k-h)). At N=floor(t^-alpha), normalized baseline growth is bounded iff alpha<=beta, including the endpoint.", "assumptions": ["Fixed finite matrix dimension and fixed real-analytic generators in a real parameter.", "Strictly positive stationary label probabilities and conditionally centered independent signs.", "Nilpotent base CP map Phi_0.", "Infinite candidates are omitted; empty minima are infinity."], "proof_pages": [13, 14], "verification_scripts": ["verify_full_feedback.py", "verify_feedback_api.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-009", "title": "Polynomial, critical ODE, and exponential phases", "theorem": "Theorem 8.4", "statement": "Below b, normalized moments converge to finite Newton-face matrix polynomials. At b they converge to a finite constant-coefficient matrix ODE profile with gamma_b=max_(k<M)(b*k-v_k). Above b, t^(alpha-b)*log||G_floor(s*t^-alpha)|| converges to s*r_b.", "assumptions": ["The stationary analytic nilpotent-base setting of Section 8.", "Fixed positive s, uniformly on compact subsets of (0,infinity).", "The spectral phase exists when b is finite."], "proof_pages": [14, 15, 16], "verification_scripts": ["verify_full_feedback.py", "verify_feedback_api.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-010", "title": "Finite arbitrary-horizon criterion and logarithmic exception", "theorem": "Theorem 8.5", "statement": "For finite beta, bounded normalized moments are equivalent to N*t^beta=O(1) for every horizon sequence iff some critical coefficient L_k beyond baseline h is nonzero. It suffices to test L_(h+1) through L_(h+M). A parallel scalar-feedback example gives a logarithmic-window counterexample when this block vanishes.", "assumptions": ["The stationary analytic nilpotent-base setting of Section 8.", "Finite preservation frontier beta.", "Sequences t->0+ and N->infinity."], "proof_pages": [16], "verification_scripts": ["verify_full_feedback.py", "verify_feedback_api.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-011", "title": "Adaptive positive mixing preserves the frontier", "theorem": "Theorem 8.6", "statement": "History-dependent label probabilities bounded above and away from zero preserve the global baseline/exponential comparison and the sharp algebraic-horizon frontier uniformly over all admissible policies. Exact critical matrix coefficients are stationary statements.", "assumptions": ["Fixed generators and nilpotent base as in Section 8.", "Conditionally fair next sign.", "Fixed positive lower and upper bounds on label probabilities."], "proof_pages": [17], "verification_scripts": ["verify_full_feedback.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-012", "title": "Complete signed precision law in the marginal subclass", "theorem": "Theorem 9.2", "statement": "If the CP moment increment stays nilpotent at every parameter, signed finite word valuations give a uniform finite moment sum and the necessary and sufficient criterion N*t^beta=O(1) for arbitrary horizon sequences. Exact protected cancellations can make beta infinite.", "assumptions": ["Fixed finite-dimensional real-analytic centered family.", "Exact spectral marginality for every parameter near zero.", "Strictly positive stationary label probabilities."], "proof_pages": [18, 19], "verification_scripts": ["verify_cancellation_frontier.py", "verify_frontier_api.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-013", "title": "Square-zero gapped pencils with hidden mixing cost", "theorem": "Theorem 10.1", "statement": "Genuine zero-anchor gapped Lie n-algebroid lifts have Q0^2=D^2=0 and Q0 D+D Q0=0 and nonzero ordered cross-product. Every pencil Q0+z*D squares to zero, but centered mixing of Q0+t*D and Q0-t*D has moment order N+t^2*N^2.", "assumptions": ["Integer n>=2.", "The explicit abelian-central plus affine ground model.", "Finite weighted-degree cutoff L>=n+3."], "proof_pages": [20, 21], "verification_scripts": ["verify_flat_pencil.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}
{"id": "NYR-V-014", "title": "Arbitrarily high hidden polynomial growth and exact tensor moments", "theorem": "Theorem 10.2", "statement": "For r independent graded gapped sectors, every generator pencil squares to zero while ||G_N|| equals sum_(k=0)^r (N)_k*e_k(lambda_1,...,lambda_r). Fixed nonzero amplitudes give moment order N^r; one fixed and r-1 small amplitudes give N*(1+N*t^2)^(r-1).", "assumptions": ["Fixed finite number r of independent nonzero odd square-zero sector lifts.", "Graded tensor extension with Koszul parity signs.", "Positive stationary label probabilities and real amplitudes."], "proof_pages": [21, 22], "verification_scripts": ["verify_flat_pencil.py"], "manuscript_path": "NYRAELUNE_Manuscript.pdf", "evidence_scope": "Manuscript proof; the listed verifiers audit explicit finite instances and do not replace the proof. This summary is not a formalized theorem."}