TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_sewing_20261029/INDEPENDENT-REVIEW.txt Original file SHA-256: a6232d863bf730a98c4d499326cd9457f866ad8896bfcaba726b46b33b1aab06 Inclusive original lines: 1-142 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 mathematical review of the spectral-weight RP implication 000002 | 29 October 2026 000003 | 000004 | Verdict: THE PROPOSED COUNTEREXAMPLE IS CORRECT FOR THE GENERAL IMPLICATION. 000005 | It refutes preservation of reflection positivity by a merely positive, 000006 | reflection-invariant spectral weight. It is not a complete TFPT/P1/P2 000007 | countermodel and does not evaluate the actual physical TFPT kernel. 000008 | 000009 | 1. Original scope read 000010 | 000011 | Read documents/archive--tfpt-45--source_extracts--05_metrology_source/ 000012 | sections/005-section-absolute-dimensionless-metrology-from-the-bounda.tex, 000013 | especially 60-105 and 136-179. The profile is explicitly 000014 | f(t)=integral_[0,infinity) exp(-s t) dnu(s), nu>=0 finite, 000015 | at 67-68 and 103. The source calls this a Bernstein cone. In standard 000016 | terminology this is a completely monotone Laplace profile, not an arbitrary 000017 | increasing Bernstein function. The proposed general Delta formula uses 000018 | exactly the positive Laplace representation displayed in the source. 000019 | 000020 | The geometric measure theorem additionally concerns a BRST-reduced 000021 | admissible geometric configuration class, a branch-compatible fixed profile, 000022 | blocked geometric exhaustions and earlier admissible OS structure. These 000023 | stronger actual-source identifications are not instantiated by the finite 000024 | example below. However, the proof at 170-173 explicitly uses the general 000025 | claim that the positive weight preserves the reflected forms. Positivity 000026 | and reflection covariance alone do not justify that step. If admissibility 000027 | already includes RP of the fully reweighted measure, that supplies an 000028 | additional premise; it is not a consequence of weight positivity. 000029 | 000030 | 2. Direct algebraic check 000031 | 000032 | Take configurations (x,y) in {+a,-a}^2, reflection theta(x,y)=(y,x), 000033 | P=[[0,1],[1,0]], D_xy=[[x,g],[g,y]], a>g>0. 000034 | Then P D_xy P=D_yx exactly. For x=y the squared eigenvalues are 000035 | (a-g)^2,(a+g)^2. For x=-y both squared eigenvalues are a^2+g^2. 000036 | All D_xy are invertible, with minimum absolute eigenvalue at least a-g. 000037 | 000038 | The base configuration weight/RP matrix K0=[[10,9],[9,10]] has eigenvalues 000039 | 19 and 1, so it is strictly RP; all entries are strictly positive as well. 000040 | It may be normalized by its positive total mass without affecting RP. 000041 | 000042 | For f(t)=exp(-t), a=2,g=1, write 000043 | S_same=exp(-1)+exp(-9), S_opp=2exp(-5), 000044 | Delta=S_same-S_opp=exp(-1)(1-exp(-4))^2>0. 000045 | All four Dirac blocks have gap at least 1. The spectral weight is positive 000046 | and reflection-invariant. Its reweighted RP matrix is 000047 | K=[[10exp(-S_same),9exp(-S_opp)], 000048 | [9exp(-S_opp),10exp(-S_same)]]. 000049 | The half-observable F(x)=sign(x) has reflected unnormalized square 000050 | F* K F=2[10exp(-S_same)-9exp(-S_opp)]. 000051 | This is strictly negative iff Delta>log(10/9). 000052 | 000053 | The proposed strict rational estimate is correct. Put z=exp(-1), so 000054 | 1/3 (1/3)(15/16)^2=75/256 000056 | >1/9>log(10/9). 000057 | The last inequality is log(1+u)0. Thus the normalized reflected 000058 | expectation is also strictly negative; the normalization constant is positive. 000059 | There is no numerical tolerance, spectral crossing or zero-mode issue. 000060 | 000061 | 3. General profile calculation 000062 | 000063 | For the source's positive Laplace representation, 000064 | Delta=2 integral exp[-s(a^2+g^2)] [cosh(2sag)-1] dnu(s). 000065 | The integral is strictly positive whenever nu gives positive mass to s>0, 000066 | equivalently whenever this finite-measure Laplace profile is nonconstant. 000067 | A constant part at s=0 cancels. All terms are finite for a>g>0. 