TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/I-FINAL-REVIEW.txt Original file SHA-256: 7ee5e27cfe8076ab8a126aa2ca96629ee39ae7f560e8cbcd08f1986c83dccae7 Inclusive original lines: 1-67 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | INDEPENDENT FINAL REVIEW OF SHARED-METRIC-SOURCE.txt 000002 | joint_origin, 2 November2026. No new model or numerical run. 000003 | 000004 | VERDICT: ACCEPTED IN THE EXPLICIT STATED SCOPE 000005 | 000006 | No mathematical error was found in the full mixed-block bound or in the 000007 | source/IR scope. GEOMETRIC-CONE.txt and the root synthesis were read in full. 000008 | 000009 | 1. THE MATRIX BOUND IS CORRECT WITHOUT COMMUTATION 000010 | 000011 | Whitening with the one positive physical K_t gives Hermitian B and C, 000012 | and epsilon<1 implies C>=c^2(1-epsilon)I>0. For a characteristic vector 000013 | x and a positive real characteristic ratio v, normalize x and put 000014 | r=v/c, b=/c, s=/c^2. 000015 | Then exactly r^2+2br-s=0, so 000016 | r=-b+sqrt(b^2+s). 000017 | For s>0 this expression increases with s and strictly decreases with b. 000018 | The extrema on |b|<=eta, 1-epsilon<=s<=1+epsilon are therefore 000019 | sqrt(eta^2+1-epsilon)-eta 000020 | and sqrt(eta^2+1+epsilon)+eta. 000021 | The stated lower bound is strictly positive. Its factors and units are 000022 | correct; in particular no factor2 is missing from the eta term. 000023 | 000024 | The Hermitian companion L=[[-2B,sqrt(C)],[sqrt(C),0]] supplies all roots, 000025 | not just a chosen real eigenvector. For an eigenpair L(x,y)=v(x,y), 000026 | C>0 excludes v=0 and gives y=sqrt(C)x/v and 000027 | (v^2 I+2vB-C)x=0. 000028 | Conversely every root of this quadratic pencil has precisely this 000029 | companion eigenvector. Hermiticity makes every root real. No product BC 000030 | or CB was interchanged in this proof, and no simultaneous eigenbasis is 000031 | needed. Thus the result is whole-class for the declared positive 000032 | nondegenerate Hermitian principal block. 000033 | 000034 | The report correctly calls v a principal characteristic ratio. It does 000035 | not claim that the bound controls arbitrary massive group velocities,+damped/non-Hermitian kernels, every higher vertex, or a nonlocal UV 000036 | causal front. Eta/epsilon are actual same-source uniform residual bounds; 000037 | they are not inferred from the native static stiffness comparison. 000038 | 000039 | 2. THE WARD AND STATE HYPOTHESES ARE KEPT AT THE RIGHT PLACE 000040 | 000041 | The source change-of-variables identity, covariance-compatible reference, 000042 | renormalization prescription and invariant vacuum are explicit hypotheses. 000043 | Differentiating that FULL Gamma covariance correctly yields the Hessian 000044 | congruence. Invertible representation factors preserve its physical zero 000045 | set. This implication does not require a new species-speed parameter. 000046 | The report does not derive whole-source/vacuum covariance from the 000047 | Clifford principal symbol, a Padm-commuting regulator or gauge Ward 000048 | identities alone. It preserves original quotient/nondegeneracy and the 000049 | causal branch, including R044/R045 cancellations. 000050 | 000051 | 3. ORIGINAL CONSUMERS AND FLOW TYPES REMAIN DISTINCT 000052 | 000053 | The native v1011 full static Kx/Ky response is retained as positive 000054 | coupled evidence. Its separate slab companion and the original v1012 000055 | Bloch/holonomy/deck comparison are not promoted to a common pole kernel. 000056 | The native SPLIT is correctly not a Lorentz counterexample. 000057 | 000058 | Ordinary Hamiltonian beta-KMS time, a thermal rest direction, and the 000059 | actual BW modular boost of a vacuum restriction are explicitly different 000060 | flows. Equal beta labels do not identify them. No replacement P1 state, 000061 | clock, temperature or physical coupling is selected. 000062 | 000063 | The common geometric kinetic and conditional all-correction Ward result 000064 | is compatible with G/H. The original controlled-IR task is not replaced 000065 | by LR tails, static isotropy, or one principal cone. The optional a->0 000066 | decision remains intact. No correction or additional test is required 000067 | before registering this scoped result.