TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/MARKED-Q-DIVISOR-CENTRAL-TWIST.txt Original file SHA-256: da1f75831a46a0c0662f931486b77e6e2085126df452b009d1a90351d413d904 Inclusive original lines: 181-340 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. 000181 | Norm_Dq(P)=mathcal P_delta_q, 000182 | delta_q=D_q-3q_0=q_1-q_2 not_equal 0 in Jac(C)[2]. (14) 000183 | 000184 | The original c moves this nonzero vector through the three nonzero elements 000185 | of V_q. Thus the q_0-normalized one-copy line is not c invariant. This is 000186 | an obstruction to that point normalization, not an intrinsic obstruction to 000187 | the source family: using q_0 as the Poincare rigidification was an additional 000188 | choice, even though q_0 itself was already source marked. 000189 | 000190 | The source has already selected a reference of the correct type, namely the 000191 | degree-one spin line S_0. From (3) and S_0^2=K_C, 000192 | 000193 | O_C(D_q)=S_0 tensor K_C=S_0^3. (15) 000194 | 000195 | Let 000196 | 000197 | mathcal R= 000198 | 000199 | be the Deligne pairing over A, and define the spin-normalized Poincare family 000200 | 000201 | P_spin=P tensor pr_A^*(mathcal R^(-1)). (16) 000202 | 000203 | This tensor factor comes from the base, so P_spin represents the same family 000204 | of degree-zero line bundles on C. Since deg(S_0)=1, bilinearity first gives 000205 | 000206 | = tensor mathcal R^(-1)=O_A. 000207 | 000208 | Norm functoriality and bilinearity of the Deligne pairing then give a 000209 | canonical cancellation: 000210 | 000211 | Norm_Dq(P_spin) 000212 | = tensor mathcal R^(-3) 000213 | = tensor ^(-3) 000214 | =O_A. (17) 000215 | 000216 | Consequently the evaluation norm of 000217 | 000218 | E_spin=S_0 tensor P_spin 000219 | 000220 | is the constant equivariant line 000221 | 000222 | mathcal N_q(E_spin)=N_q(S_0) tensor O_A. (18) 000223 | 000224 | The same statement holds for D_p because O(D_p)=O(D_q)=S_0^3. Thus the 000225 | q and p blocks remain a full-family s-exchanged pair with the conjugate 000226 | central characters computed in section 3. 000227 | 000228 | This normalization does not change the ordinary theta determinant. A line 000229 | pulled back from A enters determinant of cohomology with exponent 000230 | chi(S_0 tensor L)=0, so the factor mathcal R^(-1) has weight zero there. 000231 | It changes the length-three evaluation norm with exponent three, exactly as 000232 | needed in (17). 000233 | 000234 | The construction is c and Q covariant because S_0 and D_q were already 000235 | selected invariant objects, and the S_0-normalized universal family is unique 000236 | up to its fixed rigidification. The remaining scalar ambiguity in a lift of 000237 | S_0 does not reintroduce a phase: scaling the second entry of the Deligne 000238 | pairing acts with exponent deg(P_spin|C)=0. Therefore (17) is independent of 000239 | that scalar and obeys the same cocycles as the selected source lifts. 000240 | 000241 | The direct one-copy family correction is hence genuinely 000242 | 000243 | O(2 Theta) tensor N_q(S_0)^(-1), (19) 000244 | 000245 | with no moving Pic^0 factor. It retains the four-dimensional theta space, 000246 | the original c action, and the central relations (2). No new point or flat 000247 | character has been selected; the normalization uses the unique 000248 | c-invariant source spin line S_0. 000249 | 000250 | One distinction remains. If one first imposes vanishing at D_q via (5) and 000251 | then forms the usual two-theta line, the evaluation factor is squared: 000252 | 000253 | (D tensor mathcal N_q(E_spin)^(-1))^(-2) 000254 | =D^(-2) tensor N_q(S_0)^2. (20) 000255 | 000256 | Its beta phase is (-i)^2=-1, not +i. Thus determinant exactness alone does 000257 | not select the one-copy insertion (19). The remaining physical question is 000258 | precise: does the original puncture, boundary, sewing or conformal-block 000259 | construction attach one inverse copy of the marked evaluation norm directly 000260 | to the O(2 Theta) module? The geometry supplies that line canonically, but 000261 | the inspected source has not yet supplied this insertion rule. 000262 | 000263 | 000264 | 5. SOURCE AUDIT AND EXACT REMAINING GAP 000265 | 000266 | The September synthesis already contains every geometric input used above: 000267 | 000268 | * its section 004, lines 276--312, gives C, Q, c, s and the marked q and p 000269 | triples; 000270 | * lines 314--326 warn that the six-point transport is not yet a field 000271 | intertwiner; 000272 | * lines 378--379 require the original test-function space and 000273 | Riemann--Hilbert kernel for an actual quantization statement. 