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: 201-400
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.

000201 |           v                            v
000202 |   reflected path groupoid --Hol_A--> SU(3) family fibers
000203 | 
000204 |                  | claimed typed identification
000205 |                  v
000206 | 
000207 |   (Gamma_ij^min,nabla_F^star,e_i) --hard holonomy--> Y_f.
000208 | 
000209 | The top square is proved exactly by (4)-(5). What remains is to locate in the
000210 | original source, or derive from its selector, the vertical identification that
000211 | sends the actual v117 paths and horizontal frames to the physical
000212 | Gamma_ij^min and e_i. If such an identification includes a coordinate-indexed
000213 | family of paths, it supplies Y_f(z) and makes (8) a derived whole-source map.
000214 | If it contains only the finite canonical edge set stated in (1), the output is
000215 | the fixed hard-holonomy matrix and no nonconstant Phi field follows.
000216 | 
000217 | No physical P1 selection, new state, new time, new Majorana rule or freely
000218 | chosen Yukawa profile is claimed.
000219 | 
000220 | 7. FIRST TYPE TEST: HOM(F,F) VERSUS THE BILINEAR DUAL LEG
000221 | 
000222 | This section records the decisive correction to a generic SU(3) argument.
000223 | The actual v117 representation is not a generic complex SU(3) module. Its A4
000224 | monodromy is the complexification of the real signed-permutation rotation
000225 | representation. It therefore has a one-dimensional space of parallel
000226 | nondegenerate symmetric complex bilinear forms. In the signed horizontal
000227 | frame one may write a generator as
000228 | 
000229 |   B_0=I,  R^T B_0 R=B_0
000230 | 
000231 | for the V4 signs, the three-cycle P_c and the marked linear quarter matrix.
000232 | Thus F_v117 is self-dual as a flat complex local system. The absence of a
000233 | generic SU(3)-invariant two-tensor is not an exclusion in this actual finite
000234 | representation.
000235 | 
000236 | 7.1 What the original hard-holonomy formula types
000237 | 
000238 | The formula itself nevertheless chooses the Hom type. It uses one Hermitian
000239 | family bundle L_F, one canonical family basis {e_i}, and
000240 | 
000241 |   <e_i, H e_j>,  H=Hol_Gamma(nabla_F^star) K_f^str in End(F). (9)
000242 | 
000243 | The left and right word factors D_L,f and D_R,f are diagonal scalar path
000244 | weights. The source does not introduce separate F_L and F_R bundles, does not
000245 | declare the right endpoint to lie in F^*, and does not insert a bilinear
000246 | evaluation tensor between the two endpoints. Accordingly a common unitary
000247 | frame change acts by
000248 | 
000249 |   H -> R H R^(-1),                                             (10)
000250 | 
000251 | and the actual E0 anti-transport acts by the Hom law
000252 | 
000253 |   H -> D_E bar(H) D_E^(-1)=D_E bar(H)D_E^*.                    (11)
000254 | 
000255 | This agrees with the original v252 Dirac block. There the same generation
000256 | index set labels nu_L,nu_R,e_L,e_R,u_L,u_R,d_L,d_R, and the particle block is
000257 | explicitly the Hermitian L<->R operator
000258 | 
000259 |   S_LR=Y,  S_RL=Y^*.
000260 | 
000261 | If the same family anti-transport D_E is used on L and R, its LR block obeys
000262 | (11). This is the old particlewise source typing. It should not be replaced by
000263 | a chirality-conjugate family action merely because another E8 realization
000264 | admits one.
000265 | 
000266 | Direct sources:
000267 | 
000268 | * documents/archive--tfpt-45--source_extracts--03_em_flavor_source/
000269 |   sections/004-source.tex:115-197 and sections/005-source.tex:252-267.
000270 | * verification/v252_full_finite_triple.py:7-24,63-115,
000271 |   SHA256 820254440e178430422e50605067775656236602996307f67c9e69124ba5d163.
000272 | 
000273 | 7.2 Native marked phase test
000274 | 
000275 | In the calibrated horizontal frame the actual E0 factor is
000276 | 
000277 |   D_E=omega^2 S,  S=diag(1,-1,1),  omega^3=1.
000278 | 
000279 | It satisfies two different identities:
000280 | 
000281 |   D_E^*D_E=I,             D_E^T D_E=omega I.                  (12)
000282 | 
000283 | Therefore the two proposed laws differ already on the identity holonomy:
000284 | 
000285 |   Hom:       D_E I D_E^*=I,
000286 |   bilinear:  D_E I D_E^T=omega I.                            (13)
000287 | 
000288 | The identity path must remain the identity under functorial transport. Hence
000289 | the raw hard holonomy cannot itself obey the second law. A bilinear reading
000290 | requires an additional self-duality leg B:F->F^* (or its inverse), carrying
000291 | the compensating E0 phase. This is the smallest native-marked distinction; it
000292 | uses the actual D_E, the actual Real/quarter composition E0=Q tau_0 and no
000293 | generic representation argument.
