Established in scope
Pair index fixes relative factor without a primitive rank-one ray
In the question: Common clock sectors, quartic geometry and operational reconstruction
Extension of the existing fixed-shift theorem,no new general index mathematics. Ordinary relative Fredholm determinant on the same space; no selected physical kernel,transfer spectrum,5/4 index or EM Ward derivation.
Read the exact result, assumptions and evidence
Result
Orthogonal P,Q with Q-P in L1 and k=Tr(Q-P) obey det_F[(I-zQ)(I-zP)^-1]=(1-z)^k on|z|<1. Exactly R_prime R^-1=(I-zQ)(P-Q)/(1-z)(I-zQ)^-1. For Q=UPU* and any positive contraction C with C-P in L1,three-factor cancellation yields the same factor. Pair index therefore suffices even for indefinite or infinite-rank defects; this scalar response needs no preferred ray.
Required assumptions
- Trace-class polarization pair with actual unitary action
P,Q are orthogonal projections on the same H,Q-P in L1. The covariance theorem additionally assumes Q=UPU*,U unitary,0<=C<=I,C-P in L1. The theorem is on this common space; physical source membership is not inferred from index alone.
- Historical concrete circle, heat-transfer and source-functional passages
Historical audit contract: fullv21 contains D_Gamma=i partial_theta, u=exp(i theta) and twisted H1 domains; v31/32/42 write P_adm exp(-ell|B_Sigma|)P_adm; v42 has exp(W_k[J]) as a formal functional integral with STr. V1.06 contains the 11D C3 mode ansatz and n=1,g_top=8c3². The actual scope is audited, without assuming equality of old/new operators, physical times, traces or parameters.
Original evidence
- PAIR-INDEX-DETERMINANT.txt lines 1–114
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-admissible-source-closure-20261025/PAIR-INDEX-DETERMINANT.txt
7c7dc9d48a96a140afeb72ddb2d0e6ae2f37e73e6260426544b9a94d7abac04d
- PAIR-INDEX-DETERMINANT-REVIEW.txt lines 1–166
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-admissible-source-closure-20261025/PAIR-INDEX-DETERMINANT-REVIEW.txt
f85695ecf75c5935e469a4fedb9e72fa0920ea90d9189a33c5715526bc11acc6
- check_pair_index_determinant.py lines 1–116
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-admissible-source-closure-20261025/check_pair_index_determinant.py
cf944975609e83c78ae089f0e70d1b3fad9f9d2edda613a763af5f0babcb1904
- pair-index-results.json lines 1–81
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-admissible-source-closure-20261025/pair-index-results.json
eec63376ac978b2c8c09c55bb25f0ca919652ab77cce328c0028e9cf1b925fe2
finding:trace-class-pair-index-fixes-relative-factor-without-primitive-ray