Established in scope
Spectral heat covariance positivity does not establish time reflection positivity
In the question: Automated construction of operator maps and sources
If reflection-stable already means OS positivity,the mixture is valid and the property is a hypothesis. If it means only reflection invariance,the general inference fails. No claim about actual unspecified original covariance,no all-P1 countermodel or physical completion. Existing Boltzmann/Schur reweighting counterexample is distinct. Seed RP,whole Clifford RP,valid energy semigroups and free cusp covariances remain intact. No renewed genetic scan in the completely excluded smooth class.
Read the exact result, assumptions and evidence
Result
Original proof calls its Euclidean covariance a positive mixture of reflection-stable heat kernels but defines neither its operator function nor heat-parameter relation to geometric time. For one spectral heat covariance,exactly c_s(2a)c_s(2b)/c_s(a+b)²=exp[-(a-b)²/(2s)]<1. Reflection invariance and positive Laplace weights alone therefore do not imply OS positivity. The entire stated continuous scalar translation-invariant C1 covariance class is settled: ordinary PD and RP force c constant. From c0 c(2t)>=c(t)² and d=1-c/c0 obtain 0<=d(t)<=2^n d(t/2^n)->0. Actual energy-semigroup Gram kernels and suitable positive Stieltjes/resolvent mixtures remain RP; no P1 selection follows.
Required assumptions
- Scalar spectral heat covariance and actual time reflection
Test only the general inference for scalar translation-invariant Gaussian covariances on R with reflection tau->-tau. Single test C_s=exp[-s(-partial_tau²+omega²)],s>0,omega>=0. Class theorem additionally assumes real,even,continuous ordinarily positive definite c,c(0)>0,C1 at0 and positive reflected Hankel form. No claim that original selects this covariance. Already assumed OS-positive channels,fermionic/matrix reflections,contact/distribution kernels and purely spatial smoothing are different contracts.
- Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Original evidence
- HEAT-COVARIANCE-REVIEW.txt lines 1–140
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/HEAT-COVARIANCE-REVIEW.txt
169194868f1ccbe1ca45e4b6166a3ccb65eb286331fde105b6992442e7455989
- 002-section-closure-theorems-from-the-admissibility-complex.tex lines 267–310
Source checked at this snapshot
Original location and fingerprint
documents/archive--tfpt-45--source_extracts--04_qft_source/sections/002-section-closure-theorems-from-the-admissibility-complex.tex
61ecee7727bdec47936bbffc028d3dbac5260a025b6dbcce30124a60d635fc9c
- 001-section-primitive-core-and-carrier-interface-handoff.tex lines 345–350
Source checked at this snapshot
Original location and fingerprint
documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/001-section-primitive-core-and-carrier-interface-handoff.tex
7d6e673d99c29d950dc601283cd51c1cc0b113567eeb79c8396fec9c82a02c79
- PROOF.txt lines 1–184
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_sewing_20261029/PROOF.txt
9b6ed0c36862913dac9762c0d1a6cd1e9f9c9a5f6b4b905ac8de5dccc25424b1
finding:spectral-heat-covariance-positivity-does-not-imply-time-rp