TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/theta_lattice_closure_20261027/eta/P1-ETA-SEWING-DUAL-LINE-TEST.txt Original file SHA-256: 350063b81d790219be7d19b1f13d0b3b76a6513930b82f45d5bb97cafc10711a Inclusive original lines: 201-352 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 | This is a compact phase variable inherited from the unit winding. It is not a 000202 | nonzero element of the charged defect line and does not record the beta,c,Q 000203 | character of K or K^*. 000204 | 000205 | Source: 000206 | 000207 | Atlas paper-source:00b5d3eea8530ff18431ddda 000208 | documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000209 | 001-subsection-hadronic-admissibility-as-singlet-selection.tex:64-154 000210 | SHA256 f377e790ec73b8f5c56e1270b1bcf163022bf86e039af707dbda9fb8017a50df 000211 | 000212 | The primitive seam determinant closure defines only 000213 | 000214 | log det_adm^prim(1-U_Sigma)=-Tr_adm^prim(U_Sigma) 000215 | 000216 | and then retains -Re log det. The source itself marks this result 000217 | closed_pending_convention_adoption. The real scalar has discarded precisely 000218 | the complex orientation that would distinguish K from K^*. 000219 | 000220 | Source: 000221 | 000222 | Atlas paper-source:79aeb188204547f963615df9 000223 | documents/archive--tfpt-45--07_cmb_operational_closure_planck_pipeline/ 000224 | sections/007-section-hook-solution-sketches-v4-6-5.tex:250-270 000225 | SHA256 7b7c7d3fa47cc00ef771818c65f2a5c31c0c705d3890b8128a2d2ebe6e1a16da 000226 | 000227 | 2.5 The primitive boundary theorem deliberately stops before determinant data 000228 | ----------------------------------------------------------------------------- 000229 | 000230 | The minimal boundary package is 000231 | 000232 | T_bound=(A_+,H_+,D_+,J,Gamma,B_Sigma) 000233 | 000234 | with |SF(U_Sigma)|=1. The source explicitly says that carrier ranks and 000235 | determinant classes are recovered later, and that no determinant classes or 000236 | family geometry belong to the core tuple. This is consistent with the source 000237 | providing the winding and polarization seed while leaving the determinant-line 000238 | readout to a later functor. 000239 | 000240 | The displayed admissible bordism category does not change this conclusion. 000241 | Its objects are boundary tuples and its morphisms are unitary intertwiners. 000242 | The source does not define a symmetric monoidal amplitude functor from those 000243 | bordisms to complex lines. Such a functor would send orientation reversal to 000244 | dualization and would turn the two seam faces into K and K^*. That is the 000245 | precise positive construction suggested by the original geometry, but it is 000246 | not part of the stated theorem. 000247 | 000248 | Source: 000249 | 000250 | Atlas paper-source:d886b99793c68c5e7e90941a 000251 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000252 | sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex:250-334 000253 | SHA256 1de90a4a7379bef804cc7815d72c0f86bb398a5771e9b00997c4bb15e8815f9e 000254 | 000255 | As a separate corroborating limit, the technical companion explicitly leaves 000256 | a determinant-line orientation seed outside its closed theorem stack and 000257 | labels the canonical seam record state and trace word as conjectures. That 000258 | passage concerns the later record readout, so it is not by itself a proof of 000259 | the present ket-dual gap. 000260 | 000261 | Source: 000262 | 000263 | Atlas paper-source:3bb91e34cb0042bd09ec82dc 000264 | documents/archive--tfpt-45--source_extracts--technical_companion_source/ 000265 | sections/002-source.tex:1-35,175-270 000266 | SHA256 b923aabd635b649ab5282ba9255df4d81ba56c3114e8c1289ad9c728b1b4eef8 000267 | 000268 | 000269 | 3. C3 NEED NOT BE A PHYSICAL BULK SOURCE 000270 | ======================================== 000271 | 000272 | The already