TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/boundary_origin_20261027/source/ORIGINAL-BOUNDARY-NORM-ORIGIN.txt Original file SHA-256: 2a92d7c75aeed3fb09d85de0f9cf8be3c9fb8757b6686cf0c6724cd7b4a977a6 Inclusive original lines: 201-387 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. 000201 | sequences and symmetry bookkeeping. The Hardy/Fredholm passage and the 000202 | determinant-line identity are the analytic argument above, not a conclusion 000203 | drawn from truncation. Normal Python and python -OO give byte-identical 000204 | output. 000205 | 000206 | 3. INDEX MINUS ONE IS NOT THE INVERSE NORM DEFECT 000207 | 000208 | The old scalar circle shift has index -1. The pulled-back S_0 cable has 000209 | three sheets and index -3. These are different operators, and their 000210 | determinant orientations prevent a phase-only identification. 000211 | 000212 | The quotient by c makes the distinction exact. Set x=w^3. Then 000213 | 000214 | C/: eta^2=1-x^2, 000215 | 000216 | and q_0,q_1,q_2 map to one unramified point qbar with x=-1. The pushforward 000217 | F=rho_*S_0 has rank three and 000218 | 000219 | F|qbar=F_1 direct_sum F_omega direct_sum F_omega2, 000220 | 000221 | its regular C3 representation. Modifying the whole fiber drops the index by 000222 | three. Modifying only the invariant character F_1 drops it by one. 000223 | 000224 | But beta acts by +i on every q fiber. Hence 000225 | 000226 | beta|F_1=+i, beta|F_1^(-1)=-i, 000227 | beta|B_q=-i, beta|B_q^(-1)=+i. (10) 000228 | 000229 | The directed Fredholm line of the rank-one modification is F_1^(-1), so it 000230 | has the wrong beta and alpha-square character for N_q^(-1). The undualized 000231 | F_1 has the desired phase, but it is the opposite determinant orientation. 000232 | The full three-fiber determinant succeeds precisely because 000233 | (i^3)^(-1)=+i. 000234 | 000235 | Therefore the base index-minus-one seam defect is not already the inverse q 000236 | norm. The positive construction is the full three-cabled pullback followed 000237 | by the relative determinant quotient (7). This is a degree-three 000238 | pushforward, not an invertible identification of the old scalar operator 000239 | with the full defect. 000240 | 000241 | The exact check is check_c3_quotient_defect.py, with checked output 000242 | c3-quotient-defect.json. It verifies both indices and the determinant 000243 | orientation under normal Python and python -OO. 000244 | 000245 | 4. THE ORDINARY THETA EVALUATION SHORTCUT HAS THE OPPOSITE SIGN 000246 | 000247 | This result must be distinguished from modifying one of the two positive 000248 | theta determinants by vanishing at D_q. With 000249 | 000250 | D(E)=det H^0(E) tensor det H^1(E)^(-1), 000251 | O(Theta)=D(E)^(-1), 000252 | 000253 | and 000254 | 000255 | 0 -> E(-D_q) -> E -> E|D_q -> 0, 000256 | 000257 | one has 000258 | 000259 | D(E(-D_q))=D(E) tensor B_q^(-1). 000260 | 000261 | Therefore 000262 | 000263 | D(E)^(-1) tensor D(E(-D_q))^(-1) 000264 | = O(2 Theta) tensor B_q, (11) 000265 | 000266 | which is N_q, not N_q^(-1), after removing the ordering line. Its beta 000267 | phase is -i. Conditioning both theta factors gives N_q^2 and phase -1. 000268 | 000269 | The one inverse norm in (7) therefore comes from the determinant line of the 000270 | original multiplication shift and its relative reference. It does not come 000271 | from imposing q vanishing inside one positive theta factor. 000272 | 000273 | At D_p the analogous vanishing operation gives O(2Theta) tensor N_p and has 000274 | the desired scalar beta phase +i, but N_p is a different typed line. The 000275 | source selects the q packet for the (4,3,2) triangle. A scalar phase match 000276 | does not identify N_p with N_q^(-1). 000277 | 000278 | The exact convention audit is check_boundary_norm_conventions.py, with 000279 | checked output boundary-norm-conventions.json. 000280 | 000281 | 5. RELATION TO ETA AND PFAFFIAN SEWING 000282 | 000283 | The three-cabled proof above is a relative Fredholm determinant statement. 000284 | The general Dai--Freed theorem independently explains why a genuine Dirac 000285 | bordism amplitude contains one inverse boundary determinant line. For an 000286 | odd-dimensional spin Dirac operator on X with boundary Y, 000287 | 000288 | tau_X in Det_Y^(-1), 000289 | 000290 | and cutting introduces a pair L_Y tensor L_Y^(-1) which gluing contracts 000291 | once. See Daniel S. Freed, Determinant Line Bundles Revisited, 000292 | https://arxiv.org/html/dg-ga/9505002, equations (2)--(3) and Theorems 5/8. 