TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_time_compatibility_20261027/GSO-SOURCE-AUDIT.txt Original file SHA-256: c9dc02cfdcdae20c50baf39518d91e4e3302bb765c6ccb59a8ed9cdc151ff4b0 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 | arXiv:hep-th/9507047, 000202 | https://arxiv.org/abs/hep-th/9507047 000203 | 000204 | Yasuyuki Kawahigashi, Roberto Longo, Michael Mueger, 000205 | Multi-interval Subfactors and Modularity of Representations in Conformal 000206 | Field Theory, 000207 | arXiv:math/9903104, 000208 | https://arxiv.org/abs/math/9903104 000209 | 000210 | These theorems classify or control extensions after the local net, sectors, 000211 | and canonical endomorphism are given. They do not manufacture those inputs 000212 | from a scalar determinant or a finite parity census. 000213 | 000214 | 5. HAAG DUALITY IS NOT A HIDDEN MAXIMALITY RULE 000215 | 000216 | There are three different statements that must not be merged. 000217 | 000218 | First, ordinary interval Haag duality is a locality completeness property of a 000219 | net in a specified representation. A completely rational net can satisfy it 000220 | and still have nontrivial DHR sectors. D8 at level one is such a sectorful 000221 | net. Thus ordinary Haag duality does not imply mu=1 and does not force E8. 000222 | 000223 | Second, KLM identify the multi interval inclusion index with the global sector 000224 | index under substantive hypotheses: a local irreducible net, split property, 000225 | Haag duality, conformal covariance or the stated replacement, and finite 000226 | index. Under complete rationality, mu=1 means that only the vacuum sector 000227 | remains. That is a theorem about a net already satisfying these hypotheses. 000228 | 000229 | Third, one may impose “take a maximal local bosonic extension” as a new rule. 000230 | For D8 this leaves the two labelled E8 extensions 0+s and 0+c. It gives E8 up 000231 | to labelled chirality, but the rule and its label transport are extra source 000232 | data. No such rule occurs in the audited P1 seed definition. 000233 | 000234 | The v501 calculation is a finite lattice witness for the two interval index. 000235 | It measures mu_gauged=4 and verifies the conditional condensation arithmetic 000236 | 16 to 4 to 1. The same passage explicitly leaves continuum Haag duality, local 000237 | normality, and split for the quotient net open. It therefore cannot be used as 000238 | the missing maximality theorem. 000239 | 000240 | Exact original location: 000241 | 000242 | Atlas section: section:b18b388084f449438cbb 000243 | Atlas source: paper-source:f749b6366952d4f4f6da9013 000244 | documents/changelog/sections/006-source.tex:331-372 000245 | 000246 | Primary KLM hypothesis statement: 000247 | 000248 | https://arxiv.org/abs/math/9903104 000249 | Abstract and main setup: the multi interval result starts from a local 000250 | irreducible net satisfying split and Haag duality, with finite index in the 000251 | conformal case. 000252 | 000253 | 6. THE OPERATIONAL P1 SEED DOES NOT CONTAIN THE MISSING DATA 000254 | 000255 | The archived operational seed is the tuple 000256 | 000257 | (A_loc, tau_t, Theta, omega, [u_Sigma], D_coll), 000258 | 000259 | where A_loc is a quasilocal Euclidean observable net, omega is reflection 000260 | positive and clustering, [u_Sigma] is a nontrivial seam class, and D_coll is a 000261 | reflection regular first order elliptic collar generator. Morphisms preserve 000262 | these displayed structures. The collar completion constructs the one sided OS 000263 | Hilbert space and boundary operator. 000264 | 000265 | Exact original locations: 000266 | 000267 | Atlas section: section:da5a8d451e469c6cb1f5 000268 | Atlas source: paper-source:92324c2bfff42fe248080ffe 000269 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000270 | sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex: 000271 | 100-144 000272 | 000273 | Earlier archive copy: 000274 | Atlas section: section:a06af9918677fc215e09 000275 | Atlas source: paper-source:df15252a0b6d7f2018e3de7e 000276 | documents/archive--paper-latex--tfpt-42/sections/ 000277 | 026-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex:14-55 000278 | 000279 | None of the displayed seed entries is a spin structure family, a parity 000280 | orbifold functor, a choice of Arf quadratic refinement, a Ramond field, or a 000281 | local extension Q system. Reflection positivity of omega controls the given 000282 | observable algebra. It does not add the missing charged sectors. 000283 | 000284 | A bounded search of the original architecture, the tfpt42 archive, and the 000285 | tfpt45 operational seed sources found no separate definition of a spin 000286 | structure sum or fermion parity gauging. Later changelog and research contract 000287 | texts do contain the terms GSO, Ramond, crossed product, and holomorphy, but the 000288 | exact passages above mark their source identification or continuum 000289 | realization as conditional or open. 000290 | 000291 | 7. THE ACTIVE CONTRACT ALREADY STATES THE SAME RESIDUE 000292 | 000293 | The active externalization contract separates the free bilinear current limit 000294 | from four uncovered items: 000295 | 000296 | N1 the GSO extension SO(16)_1 to E8 as a net statement, 000297 | N2 the twist or spin field sector, 000298 | N3 the one sided chiral edge net, 000299 | N4 the OS or holomorphy selector. 