proved auxiliary construction uses the existing curve 000273 | 000274 | C: eta^2=1-w^6, 000275 | z=eta/(1-w^3), div(z)=D_q-D_p, 000276 | 000277 | and the selected spin family S_0. On the three unramified discs over |z|<1, 000278 | multiplication by z has cokernel 000279 | 000280 | V=direct_sum_a S_0|q_a 000281 | 000282 | and Fredholm line det(V)^(-1). This construction is a determinant functor on 000283 | the auxiliary marked family. A determinant functor does not require C to be 000284 | literal physical spacetime or a physical bulk. For the abstract complex 000285 | module, the C3 curve is allowed to be the functorial coefficient geometry 000286 | which realizes the already fixed line and its c,Q,beta action. 000287 | 000288 | The stronger physical statement needs only a local natural transformation, 000289 | not an identification of all P1 and C3 Hilbert spaces: 000290 | 000291 | actual cyclic P1 source sector 000292 | -> three-sheet S_0 Hardy source sector, 000293 | 000294 | intertwining the actual seam shift with M_z and carrying the resulting 000295 | relative determinant functor to K^*. Equivalently, one can specify directly a 000296 | sewing covector 000297 | 000298 | A_Sigma in Hom(K,C)=K^* (11) 000299 | 000300 | with the established beta,c,Q marks. Either construction makes the auxiliary 000301 | curve a coefficient and determinant-line realization. Neither requires it to 000302 | be a new physical boundary component or bulk field. 000303 | 000304 | What remains absent from the original formulas checked here is exactly this 000305 | natural transformation or the typed covector (11). Requiring a physical C3 000306 | bulk source would be stronger than necessary. Claiming the original P1 000307 | amplitude already equals (11) would be stronger than the source. 000308 | 000309 | Sources for the auxiliary construction: 000310 | 000311 | research-20261027-boundary-origin/source/ 000312 | ORIGINAL-BOUNDARY-NORM-ORIGIN.txt:21-69,73-164,347-386 000313 | SHA256 2a92d7c75aeed3fb09d85de0f9cf8be3c9fb8757b6686cf0c6724cd7b4a977a6 000314 | 000315 | research-20261027-boundary-origin/originals/ 000316 | ACTUAL-REFERENCE-SPIN-TEST.txt:14-68,122-151,195-259 000317 | SHA256 df6b3894ee736e3a0d503eb1d90e7b7e4e462af966eb2d6718e82e24516fac72 000318 | 000319 | 000320 | 4. SHARP CONCLUSION 000321 | =================== 000322 | 000323 | For module reconstruction the requested inverse line is already canonical: 000324 | 000325 | negative-sea CAR defect V 000326 | -> charged ket line det(V) 000327 | -> CAR dual line det(V)^(-1) 000328 | -> beta character +i and the established c,Q actions. 000329 | 000330 | The eta term is not needed to select this line. It must also not be identified 000331 | with tau_rel: its source factor is the square root in (8),(9), subject to the 000332 | actual Pfaffian data. 000333 | 000334 | For a physical P1 transition amplitude, the smallest remaining premise is: 000335 | 000336 | the outgoing seam response is a complex-linear covector on the charged 000337 | negative-sea defect, or is naturally carried to that covector by the local 000338 | cyclic source-sector map. (12) 000339 | 000340 | Under (12), the original shift fills V, the response annihilates those three 000341 | added modes, and the scalar transition is the canonical pairing K^* tensor K 000342 | to C. This gives the dual orientation without changing the sea convention and 000343 | without promoting C3 to a physical bulk source. The exact original formulas 000344 | audited above provide the winding, eta scalar, positivity and determinant 000345 | phase, but they do not state (12). 000346 | 000347 | Existing orientation review reused: 000348 | 000349 | research-20261027-source-unification/review/ 000350 | PROJECTOR-CAR-DEFECT-LINE-REVIEW.txt 000351 | SHA256 a74aad8f91c90bcbf26e6afbbe7567bd28a540a29cb71008422e44cdccd7f457 000352 |