000293 | 000294 | The primitive TFPT action contains +(i*pi/2) Delta eta_Sigma, suggesting the 000295 | real/Majorana square-root, hence Pfaffian, refinement. That physical 000296 | application remains conditional because the September source itself treats 000297 | the same-chirality real Spin--Dirac symbol and Pfaffian structure as a premise, 000298 | not a completed identification of the raw seam operator. Yonekura likewise 000299 | states that the Majorana case requires a square-root refinement but does not 000300 | carry it out in his Dirac derivation: https://arxiv.org/html/1607.01873, 000301 | sections 1.2, 3.2 and 4.3. 000302 | 000303 | Accordingly, two levels must remain separate: 000304 | 000305 | * (7) is an exact complex relative Fredholm line for the original 000306 | multiplier on the source-selected three-sheet cable; 000307 | 000308 | * identifying this line with the physical eta/Pfaffian boundary amplitude 000309 | still requires the actual Spin--Dirac/Pfaffian field map. 000310 | 000311 | 6. ORIGINAL SOURCE CHECK 000312 | 000313 | The relevant original passages are precise. 000314 | 000315 | 1. fullv21 section001, lines 225--250, cuts the double cover along the seam, 000316 | declares the unitary seam gluing U_Gamma(theta)=exp(i theta), and computes 000317 | one scalar spectral-flow crossing. After the already reviewed orientation 000318 | correction, its Hardy compression has index -1. 000319 | 000320 | 2. The September genus-two reconstruction gives 000321 | w^3=(z^2-1)/(z^2+1), eta=z(1-w^3), eta^2=1-w^6 and the original Q lift. 000322 | These printed equations are exactly (1)--(2), so the three-cabled map is 000323 | recovered from the original coordinate rather than introduced as a new 000324 | model. 000325 | 000326 | 3. The same September source warns that the finite point transport is not by 000327 | itself a field intertwiner or Calderon projector. This warning is still 000328 | active: equations (3)--(7) become the physical source only after the actual 000329 | boundary field is identified with the S_0 cable. 000330 | 000331 | 4. The TFPT45 primitive core has a declared relative split 000332 | (D_ref,D_rel) and the action 000333 | 000334 | tr f(D_rel/chi)-tr f(D_ref/chi)+(i*pi/2) Delta eta. 000335 | 000336 | It does not identify its two coefficient cables with 000337 | (S_0 tensor L,L). That is the precise remaining reference premise in 000338 | (7). 000339 | 000340 | 5. The Riemann--Hilbert free-fermion formula uses separate bases at z=0 and 000341 | z=infinity and retains a diagonal normalization ambiguity. It confirms 000342 | the incoming q and outgoing p typing, but it does not by itself prove the 000343 | (S_0 tensor L,L) relative boundary pair. See Gavrylenko--Marshakov, 000344 | https://arxiv.org/html/1605.04554, equations (4.1)--(4.3), (5.2)--(5.3), 000345 | and the normalization discussion after (4.1). 000346 | 000347 | 7. VERDICT AND NEXT DECISIVE TEST 000348 | 000349 | PROVED FROM THE EXISTING SOURCE GEOMETRY: 000350 | 000351 | * the old seam multiplier is the same function z whose zeros are exactly 000352 | D_q and whose poles are D_p; 000353 | 000354 | * its pullback to the three unramified q sheets has cokernel 000355 | direct_sum S_0|q_a, index -3 and Fredholm line B_q^(-1); 000356 | 000357 | * the half-form identity e^2=dz gives the standard Hardy norm on every open 000358 | sheet without a fitted measure; boundary ramification adds no point term; 000359 | 000360 | * relative to the otherwise identical cable L, with the spin twist S_0 000361 | present only in the relative member, winding n gives exactly N_q^(-n) 000362 | for every degree-zero L; 000363 | 000364 | * the original primitive spectral flow one sets n=1 and therefore fixes 000365 | the literal single inverse norm, beyond the finite clock congruence; 000366 | 000367 | * the c-invariant rank-one quotient channel has index -1 but its directed 000368 | Fredholm line has the opposite beta phase, so it is not N_q^(-1); 000369 | 000370 | * the ordinary mixed theta-determinant construction gives the opposite line 000371 | N_q and therefore is not the origin. 000372 | 000373 | REMAINING PHYSICAL PREMISE: 000374 | 000375 | Prove on the actual boundary Cauchy-data space that 000376 | 000377 | (D_rel coefficient cable, D_ref coefficient cable) 000378 | ~= (S_0 tensor L, L) on pi^(-1)(disc) for one common L, (12) 000379 | 000380 | with multiplication by the original z, and with the c,Q actions preserved. 000381 | Equation (12), not another line-power choice or a rank-one character 000382 | projection, is now the smallest decisive operator test. 000383 | 000384 | If (12) holds, (7) derives the single inverse N_q factor directly from the 000385 | original boundary operation. If D_ref induces a different coefficient line, 000386 | the relative factor is its actual defect determinant and need not be N_q^-1. 000387 | This is a concrete falsifiable distinction.