TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/dynamic_reconstruction_20261027/PETERSEN-POLAR-DECK-INTERTWINER.txt Original file SHA-256: 0d2f283a3c6568bd77cc24f081c61dc706a7faef209138335336a1fbbd8007cd 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 | It satisfies U U^T = I and the claimed conjugation exactly. 000202 | 000203 | The shell charge multiplication operator is 000204 | 000205 | Y_shell = diag(6,-4,1). 000206 | 000207 | It is not discarded. In the target frame it becomes 000208 | 000209 | [0, 3sqrt(2)/2, 3sqrt(2)/2 ] 000210 | Y_target = [3sqrt(2)/2, 3/2+5sqrt(5)/3, -7/6 ]. 000211 | [3sqrt(2)/2, -7/6, 3/2-5sqrt(5)/3 ] 000212 | 000213 | This matrix has the original eigenvalues 6, -4 and 1. The marked hypercharge 000214 | function itself maps more simply as W(q/3) = e+. 000215 | 000216 | 5. Exact deck obstruction 000217 | -------------------------- 000218 | 000219 | The v487 deck operator swaps the surviving pair: 000220 | 000221 | [1 0 0] 000222 | Delta = [0 0 1]. 000223 | [0 1 0] 000224 | 000225 | Its spectral marks are 000226 | 000227 | Delta e0 = e0, 000228 | Delta e+ = e+, 000229 | Delta e- = -e-. 000230 | 000231 | The transported Petersen polar sign is instead 000232 | 000233 | [1 0 0] 000234 | K = W J_P W^-1 = [0 0 -1] 000235 | [0 -1 0], 000236 | 000237 | so that 000238 | 000239 | K e0 = e0, 000240 | K e+ = -e+, 000241 | K e- = e-. 000242 | 000243 | Equivalently, 000244 | 000245 | K = Z Delta, Z = diag(1,-1,-1) = 2|e0>= 0, 000286 | 000287 | W P^(2n) W^-1 = M_lazy^(2n), 000288 | W P^(2n+1) W^-1 = K M_lazy^(2n+1). 000289 | 000290 | The old sixth power match is the n=3 member of an exact even time theorem. 000291 | 000292 | Define alternating frames 000293 | 000294 | W_n = K^n W. 000295 | 000296 | Then 000297 | 000298 | W_(n+1) P W_n^-1 = M_lazy 000299 | 000300 | for every n. The hypercharge mark is carried as 000301 | 000302 | W_n(q/3) = (-1)^n e+. 000303 | 000304 | Thus the sign of the Petersen hypercharge mode becomes orientation data of the 000305 | time frame. The full charge multiplication operator is transported as 000306 | 000307 | Y_n = K^n Y_target K^n, 000308 | 000309 | which has period two. A fixed positive family transfer and a fixed charge 000310 | matrix cannot both be obtained from this dictionary without retaining that 000311 | orientation bit. 000312 | 000313 | 7. Canonical bipartite double cover 000314 | ----------------------------------- 000315 | 000316 | The same result can be made static by retaining the orientation sheet. On 000317 | H_pi direct sum H_pi define 000318 | 000319 | [0 P] 000320 | P_hat = [P 0]. 000321 | 000322 | This is the radial action induced by the canonical bipartite double cover: 000323 | every step exchanges the two orientation sheets. Let 000324 | 000325 | [I 0 ] 000326 | G = [0 J_P]. 000327 | 000328 | Using P J_P = J_P P = |P| gives 000329 | 000330 | [ 0 |P|] 000331 | G P_hat G = [|P| 0 ]. 000332 | 000333 | Therefore 000334 | 000335 | W_hat = diag(W,W) G 000336 | 000337 | is a fixed isometry with 000338 | 000339 | [ 0 M_lazy] 000340 | W_hat P_hat W_hat^-1 = [M_lazy 0 ]. 000341 | 000342 | So the orientable double cover does remove the signed one step obstruction to 000343 | positive transport. It does not preserve the plain deck mark. If tau is the 000344 | sheet exchange, then 000345 | 000346 | W_hat tau W_hat^-1 = [0 K; K 0], 000347 | 000348 | while a plain target sheet exchange would contain I, and the v487 internal 000349 | deck would contain Delta. The exact transported deck is the product of sheet 000350 | exchange, native deck and Perron reflection: 000351 | 000352 | tau tensor K = tau tensor (Z Delta). 000353 | 000354 | 8. Scope and consequence 000355 | ------------------------- 000356 | 000357 | The strongest source based conclusion is positive and precise: 000358 | 000359 | * signed Petersen transport and the lazy walk are the same reversible 000360 | contraction after retaining a canonical period two orientation frame; 000361 | * all even powers agree under one fixed marked isometry; 000362 | * the bipartite double cover makes the equivalence static; 000363 | * the shell charge and all three marked modes are explicitly transported. 000364 | 000365 | The literal identification of the Petersen polar sign with the v487 deck is 000366 | excluded. Their nontrivial parity assignments are opposite. The smallest 000367 | additional source statement that would close the natural route is therefore: 000368 | 000369 | geometric deck = orientation-sheet exchange times Z Delta, 000370 | 000371 | or an independently derived reason that the OS or clock lift includes the 000372 | Perron-complement reflection Z. The inspected source supplies Delta but does 000373 | not supply this extra reflection as part of the deck action. Interpreting the 000374 | period two frame as physical time or promoting Y_target to a charged local 000375 | field remains a separate identification. 000376 | 000377 | 9. Operator test: the shell charge is transported but not conserved 000378 | -------------------------------------------------------------------- 000379 | 000380 | The preceding formula transports the multiplication operator attached to the 000381 | literal shell hypercharge 000382 | 000383 | Y_shell = diag(6,-4,1). 000384 | 000385 | It does not turn that observable into a conserved charge of the Markov step. 000386 | Directly, 000387 | 000388 | [ 0 10 0 ] 000389 | [Y_shell,P] = [-10/3 0 -10/3 ] 000390 | [ 0 5/3 0 ], 000391 | 000392 | which has rank two. In the alternating target frames let 000393 | 000394 | Y_n = K^n W Y_shell W^-1 K^n. 000395 | 000396 | Using W_(n+1) P W_n^-1 = M_lazy gives the exact defect identity 000397 | 000398 | Y_(n+1) M_lazy - M_lazy Y_n 000399 | = W_(n+1) [Y_shell,P] W_n^-1. 000400 |