Established in scope
Whole weak cap converges across polarization changes
In the question: Automated construction of operator maps and sources
Complete stated constant weak cap comparison in coupled ground-energy normal ordering. No derivation of original P0,fields,normalization,Zrel or physicaltime and no nonconstant-source theorem. Large-time limit can vanish; strict positivity is finite-time. HS equivalence gives a common Fock vector, not an automatic original positive-time OS insertion or global Thouless chart. Orthogonal-line control refutes only that universal chart inference, not a pair of full TFPT sources. Original reference/bulk/phase prescriptions remain required.
Read the exact result, assumptions and evidence
Result
For the actual constant weak source, Zp=exp(ell E_p)det_RanP0(P0 exp(-ell h_p)P0) is the old Slater expectation of exp(-ell H_exc,p), with0<Zp<=1. For v=||V|| and g=|p|>2v, negative eigenvectors yield Tr(Ph P_wrong)<=48v²/(2g-v)². Ground overlap and spectral resolution give the same bound for1-Zp uniformly inell. Thus the product over all quarter modes is strictly positive and finite at every finite ell>0. Finitely many crossings or overlap zeros do not destroy the finite-time cap. Zero-mode occupations leave E_p and H_exc,p unchanged and do not affect the cap at fixedP0; only a particular ground vector/Thouless chart needs them. ConstantQ preserves Fourier modes but not the old scalar-pair reduction. The actual free-reference ratio retains exp[-ell(E_p-E_ref,p)]Zp. Full96 numerical controls confirm that factor and recover the t0 scalar formula; convergence is analytic.
Required assumptions
- 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.
- Actual constant weak quarter source with old boundary space and coupled normal energy
Given full96 source h_p=p gamma_F+V with V=tQ gamma_F-i gamma_F D_F selfadjoint, finite, constant and independent of quarter p. Retain old rank48 P0,p=1_(p gamma_F<0), existing exterior/CAR caller and coupled ground energy E_p=Tr(h_p 1_(h_p<0)) for the cap comparison. No original physical selection of P0,V,ell,source measure or energy subtraction is assumed.
Original evidence
- ROOT-C58-WHOLE-WEAK-CAP-AND-REFERENCE.txt lines 1–99
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ROOT-C58-WHOLE-WEAK-CAP-AND-REFERENCE.txt
6f4aa06a5fa5684b151ac6351136643502d4e8ce7819dd1289f4349ec78d8b4f
- C58-THOULESS-AND-CAP-REVIEW.txt lines 1–200
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C58-THOULESS-AND-CAP-REVIEW.txt
8539b8b4a901ae9cac751f299a2753ee06f75c25f95cdb74dc4a17b3b5f2a036
- C58-THOULESS-AND-CAP-REVIEW.txt lines 201–224
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C58-THOULESS-AND-CAP-REVIEW.txt
8539b8b4a901ae9cac751f299a2753ee06f75c25f95cdb74dc4a17b3b5f2a036
- C58-ORIGINAL-APS-AND-SEWING-REVIEW.txt lines 1–127
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C58-ORIGINAL-APS-AND-SEWING-REVIEW.txt
6efaea0323cd55e250bec2d1423151fb9bc05555c139907fd5944dcd525da34a
- C58-ACTUAL-WEAK-CAP-CHECK.json lines 1–81
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C58-ACTUAL-WEAK-CAP-CHECK.json
5b1a67671ab6abf656e55012641a70eec796465ad064e7e0a3eb904afd26b60f
finding:actual-weak-cap-converges-on-whole-quarter-circle