TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-compiler-closure-20261023/COVERS.txt Original file SHA-256: cdd04f5f38819b60ddd8488eb1c3c316c6849829387205e78ec5e4bf767f80bf 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 | exterior2 V4, rank6: 000202 | chi=(X+1)^2 Phi_6^2, m=(X+1)Phi_6; 000203 | 000204 | quartic P5, rank5: 000205 | chi=Phi_1^3 Phi_3, m=X^3-1. (6.3) 000206 | 000207 | For P5 the checker acts on the exact five sparse supports of 000208 | process._code_basis: sum z_i^4, the three paired-square sums, and 000209 | z0z1z2z3. It does not fit a new invariant five-space. The original twenty 000210 | erasures partial_a P_i are linearly independent and span Sym3 V4. In the 000211 | existing bar(V4) tensor P5 reading their complex characteristic polynomial is 000212 | 000213 | chi_c,erasure20=(X+i)^8 (X^2-iX-1)^6, 000214 | m_c,erasure20 =(X+i) (X^2-iX-1). (6.4) 000215 | 000216 | Its rational realification therefore has characteristic polynomial 000217 | Phi_4^8 Phi_12^6 and minimal polynomial Phi_4 Phi_12. The multiplicity six 000218 | means the erasure carrier contains Phi_12 copies, but it supplies no preferred 000219 | one of them. 000220 | 000221 | For the curve, use exactly the reduced Burau generators of v597/v610/v613 at 000222 | T=omega and rationalize each Eisenstein coordinate in the integral basis 000223 | (1,omega). The predecessors identify this rank-three Eisenstein module with 000224 | the H1 character carrier. The certificate below works in its retained native 000225 | Z[omega]^3 basis; it does not independently construct a geometric integral 000226 | cycle basis. The matrix r becomes a rank-six integral rational matrix with 000227 | 000228 | chi_r,H1=Phi_6 Phi_12, 000229 | m_r,H1 =Phi_6 Phi_12, (6.5) 000230 | r^4=D_omega, r^9=Q, r^12=1. 000231 | 000232 | Over Q(omega), its characteristic polynomial is 000233 | 000234 | X^3+omega X^2+omega^2 X+1. 000235 | 000236 | This is the exact H1 realization of the v610 combined character orders 000237 | (12,12,6). Equations (6.1) and (6.5) show both the common component and the 000238 | full obstruction: 000239 | 000240 | common: one rational Phi_12 module, dimension4; 000241 | E8 remainder: Phi_4^2, dimension4; 000242 | H1 remainder: Phi_6, dimension2. (6.6) 000243 | 000244 | The coprime factors make both Phi_12 kernels canonical. No carrier 000245 | isomorphism extending the common component exists because the complementary 000246 | cyclotomic factors and dimensions differ. 000247 | 000248 | The same factor test completely settles the retained descendants. Quartic 000249 | P5 has no rational irreducible factor in common with H1. Exterior2 V4 shares 000250 | Phi_6, but carries two copies while H1 carries one; the existing data do not 000251 | select one exterior copy. Erasure20 shares Phi_12, but carries six copies and 000252 | likewise supplies no selected copy. Thus only real8 versus H1 has a common 000253 | irreducible of multiplicity one on both sides, which is why it yields the 000254 | canonical single-copy subspaces used below. 000255 | 000256 | 7. Explicit maximal Phi12 transport 000257 | 000258 | Let U_E=ker Phi_12(c) in the actual E8 lattice and let 000259 | U_B=ker Phi_12(r) in the rationalized reduced-Burau Z^6 lattice. In the 000260 | original coordinate orders, take the columns of 000261 | 000262 | B_E = [ -1 0 -1 0 ] B_C = [ 0 1 0 0 ] 000263 | [ 0 0 0 1 ] [ 0 1 0 1 ] 000264 | [ 1 0 0 0 ] [ 1 0 1 0 ] 000265 | [ 0 -1 0 -1 ] [ 0 0 1 0 ] 000266 | [ 0 0 1 0 ] [ 0 1 0 1 ] 000267 | [ 0 1 0 0 ] [ 0 0 0 1 ] 000268 | [ 0 0 0 0 ] 000269 | [ 0 0 0 0 ]. 000270 | 000271 | Both column lattices are saturated: the gcd of their maximal minors is1. 000272 | On the E8 side the Phi_12 subspace has last complex coordinate zero, so the 000273 | half-integral E8 coset does not meet it; all of its integer points have even 000274 | coordinate sum. Thus B_E spans the actual U_E intersection E8, not merely an 000275 | ambient Z^8 substitute. 000276 | In both bases the order-twelve generator is 000277 | 000278 | A12 = [0 0 0 -1] 000279 | [1 0 0 0] 000280 | [0 1 0 1] 000281 | [0 0 1 0], Phi_12(A12)=0. (7.1) 000282 | 000283 | Therefore 000284 | 000285 | T12(B_E a)=B_C a, a in Z^4, (7.2) 000286 | 000287 | is an explicit integral Z[C12]-module isomorphism. It preserves more than 000288 | the generator. In the same bases, 000289 | 000290 | J_E = Q_C = [ 0 1 0 1] sigma_E = D_C = [-1 0 -1 0] 000291 | [ 0 0 1 0] [ 0 -1 0 -1] 000292 | [ 0 -1 0 0] [ 1 0 0 0] 000293 | [-1 0 -1 0], [ 0 1 0 0]. 000294 | 000295 | Thus the actual carrier action satisfies exactly 000296 | 000297 | T12 c = r T12, 000298 | T12 J = Q T12, J=c^9, Q=r^9, 000299 | T12 sigma = D_omega T12, sigma=c^4, D_omega=r^4. (7.3) 000300 | 000301 | This is the maximal common transport component asked for: Phi_12 occurs once 000302 | on each of real8 and curve H1, and the remaining irreducibles do not match. 