TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/EXISTING-C3-SOURCE-DICTIONARY.txt Original file SHA-256: 9653148d8235dff822254d2b5be96da73f17c374741e1d6275d0e3e649826817 Inclusive original lines: 191-380 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. 000191 | and contains invertible matrices. For example 000192 | 000193 | Z0 = [ 1 4 -2] 000194 | [-2 1 4] 000195 | [ 0 0 9] 000196 | 000197 | satisfies (3.1) and det Z0=81. This is the expected bare C3 equivalence. 000198 | 000199 | Now retain the original compiler on the same source domain: 000200 | 000201 | Z P_nat = P_* Z, 000202 | Z Q = Q Z. (3.2) 000203 | 000204 | The combined 18 linear equations have rank 9 in the 9 entries of Z. Hence 000205 | 000206 | Z=0. (3.3) 000207 | 000208 | There is not merely no unitary or integral solution. There is no nonzero 000209 | Q equivariant intertwiner at all. 000210 | 000211 | The first commutators already display the mismatch: 000212 | 000213 | charpoly([Q,P_nat]) = x(x-2)(x+2), rank 2, 000214 | charpoly([Q,P_*]) = x^3-4x-20, rank 3. 000215 | 000216 | A simultaneous similarity which fixes Q would preserve these data. Thus a 000217 | change to a Fourier basis cannot repair the source dictionary while Q is 000218 | kept fixed. 000219 | 000220 | 000221 | 4. Quarter carrier test 000222 | ----------------------- 000223 | 000224 | The marked v117 dictionary sends U to -G. Retaining this quarter operator 000225 | and trying to replace P_* by the native P_nat gives 000226 | 000227 | Z P_nat = P_* Z, 000228 | Z G = G Z. (4.1) 000229 | 000230 | Again the coefficient matrix has rank 9, hence only Z=0. 000231 | 000232 | The intrinsic distinction can also be seen without solving for Z. 000233 | 000234 | For the v117 pullback pair (P_*,-G): 000235 | 000236 | (P_* (-G))^4=I, 000237 | charpoly(P_* (-G))=(x-1)(x^2+1), 000238 | the invariant symmetric forms are C N, 000239 | 000240 | where 000241 | 000242 | N = [1 0 0] 000243 | [0 5 -2] 000244 | [0 -2 1] 000245 | 000246 | is positive definite. The generated group is the exact order 24 W(A3). 000247 | 000248 | For the attempted native pair (P_nat,-G): 000249 | 000250 | charpoly(P_nat (-G))=(x-1)(x+1)^2, 000251 | 000252 | and the -1 eigenspace has dimension one although its algebraic multiplicity 000253 | is two. Thus the minimal polynomial also contains (x+1)^2 and the product is 000254 | not of finite order. The simultaneous invariant symmetric 000255 | form equations 000256 | 000257 | P_nat^T H P_nat=H, 000258 | (-G)^T H (-G)=H 000259 | 000260 | have only H=0. Hence this attempted pair is not a different unitary chart of 000261 | the v117 S4 representation. 000262 | 000263 | The v417 seam quarter D gives the same qualitative warning in the original 000264 | native basis: [D,P_nat] is nonzero and v417 explicitly records that these two 000265 | operators do not form a common mu12 clock. The exact C12 clock exists on the 000266 | larger marked cover and Phi12 module, not on this rank three pair by scalar 000267 | phase repair. 000268 | 000269 | 000270 | 5. Direct test of the displayed TFPT45 puncture formula 000271 | ------------------------------------------------------- 000272 | 000273 | The current extract at archive source 03, section 004, lines 2 to 110 sets 000274 | 000275 | P = [0 1 0] D=diag(1,omega,omega^2) 000276 | [0 0 1] 000277 | [1 0 0] 000278 | 000279 | and declares the first three puncture monodromies to be 000280 | 000281 | D, P D P^{-1}, P^2 D P^{-2}, 000282 | 000283 | with the fourth defined as the inverse of their product. Exact reduction by 000284 | omega^2+omega+1=0 gives 000285 | 000286 | P D P^{-1} = omega D, 000287 | P^2 D P^{-2} = omega^2 D, 000288 | 000289 | up to the equivalent inverse orientation convention, and therefore 000290 | 000291 | D (PDP^{-1}) (P^2DP^{-2}) = D^3 = I. 000292 | 000293 | Consequently 000294 | 000295 | rho(gamma_{-i})=I, 000296 | 000297 | whose spectrum is (1,1,1), not the stipulated (1,omega,omega^2). The same 000298 | formula also assigns the geometric quarter turn to P although P^3=I. Four 000299 | applications act by conjugation with P^4=P, which does not restore D. 000300 | 000301 | This is a literal internal counterexample to that displayed rigidity proof. 