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: 238-335 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. 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 |