TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HOLONOMY-E0-SOURCE-REVIEW.txt Original file SHA-256: 58933c7051e1b008adecf553a607a93be768e0bbcfec17c6a50428a190f0982b Inclusive original lines: 401-497 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 | Pi_triangle = Hol_Gamma12 Hol_Gamma23 Hol_Gamma31 (16) 000402 | 000403 | and calls (16) the "ordered closed family holonomy". With the ordinary 000404 | parallel-transport convention Hol_ij:F_j->F_i, its factors are composable and 000405 | their product is an endomorphism of F_1. Hence, if nabla_F^star is identified 000406 | with the v117 flat local system, functoriality puts Pi_triangle in the actual 000407 | v117 monodromy image. 000408 | 000409 | There is a structurally richer formula earlier in the source, 000410 | 000411 | (Y_f)_ij=, 000412 | 000413 | but K_f^str is absent both from the diagonal path formula and from (16). The 000414 | displayed definition of Pi_triangle therefore does not yet contain a 000415 | structural-kernel insertion or a separate open-leg phase. 000416 | 000417 | Sources: 000418 | 000419 | * documents/archive--tfpt-45--source_extracts--03_em_flavor_source/sections/ 000420 | 004-source.tex:164-200, 000421 | SHA256 107d525221eb816043dbf4fb1a3c7e8244d3e24a5a5bcbfe1c2c34d8ee192f4a. 000422 | * the same source tree, 005-source.tex:252-299, 000423 | SHA256 786d2619e775775d70adf177794138485b261968042a880b2f0f617bb0111814. 000424 | 000425 | 8.2 Exact smallest character test 000426 | 000427 | The exact v117 checker enumerates the full image =W(A3)=S4 in its 000428 | three-dimensional twisted reflection representation. Its character values are 000429 | 000430 | tr(g) in {3,-1,0,1}. (17) 000431 | 000432 | The orientation-preserving A4 subgroup has only {3,-1,0}; allowing the full 000433 | S4 extension adds 1 but no complex trace. Consequently every closed v117 000434 | triangle satisfies 000435 | 000436 | arg tr(Pi_triangle) in {0,pi} 000437 | 000438 | when the trace is nonzero; for trace zero the argument is undefined. In 000439 | particular it cannot equal pi/3 plus a small real correction. Conjugating the 000440 | representation, or converting Hom to a bilinear tensor with the parallel 000441 | self-duality B from Section 7, does not change this closed-loop trace. 000442 | 000443 | This is an exact character obstruction, not a numerical fit. It uses the 000444 | already enumerated 24 matrices in verification/v117_monodromy_weyl_a3.py: 000445 | 161-182, SHA256 000446 | 55eca6a64144f6e49789f3f1bbe84e256764e7ce92b207044be693f8684d5df5. 000447 | An independent read-only replay of those exact generators returned the S4 000448 | trace counts 3^1,(-1)^9,0^8,1^6 and, for the subgroup generated by M0 and 000449 | U M0 U^(-1), the A4 counts 3^1,(-1)^3,0^8. 000450 | 000451 | 8.3 Where the nonreal phase enters in the current sources 000452 | 000453 | The proof sketch following (16) says that a Z6 transport grammar fixes the 000454 | base phase pi/3 and that a nonabelian area term supplies the quadratic 000455 | correction. It does not give an operator that inserts the Z6 factor into 000456 | (16). The later CKM closure section simply reuses the same pure-holonomy 000457 | definition and the same asymptotic. 000458 | 000459 | The separate red-team account is more explicit about the status. It locates 000460 | pi/3 in the hexagonal unit rho=exp(i pi/3), a Z6 phase fiber over the Z4 seam, 000461 | and states that only the leading term is character-covariant. It also records 000462 | the invariant core Im(rho) as selected rather than derived and keeps the 000463 | physical CP derivation open. This supports a separate phase-fiber reading; it 000464 | does not turn the pure v117 trace in (16) into a nonreal number. 000465 | 000466 | Sources: 000467 | 000468 | * documents/archive--tfpt-45--source_extracts--03_em_flavor_source/sections/ 000469 | 007-subsection-ckm-and-pmns-closure.tex:3-47, 000470 | SHA256 f9bb72dfc8d1c5bbce80a2a174c65b23eed35e827423d5b76bcf778e4b4f0f9f. 000471 | * documents/tfpt_5_redteam/sections/002-section-outcome.tex:188-240, 000472 | SHA256 f8def5572fca9e8295f9254eeb8935c614368e817f3d79d2f2fb96cc298b3e94. 000473 | 000474 | 8.4 Precise consequence for the proposed original identification 000475 | 000476 | The following three statements cannot all hold at once: 000477 | 000478 | (a) nabla_F^star is exactly the finite v117 W(A3) local system; 000479 | (b) Pi_triangle is exactly the pure closed product (16), with no additional 000480 | typed transport, structural-kernel or phase-fiber leg; 000481 | (c) arg tr(Pi_triangle) is genuinely nonreal, in particular on the branch 000482 | pi/3+3 lambda_C^2+O(lambda_C^4). 000483 | 000484 | The original text establishes the syntax of (b) and gives (c) only as a proof 000485 | sketch/asymptotic tied verbally to the Z6 grammar. It does not establish the 000486 | direct identity (a). Therefore this test does not refute a broader physical 000487 | SU(3)_F connection or a correctly typed Z6/open-leg completion. It does refute 000488 | the immediate identification of that nonreal closed-triangle phase with the 000489 | trace of the bare v117 monodromy. 000490 | 000491 | The self-duality result of Section 7 remains useful for reconciling Hom and 000492 | bilinear Yukawa typing. It cannot repair this trace obstruction: an 000493 | intertwining pairing changes the presentation of the same monodromy and leaves 000494 | its character real. The first missing exact datum is now sharply located: the 000495 | source must either supply a connection with genuinely complex closed-loop 000496 | character or display the additional typed Z6/structural/open-leg factor inside 000497 | the phase observable. The asymptotic sketch alone supplies neither.