TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/AFFINE-DEFECT-INDEPENDENT-REVIEW.txt Original file SHA-256: 0870f40d5b6d65b3d62a435e6c11706bba84dbf1fb21706bd6c6aa6cc058f142 Inclusive original lines: 201-337 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 | 4. The decisive lift phase 000202 | -------------------------- 000203 | 000204 | Order the four sectors as (0,s,v,c). The geometric torus character is 000205 | 000206 | t_geo=exp(2 pi i e0.P)=(+,-,+,-). (14) 000207 | 000208 | It is + on both integral sectors 0,v and - on both half integral sectors 000209 | s,c. Since W_aff pairs 0 with v and s with c, it commutes with t_geo. 000210 | 000211 | The standard E8 defect lift is different. Smith, Lin, Tachikawa and Zheng, 000212 | equation (5.3), obtain 000213 | 000214 | Z[E8]^a=chi_0-chi_s, 000215 | Z[E8]_a^a=chi_c-chi_v. (15) 000216 | 000217 | Thus their chosen lift on the four sectors is 000218 | 000219 | g_hat=(+,-,-,+). (16) 000220 | 000221 | It anticommutes with W_aff on both pairs in (3): 000222 | 000223 | g_hat W_aff=-W_aff g_hat. (17) 000224 | 000225 | Equations (14) and (16) differ by the character 000226 | 000227 | d_defect=(+,+,-,-), (18) 000228 | 000229 | which is -1 on the complete g twisted space H_v+H_c. This is the defect or 000230 | quantum Z2 lift, not another torus character. Omitting (18) would incorrectly 000231 | identify the spatial NS/R sector sign with fermion parity. 000232 | 000233 | The modular S and T convention fixes the relative v versus c sign in (15). 000234 | The fermionization formula additionally chooses the spin invertible theory 000235 | and the Arf sign. Stacking with Arf exchanges the two Ramond parity labels, 000236 | as stated after equation (5.5) of the primary paper. Hence this phase is 000237 | standard and consistent, but it is not derived from the original NS CAR 000238 | operator alone. 000239 | 000240 | 5. What fixes the R sector action of W_aff 000241 | ------------------------------------------ 000242 | 000243 | The relevant intertwining spaces are one dimensional. Once the abelian 3 000244 | cocycle and one nonzero intertwiner in every fusion channel are normalized, 000245 | associativity and braided locality determine all products. Rescaling those 000246 | intertwiners by a normalized 2 cochain gives an equivalent abelian 000247 | intertwining algebra. Absolute phases of individual spinor vertex operators 000248 | are therefore gauge, while monodromy, conformal weight and the signs in a 000249 | fixed modular convention are invariant data. 000250 | 000251 | The ordered CAR cocycle fixes W_aff on H_0+H_v. Extending the R lift while 000252 | preserving the full four sector algebra still permits multiplication by the 000253 | discriminant character that is + on 0,v and - on s,c. Equivalently, the two 000254 | extensions agree on the entire NS theory and differ by one sign on the 000255 | entire Ramond theory. If only a direct sum Hilbert space unitary were asked 000256 | for, an arbitrary common U(1) phase on the Ramond block would be possible; 000257 | compatibility with all intertwining products reduces it to this character 000258 | choice. 000259 | 000260 | The standard defect convention (15), a chosen Arf theory and a chosen basis 000261 | of Ramond ground intertwiners select one representative. The source NS 000262 | winding, equation (7), and its neutral Cartan word do not distinguish the 000263 | two representatives. A claimed equality with one physical source operator 000264 | must therefore cite the original spin/defect lift or calculate the mixed NS 000265 | and Ramond cocycle explicitly. 000266 | 000267 | 6. Source words and exact scope 000268 | ------------------------------- 000269 | 000270 | On H_0+H_v the construction has been normalized to the original W1. Every 000271 | old CAR word, including the bounded 2 pi Cartan holonomy four word, is 000272 | therefore unchanged. The residual sign in Section 5 acts only on the Ramond 000273 | block and cannot alter that NS equality. 000274 | 000275 | On H_0+H_s the same construction is not an ordinary local odd field 000276 | embedding. Equations (12) and (13) say that the image lies in the g twisted 000277 | module H_v+H_c. This is precisely the typed alternative allowed by the old 000278 | odd sector exclusion. It neither revives the rejected global odd Phi map nor 000279 | identifies an odd CAR field with an ordinary bosonic E8 vacuum field. 000280 | 000281 | The final status is therefore: 000282 | 000283 | EXISTENCE PROVED: the four sector abelian intertwining extension, affine 000284 | W_aff, its all sector Ward identity, and its Delta twisted field functor. 000285 | 000286 | PHASE CONDITIONAL: the concrete Ramond lift and spinor vertex phases are 000287 | fixed only after choosing the normalized 3 cocycle, defect junction and 000288 | Arf convention. The standard choice is (16), not (14). 000289 | 000290 | PHYSICAL IDENTIFICATION OPEN: no theorem here selects that lift from raw 000291 | P1, identifies it with an admissible physical intervention, or supplies a 000292 | bounded local net defect endpoint on every interval. 000293 | 000294 | The smallest decisive remaining test is a mixed sector cocycle matrix 000295 | element. Choose one normalized spinor ground state xi_s and one vector 000296 | fermion field psi_v. Compare 000297 | 000298 | W_aff Y(psi_v,z) xi_s 000299 | 000300 | with the original source defect transport including its Klein factor and 000301 | the standard lift (16). Equality fixes the remaining Ramond sign. A mismatch 000302 | cannot be repaired by the already fixed NS winding without changing the 000303 | spin/Arf lift. 000304 | 000305 | Primary and exact local sources 000306 | ------------------------------- 000307 | 000308 | * C. H. Lam and H. Shimakura, Orbifold construction of holomorphic vertex 000309 | operator algebras associated to inner automorphisms, 000310 | arXiv:1501.05094v2, Proposition 3.1 and equations (3.1),(3.2). 000311 | 000312 | * H. Li, Local systems of twisted vertex operators, vertex operator 000313 | superalgebras and twisted modules, arXiv:q-alg/9504022. 000314 | 000315 | * P. Boyle Smith, Y.-H. Lin, Y. Tachikawa and Y. Zheng, 000316 | Classification of chiral fermionic CFTs of central charge <=16, 000317 | SciPost Phys. 16, 058 (2024), equations (2.2),(5.1),(5.3)--(5.6). 000318 | 000319 | * Dong and Lepowsky, Generalized Vertex Algebras and Relative Vertex 000320 | Operators, Theorem 12.24: the direct sum of lattice modules is an 000321 | abelian intertwining algebra governed by the discriminant quadratic form. 000322 | 000323 | * C. Dong and F. Xu, Conformal Nets Associated with Lattices and Their 000324 | Orbifolds, arXiv:math/0411499. Its net and soliton results explain the 000325 | analytic upgrade required beyond the formal Delta module, but apply 000326 | directly to lattice isometries rather than the present inner torus twist. 000327 | 000328 | * S. Carpi and L. Tomassini, Energy bounds for vertex operator algebra 000329 | extensions, arXiv:2303.14097. Energy bounds are one required analytic 000330 | input; they do not fix the Ramond cocycle phase or defect localization. 000331 | 000332 | * experiments/theory-contracts/seam-joint-ward-state-20261014/PROOF.txt, 000333 | Sections 16,17 and21. 000334 | 000335 | * experiments/theory-contracts/canonical-seam-fermionization-20261025/ 000336 | NEUTRAL-NET-AND-DEFECT.txt. 000337 |