TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ORIGINAL-RELEVANCE.txt Original file SHA-256: 516d23005f6b55e0471279ab7b1ab85d4f7f8b56282423c0aae4fabf51ef4362 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 | 000192 | v510 then proves three further facts already in the record: the cylinder 000193 | double of a Moebius band has a free glide deck; the deck-commuting real 000194 | structures on the marked base sphere form the diagonal and antidiagonal 000195 | families; and the mu4 marks lie on the equatorial, free-deck fixed circle. 000196 | It also states explicitly that freeness fixes the deck class but does not fix 000197 | the modulus. Its actual functions b_m1_moebius, b_m3_classification, 000198 | b_m4_per_config and b_m7_chain implement those claims. 000199 | 000200 | verification/v510_seam_bit_freedom.py:662-711,747-872,948-968 000201 | SHA256 5b174a8914b14308621e0be98fc9d44e23b89713400ea5c09e1cae17d3546193 000202 | 000203 | paper-source:3ff6da26622ba2984cb0e64c 000204 | documents/changelog/sections/005-source.tex:186-242 000205 | SHA256 f0f5a3c3add4d0a0ec714e3ef03fd838e4a47708ccc34244c25398da738e6987 000206 | 000207 | Therefore the present advance is narrower and more concrete. It lifts the 000208 | already known base-sphere geometry to the actual parity cover (1), finds the 000209 | two dividing affine Real lifts hQkappa and hQ^3kappa, identifies their common 000210 | free translation delta=hQ^2, and computes the genuine flat-spin central 000211 | commutator and bordered Dirac graph. It does not rediscover the free-cover 000212 | principle, the order-four clock, or the selection of tau=i. 000213 | 000214 | 1.6 The existing concrete flat Pin annulus has a different modulus 000215 | ------------------------------------------------------------------- 000216 | 000217 | The broad A4 axiom does not fix a conformal modulus. A later concrete flat 000218 | Pin realization does. In its coordinates, 000219 | 000220 | M_flat=(R x [-1,1])/((x,w)~(x+1,-w)). 000221 | 000222 | Its orientable cover has x-period 2 and transverse width 2. The full 000223 | two-boundary annulus therefore has 000224 | 000225 | L_flat=2, W_flat=2, m_flat=W_flat/L_flat=1. (4a) 000226 | 000227 | The elliptic dividing annulus has t in [0,1/2] and s modulo 1, hence 000228 | 000229 | L_E=1, W_E=1/2, m_E=W_E/L_E=1/2. (4b) 000230 | 000231 | Uniform rescaling preserves W/L, so (4a) and (4b) are not the same conformal 000232 | surface by a unit change. The boundary-to-crosscap half strip used in the 000233 | flat calculation has width 1 and circumference 2, and thus modulus 1/2. It 000234 | is only half of the orientation annulus and carries a crosscap condition at 000235 | its inner edge. It is not the same full two-boundary Dirac problem as the 000236 | elliptic annulus. 000237 | 000238 | research-20261026-fundamental-rethink/primitive/ 000239 | CROSSCAP-NORMAL-TIME.txt:36-64,66-100 000240 | SHA256 291a56fb17bb2eff3e1969f5aef547b281b3cd9c28d6be3494d683d9e7e40bce 000241 | 000242 | research-20261026-p1-reflection/NS-COVER-FULL-DIRAC-RP.txt:4-45 000243 | SHA256 a18c7589ac87b0ea875af7a2a46790989c80c836369b9ee001b0417dda6ab154 000244 | 000245 | The elliptic annulus is consequently a legitimate canonical realization of 000246 | the broad A4 topology. Equality with the later fixed flat full Dirac source 000247 | would require an additional conformal or operator transport. Matching the 000248 | circle B_Sigma after a unit conversion does not identify the transverse 000249 | spectrum, Calderon projector or Green kernel. 000250 | 000251 | 2. Intrinsic selection inside the supplied C2 object 000252 | ==================================================== 000253 | 000254 | 2.1 The dividing orbit is a group invariant 000255 | -------------------------------------------- 000256 | 000257 | All antiholomorphic involutions generated by the supplied marks are 000258 | 000259 | rho_(a,b)=h^a Q^b kappa, 000260 | a in {0,1}, b in Z/4. (5) 000261 | 000262 | Their complete fixed-locus census has exactly two dividing members, those 000263 | in (2). Q and kappa exchange them and h preserves each. Hence the set (2), 000264 | viewed as an unordered pair or two-object groupoid, is invariant under the 000265 | whole marked group. 000266 | 000267 | There is no invariant choice of one member, and none is required. A single 000268 | chart is a presentation. Full quarter transport uses both: 000269 | 000270 | Q: A_+=E/rho_+ -> A_-=E/rho_- . (6) 000271 | 000272 | This is stronger than choosing a favorable real structure after inspecting 000273 | the answer. Dividing type is an intrinsic topological property, and the 000274 | complete eight-element census leaves one group orbit. 