000274 | 000275 | Its following section 005, lines 1--13, gives the different determinant 000276 | 000277 | det Psi(z)=1-z^4 000278 | 000279 | and its primitive winding, while explicitly leaving the contour and boundary 000280 | operator map open. That Fuchs fundamental-matrix determinant is not the 000281 | finite evaluation determinant (4), so it does not already insert N_q^(-1) 000282 | into the source state space. 000283 | 000284 | The targeted Atlas source v519 carries exact CAR reflection and spin phases, 000285 | but on a different freely chosen sixteen-Majorana Fock object. It supplies 000286 | no functor from (C,S_0,D_q) to (1) and cannot be reused as that missing map. 000287 | The inspected Atlas anchors are paper-source:c3a096bf0487b5234a147dce for 000288 | the September determinant/winding passage, and v519 sources 000289 | source:588e55951801405a5c25 and source:bd323276b179791d1d6c for the separate 000290 | CAR object. 000291 | No explicit q-divisor determinant twist was found in the bounded original 000292 | source search. This is a statement about the inspected source family, not 000293 | an assertion that every archived paper has been exhausted. 000294 | 000295 | The sharp result is therefore: 000296 | 000297 | * MATHEMATICALLY PROVED: N_q(S_0)^(-1) is a canonical line constructed from 000298 | existing marked source data and restores exactly the +i central 000299 | character without changing c; 000300 | * MATHEMATICALLY PROVED: its s partner has the conjugate -i character; 000301 | * MATHEMATICALLY PROVED: q_0 normalization creates the moving flat class 000302 | delta_q=q_1-q_2, while the source-selected S_0 normalization cancels it 000303 | canonically and leaves the ordinary theta determinant unchanged; 000304 | * CANDIDATE BRIDGE: the actual field/theta construction uses the one-copy 000305 | line (19), rather than the different boundary-first-then-square operation 000306 | (20); 000307 | * STILL OPEN: the original Calderon, Riemann--Hilbert, sewing or conformal- 000308 | block construction selects this boundary factor and identifies the 000309 | resulting four-dimensional linear module with the native one. 000310 | 000311 | 000312 | MATHEMATICAL REFERENCES 000313 | 000314 | * The Stacks Project, Tag 0FJ9, determinant of finite locally free modules: 000315 | https://stacks.math.columbia.edu/tag/0FJ9 000316 | 000317 | * The Stacks Project, Tag 0FJW, determinant functor for perfect complexes 000318 | and exact/pullback functoriality: 000319 | https://stacks.math.columbia.edu/tag/0FJW 000320 | 000321 | * The Stacks Project, Tag 0BD2, canonical norm for a finite locally free 000322 | morphism and base-change compatibility: 000323 | https://stacks.math.columbia.edu/tag/0BD2 000324 | 000325 | * Pierre Deligne, Le determinant de la cohomologie, Contemporary 000326 | Mathematics 67 (1987), 93--177, for determinant and Deligne-pairing 000327 | functoriality: 000328 | https://publications.ias.edu/book/export/html/404 000329 | 000330 | 000331 | LOCAL SOURCE ANCHORS 000332 | 000333 | [OMITTED: non-scientific conversation or execution metadata] 000334 | [OMITTED: non-scientific conversation or execution metadata] 000335 | [OMITTED: non-scientific conversation or execution metadata] 000336 | [OMITTED: non-scientific conversation or execution metadata] 000337 | [OMITTED: non-scientific conversation or execution metadata] 000338 | [OMITTED: non-scientific conversation or execution metadata] 000339 | [OMITTED: non-scientific conversation or execution metadata] 000340 | [OMITTED: non-scientific conversation or execution metadata]