000302 | It does not affect the corrected v117 package: v117 uses the order four U, 000303 | the noncommuting M, and verifies all four conjugate local classes plus their 000304 | product identity exactly. 000305 | 000306 | The follow-on claim of a unique flat Hermitian connection in lines 113 to 160 000307 | therefore does not follow from the displayed D,P construction. It would need 000308 | to be restated using the actual v117 matrices and with the recorded analytic 000309 | identification status retained. 000310 | 000311 | 000312 | 6. What is proved, excluded, and still open 000313 | ------------------------------------------- 000314 | 000315 | PROVED 000316 | 000317 | * Native sigma on the three family slots and the v117 cusp M are isomorphic 000318 | as bare regular C3 modules. 000319 | * The Fermat deck on natural H10 and full H1 is not that regular module. 000320 | * The existing C12 Phi12 bridge is rank four with C3 multiplicities 000321 | (0,2,2), not the v117 rank three carrier. 000322 | * The exact P_* pullback gives the marked v117 pair and transports Q. 000323 | * P_nat cannot replace P_* while Q or the quarter operator -G is retained: 000324 | both simultaneous intertwiner spaces are zero. 000325 | * The D,P puncture formula printed in the current TFPT45 extract makes the 000326 | fourth local monodromy trivial and cannot establish its stated theorem. 000327 | 000328 | EXCLUDED IN THIS SCOPE 000329 | 000330 | * Identifying all order three operators merely from their eigenvalues. 000331 | * Repairing the Fermat deck character with a scalar cubic phase. 000332 | * Claiming that the C12 Phi12 intertwiner supplies the rank three P2 000333 | connection or the fixed Q action. 000334 | * Calling the native slot permutation and v117 M the same marked operator 000335 | while the original compiler and quarter action stay fixed. 000336 | 000337 | OPEN, NOW SHARPLY FORMULATED 000338 | 000339 | A direct origin bridge would have to provide a source map between the 000340 | native E8 family slot module and the v117 puncture local system which also 000341 | states how Q and the quarter action transform. For the currently fixed Q 000342 | and quarter representatives that map does not exist. Progress therefore 000343 | requires a different source derived action or a proved transport of the 000344 | whole marked tuple. A new scalar phase, an unmarked Fourier transform, or 000345 | the existing rank four C12 bridge cannot do it. 000346 | 000347 | 000348 | 7. Reproduction and checked sources 000349 | ----------------------------------- 000350 | 000351 | Executed successfully on 26 October 2026: 000352 | 000353 | python3 -B verification/v117_monodromy_weyl_a3.py 000354 | 5 passed, 0 failed. 000355 | 000356 | python3 -B verification/v417_eisenstein_cp_operator.py 000357 | 5 passed, 0 failed. 000358 | 000359 | python3 -B experiments/theory-contracts/source-native-clock-flavor-bridge-20260922/checker.py --out /tmp/c3-source-dictionary.json 000360 | PASS_SCOPED_NATIVE_CLOCK_FLAVOR_EXACT_AUDIT, group order 24. 000361 | 000362 | python3 -B experiments/theory-contracts/e-series-compiler-closure-20261023/cover_dictionary.py 000363 | ALL COVER DICTIONARY CHECKS PASSED. 000364 | 000365 | The new calculations in Sections 3 to 5 used exact SymPy linear algebra over 000366 | Q and Q(i), plus polynomial reduction by omega^2+omega+1. No floating point 000367 | comparison enters the verdict. 000368 | 000369 | Source locations and current SHA256 pins: 000370 | 000371 | documents/archive--tfpt-45--source_extracts--03_em_flavor_source/ 000372 | sections/004-source.tex:2-160 000373 | 107d525221eb816043dbf4fb1a3c7e8244d3e24a5a5bcbfe1c2c34d8ee192f4a 000374 | 000375 | verification/v117_monodromy_weyl_a3.py:1-43,52-56,129-182 000376 | 55eca6a64144f6e49789f3f1bbe84e256764e7ce92b207044be693f8684d5df5 000377 | 000378 | verification/v417_eisenstein_cp_operator.py:1-40,49-58,77-83,126-140 000379 | c5f1fe26b9e6fd30bf2ad45354de6819a184c4a86b491f19ab21cb53b21c2b7e 000380 |