TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/theta_lattice_closure_20261027/direct/THETA-E8-LATTICE-COVARIANCE.txt Original file SHA-256: 81132e7d111ea83bb5ac7e360839083433fc9efb3c2a1e74dab0e5ce1866424e 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 | 000202 | The selected q divisor, spin normalization and inverse q norm factor give 000203 | actual operators gamma and alpha on W_q with 000204 | 000205 | gamma^3=I, 000206 | alpha^2=iI, 000207 | alpha gamma alpha^(-1)=gamma^(-1). (3.1) 000208 | 000209 | The exact native lift census places gamma in G31 and every permitted alpha 000210 | in 000211 | 000212 | zeta8 G31. (3.2) 000213 | 000214 | Consequently 000215 | 000216 | gamma Lambda=Lambda, 000217 | beta Lambda=Lambda, beta=alpha^2=iI, 000218 | alpha Lambda=zeta8 Lambda. (3.3) 000219 | 000220 | There is no allowed rephasing that turns the last relation into invariance 000221 | while retaining (3.1). If alpha=zeta8 g with g in G31, then 000222 | 000223 | (zeta8^(-1)alpha)^2=g^2=I, (3.4) 000224 | 000225 | whereas alpha^2=iI. The phase correction that was legitimate for a theta 000226 | coordinate change is therefore illegitimate for the fixed geometric 000227 | operator. 000228 | 000229 | The union Lambda union zeta8 Lambda must not be called a lattice. The exact 000230 | object is the two object system 000231 | 000232 | object 0: Lambda, 000233 | object 1: zeta8 Lambda, 000234 | alpha: object 0 <-> object 1, 000235 | alpha^2=i on each object. (3.5) 000236 | 000237 | For the frozen representative used in the exact check, 000238 | 000239 | gamma=sigma=(0 1 2), 000240 | alpha=zeta8 V, 000241 | V=(X tensor I)U. (3.6) 000242 | 000243 | Here V^2=I, V sigma V^(-1)=sigma^(-1), and both V and iV are 000244 | integral isometries of Lambda. Hence the two arrows in (3.5) are concrete: 000245 | 000246 | x in Lambda: alpha(x)=zeta8 Vx in zeta8 Lambda, 000247 | zeta8 x in zeta8 Lambda: alpha(zeta8 x)=i Vx in Lambda. (3.7) 000248 | 000249 | Their composite is multiplication by i, exactly as (3.1) requires. 000250 | 000251 | The failure of a common fixed embedding is sharp. In the physical 000252 | coordinates Lambda lies in 2^(-1/2)Q(i)^4. If a nonzero vector belonged to 000253 | both Lambda and zeta8 Lambda, cancellation of the common 2^(-1/2) factor 000254 | would put zeta8 times a nonzero Q(i) vector back in Q(i)^4, impossible because 000255 | zeta8 is not in Q(i). Hence 000256 | 000257 | Lambda intersect zeta8 Lambda={0}. (3.8) 000258 | 000259 | Their generated abelian group has rank sixteen in an eight dimensional real 000260 | space and is not discrete. In particular, no common discrete overlattice in 000261 | this fixed embedding contains both. This does not exclude a different 000262 | alpha-stable E8 embedding or a conjugated representation. 000263 | 000264 | This is the integral counterpart of the earlier rule that transition groups 000265 | transport marks and need not preserve one fixed marked object. 000266 | 000267 | 000268 | 4. THE MINIMAL COVER FOR THE ACTUAL FIXED IMAGE HAS DEGREE TWO 000269 | ================================================================ 000270 | 000271 | Let F be the actual matrix image of the retained marked theta transports in 000272 | C2, in the fixed Schrodinger frame above. The coset decomposition (2.1) 000273 | defines a character 000274 | 000275 | chi:F -> C2/G31 = Z/2. (4.1) 000276 | 000277 | For this construction a single embedded Lambda descends under F exactly when 000278 | chi|F is zero. If chi is nonzero, its kernel has index two. Even elements 000279 | preserve one lattice branch and odd elements exchange the two branches in 000280 | (3.5). Therefore the coset cover 000281 | 000282 | F/ker chi (4.2) 000283 | 000284 | is both sufficient and minimal for the actual fixed image: any equivariant 000285 | fiber system must distinguish the two values of chi. Since 000286 | 000287 | chi(alpha)=1, (4.3) 000288 | 000289 | the retained source image really requires the two branches. Its restriction 000290 | to ker chi, including gamma and alpha^2=i, acts on either single E8 lattice. 000291 | 000292 | This is a statement about the displayed theta-frame construction and the 000293 | fixed source alpha. It does not assert that no differently embedded or 000294 | conjugated E8 lattice can be invariant under some other representation. 000295 | 000296 | The same-spin Fredholm representative is 000297 | 000298 | H_theta tensor N_q^(-1) tensor B_O^(-1). (4.4) 000299 | 000300 | The ordering line B_O^(-1) changes alpha by its retained sign and carries odd 000301 | Fredholm parity. On the underlying ungraded four dimensional vector space 000302 | the sign does not change the lattice conclusions above. The odd parity is 000303 | not erased and remains part of any physical sewing statement. 