000303 | The displayed T12 is a concrete certificate intertwiner. The source data do 000304 | not choose it uniquely among the automorphisms of the common cyclotomic 000305 | module, so it is not promoted to a new physical identification. 000306 | 000307 | The existing integral forms put a further exact boundary on (7.2). On U_E, 000308 | the E8 Euclidean form and the existing complex structure give the alternating 000309 | form E_E(u,v)=. In the B_E basis, 000310 | 000311 | E_E = [ 0 1 0 2] 000312 | [-1 0 1 0] 000313 | [ 0 -1 0 1] 000314 | [-2 0 -1 0], det E_E=9, type (1,3). (7.4) 000315 | 000316 | On the retained Eisenstein Burau lattice, use the actual invariant Hermitian 000317 | matrix J_B, called J in v597. The primitive alternating form 000318 | 000319 | E_C=Tr_(Q(omega)/Q)((1+2omega)J_B)/6 000320 | 000321 | is integral, unimodular on the full retained rank-six lattice and r-invariant. 000322 | Its restriction in the B_C basis is 000323 | 000324 | E_C|Phi12 = [0 -1 -1 -2] 000325 | [1 0 -1 -1] 000326 | [1 1 0 -1] 000327 | [2 1 1 0], det=4, type (1,2). (7.5) 000328 | 000329 | The notation in the trace line means the field trace from Q(omega) to Q. 000330 | The different elementary-divisor types prove that no unimodular Phi_12 000331 | intertwiner can also preserve these retained invariant integral alternating 000332 | forms. This calculation does not identify E_C with the geometric topological 000333 | intersection pairing on an independently constructed cycle lattice. The 000334 | unpolarized integral clock transport (7.2)-(7.3) remains exact. 000335 | 000336 | 8. The two roles called J 000337 | 000338 | The comparison exposes one necessary distinction. The all-J peripheral 000339 | lattice monodromy defines the C4 kernel cover, where its image is the deck map 000340 | 000341 | J_peripheral <-> Delta_4=(0,1). 000342 | 000343 | The compiler relation J=c^9, and the actual Phi_12 intertwiner above, give 000344 | 000345 | J_compiler=c^9 <-> Q_4=(1,0), 000346 | 000347 | the plain lift of x->ix. These are different automorphisms: Delta_4 covers 000348 | the identity on the base, while Q_4 covers q. More strongly, every lift 000349 | (1,k) of q has ninth power with base component1, so its ninth power can never 000350 | be a deck transformation with base component0. Consequently the common 000351 | cover tower and the carrier intertwiner are each exact, but there is no single 000352 | equivariant action that identifies peripheral J and compiler J as the same 000353 | cover automorphism. This is an exact scoped mismatch, not a missing choice of 000354 | phase. 000355 | 000356 | 9. Root and covered-seam permutation characters 000357 | 000358 | The v623 action on its 48 sites consists of four twelve-cycles. Hence 000359 | 000360 | chi_L,48=(X^12-1)^4, m_L=X^12-1. 000361 | 000362 | The actual compiler on the 240 E8 roots has nineteen twelve-cycles and three 000363 | four-cycles: 000364 | 000365 | chi_c,roots=(X^12-1)^19 (X^4-1)^3, m_c,roots=X^12-1. 000366 | 000367 | Thus equal minimal polynomial and the same marked power relations do not make 000368 | the permutation carriers equal. Their cyclotomic multiplicities for 000369 | d=(1,2,3,4,6,12) are respectively 000370 | 000371 | E8 roots: (22,22,19,22,19,19), 000372 | seam48: ( 4, 4, 4, 4, 4, 4). 000373 | 000374 | As an abstract Q[C12]-module the whole seam48 can be embedded in the root 000375 | permutation module by choosing four of the nineteen regular twelve-orbits. 000376 | No existing mark selects four such root orbits, so that embedding is not a 000377 | source-derived dictionary and is not made here. 000378 | 000379 | 10. Exact scope 000380 | 000381 | The result now identifies the complete common-cover tower, the unique marked 000382 | common-cover clock, and the maximal actual real8/Burau-H1-model transport 000383 | component. It also proves the precise retained-integral-form, complementary-spectrum, 000384 | peripheral-role and permutation-character obstructions. The quartic, 000385 | exterior and erasure carriers retain their original actions and are not 000386 | silently equated to H1. 000387 | 000388 | Nothing here promotes the older Q/V operator dictionary, selects a physical 000389 | seam, or derives physical time. The common covers are forced fibre products 000390 | of supplied covers, and all representation comparisons use the supplied 000391 | matrices and bases. 000392 | 000393 | Executable certificate 000394 | 000395 | python3 experiments/theory-contracts/e-series-compiler-closure-20261023/cover_dictionary.py 000396 | 000397 | The certificate checks the five Riemann-Hurwitz and peripheral records, all 000398 | cover morphisms, both CRT minimality statements, the actual E8 root action of 000399 | J, sigma and c, c^4=sigma, c^9=J, every common-cover lift candidate, unique 000400 | k=4 selection and all descents. It also reconstructs the actual V4,