TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/GEOMETRY.txt Original file SHA-256: be6dca12fe366f6aeba7953dd7ba1c100ddfe40d5faca43d8ea1a883fe85c4ce Inclusive original lines: 240-393 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. 000240 | rho(z)=c bar z+d. 000241 | 000242 | For the normalized anti-linear lift on alpha, the square is the character 000243 | of the lattice translation made by rho^2: 000244 | 000245 | j_rho^2=chi(c bar d+d). (18) 000246 | 000247 | For a=0, d=T and the four translations in (18) are 000248 | 000249 | 0, 1, 1+i, i, 000250 | 000251 | so the lift squares are 000252 | 000253 | +1, -1, +1, -1. (19) 000254 | 000255 | For a=1, d=0 and all four lift squares are +1. In particular, 000256 | 000257 | j_(h Q kappa)^2=j_(h Q^3 kappa)^2=+1. (20) 000258 | 000259 | The two fixed-point-free anti-linear Real lifts j_(Qkappa) and 000260 | j_(Q^3kappa) are instead quaternionic on alpha, with square -1. The deck 000261 | factor h therefore simultaneously 000262 | creates the dividing fixed set and removes the Kramers obstruction. This 000263 | is the cover-level counterpart of v519's Kramers-free family D. 000264 | 000265 | The spin line itself passes the kernel test: 000266 | 000267 | h0(E,alpha)=0. (21) 000268 | 000269 | On A_+, both boundary components run in the B direction; on A_- they run 000270 | in the A direction. Equation (17) gives holonomy -1 in either direction. 000271 | Thus both annulus boundaries carry the bounding NS spin structure, with 000272 | 000273 | spec B_Sigma={2 pi(n+1/2):n in Z}, 000274 | ker B_Sigma=0. (22) 000275 | 000276 | The anti-linear Real lifts j_rho in (18)-(20) must now be kept separate from 000277 | the complex-linear pullbacks on the full two-component Dirac coefficients. 000278 | Denote the latter by U_rho, U_delta and U_g. On the universal square cover 000279 | write 000280 | 000281 | rho z=-bar z, delta_E z=z+T, 000282 | T=(1+i)/2. (23a) 000283 | 000284 | Although rho and delta_E commute on E, their displayed universal lifts do 000285 | not commute literally: 000286 | 000287 | rho delta_E(z)=-bar z-bar T, 000288 | delta_E rho(z)=-bar z+T, 000289 | 000290 | and the two right hand sides differ by the A-cycle 1. Because the NS--NS 000291 | character (17) is invariant, the complex-linear coefficient pullbacks obey 000292 | 000293 | U_rho^2=+I, 000294 | U_delta^2=chi(2T)=chi(1+i)=+I, 000295 | U_rho U_delta=-U_delta U_rho. (23b) 000296 | 000297 | Thus the bare half-period translation on the closed torus has square +I. 000298 | The order-four element is instead the combined Schottky glide. Choose the 000299 | representative 000300 | 000301 | g(z)=delta_E rho(z)=-bar z+T. (23c) 000302 | 000303 | It obeys 000304 | 000305 | g^2(z)=z+i, 000306 | U_g^2=chi(i)=-I, (23d) 000307 | 000308 | equivalently (U_rho U_delta)^2=-I by (23b). This is the 000309 | central-commutator mechanism missing from the earlier argument. It uses 000310 | both NS holonomies, chi(1)=chi(i)=-1; a single core-holonomy number would 000311 | not determine it. 000312 | 000313 | A concrete two-component boundary realization makes the Pin statement 000314 | equally explicit. With gamma_t=sigma_t, gamma_s=sigma_s and 000315 | 000316 | sigma_t=[[0,1],[1,0]], sigma_s=[[0,-i],[i,0]], 000317 | D=sigma_t partial_t+sigma_s partial_s, 000318 | 000319 | the reflected complex-linear pullback U_g flips the t symbol. Hence 000320 | 000321 | A=sigma_s U_g (23e) 000322 | 000323 | is Dirac covariant, and (23d) gives A^2=-I. Its two eigenspaces have 000324 | eigenvalues c=+i and c=-i. On either eigenspace the actual quotient descent 000325 | operator c^-1 A has square +I. The fiber matrix c sigma_s also has square 000326 | -I, but that algebraic square is not U_delta^2 and must not be used to 000327 | replace the central computation (23b)-(23d). 000328 | 000329 | The corresponding Real lift j_g is anti-linear. Since sigma_s is imaginary, 000330 | (sigma_s j_g)^2=+I in this convention; an anti-linear operator has no complex 000331 | +/-i eigenspace decomposition. Therefore the c=+/-i sectors above belong 000332 | only to the complex-linear full-Dirac operator A. They are not eigensectors 000333 | of the OS Real structure. 000334 | 000335 | Equations (23b)-(23e) are precisely compatible with the original A4 000336 | order-four gluing declaration. They do not identify the flat alpha lift of 000337 | Q with the independent Z8 spin-carrier matrix used in v519. Establishing 000338 | that carrier identification would require the still-open field map. 000339 | 000340 | The convention label Pin plus or Pin minus depends on the Clifford sign. 000341 | The invariant statement used here is the explicit square A^2=-I together 000342 | with the involutive descended quotient sector. It concerns neither the OS 000343 | Real lift in (20) nor the quarter clock Q. Bare Schottky rho fixes the two 000344 | annulus boundaries; the free glide delta_A swaps them. Hence the cut Real 000345 | structure, the Moebius deck and the quarter clock occupy three distinct 000346 | operator slots. 000347 | 000348 | The normally twisted bundle alpha tensor p*O(1) still has its previously 000349 | proved two holomorphic zero modes and QD=-DQ. It is therefore not the 000350 | bundle used for the nullmode-free annulus kernel. No zero-mode projector is 000351 | silently inserted here. 000352 | 000353 | 6. The actual bordered Calderon graph 000354 | ====================================== 000355 | 000356 | The correct operator for the half A_+ is the two-boundary Cauchy-data or 000357 | Calderon projector, not the closed-torus Green kernel evaluated on one 000358 | circle. 000359 | 000360 | On the product cylinder (12), let B_Sigma e_n=lambda_n e_n with 000361 | 000362 | lambda_n=2 pi(n+1/2). 000363 | 000364 | A harmonic chiral spinor for D=partial_t+B_Sigma has boundary values related 000365 | mode by mode by 000366 | 000367 | f_1,n=exp(-W lambda_n) f_0,n, 000368 | W=1/2. (24) 000369 | 000370 | The orthogonal projector onto this Cauchy graph is therefore 000371 | 000372 | 1 000373 | C_n = ------------------------- 000374 | 2 cosh(W lambda_n) 000375 | 000376 | [[ exp(W lambda_n), 1 ], 000377 | [ 1, exp(-W lambda_n) ]], (25) 000378 | 000379 | with 000380 | 000381 | W lambda_n=pi(n+1/2). 000382 | 000383 | This formula uses the product trivialization to identify the two boundary 000384 | spin lines. Reversing the outward-normal convention conjugates (25) by a 000385 | fixed diagonal unitary. It does not change (26), the mode weights or the 000386 | nonzero magnitude in (27). 000387 | 000388 | Direct multiplication gives 000389 | 000390 | C_n^2=C_n, C_n*=C_n, spec C_n={0,1}. (26) 000391 | 000392 | Thus the existing geometry supplies a canonical positive one-particle 000393 | polarization on the two boundary circles, with no free temperature or width.