000275 | 000276 | 2.2 The Moebius deck is conditionally forced 000277 | --------------------------------------------- 000278 | 000279 | In the square uniformization E=C/(Z+iZ), with T=(1+i)/2, 000280 | 000281 | h z=-z+T, 000282 | Q^2 z=-z, 000283 | delta=hQ^2: z -> z+T. (7) 000284 | 000285 | The first two involutions have fixed points. The third is a nonzero 000286 | two-torsion translation and is fixed-point free. It commutes with both 000287 | members of (2). On A_+ it descends to 000288 | 000289 | delta_A(t,s)=(1/2-t,s+1/2), (8) 000290 | 000291 | which is free, swaps the two annulus boundaries and has Moebius quotient. 000292 | 000293 | Thus, among the involutions in the supplied marked h,Q action, the original 000294 | orientable-double requirements select delta uniquely. A4 alone does not 000295 | select (7) among arbitrary abstract annulus presentations. The uniqueness 000296 | becomes applicable only after (4) says that the original deck is represented 000297 | inside this marked action. 000298 | 000299 | The three relevant operations remain distinct: 000300 | 000301 | rho_+/- Schottky real structures fixing the annulus boundaries; 000302 | delta_A free Moebius deck swapping those boundaries; 000303 | Q quarter transport exchanging the two annulus charts. (9) 000304 | 000305 | The original collar reflection and its boundary trace are not automatically 000306 | rho, delta_A or their product. Map (4) must state that operator slot and its 000307 | domain explicitly. 000308 | 000309 | 3. Spin minimality 000310 | ================== 000311 | 000312 | 3.1 The four flat characters 000313 | ---------------------------- 000314 | 000315 | Let A=1 and B=i be the primitive cycles of E. A flat spin line is specified 000316 | by 000317 | 000318 | (epsilon_A,epsilon_B), epsilon_A,epsilon_B in {+1,-1}. (10) 000319 | 000320 | Because Q maps A to B and B to -A, Q equivariance gives 000321 | 000322 | epsilon_A=epsilon_B. (11) 000323 | 000324 | There are therefore only two Q-equivariant flat spin characters: 000325 | 000326 | (+,+), the trivial odd spin line with one holomorphic zero mode; 000327 | (-,-), the nontrivial even spin line alpha with no zero mode. (12) 000328 | 000329 | No nullity preference has been used in obtaining this pair. 000330 | 000331 | 3.2 The original order-four condition chooses alpha after genuine transport 000332 | ---------------------------------------------------------------------------- 000333 | 000334 | For rho=rho_+=-conjugation and delta_E(z)=z+T, the annulus glide is represented 000335 | on the square cover by 000336 | 000337 | g(z)=delta_E rho(z)=-bar z+T, 000338 | g^2(z)=z+i=z+B. (13) 000339 | 000340 | On the complex-linear full-Dirac coefficient space let U_g denote the 000341 | character lift of this pullback. Then 000342 | 000343 | U_g^2=epsilon_B. (14) 000344 | 000345 | Use the fixed Euclidean two-component Clifford convention 000346 | 000347 | sigma_t=[[0,1],[1,0]], 000348 | sigma_s=[[0,-i],[i,0]], 000349 | sigma_s^2=+I, 000350 | A_g=sigma_s U_g. (15) 000351 | 000352 | The coordinate reflection flips the t symbol, so A_g is Dirac covariant and 000353 | 000354 | A_g^2=epsilon_B. (16) 000355 | 000356 | If (4) is an actual Spin or Pin bundle transport, the original A4 lift and 000357 | A_g may differ only by the central sign +/-1. A4 requires the transported 000358 | glide lift to have square -I. Equations (11) and (16) then force 000359 | 000360 | epsilon_B=-1, 000361 | epsilon_A=-1, 000362 | alpha=(-,-) uniquely. (17) 000363 | 000364 | The genus-one parity theorem now gives h0(E,alpha)=0. Zero-mode absence is a 000365 | consequence of Q equivariance plus the transported A4 relation, not a third 000366 | selection rule. 000367 | 000368 | This conclusion must not be overstated. If (4) were only a projective 000369 | complex operator identification allowing an independent phase i, one could 000370 | replace A_g by i A_g and manufacture square -I even for (+,+). The original 000371 | source does not provide the C2 Spin-bundle transport which forbids that 000372 | extra phase. Spin minimality is therefore exact once the missing genuine 000373 | bundle map is supplied, and it is not proof that the source already supplied 000374 | that map. 000375 | 000376 | The anti-linear Real lifts are a different slot. Their squares classify 000377 | real versus quaternionic structures, but an anti-linear lift has no complex 000378 | +/-i eigenspace decomposition. The c=+/-i pair belongs to the complex-linear 000379 | operator A_g in (15), not to the OS Real structure. 000380 |