TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/EQUATORIAL-LIFTS.txt Original file SHA-256: 434eaab2fa67fbf157967a78f4b3f742cb7731d61c9632cdb06f12773c0ba1dd Inclusive original lines: 381-466 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. 000381 | 000382 | The existing source calculation identifies the unique Q-fixed nontrivial 000383 | elliptic spin class as 000384 | 000385 | alpha=O(T-O), T=(1+i)/2. 000386 | 000387 | See 000388 | 000389 | experiments/theory-contracts/original-admissible-source-closure-20261025/ 000390 | ORIGINAL-BASES-AND-ELLIPTIC-SPIN.txt:60-65,72-101 000391 | SHA256 bf6b2336e24a39aef26b9907c24f033a47a8a1318f8f19ae561e4c477680ce67. 000392 | 000393 | As a flat character on the square torus this is 000394 | 000395 | chi_alpha(1)=chi_alpha(i)=-1. (21) 000396 | 000397 | Both Q and S pull the elliptic differential dx/y to i dx/y. Their integral 000398 | H1 action is the quarter rotation, so alpha is invariant under both. It is 000399 | also invariant under h and under the anti-holomorphic maps above as a 000400 | two-torsion class. 000401 | 000402 | By (19), every j_+ boundary oval represents the A cycle and every j_- 000403 | boundary oval represents the B cycle. Equation (21) therefore gives 000404 | 000405 | hol_alpha(Fix(j_+) component)=-1, 000406 | hol_alpha(Fix(j_-) component)=-1. (22) 000407 | 000408 | Thus all four smooth oval components carry NS spin. Q exchanges the two 000409 | horizontal and vertical presentations without a new spin choice. 000410 | 000411 | This is compatibility, not yet the original kernel theorem. At each branch 000412 | vertex p^{-1}(Gamma) has four incident half-edges. The flat spin line does 000413 | not select a self-adjoint vertex condition, a Calderon graph subspace, or a 000414 | state. Also, v519's OS anti-linear operator is family D, whereas j_+/- are 000415 | family A. Separate Real linearizations are required for the two roles. The 000416 | old source has not supplied their common Fock or boundary-domain transport. 000417 | 000418 | 5. Precise correction to the preceding reports 000419 | =============================================== 000420 | 000421 | The following results remain valid: 000422 | 000423 | * the eight-map h^a Q^b kappa census; 000424 | * the dividing nature of hQkappa and hQ^3kappa; 000425 | * their annulus quotients, the free translation hQ^2, and the exact spin 000426 | central commutator already calculated there; 000427 | * their identification with the v519 mark-swapping midpoint OS axes. 000428 | 000429 | The following earlier wording is too broad and is superseded here: 000430 | 000431 | * "the complete real-form census has eight members" must read "the 000432 | complete census inside has eight members"; 000433 | * "the unique dividing orbit in the full marked group" is false after the 000434 | already present base mirror x -> 1/x is lifted; 000435 | * a dividing fixed oval of j_+ or j_- is not by itself the lift of the 000436 | complete original equatorial seam; 000437 | * x -> 1/bar x is not the v519 OS reflection. It is the pointwise fixed 000438 | Euclidean seam structure on which the family-D OS reflections act. 000439 | 000440 | The new positive bridge is exact but bounded: one and the same existing C2 000441 | curve, with no new cover and no fitted spin character, realizes the complete 000442 | two-family v510/v519 base classification and the NS restriction on the new 000443 | equatorial oval components. The missing physical arrow is now the boundary 000444 | domain and state on the ramified seam graph, or an original prescription 000445 | that replaces that graph by a smooth carrier. 000446 | 000447 | 6. Replayed finite certificate 000448 | =============================== 000449 | 000450 | The smallest exact SymPy replay checked the curve identities for S and both 000451 | j signs, S^2=h, j_+^2=j_-^2=1, formula (15), the 12+4 count, and the NS 000452 | cycle labels. Its observed output was: 000453 | 000454 | r_curve_residue_mod_curve=0 000455 | j_sign +1 curve_residue_mod_conj_curve=0 000456 | j_sign -1 curve_residue_mod_conj_curve=0 000457 | r2=h, j_plus_square=1, j_minus_square=1 000458 | typeB b=0: square=1, order=2, base=1/barx, 2 ovals 000459 | typeB b=1: square=hQ^2, order=4, base=i/barx 000460 | typeB b=2: square=1, order=2, base=-1/barx, empty 000461 | typeB b=3: square=hQ^2, order=4, base=-i/barx 000462 | strict_real_count=12, antiholomorphic_order4_count=4 000463 | j_plus boundary class=A, chi_alpha(A)=-1 000464 | j_minus boundary class=B, chi_alpha(B)=-1 000465 | 000466 | No new checker file or auxiliary model was introduced.