000068 | 000069 | For an arbitrary such profile the specific base ratio 9/10 need not violate 000070 | RP: its violation still requires Delta>log(10/9). The general invalidity of 000071 | the implication nevertheless persists for every nonconstant profile: choose 000072 | K0=[[1,r],[r,1]], exp(-Delta)=0, 000084 | with B_k supported on the positive half and limits justified in a topology 000085 | that preserves the relevant integrals. For an RP reference measure, 000086 | integral theta(F) F w dmu 000087 | =sum_k c_k integral theta(B_k F) (B_k F) dmu >=0. 000088 | For a genuine positive reweighted measure also retain w>=0 and a finite 000089 | positive normalizer. In finite split spaces the same sufficient statement 000090 | is the Schur product theorem: K0>=0 and weight matrix W>=0 imply 000091 | K0 entrywise-times W>=0. Entrywise positivity of W is insufficient. 000092 | 000093 | Factorization w=theta(B)B is a simple special case; a reflection-splitting 000094 | action has this form after exponentiation. The actual nonlocal spectral 000095 | trace requires its own such factorization, cone membership, or a direct 000096 | weighted RP proof. The example's W has same-diagonal value exp(-S_same) 000097 | smaller than its off-diagonal value exp(-S_opp), so W is not positive 000098 | semidefinite and the proposed preservation inference cannot be used. 000099 | 000100 | Primary comparison checked online: Jobst Ziebell, theta-splitting densities 000101 | and reflection positivity, Letters in Mathematical Physics 114,53 (2024), 000102 | Definition3.4 and Theorem3.5; Lemma2.4 gives the Schur product argument. 000103 | https://link.springer.com/article/10.1007/s11005-024-01799-8 000104 | Its theorem uses an explicitly theta-splitting density and its stated 000105 | Gaussian reference assumptions. It is not a general positive-weight theorem. 000106 | The elementary finite split proof above is given separately and does not 000107 | import a Gaussian hypothesis into this counterexample. 000108 | 000109 | 5. Final source boundary 000110 | 000111 | Established: the precise generic inference in the original proof is false 000112 | under the advertised positivity/covariance reasons alone. Gappedness, 000113 | primitivity of K0 and reflection covariance of D do not repair it. 000114 | Not established: that F is a TFPT admissible observable; that this finite 000115 | configuration space is the original BRST-reduced geometric class; that the 000116 | chosen profile is the actual branch fixed point; or that the actual TFPT 000117 | measure violates RP. No global P1/P2 exclusion, no replacement field theory 000118 | and no full fundamental solution is claimed. The exhaustion and projective 000119 | consistency parts of the original theorem were not tested here. 000120 | 000121 | Independent analytic/source review only. No original source changed and 000122 | no suite or new model-search harness was run. Only this report was written. 000123 | 000124 | 6. Completed canonical wording and checker review 000125 | 000126 | Subsequently read the full canonical PROOF.txt, the 123-line 000127 | check_spectral_weight_rp.py and STAND-EINFACH-DE.txt under 000128 | experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ 000129 | source_sewing_20261029/. No additional execution was requested or performed. 000130 | The checker implements the reviewed spectra, reflection covariance, gap, 000131 | polynomial Delta identity, rational sign bound and normalized observable 000132 | correctly. Matrix heat exponentials are marked as a numerical reproduction, 000133 | not the proof of the strict sign. The reference check concerns a fixed 000134 | source-independent denominator, not reference-dependent numerator terms. 000135 | 000136 | The canonical text keeps the finite counterexample separate from the full 000137 | BRST/geometric P1 class, retains the old microscopic/CAR/KMS/HS source path, 000138 | and explicitly allows a direct neutral local embedding without first 000139 | constructing the entire ambient geometric measure. Tr f(D^2), a fermionic 000140 | kinetic determinant and a source normalization are not identified. 000141 | No scientific correction is required. Canonical REVIEW-ADDENDUM.txt records 000142 | this final source-text/code review and its limited scope.