TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SEAM-GRAPH-REVIEW.txt Original file SHA-256: d5701f6475333f77ac7870a7917518bbe69f637fe6727e2eeb90e59edf3345c8 Inclusive original lines: 191-365 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 | 000192 | G0= 000193 | 000194 | has order 16, with eight holomorphic elements and an antiholomorphic coset of 000195 | eight real involutions. Strictly speaking, those eight real maps form a 000196 | coset, not a subgroup. 000197 | 000198 | Now adjoin r. Set t_x=r Q^(-1), the half translation z->z+1/2. Conjugation by 000199 | Q gives t_y:z->z+i/2. The four half translations form V4, and 000200 | 000201 | H==V4 semidirect C4, 000202 | |H|=16. (8) 000203 | 000204 | The deck h is central. The lifted dihedral relation is 000205 | 000206 | r Q r^(-1)=h Q^(-1), (9) 000207 | 000208 | so after quotienting by , relation (9) becomes the ordinary D4 reflection 000209 | relation. Hence H is a central deck extension of the base D4 group, not D4 000210 | itself. This is the required scope correction: the old canonical H0 of order 000211 | eight is a proper abelian subgroup of the full lifted action. 000212 | 000213 | Finally 000214 | 000215 | G= 000216 | 000217 | has order 32. Orientation separates it into 16 holomorphic and 16 000218 | antiholomorphic maps. 000219 | 000220 | 4. Real-form census computed from fixed equations 000221 | ================================================== 000222 | 000223 | Every antiholomorphic affine map on the square torus is represented in real 000224 | coordinates by 000225 | 000226 | v -> M v+d, det(M)=-1. (10) 000227 | 000228 | The checker first computes its order in the finite affine group. For an 000229 | involution it solves 000230 | 000231 | (I-M)v=d mod Z^2. (11) 000232 | 000233 | This is a rank-one torus equation. A primitive integer vector in the left 000234 | kernel of I-M detects whether (11) is soluble. If it is, the nonzero Smith 000235 | invariant 000236 | 000237 | gcd(entries of I-M) 000238 | 000239 | is the number of fixed circles. Thus the program classifies an involution as 000240 | 000241 | 0 circles: empty, 000242 | 1 circle: nondividing, 000243 | 2 circles: dividing. 000244 | 000245 | This procedure independently yields for the old antiholomorphic coset: 000246 | 000247 | nondividing 4, empty 2, dividing 2. (12) 000248 | 000249 | For the eight maps added by r it yields: 000250 | 000251 | dividing 2, empty 2, antiholomorphic order four 4. (13) 000252 | 000253 | Combining (12) and (13), the full set of 16 antiholomorphic maps contains 000254 | 000255 | 4 nondividing real involutions, 000256 | 4 empty real involutions, 000257 | 4 dividing real involutions, 000258 | 4 antiholomorphic order-four maps. (14) 000259 | 000260 | Consequently the phrase "16 real forms" would be wrong. There are 16 000261 | antiholomorphic affine symmetries, but only 12 are involutions and hence real 000262 | structures. 000263 | 000264 | 5. Old OS forms versus marked-equator forms 000265 | ============================================ 000266 | 000267 | The two dividing members of the old coset are 000268 | 000269 | rho_+=hQkappa, 000270 | rho_-=hQ^3kappa. 000271 | 000272 | In affine coordinates their fixed circles lie at 000273 | 000274 | t=0,1/2 or s=0,1/2. 000275 | 000276 | They avoid all four branch points (5). These are the previously classified 000277 | dividing Schottky/OS forms whose halves are smooth annuli. 000278 | 000279 | The new dividing members are 000280 | 000281 | S=rkappa, 000282 | hS. 000283 | 000284 | Their fixed circles are (4), and each real structure fixes all four branch 000285 | points. Together they give the full marked pullback (3). The distinction is 000286 | not cosmetic: 000287 | 000288 | rho_+/-: smooth dividing OS boundaries, no marked branch point fixed; 000289 | 000290 | S,hS: marked-equator lifts, all four branch points fixed, union is a graph. 000291 | 000292 | The old NS spin character on E can still be restricted to these circles, but 000293 | that does not identify the complete graph with one old NS circle. A boundary 000294 | Dirac operator on one selected component and a graph Dirac operator with 000295 | vertex conditions are different closed operators and have different domains. 000296 | 000297 | 6. Consequence for the source bridge 000298 | ==================================== 000299 | 000300 | The earlier source audit remains valid but its target now needs one sharper 000301 | typing condition. A proposed P1-to-elliptic map must say whether the original 000302 | smooth seam is transported to 000303 | 000304 | a. one old OS oval of rho_+ or rho_-, 000305 | b. one selected component of Fix(S) or Fix(hS), 000306 | c. the normalization of all four marked ovals, or 000307 | d. the full V4/E8 graph with explicit vertex conditions. 000308 | 000309 | These targets are not interchangeable. In particular, identifying the old 000310 | smooth circle with (d) is false before any field or state question is asked. 000311 | Selecting (b), (c), or a smoothing of (d) adds boundary data and must be 000312 | derived from the original source rather than hidden in a cover dictionary. 000313 | 000314 | This does not invalidate the elliptic annulus, its dividing OS cuts, or the 000315 | quarter boundary calculation. It separates two real structures that were 000316 | previously close enough to confuse: 000317 | 000318 | the OS/Schottky cut rho_+/- produces a smooth annulus boundary; 000319 | the lifted marked equator S produces the ramified seam graph. 000320 | 000321 | The smallest next test is therefore no longer a comparison of group orders. 000322 | It is the domain-level question whether the original P1 seam trace lands on 000323 | one smooth rho oval, or whether the original theory supplies graph vertex 000324 | conditions at the four ramification points. Without that relation, the full 000325 | marked pullback cannot be used as the original one-circle source. 000326 | 000327 | 7. Independent checker 000328 | ======================= 000329 | 000330 | Files: 000331 | 000332 | research-20261026-elliptic-source-continuation/ 000333 | check_independent_seam_graph.py 000334 | independent-seam-graph-results.json 000335 | 000336 | The checker performs 21 exact tests. It: 000337 | 000338 | * verifies algebraically that r preserves y^2=x^4-1 and r^2=h; 000339 | * closes the old and enlarged affine groups by breadth-first composition; 000340 | * computes element orders rather than assigning them from names; 000341 | * computes every fixed-component count from equation (11); 000342 | * separates the original and newly added antiholomorphic cosets; 000343 | * verifies that the old dividing forms avoid all branch marks; 000344 | * identifies the new dividing forms as exactly S and hS; 000345 | * constructs the seam graph, incidence matrix, rank, b1, and torus face 000346 | count. 000347 | 000348 | Observed result: 000349 | 000350 | PASS 21/21 000351 | RESULT_SHA256 e7cb292768c5432840f188c865c81924350fbbc2eb14defacb100be3d82ece6f 000352 | 000353 | Scope 000354 | ===== 000355 | 000356 | PROVED: the r extension, the 32-element affine group, the 12-real-involution 000357 | census, the distinction between old OS forms and new marked-equator forms, 000358 | and the V4/E8/F4, b1=5 seam graph. 000359 | 000360 | REFUTED: the direct identification of the full marked equator pullback with 000361 | the old smooth one-circle NS source. 000362 | 000363 | OPEN: which smooth oval or graph domain the original P1 boundary trace 000364 | actually selects. That is an operator-domain source question, not a missing 000365 | finite group calculation.