TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/TENSOR-BOUND-REVIEW.txt Original file SHA-256: 8e8455cb57da79227a7fa3fbbba4dd88ca4a2606591eb32603b716b05d8bf3cf Inclusive original lines: 1-123 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 | I: INDEPENDENT STRUCTURAL REVIEW OF SHARED-METRIC-SOURCE 000002 | 2 November2026. relational_logic. Analytic review only; no new harness. 000003 | Reviewed SHARED-METRIC-SOURCE.txt in full, especially sections3–4. 000004 | 000005 | VERDICT 000006 | 000007 | The mixed-term tensor test, whitening, two-sided velocity estimate and 000008 | Hermitian companion argument are correct under the displayed assumptions. 000009 | No simultaneous diagonalization of B and C is needed. Native epsilon/eta 000010 | values are explicitly NOT supplied. The Hamiltonian KMS / BW wedge-KMS 000011 | distinction is also correct and retained. 000012 | 000013 | One wording precision is useful before persistence: section3's last 000014 | 'necessary and sufficient test of that ... block' should explicitly name 000015 | COEFFICIENT TENSOR FACTORIZATION. General common physical pole geometry 000016 | in section2 is a weaker/broader statement for possibly nonlocal kernels, 000017 | physical subchannels and dressed form factors. It should not be silently 000018 | identified with a constant local two-derivative coefficient factorization. 000019 | 000020 | Suggested replacement for that sentence: 000021 | 'This is a necessary and sufficient test of the stated coefficient-tensor 000022 | factorization on the nondegenerate Hermitian quadratic block. It is a 000023 | sufficient common-principal-cone test there; it is not asserted as a 000024 | universal necessary test for all dressed physical pole geometries or 000025 | as a sufficient test of full IR universality.' 000026 | 000027 | This clarification does not require changing the bound or introducing 000028 | another assumption on the whole original source. 000029 | 000030 | 1. THE PRECISE QUADRATIC CLASS 000031 | 000032 | The source retains the full homogeneous quadratic symbol 000033 | K_t omega^2+2 SUM_a K_ts^a omega k_a-SUM_ab K_s^(ab)k_a k_b. 000034 | It explicitly keeps the same-order mixed terms. K_t>0 is on the ACTUAL 000035 | retained physical block. After congruence by K_t^(-1/2), for k=|k|n this is 000036 | 000037 | Q_n(v)=v^2 I+2v B(n)-C(n), v=omega/|k|. (1) 000038 | 000039 | B(n)=B(n)* and C(n)=C(n)* are essential for the real-root argument. 000040 | The bound ||C(n)-c^2 I||<=epsilon c^2 with epsilon<1 already implies 000041 | C(n)>=(1-epsilon)c^2 I>0. 000042 | No separate commuting-basis or scalar-species assumption is used. 000043 | 000044 | In the stated common orthogonal frame, B(n)=0 and C(n)=c^2 I for ALL 000045 | directions n are equivalent to the complete tensor factorization there: 000046 | polarization in n recovers every symmetric spatial coefficient, including 000047 | mixed spatial directions. In a nonorthogonal frame use the full general 000048 | K^(mu nu)=g^(mu nu)Z identity as printed, not B=0 by coordinate fiat. 000049 | 000050 | The full coefficient test is not inferred from one determinant pole, 000051 | one polarization or static K_x=K_y. A block with a cancelled kinetic 000052 | coefficient is not in the invertible class. The report explicitly preserves 000053 | these distinctions, R044/R045 and nonpropagating-clock typing. 000054 | 000055 | 2. COMPANION PROOF WITH ALL NONCOMMUTING MATRICES RETAINED 000056 | 000057 | Set S=sqrt(C), the positive invertible square root. The companion 000058 | T=[[-2B,S],[S,0]] 000059 | is Hermitian. It has no zero eigenvalue: T(x,y)=0 implies Sx=0, hence 000060 | x=0 and Sy=0, hence y=0. For every nonzero v, 000061 | 000062 | Q_n(v)x=0 000063 | iff T (x,Sx/v)=v(x,Sx/v). (2) 000064 | 000065 | Conversely the bottom companion equation gives y=Sx/v and the top 000066 | equation gives (1), without interchanging B and C. Thus all characteristic 000067 | roots are real and nonzero. There are exactly d positive and d negative 000068 | roots counted with multiplicity for a d-dimensional block: continuously 000069 | replace B by tB; no eigenvalue can crosszero, and at t=0 the companion 000070 | has eigenvalues +/-sqrt(spec C). This supplies the existence/inertia 000071 | detail behind section4's sentence. 000072 | 000073 | Within THIS exact positive Hermitian quadratic class, if ALL complete 000074 | characteristic roots are exactly +/-c, the converse factorization can 000075 | also be proved: the Hermitian companion has T^2=c^2 I; its lower-right 000076 | block is C=c^2 I and its offdiagonal block is -2B sqrt(C)=0, so B=0. 000077 | That restricted equivalence needs all physical characteristic branches, 000078 | with no hidden omitted block. It is not a replacement for the broader 000079 | source/pole typing in section2. 000080 | 000081 | 3. THE ERROR BOUND IS CORRECT 000082 | 000083 | For any normalized x in ker Q_n(v), v>0, put 000084 | r=v/c, b=/c, s=/c^2. 000085 | Hermiticity gives real b,s; the actual residual hypotheses give 000086 | |b|<=eta, 1-epsilon<=s<=1+epsilon. 000087 | Taking the scalar expectation of Q_n(v)x=0 yields 000088 | r^2+2br-s=0. 000089 | Its positive root -b+sqrt(b^2+s) increases with s and decreases with b 000090 | because s>0. Therefore exactly 000091 | 000092 | sqrt(eta^2+1-epsilon)-eta <=v/c 000093 | <=sqrt(eta^2+1+epsilon)+eta. (3) 000094 | 000095 | The lower endpoint is strictly positive. The eta=0 and epsilon=eta=0 000096 | specializations in the source are correct. This is an enclosure for the 000097 | principal characteristic ratios of the stated Hermitian block, not a 000098 | claim about finite-momentum massive group velocities, damped complex 000099 | thermal poles or the UV front of a nonlocal effective kernel. The source 000100 | already says the relevant latter distinction explicitly. 000101 | 000102 | 4. ACTUAL SOURCE AND STATE SCOPE 000103 | 000104 | The residual bounds are stated as results that an ACTUAL complete source 000105 | calculation must supply. The source does not assign epsilon or eta from 000106 | native static stiffnesses or from separate Bloch/holonomy/deck variables. 000107 | Nothing in(3) estimates their IR flow or the full source's higher vertices. 000108 | Hence the distinction between a controlled quadratic cone and sufficient 000109 | controlled IR universality remains sound. 000110 | 000111 | The H finite-beta KMS source uses its given Hamiltonian time; it can have 000112 | a rest-frame response. A vacuum restricted to a wedge can instead be KMS 000113 | for a BW-identified geometric boost. Equal beta notation or KMS status 000114 | does not identify these dynamics or prove that the finite-temperature 000115 | Hamiltonian state is that vacuum. Section6 states this correctly. 000116 | 000117 | No physical, reference, metric, coupling or state selection was added by 000118 | this review. Apart from explicitly naming factorization in the section3 000119 | criterion, no mathematical correction to sections3–4 is needed. 000120 | 000121 | ROOT RESOLUTION: section3 now explicitly limits necessary/sufficient 000122 | to the stated coefficient-tensor factorization, not every dressed quantum 000123 | pole kernel or full IR universality. No mathematical scope was enlarged.