TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/NEUTRAL-NET-AND-DEFECT.txt
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Inclusive original lines: 209-310
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000209 |   arXiv:2303.16917v2, Sections 2.1 and 5.1, Appendix A.
000210 | 
000211 | Their non-anomalous Z2 symmetry g of (E8)_1 has invariant affine algebra
000212 | (D8)_1. In D8 discriminant notation {0,v,s,c}, their equations (5.3)--(5.6)
000213 | give exactly
000214 | 
000215 |   H_E8,untwisted = H_0 direct-sum H_s,
000216 |   H_E8,g-twisted = H_c direct-sum H_v,                  (20)
000217 | 
000218 | and after the spin/Arf fermionization
000219 | 
000220 |   H_F,NS = H_0 direct-sum H_v,
000221 |   H_F,R  = H_s direct-sum H_c,
000222 |   F[(E8)_1;g]=16 free Majorana-Weyl fermions.           (21)
000223 | 
000224 | The paper also states that the unrefined spinor characters chi_s and chi_c
000225 | are equal as q-series but differ after flavor fugacities are turned on. This
000226 | is decisive here: equality of unrefined characters cannot transport the
000227 | eight named source Cartans or the quarter background.
000228 | 
000229 | 6. Monodromy: why v+c is the correct twisted sector
000230 | ----------------------------------------------------
000231 | 
000232 | Choose representatives
000233 | 
000234 |   v=(1,0,0,0,0,0,0,0),
000235 |   s=(1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2),
000236 |   c=(-1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2).
000237 | 
000238 | Modulo D8,
000239 | 
000240 |   v+s=c,
000241 |   h_v=v.v/2=1/2,
000242 |   h_s=h_c=1.                                            (22)
000243 | 
000244 | The full lattice monodromies with the extending current s are
000245 | 
000246 |   M(s,v)=exp(2 pi i s.v)=-1,
000247 |   M(s,c)=exp(2 pi i s.c)=-1.                            (23)
000248 | 
000249 | Thus neither v nor c is an ordinary local module of the bosonic extension
000250 | 0+s. Together they form its g-twisted sector v+c. The old monodromy -1 is
000251 | retained and correctly typed as defect monodromy. Fermionization combines
000252 | this defect sector with the spin invertible theory and the Z2 quotient; in
000253 | the resulting spin theory the v field has its correct graded locality.
000254 | 
000255 | This does not put v and s into one ordinary local lattice VOA. It therefore
000256 | does not contradict ODD-SECTOR-REFIELDING-AND-NAHT.txt. It uses the honest
000257 | alternative already left open there: an outer twisted sector followed by
000258 | spin fermionization.
000259 | 
000260 | 7. Neutral answer-word test in the four-module completion
000261 | ---------------------------------------------------------
000262 | 
000263 | For each coordinate define the exponentiated Cartan character
000264 | 
000265 |   chi_k(p)=exp(2 pi i p_k).
000266 | 
000267 | It has the module values
000268 | 
000269 |   chi_k(0)=chi_k(v)=+1,
000270 |   chi_k(s)=chi_k(c)=-1.                                 (24)
000271 | 
000272 | Fusion with v pairs
000273 | 
000274 |   0 <-> v,       s <-> c.                              (25)
000275 | 
000276 | Therefore the exponentiated Cartan sign is preserved by the defect pairing.
000277 | An intertwiner realizing (25) obeys the necessary neutral four-word test
000278 | 
000279 |   W_v* U_k W_v U_k*=I                                  (26)
000280 | 
000281 | on both pairs. The cyclic channel permutation contained in the actual W1
000282 | does not change this 2pi test, because every U_k has the same sign on each of
000283 | the four D8 modules.
000284 | 
000285 | Equation (26) is a real positive improvement over the old map 0+v -> 0+c,
000286 | which changed the sign. It shows that the complete untwisted/twisted sector
000287 | assembly can preserve the original bounded neutral answer word.
000288 | 
000289 | It proves only the exponentiated 2pi word. Continuous Cartan currents and
000290 | general flavor fugacities retain more information. The primary paper's
000291 | chi_s versus chi_c warning shows why those refined responses still require
000292 | an explicit background and intertwiner calculation.
000293 | 
000294 | 8. The actual fixed-frame obstruction
000295 | --------------------------------------
000296 | 
000297 | The standard formulas above refer to one chosen D8 embedding. The current
000298 | TFPT data contain two:
000299 | 
000300 |   D8_can = the canonical order-two intermediate of the marked D5+A3 glue,
000301 |   D8_F   = M(D8_source), the actual folded CAR even lattice in the target
000302 |            E8 frame.                                    (27)
000303 | 
000304 | The exact checker
000305 | 
000306 |   experiments/theory-contracts/canonical-seam-fermionization-20261025/
000307 |   check_lattice_source_frame.py
000308 | 
000309 | finds
000310 |