TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/LATTICE-AND-SOURCE-FRAME.txt Original file SHA-256: 77e0687412cb79bf1d51d079946484b6e8be337adf8a72a85bf2e6079b045ae7 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 | stronger than comparing root-system types. 000202 | 000203 | The earlier fold audit already stated the same structure in a coarser form: 000204 | the M-transported deck-fixed D8 contains 48 roots which are integer in the 000205 | old target frame and 64 which are spinor roots in that frame. Equation (6) 000206 | now identifies those 48 as the exact intersection with the marked 000207 | order-two intermediate. 000208 | 000209 | 3. Cartan action 000210 | ---------------- 000211 | 000212 | Both D8s have full rank, so both use the same eight-dimensional target 000213 | Cartan vector space. This does not make their charged root actions equal. 000214 | 000215 | The canonical marked D8 is the +1 root sector of 000216 | 000217 | h_can=e_0, 000218 | sigma_can(E_r)=exp(2 pi i h_can.r) E_r. (7) 000219 | 000220 | Integer D8 roots have phase +1 and the 128 spinor roots have phase -1. 000221 | 000222 | The folded D8 transported by M is the +1 root sector of 000223 | 000224 | h_fold=M e_0=(1/2,1/2,0,0,0,1/2,1/2,0), 000225 | sigma_fold(E_r)=exp(2 pi i h_fold.r) E_r. (8) 000226 | 000227 | Both (7) and (8) fix all eight Heisenberg Cartan currents, as every torus 000228 | character does. On the 240 charged root currents their joint sign counts 000229 | are 000230 | 000231 | (sigma_can,sigma_fold) root count 000232 | (+1,+1) 48 000233 | (+1,-1) 64 000234 | (-1,+1) 64 000235 | (-1,-1) 64. (9) 000236 | 000237 | Thus the charged action differs on 128 roots. The two character vectors 000238 | are not congruent modulo the self-dual E8 lattice. Their difference has 000239 | mixed integral and half-integral coordinates and is not an E8 vector. 000240 | 000241 | At the current level the actual dictionaries are correspondingly 000242 | 000243 | H_target=O H_Gaussian, 000244 | H_target=M j_fold. 000245 | 000246 | M is a half-integral Hadamard-type E8 automorphism, not a signed permutation 000247 | of the eight elementary CAR charges. Replacing M by a new rotation which 000248 | makes (5) equal (2) would change the marked dictionary being tested. 000249 | 000250 | 3a. Coordinate equivalence versus a simultaneous marking 000251 | ---------------------------------------------------------- 000252 | 000253 | Let W be an E8 Weyl automorphism. If one replaces every target datum by its 000254 | W transport, 000255 | 000256 | R_can -> W R_can, R_fold -> W R_fold, 000257 | O -> W O, M -> W M, 000258 | d -> W d, 000259 | 000260 | and transports the D5+A3 flag, charged field labels, cocycle choices and 000261 | local source inclusion by the same W, then all inner products, OPEs and time 000262 | pairings are unchanged. This is a passive coordinate change. In 000263 | particular, the statement R_can=R_fold is invariant under such a common 000264 | transport: if it is false before W, it remains false after W. 000265 | 000266 | A different operation chooses W so that W R_fold=R_can while O, the marked 000267 | D5+A3 flag or d is kept at its old value. That changes the relative 000268 | position of source and compiler data. Such a map certainly exists here: 000269 | W=M^T sends R_fold=M R_can back to R_can when it is applied to the source 000270 | side alone. This does not prove that the original O/M dictionary contains 000271 | that re-marking or that the source fields and time intertwine after it. The 000272 | negative result in (6) is exactly this relative, simultaneously marked 000273 | statement. It is independent of the common global coordinate convention 000274 | and it does not exclude TFPT as a whole. 000275 | 000276 | 4. The actual quarter-time vector 000277 | --------------------------------- 000278 | 000279 | In the folded CAR component frame the source shift is 000280 | 000281 | delta_(alpha,k)=(k+alpha)/8-1/2. 000282 | 000283 | The existing dictionary transports it, without a fit, to 000284 | 000285 | d_alpha=M delta_alpha. (10) 000286 | 000287 | For the two actual quarters this gives 000288 | 000289 | alpha=1/4: 000290 | delta=(-15,-11,-7,-3,1,5,9,13)/32, 000291 | d=(-1/2,0,1/16,-1/2,0,-3/16,-1/4,-1/4); 000292 | 000293 | alpha=3/4: 000294 | delta=(-13,-9,-5,-1,3,7,11,15)/32, 000295 | d=(-1/2,0,3/16,-1/2,0,-1/16,-1/4,-1/4). (11) 000296 | 000297 | Orthogonality of M gives, for every CAR root beta, 000298 | 000299 | d_alpha.(M beta)=delta_alpha.beta. (12) 000300 | 000301 | The checker verifies (12) root by root. Hence the phase multiset on 000302 | R_fold is exactly the original folded-source time multiset. 000303 | 000304 | The same d_alpha has a different phase multiset on R_can. A short part of 000305 | the exact mod-one census, identical in these displayed multiplicities for 000306 | both quarter choices, is 000307 | 000308 | phase on R_can on R_fold 000309 | 0 10 0 000310 | 1/2 18 8 000311 | 1/4 17 8 000312 | 3/4 17 8 000313 | 3/8 0 8 000314 | 5/8 0 8. (13) 000315 | 000316 | The full phase tables are emitted by the checker. On the two witness roots, 000317 | 000318 | alpha d.r_can d.r_fold 000319 | 1/4 7/16 13/16 000320 | 3/4 5/16 11/16. (14) 000321 | 000322 | Equations (11)-(14) rule out rescuing the equality by retaining only the 000323 | same eight-dimensional Cartan space or the same abstract D8 type. The 000324 | actual source time sees the embedding difference. 000325 | 000326 | There is also a precise zero-energy spinor line which must be retained in 000327 | any state/time comparison. In the folded source coordinates set 000328 | 000329 | p=(1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2). (15) 000330 | 000331 | It is an E8 root: p^2=2 and it has four minus signs. With 000332 | s=(1/2,...,1/2), 000333 | 000334 | p-s=(0,0,0,0,-1,-1,-1,-1) in D8, 000335 | 000336 | so p lies in the standard even-parity spinor neighbour D8+s. It is not an 000337 | integer CAR charge. It is also not in the coset called c in 000338 | ODD-SECTOR-REFIELDING-AND-NAHT.txt, where c was deliberately represented as 000339 | m+s with sum m odd. That file uses the other spinor convention. 000340 | 000341 | For either alpha=1/4 or 3/4, the relation 000342 | 000343 | delta_(k+4)=delta_k+1/2 000344 | 000345 | gives 000346 | 000347 | delta.p=(1/2)[sum_(k=0)^3 delta_k-sum_(k=4)^7 delta_k] 000348 | =-1, 000349 | p^2/2+delta.p=1-1=0. (16) 000350 | 000351 | The fixed dictionary M sends this vector to 000352 | 000353 | q=M p=(1,0,0,1,0,0,0,0)=e_1+e_4 (17) 000354 | 000355 | in one-based target notation. By orthogonality and d=M delta, 000356 | 000357 | q^2/2+d.q=p^2/2+delta.p=0. (18) 000358 | 000359 | The root q belongs to R_can but not to R_fold=M(D8_CAR). Thus this same 000360 | vector simultaneously exhibits the frame distinction and the extra sector 000361 | introduced by extending the integer CAR charge lattice to E8. 000362 | 000363 | Thus the M-transported zero-time-ground line is the lattice charge line 000364 | 000365 | C e^(e_1+e_4). 000366 | 000367 | This is the same equal-energy target root primary already recorded in 000368 | FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt, not a newly discovered 000369 | degeneracy. Equation (15) supplies its exact source-spinor preimage. The 000370 | bare folded CAR integer charge lattice does not contain p, whereas the full 000371 | E8 extension does. Consequently an E8 extension adds a zero-energy charge 000372 | line at the actual quarter background. A claim that the extended theory 000373 | has the same source state and time therefore needs an explicit sector or 000374 | ground-state selection rule. This is a state/time compatibility condition, 000375 | not a rejection of the E8 extension. 000376 | 000377 | 5. Why the odd-sector continuation stops here 000378 | ---------------------------------------------- 000379 | 000380 | For one fixed D8, write its four discriminant classes as 000381 | 000382 | 0, v, c, v+c. 000383 | 000384 | The quadratic form makes v the unique nonisotropic class and c,v+c the two 000385 | isotropic spinor classes. The usual neighbour statements are 000386 | 000387 | D8 union (D8+v) is the odd unimodular lattice Z8, 000388 | D8 union (D8+c) and D8 union (D8+v+c) are E8 neighbours. 000389 | 000390 | This is the standard discriminant-form/overlattice correspondence. Nikulin's 000391 | classification of integral symmetric bilinear forms supplies the general 000392 | theorem that even overlattices correspond to isotropic discriminant 000393 | subgroups. Here equations (1) and (2) prove the required special case 000394 | directly. 000395 | 000396 | The known ODD-SECTOR-REFIELDING-AND-NAHT result starts with the CAR D8 and 000397 | defines a global unitary which is the identity on that even module and is 000398 | parity-dependent on its odd vector coset. It already proves that the pulled 000399 | back target Cartan observable is not local in the unchanged CAR inclusion. 000400 | The new comparison adds an earlier obstruction: the even module fixed by