000304 | 000305 | 000306 | 5. THE OLD C12 CLOCK LIVES INSIDE EACH LATTICE 000307 | ============================================== 000308 | 000309 | The square beta=iI is the native Gaussian quarter J, and gamma is the native 000310 | cubic sigma. They commute because beta is central. Therefore 000311 | 000312 | [OMITTED: non-scientific conversation or execution metadata] 000313 | 000314 | has order twelve and preserves each lattice in (3.5). It is the native 000315 | compiler relation 000316 | 000317 | [OMITTED: non-scientific conversation or execution metadata] 000318 | 000319 | [OMITTED: non-scientific conversation or execution metadata] 000320 | [OMITTED: non-scientific conversation or execution metadata] 000321 | 000322 | alpha sigma alpha^(-1)=sigma^(-1). (5.3) 000323 | 000324 | [OMITTED: non-scientific conversation or execution metadata] 000325 | [OMITTED: non-scientific conversation or execution metadata] 000326 | [OMITTED: non-scientific conversation or execution metadata] 000327 | [OMITTED: non-scientific conversation or execution metadata] 000328 | [OMITTED: non-scientific conversation or execution metadata] 000329 | [OMITTED: non-scientific conversation or execution metadata] 000330 | [OMITTED: non-scientific conversation or execution metadata] 000331 | [OMITTED: non-scientific conversation or execution metadata] 000332 | 000333 | 000334 | 6. CONSEQUENCES FOR THE FIELD CONSTRUCTION 000335 | ========================================== 000336 | 000337 | There are now two valid uses of the result. 000338 | 000339 | Abstract integral reconstruction. 000340 | 000341 | The theta module, its selected central i and its stabilizer root shell 000342 | recover the even unimodular rank eight lattice class E8. No physical C3 000343 | propagation is required. 000344 | 000345 | Fixed native field reconstruction. 000346 | 000347 | Choose one branch Lambda. G31 preserves it as a lattice, and every 000348 | individual lattice isometry admits a lift to V_Lambda. The complete 000349 | group of lifts is a central extension; the integral construction does 000350 | not choose a splitting or a common cocycle lift. The explicit shell 000351 | isometry identifies V_Lambda abstractly with the E8 lattice VOA supplied 000352 | by the marked generator theorem. Likewise the lattice isometry alpha 000353 | implies that there exists a lattice VOA isomorphism 000354 | 000355 | V_Lambda -> V_(zeta8 Lambda), (6.1) 000356 | 000357 | lifting alpha. Such a lift is unique only up to the usual character 000358 | gauge of the lattice central extension. The integral checker does not 000359 | select that cocycle lift or identify it with an already marked source 000360 | field operator. In particular, it gives no automorphism of V_Lambda. 000361 | 000362 | Equation (6.1) is a lattice and field transition. It does not create a 000363 | single connected local VOA containing both branches, and the direct sum of 000364 | the two VOAs is not automatically the primitive physical source. A sewing 000365 | or descent theorem would be required for that stronger statement. 000366 | 000367 | The integral result therefore strengthens the finite auxiliary route but 000368 | does not alter the remaining physical boundary identified in 000369 | WEAKEST-DIRECT-E8-SOURCE-ROUTE.txt. 000370 | 000371 | 000372 | 7. EXACT CHECK, SCOPE AND SOURCES 000373 | ================================ 000374 | 000375 | New exact certificate: 000376 | 000377 | research-20261027-fundamental-next/lattice/ 000378 | check_theta_lattice.py 000379 | theta-lattice.json 000380 | 000381 | It verifies the support counts 16+96+128, rank eight, |det B|=16, the even 000382 | unimodular physical Gram matrix, the literal standard E8 shell equality, and 000383 | all nine integral Clifford generators in (2.2). It also checks the frozen 000384 | sigma, V and alpha equations in (3.6)-(3.7), failure of alpha to preserve one 000385 | Lambda, and the order twelve identities (5.2). The code checks a finite 000386 | algebraic construction; the covariance and source scope are the mathematical 000387 | statements of this note. 000388 | 000389 | Established native checks used here: 000390 | 000391 | verification/v633_orbit60_quotient.py 000392 | The sixty Gaussian lines, their 240 mu4 roots, their Z[i] span and the 000393 | Hermitian unimodular E8 lattice. 000394 | 000395 | verification/v783_two_qubit_clifford.py 000396 | Exact equality of the sixty Gaussian lines and sixty stabilizer rays, 000397 | the support census 4+24+32, and 000398 | C2=G31 disjoint union zeta8 G31 with the explicit corrected Hadamard. 000399 | 000400 | experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/