TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/theta_lattice_closure_20261027/direct/WEAKEST-DIRECT-E8-SOURCE-ROUTE.txt Original file SHA-256: fb13b19762d7e8a327547859a56a335f717d1711f0d4ad3d924c7b26b7ab6583 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 | action, the quarter square and the inversion relation simultaneously. 000202 | 000203 | 000204 | 3. WHY THIS DOES NOT NEED A PHYSICAL C3 DIRAC MAP 000205 | ================================================= 000206 | 000207 | The only analytic information used in (2.12) is the original circle Hardy 000208 | orientation and its multiplier z. The cable is a functorial algebraic base 000209 | change along the already marked P2 map. The genus two curve serves the same 000210 | role as a splitting field in algebra: it exposes the finite action without 000211 | claiming that physical propagation occurs on that field. 000212 | 000213 | This distinction is decisive because the two possible claims have different 000214 | domains. 000215 | 000216 | Auxiliary claim: 000217 | the determinant of the pulled back multiplier realizes the finite marked 000218 | representation. 000219 | 000220 | Physical claim: 000221 | the primitive P1 bulk harmonic Cauchy data are unitarily identical to the 000222 | spin Dirac Cauchy data on the three sheeted curve. 000223 | 000224 | The first claim follows from (2.2) through (2.12). The second needs the 000225 | missing bulk operator and domain intertwiner. It is stronger and is not 000226 | used in the proof of the first. 000227 | 000228 | There is also no hidden choice of three copies. The degree of pi is three, 000229 | c permutes the three fibers transitively, and the same original M_z acts on 000230 | all of them. Nor is there a new physical polarization. Treating the 000231 | holomorphic pullback as an auxiliary determinant functor fixes the complex 000232 | Hardy space algebraically. Claiming it is the physical Calderon range would 000233 | be the additional assertion that remains unproved. 000234 | 000235 | 000236 | 4. THE WEAKEST LOCAL E8 FIELD THEOREM 000237 | ===================================== 000238 | 000239 | Retain the compiler lattice 000240 | 000241 | K=D5+A3, A=K*/K=C4 x C4, h=(s,f), 000242 | U=V_K=V_D5 tensor V_A3. (4.1) 000243 | 000244 | Let W be a connected local unitary VOA extension of U with the same conformal 000245 | vector and finite dimensional weight spaces. Do not prescribe its support, 000246 | multiplicities, holomorphicity, finite index or minimality. Assume only 000247 | 000248 | Hom_U(M_h,W) is nonzero. (4.2) 000249 | 000250 | Regularity of U decomposes W over the sixteen simple currents M_a. The 000251 | connected separable algebra object B associated with W has a simple regular 000252 | B module. Invertibility of M_a and free module adjunction give 000253 | 000254 | n_a=dim Hom_U(M_a,W) 000255 | =dim Hom_B(M_a tensor B,B) in {0,1}. (4.3) 000256 | 000257 | Thus the support S={a:n_a=1} is a subgroup. Locality and the common 000258 | conformal vector make S isotropic for 000259 | 000260 | q(a,b)=(5a^2+3b^2)/8 mod 1. (4.4) 000261 | 000262 | Condition (4.2) gives H= subset S. The native discriminant pairing gives 000263 | 000264 | H^perp={(a,b):5a+3b=0 mod 4}=H. (4.5) 000265 | 000266 | Hence 000267 | 000268 | H subset S subset H^perp=H, 000269 | 000270 | so S=H and every sector occurs once. The coherent local extension theorem 000271 | then yields the unique marked extension 000272 | 000273 | W=M_0 + M_h + M_2h + M_3h = V_E8. (4.6) 000274 | 000275 | Its weight one branching is literal: 000276 | 000277 | 248 000278 | =(45,1)+(1,15)+(16,4)+(10,6)+(bar16,bar4) 000279 | =60+64+60+64. (4.7) 000280 | 000281 | This proves the full lattice VOA, not merely its 248 dimensional weight one 000282 | slice. The vacuum, conformal vector, unitary structure, L0, local vertex 000283 | products, normalized Q system, vacuum functional, vacuum correlations and 000284 | strongly local conformal net follow in this same marked class. 000285 | 000286 | The hypotheses of this theorem contain no genus two curve and no C3 bulk 000287 | operator. The weakest remaining source premise is exactly: 000288 | 000289 | the primitive construction realizes U as a connected local unitary 000290 | same conformal extension and realizes the marked charged module M_h. 000291 | (4.8) 000292 | 000293 | If the already accepted phrase "field construction in the scoped marked 000294 | class" includes (4.8), then the chiral E8 field and its canonical vacuum are 000295 | already closed. The C3 physical identification should be removed from that 000296 | critical path. 000297 | 000298 | 000299 | 5. HOW THE FINITE MARKS ACT ON THE FULL FIELD 000300 | ============================================== 000301 | 000302 | The finite module is not being used here to choose the E8 lattice. The 000303 | original compiler has already fixed that integral lattice, its D5+A3 000304 | embedding and its cocycle convention. The purpose of (2.12) in this proof is 000305 | narrower: it proves that the auxiliary geometric marks belong to the same 000306 | finite native N64, sigma and quarter class. 