000300 | 000301 | It further says that commuting the crossed product with the scaling limit is 000302 | open and that the reconstructed net has not been proved to satisfy the 000303 | holomorphic c=8 classification hypotheses. 000304 | 000305 | Exact original location: 000306 | 000307 | Atlas section: section:f97072f658fe6bcec8d8 000308 | Atlas source: paper-source:67ce0c0c0856e9caaefdeadb 000309 | documents/tfpt_research_contracts/sections/020-source.tex:14-58, 87-92 000310 | 000311 | The compressed local field contract names the same missing mathematical 000312 | object. One normalized simple current field must satisfy convergence with 000313 | energy bounds, locality relative to the D8 current net, correct braiding, and 000314 | Q system closure. Its lattice vector and character shadow are known, while the 000315 | net statements remain open. 000316 | 000317 | Exact original location: 000318 | 000319 | Atlas section: section:aeb16d254373e2e88a13 000320 | Atlas source: paper-source:4c76025b97bcb46ca36b8c34 000321 | documents/tfpt_research_contracts/sections/017-source.tex:1-48, 112-135 000322 | 000323 | 8. MODULAR SEWING, THE HOLOMORPHIC KMS THEOREM, AND THE SINGLE VACUUM LINE 000324 | 000325 | The newer finite modular sewing result is exact: in the already classified 000326 | extension class of U=V_D5 tensor V_A3, a one dimensional chiral vacuum line 000327 | fixed by cycle exchange forces a Lagrangian support of order four, hence one 000328 | of the two E8 extensions. This is an efficient sufficient test. 000329 | 000330 | It does not follow from a unique Hilbert space vacuum vector. It requires an 000331 | actual labelled torus conformal block and its cycle exchange action. A scalar 000332 | paired partition function, a determinant line, or a state at one fixed torus 000333 | does not supply that datum. 000334 | 000335 | The holomorphic KMS theorem likewise starts by assuming an inclusion 000336 | 000337 | A = B^Z4 subset B, 000338 | [B:A]=4, 000339 | A(I)' intersect B(I)=C, 000340 | mu_B=1, 000341 | c_B=8. 000342 | 000343 | It then proves uniqueness of the KMS state extension. It cannot be reversed to 000344 | construct B or to infer mu_B=1 from the finite CAR KMS state. Its own final 000345 | section says that the concrete TFPT scaling limit, nonbilinear spin fields, 000346 | relative locality, and commutative Q system remain to be constructed. 000347 | 000348 | Exact original location: 000349 | 000350 | Atlas source: paper-source:5fb763e4d65ef02e182c8a67 000351 | documents/articles--2026-08-29--holomorphic_kms_extension_en/ 000352 | sections/001-section-setting-and-correction-of-terminology.tex:4-36, 43-77 000353 | sections/002-subsection-the-frozen-car-covariance.tex:68-103 000354 | sections/003-section-what-this-closes-and-does-not-close.tex:1-92 000355 | 000356 | Thus neither the v490 census, the conditional modular sewing test, nor the KMS 000357 | uniqueness theorem adds the missing spin sum. 000358 | 000359 | 9. SMALLEST DECISIVE COMPLETION TEST 000360 | 000361 | A future claim that P1 itself selects E8 should pass the following test before 000362 | any character or dimension comparison is used. 000363 | 000364 | Input constructed from P1: 000365 | 000366 | * four source amplitudes A,B,C,D on the same torus, with explicit spatial 000367 | and temporal spin labels; 000368 | * a common state and time normalization; 000369 | * the action of fermion parity on each amplitude; 000370 | * the Arf or quadratic refinement phase; 000371 | * the retained D8 charge marking. 000372 | 000373 | Required identity: 000374 | 000375 | * prove that the P1 operation is exactly the normalized sum 000376 | (A+B+C+/-D)/2; 000377 | * prove that its output has the refined character chi_0+chi_s or 000378 | chi_0+chi_c; 000379 | * prove sewing under S and T, locality of the added spin fields, and 000380 | compatibility with the original reflection and adjoint. 000381 | 000382 | Kill controls: 000383 | 000384 | * parity projection only must stop at the even D8 observable algebra; 000385 | * changing the Arf sign must exchange the labelled s and c completions; 000386 | * deleting the Ramond blocks must not yield E8; 000387 | * a scalar determinant or unrefined character match must not count as a 000388 | labelled field identification. 000389 | 000390 | Equivalent local net completion: 000391 | 000392 | Construct one normalized spin or simple current field in the P1 scaling 000393 | limit and prove the four G1 to G4 properties in the active contract. The 000394 | existing extension uniqueness then supplies all 128 spinor currents and the 000395 | holomorphic E8 net. This route avoids adding a separate arbitrary GSO rule, 000396 | but it still requires a source derived local field. 000397 | 000398 | FINAL STATUS 000399 | 000400 | PROVED OR SOURCE VERIFIED: