Conditional result
The same two-face collar fixes the current phase while common and relative boundary transports remain distinct
In the question: Historical: common source, state and event selection
Given scalar quarter circle and full product collar with actual charged CAR quantization; no P1-derived statistics or physical source selection. No novelty is claimed for general current algebra. Infinite analytic formulas plus exact finite controls, not finite CCR representations. Relative endpoint transport is an explicit frame/operator family, not a proven original boundary intervention with fixed reference/domain/Padm. Phase distinction needs full source access or coherent control; one-face conjugation channels can agree if the opposite face never recouples. Normal distance is not physical time; original time, zero mode and absolute reference remain separately retained. No native3+2 selector, pointer/record law,4D determinant-line theorem or T1-T8 closure. Existing4/5/20/carry/E8/eight-preimage structure remains.
Read the exact result, assumptions and evidence
Result
UR.COLLAR.FACE_CURRENT.01: In the existing scalar quarter circle, the original G_ell Cauchy graph under the explicit charged-CAR/quasifree premise gives exactly the original level1 current cocycle on its second face, its negative on the first, and zero for the common lift. This holds for all smooth real profiles by sum_r(n_r-n_(r+k))=k, not a mode truncation. The same n gives beta=2ell, full thermal current response, the charge-retaining Hall/winding partition and a controlled limit to the original negative sea. Common g-direct_sum-g transport and I-direct_sum-g have the same second-face covariance but distinct full G and coherent phases. Even four-mark-fixed quarter-periodic sin4theta,1-cos4theta give exp(2ist) on one face versus1 jointly. A concrete geometric repair g_chi=exp(i chi(r)phi(theta)) gives L_phi=g_chi Lg_chi*=partial_r+B_alpha-i chi_prime phi-chi phi_prime and T_phi=g_phi T in the actual scalar transport block; its boundary graph is independent of the interior frame interpolation. This does not select physical execution.
Required assumptions
- Complete data of both product-collar faces
A selfadjoint B occurs in an exact normal-independent product equation. The full orthogonal L2 Cauchy graph projection of both labeled faces, their actual fiber identification and independently known positive normal length ell are supplied. Selection of these data from P1/P2 is not presumed.
- Given charged CAR quantization of the full two-face Cauchy graph
As in section4 of the original collar proof, additionally assume charged CAR(H direct_sum H) and the pure gauge-invariant quasifree state with occupation G_ell. The second labeled face has n=(1+exp(2ell B))^-1. This is a field/statistics premise, not a consequence of P1; reduction does not select or physically execute a one-face operation. The long-collar limit is used only within this declared operator/CAR class.
- Fixed scalar Hardy model with quarter time
H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2.
- Charged CAR algebra and original negative sea
Assume the actual field algebra is charged CAR(H) and its reference state is the pure quasifree sea with occupation 1−C. Relative charge and positive excitation time are normal ordered against this sea. This physical identification is an additional premise.
- Original single-circle density and geometric stress
Same stipulated chiral CAR source, negative-D sea, actual quarter domains, eightfold F and existing M/O. Local density J_m=sum:a_r* a_(r+m): and its half-density geometric stress with alpha ground energy follow from this source. Original scalar phases fold to correlated generally offdiagonal channel profiles, not independently selected E8 root weights. NS/quartic reference remains conditional. Global Fourier modes control existing fields; no P1-selected local profile/detector/coupling. No isolated b->-b invariance for these new responses; transport positive QWZ occupation with charges/currents/V/b together.
- 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
- PROOF.md lines 1–196
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/collar-current-source-bridge-20261021/PROOF.md
2dae051bb6a3ee77cc96bc0d63470ac4875950b46057865d1177e4ed3322e5ac
- checker.py lines 1–132
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Original location and fingerprint
experiments/theory-contracts/collar-current-source-bridge-20261021/checker.py
c7eb757690bd4ae238812f47bb99449f939a946f74dc89ce00ff5011fad6770a
- results.json lines 1–15
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Original location and fingerprint
experiments/theory-contracts/collar-current-source-bridge-20261021/results.json
b7368c956426f1dfa8dbeffc4239805829cac657acd209eadcd120e958578216
- PROOF.txt lines 96–143
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Original location and fingerprint
experiments/theory-contracts/seam-full-collar-reconstruction-20261015/PROOF.txt
9f4effed34316798c6a35e1ac0ec2832cd3720180f476cb8747bb32359b56e6a
- ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt lines 33–161
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Original location and fingerprint
experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt
62a9eeb4519b59d4514445f9d4dd5d2059e29179956a4c97ec94b7bae9ae3094
- ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt lines 36–99
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Original location and fingerprint
experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt
2abf4b44f4cb4ca713d4e520676ef8db0849e5142962521a9592550f1c16ddbe
- 017-source.tex lines 239–285
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Original location and fingerprint
documents/tfpt_research_contracts/sections/017-source.tex
492599a0d20043a5589ca610ce7e1ebceffb8de043eae8eb406e4469d3e4be83
- contract_index.json lines 1–61
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Original location and fingerprint
experiments/theory-contracts/collar-current-source-bridge-20261021/contract_index.json
80de96c729cf3e744da3fdc8ae5da56f3588692de12a8f62dd7602036d92e108
finding:two-face-collar-fixes-face-current-phase-and-distinguishes-relative-endpoint-lift