Conditional result
Weak determinant class carries index one independently of internal rank
finding:matrix-clutch-degree-connects-weak-determinant-class-without-ray-selector
On the negative vector Fourier sea Q-P is trace class and Tr(Q-P)=sum_j j tr(g_j* g_j)=wind detg for smooth g. Thus det_rel=(1-z)^(wind detg). uI_r has indexr,while an actual degree-one U2 clutch has indexone. Originals v31/v32/v42 already contain the weak degree-one line and its assignment to total seam twist. S(U3×U2) is a fiber stabilizer and does not force a globally trivial determinant line. A common realization on the same source covariance therefore yields1-z without a rank-one choice.
Scope: Analytic matrix-index theorem proved; original class is reused as a branch statement,not newly derived from P1/P2. Normal S2 clutch equator is not automatically the tangential field-source circle. Ordinary admissible EM trace and relative pair response still require an original intertwiner.
Required premises and original source excerpts
- 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.
premise:trace-class-polarization-pair-with-restricted-unitary - Smooth matrix clutch on original negative Fourier sea
H=L2(S1,C^r),P=1_(n<0) tensor I_r and U=M_g for smooth g:S1→U(r). The existing bosonic branch has c1(det E2)=1,c1(det E3)=0 on the normal S2. A common physical identification of that clutch equator with the tangential source circle,C_Sigma and response is an additional map to prove.
premise:smooth-matrix-clutch-on-original-negative-fourier-sea - 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.
premise:historical-original-circle-and-transfer-source-passages - 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.
premise:typed-source-structures
- experiments/theory-contracts/original-admissible-source-closure-20261025/PAIR-INDEX-DETERMINANT.txt, lines 79–151
Source SHA256: 7c7dc9d48a96a140afeb72ddb2d0e6ae2f37e73e6260426544b9a94d7abac04d
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- experiments/theory-contracts/original-admissible-source-closure-20261025/ADMISSIBLE-INDEX-CARRIER-SELECTOR.txt, lines 130–240
Source SHA256: 3364dc34b8152e1edbfa0711b97a95f14c3069cf40ce313ac23081e99cd397fe
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- documents/archive--paper-latex--tfpt-42/sections/012-subsection-bosonic-relative-index-and-higgs-selection.tex, lines 3–82
Source SHA256: 8140f7618b51292b8acf79fdc999cac9482842626ed07db4edb54dc91c9499d6
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- documents/archive--paper-latex--tfpt-42/sections/005-subsection-internal-carrier-stabilizer-and-one-family-pa.tex, lines 72–98
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- documents/archive--paper-latex--tfpt-31/sections/007-subsection-occupancy-and-boundary-to-bulk-bridge.tex, lines 142–169
Source SHA256: 9c441d14de6a6c9586a4d2b077b160a4c5ce020d82243ff832300ff9554e6397
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- documents/archive--paper-latex--tfpt-32/sections/007-subsection-occupancy-and-boundary-to-bulk-bridge.tex, lines 122–149
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