000294 | 
000295 | 7.3 The actual self-duality leg and its one remaining phase
000296 | 
000297 | For J_E(v)=D_E bar(v), a symmetric form B=cI is E0-real precisely when
000298 | 
000299 |   B(J_Ev,J_Ew)=bar(B(v,w))
000300 | 
000301 | or equivalently
000302 | 
000303 |   D_E^T B D_E=bar(B),  c omega=bar(c).                        (14)
000304 | 
000305 | Thus the real-type representation does permit a repair. Equation (14) fixes
000306 | the phase of c up to a nonzero real scale; for example
000307 | 
000308 |   c=exp(-i pi/3)
000309 | 
000310 | works; equivalently c=-omega. With such an E0-real b:F->F^*, set
000311 | 
000312 |   K=H B^(-1) : F^* -> F.
000313 | 
000314 | Equation (14) implies
000315 | 
000316 |   D_E^(-1)B^(-1)=bar(B)^(-1)D_E^T,
000317 | 
000318 | and therefore
000319 | 
000320 |   K^E=H^E B^(-1)
000321 |      =D_E bar(H)D_E^(-1)B^(-1)
000322 |      =D_E bar(K)D_E^T.                                     (15)
000323 | 
000324 | Thus converting H by the E0-real b leg makes the Hom and bilinear
000325 | transformation rules exactly equivalent. If instead B_0=I is held
000326 | fixed in the displayed signed frame, E0 transports it with the nontrivial
000327 | character omega, and the two matrix laws differ by exactly that character.
000328 | 
000329 | The decisive source question is therefore no longer whether a self-duality
000330 | exists. It does. The question is whether the original source fixes the phase
000331 | in (14), and inserts that particular b leg into the Yukawa evaluation.
000332 | 
000333 | The currently read sources do not do so:
000334 | 
000335 | * The sign-line source fixes the three two-torsion local systems and their
000336 |   unitary flat structures, but explicitly says that equality with a particular
000337 |   original flat fiber basis still needs constant frame calibration.
000338 | * Its Real/quarter review retains two lift choices and states that complex
000339 |   frame phases change the antilinear matrix without changing the line class.
000340 |   It therefore does not canonically choose the scalar c in (14).
000341 | * The algebraic G construction fixes a useful normalized fundamental matrix,
000342 |   but the invariant complex bilinear is still unique only up to a nonzero
000343 |   scalar. The positive Hermitian metric fixes adjoints and norms; it does not
000344 |   fix this complex bilinear phase.
000345 | 
000346 | Relevant source boundaries:
000347 | 
000348 | * experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/
000349 |   SIGN-LINE-COVER-DICTIONARY.txt:7-40,56-87,127-136,
000350 |   SHA256 555f75a2d3c67347fa5f8e3c4f7f46e79d94cb4172176c84010224585dc842e7.
000351 | * aps_source_review/C3-REAL-QUARTER-COMPATIBILITY-REVIEW.txt:21-57,59-84,
000352 |   SHA256 e118c54415cb2b80174a4950b146e8050f787bc1f536f3fb0068054f6420a242.
000353 | 
000354 | 7.4 The existing Yukawa top form is not yet this bridge
000355 | 
000356 | The carrier formalization contains an alternating top form
000357 | 
000358 |   omega_Y : Lambda^3 E -> K
000359 | 
000360 | and its contraction E->(Lambda^2 E)^*. This is a genuine invariant
000361 | three-tensor and is compatible with a three-dimensional self-duality story.
000362 | However, its own interface states explicitly that deriving the canonical form
000363 | from the TFPT boundary kernel is still the upstream physics question. It does
000364 | not identify E with the v117 family local system, does not provide a selected
000365 | vector/Higgs evaluation that converts an endomorphism H into a two-leg form,
000366 | and does not fix the E0 phase (14). A top form alone cannot supply that missing
000367 | one-leg contraction.
000368 | 
000369 | See:
000370 | 
000371 | * experiments/lean4-carrier-rigidity/TfptCarrier/
000372 |   YukawaTrilinearForm.lean:76-115,
000373 |   SHA256 c64e60913792d5627eae8ac58f4d69336598f07bb9593b5ddd43a660618d2d91.
000374 | * the same directory, BoundaryYukawaKernelInterface.lean:15-57,69-93,
000375 |   SHA256 63cbb56007de24900f1cd632751b85ccbd4cafbab2e368a689b7987e4c7cf1ac.
000376 | 
000377 | 7.5 Refined verdict
000378 | 
000379 | The original hard-holonomy and v252 source are presently typed as Hom(F,F)
000380 | with the Hermitian adjoint on the reverse LR block. The new chirality-dependent
000381 | E8 action, which yields D_E bar(Y)D_E^T, is a different representation choice
000382 | unless its self-duality/evaluation leg is supplied.
000383 | 
000384 | The actual finite v117 local system makes that bridge mathematically possible:
000385 | its parallel symmetric form is unique up to scale, and E0 fixes the required
000386 | phase condition (14). But the original sources read here do not yet select
000387 | that phase or insert the leg into the hard-holonomy Yukawa rule. Therefore the
000388 | omega in the bilinear law is neither an inconsistency nor already a derived
000389 | physical phase. It is the exact marker of the still missing original pairing
000390 | leg.
000391 | 
000392 | 8. Closed-triangle character test: the v117 identity is too small for the
000393 | displayed CKM phase
000394 | 
000395 | 8.1 The three paths are asserted to form a genuine closed loop
000396 | 
000397 | The original flavor source first defines Gamma_ij^min as canonical admissible
000398 | edges of one D4-invariant geodesic spine. It then states, without an extra
000399 | endpoint or evaluation operator,
000400 |