TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/SAME-SOURCE-AFFINE-DEFECT.txt Original file SHA-256: 514356e9ca01f8aa02399020a503f6cd782c879b00220274b7c4377cbbad7255 Inclusive original lines: 401-565 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000401 | 000402 | 000403 | 8. Transport to the actual fixed M frame 000404 | ---------------------------------------- 000405 | 000406 | The source result is transported by the already fixed orthogonal dictionary 000407 | M. Set 000408 | 000409 | D8_F = M(D8_CAR), 000410 | R_F = M R M^(-1), 000411 | h_F = M e0 000412 | = (1/2,1/2,0,0,0,1/2,1/2,0), 000413 | d = M delta. 000414 | 000415 | Then the actual target-frame affine map is 000416 | 000417 | q -> R_F q+h_F, 000418 | 000419 | and the transported charge is 000420 | 000421 | Q_F(q)=(M 1).q. 000422 | 000423 | For 000424 | 000425 | K_F(q)=N+q^2/2+d.q 000426 | 000427 | one has identically 000428 | 000429 | K_F(R_F q+h_F)-K_F(q)=(Q_F(q)+alpha)/8. 000430 | 000431 | All charge, oscillator, cocycle and sector words above transport together by 000432 | M. This is a simultaneous coordinate transport of the actual source frame. 000433 | It does not assert 000434 | 000435 | D8_F = canonical compiler D8. 000436 | 000437 | The canonical compiler D8 remains a distinct marked audit frame. Its 000438 | previously proved fixed-frame inequality with D8_F is neither used nor turned 000439 | into a physical obstruction here. 000440 | 000441 | 000442 | 9. What is closed and what remains open 000443 | --------------------------------------- 000444 | 000445 | Closed mathematically: 000446 | 000447 | A. A unitary W_ext exists on the full four-module D8 abelian intertwining 000448 | algebra and its Hilbert completion. 000449 | 000450 | B. It restricts exactly to the old CAR W1 on NS states, including phases. 000451 | 000452 | C. Its eighth power, sector exchange, E8 twist, cocycle transport and Ward 000453 | identities are exact. 000454 | 000455 | D. The construction lives in the actual fixed M frame and needs no global 000456 | odd Phi map. 000457 | 000458 | E. The Cartan and standard spin-sector lifts and their four-word signs are 000459 | explicitly separated, while the apparent kappa freedom is removed by 000460 | the intertwiner-basis gauge U_tau. 000461 | 000462 | F. Consequently the finite-energy four-sector continuation in the actual 000463 | M frame is unique up to intertwiner basis, with no remaining kappa 000464 | choice in the unmarked mathematical object. 000465 | 000466 | Still required for a physical TFPT source statement: 000467 | 000468 | 1. Primitive P1 or the original seam data must select the four-module 000469 | extension H_* and identify its local observable net with the physical 000470 | source. The present theorem starts once H_* is admitted. 000471 | 000472 | 2. The physical construction must select the geometric Cartan operator or 000473 | the standard defect lift on v+c if a compiler Gaussian or Arf word is to 000474 | be compared with it. NS data and the untwisted E8 character do not make 000475 | those two operators equal. No separate physical selection of kappa is 000476 | required unless a particular v-to-c action of X is additionally marked 000477 | and held fixed. 000478 | 000479 | 3. A local defect field, if required, needs a localization theorem beyond 000480 | this global finite-energy intertwiner. 000481 | 000482 | 4. The physical state must decide how the additional E8 half-charge 000483 | zero-energy line participates. Ward covariance alone does not select a 000484 | state. 000485 | 000486 | 5. The earlier P1-to-QWZ and QWZ-to-this-specific-CAR-fold selection remains 000487 | the upstream source question. The present result proves the affine 000488 | completion after that specific CAR fold has been selected. 000489 | 000490 | Thus the smallest precise conclusion is: 000491 | 000492 | The full affine W1 defect is mathematically closed on the stipulated 000493 | same-source four-module D8 system. Physical P1 selection and the operational 000494 | choice of geometric versus standard twist lift remain separate inputs. 000495 | 000496 | 000497 | 10. Verification and sources 000498 | ---------------------------- 000499 | 000500 | Exact checker: 000501 | 000502 | experiments/theory-contracts/canonical-seam-fermionization-20261025/ 000503 | check_same_source_affine_defect.py 000504 | 000505 | The checker verifies: 000506 | 000507 | * ordered-CAR cyclic cocycle covariance; 000508 | * associativity of the four-sector extension on generators; 000509 | * exact restriction to the old Z8 cocycle; 000510 | * exact restriction to the fixed E8 cocycle; 000511 | * both forced kappa choices; 000512 | * their exact conjugacy by U_tau, pointwise identity on 0+v and 0+s; 000513 | * invariance of fully transported mixed words and the fixed-X scalar 000514 | W_kappa X W_kappa^* X^*=-i kappa I; 000515 | * the four-sector affine action; 000516 | * W_ext^8=E_1 with its charge-dependent cocycle phase; 000517 | * Cartan and standard lift signs and their W1 commutation laws; 000518 | * arbitrary-charge quarter Ward identities, including p_*; 000519 | * the transported Ward identity in the actual M frame. 000520 | 000521 | Observed checker verdict: 000522 | 000523 | FULL_FOUR_MODULE_AFFINE_DEFECT_UNIQUE_UP_TO_INTERTWINER_BASIS 000524 | 000525 | Existing project inputs: 000526 | 000527 | experiments/theory-contracts/seam-joint-ward-state-20261014/PROOF.txt 000528 | experiments/theory-contracts/seam-joint-ward-state-20261014/ 000529 | operator_transport_check.py 000530 | experiments/theory-contracts/mixed-current-origin-20261014/ 000531 | ATLAS-KMS-LOCALITY-AND-SOURCE-WINDING.txt 000532 | experiments/theory-contracts/mixed-current-origin-20261014/ 000533 | ODD-SECTOR-REFIELDING-AND-NAHT.txt 000534 | experiments/theory-contracts/v16-operator-pullback-20261014/ 000535 | fold-results.json 000536 | experiments/theory-contracts/canonical-seam-fermionization-20261025/ 000537 | LATTICE-AND-SOURCE-FRAME.txt 000538 | 000539 | Relevant primary references for the formal interpretation: 000540 | 000541 | H. Li, Twisted modules and pseudo-endomorphisms, 000542 | https://arxiv.org/abs/1004.0864 000543 | 000544 | H. Li, Local systems of twisted vertex operators, vertex operator 000545 | superalgebras and twisted modules, 000546 | https://arxiv.org/abs/q-alg/9504022 000547 | 000548 | C. H. Lam and H. Shimakura, Orbifold construction of holomorphic vertex 000549 | operator algebras associated to inner automorphisms, Proposition 3.1, 000550 | https://arxiv.org/abs/1501.05094 000551 | 000552 | B. Bakalov and V. Kac, Twisted modules over lattice vertex algebras, 000553 | https://arxiv.org/abs/math/0402315 000554 | 000555 | A. Carey and E. Langmann, Loop groups, anyons and the Calogero-Sutherland 000556 | model, 000557 | https://arxiv.org/abs/math-ph/9805010 000558 | 000559 | J. Smith, Z. Lin, Y. Tachikawa and Y. Zheng, Classification of chiral 000560 | fermionic CFTs of central charge less than or equal to 16, 000561 | https://arxiv.org/abs/2303.16917 000562 | 000563 | C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex 000564 | Operators, Theorem 12.24. 000565 |