000307 | 000308 | There is a stronger auxiliary integral reconstruction, recorded separately 000309 | in THETA-E8-LATTICE-COVARIANCE.txt. The support weighted shell of all sixty 000310 | theta stabilizer rays spans an even unimodular rank eight lattice and hence 000311 | recovers the abstract E8 lattice class. Theta frame changes preserve its 000312 | complex homothety class. The actual source alpha lies in zeta8 G31 and 000313 | exchanges Lambda with zeta8 Lambda, so it does not select one alpha invariant 000314 | embedded lattice. This strengthens the auxiliary branch without changing 000315 | the field argument below. 000316 | 000317 | On the field side an epsilon preserving lattice lift has the actual action 000318 | 000319 | ghat[h_1(-n_1)...h_r(-n_r)e^lambda] 000320 | =(gh_1)(-n_1)...(gh_r)(-n_r) 000321 | eta_g(lambda)e^(g lambda). (5.1) 000322 | 000323 | Because g is an E8 lattice isometry and eta_g is the fixed cocycle carry, 000324 | this is a lattice VOA automorphism. Therefore 000325 | 000326 | ghat Y(v,x) ghat^(-1)=Y(ghat v,x), 000327 | ghat 1=1, 000328 | ghat omega=omega, 000329 | [ghat,L_n]=0. (5.2) 000330 | 000331 | Thus the lattice form, vertex products, locality, vacuum and conformal 000332 | grading are preserved by the actual native lifts. The finite intertwiner in 000333 | (2.12) identifies the marked action with these already existing native lifts. 000334 | It is not used to pull an integral lattice or a field algebra back to the 000335 | curve. 000336 | 000337 | This avoids the earlier circular move. The E8 lattice and fields come from 000338 | the compiler and the local extension theorem. The curve proves the finite 000339 | marking class. Their conjunction is made in the fixed native chart, where 000340 | both sets of operators are explicit. 000341 | 000342 | 000343 | 6. THE DECISIVE TYPE CHECK 000344 | ========================== 000345 | 000346 | The auxiliary theta module cannot by itself supply the local generator in 000347 | (4.2). 000348 | 000349 | W_aux has complex dimension 4 and carries the finite Heisenberg action. 000350 | 000351 | The marked local E8 sector is 000352 | 000353 | M_h=(16,4), 000354 | 000355 | whose lowest weight space has dimension 64 and whose full module contains 000356 | infinitely many oscillator descendants. 000357 | 000358 | The theta functor supplies neither the Heisenberg oscillator Fock space of 000359 | the lattice VOA, nor the twisted group algebra and its cocycle, nor vertex 000360 | operators, OPE locality, positive energy L0 or a vacuum functional. Even the 000361 | separate abstract E8 root shell reconstruction supplies none of these local 000362 | field structures. The inverse q norm is a one dimensional central character 000363 | correction. It is not the charged module M_h. 000364 | 000365 | Consequently the finite auxiliary reconstruction cannot prove (4.8) and 000366 | cannot close SEAM.EQUIV.01. This is a type obstruction, not a missing 000367 | numerical check. 000368 | 000369 | 000370 | 7. FINAL CAUSAL DIAGRAM 000371 | ======================= 000372 | 000373 | The smallest noncircular structure is 000374 | 000375 | original P1/P2 compiler 000376 | | 000377 | fixed K=D5+A3, h and E8 glue 000378 | / \ 000379 | / \ 000380 | auxiliary marked P2 curve local realization (4.8) 000381 | | | 000382 | circle M_z -> defect -> theta marked generator theorem 000383 | | | 000384 | finite N64,sigma,alpha V_E8 000385 | | | 000386 | native finite mark class fields, vacuum, L0, local net 000387 | \ / 000388 | fixed native chart 000389 | 000390 | The left branch does not require physical C3 propagation. The right branch 000391 | does not require the genus two curve. Their common native chart provides the 000392 | exact finite action on the already reconstructed field. 000393 | 000394 | What remains open for the unconditional physical origin is one weaker and 000395 | better typed arrow than a full P1 to C3 Dirac equivalence: 000396 | 000397 | raw RP seam source 000398 | -> a connected local same conformal U extension containing M_h, 000399 | 000400 | or, equivalently after the theorem,