TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_unification_20261027/analytic/C3-CAUCHY-HARDY-ARROW.txt Original file SHA-256: 3c8c2b55ca3a58469aaaa61be3e2d74b2075cc33d132fef799790fad22c1bdc2 Inclusive original lines: 201-400 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 | (ii) the ambient smooth six-edge matching, 000202 | (iii) a self-adjoint graph operator on that singular boundary. (18) 000203 | 000204 | A self-adjoint six-edge vertex condition may exist, but it has to be stated 000205 | and proved as the graph closure of a named core. It is not automatically the 000206 | abstract B_Sigma of the smooth P1 seam. Choosing the normalization instead is 000207 | mathematically clean, but it deliberately separates the vertices. L2 closure 000208 | alone does not decide between these operator domains. The derivative test 000209 | above therefore rules out only the naive constant-generator identification 000210 | on the natural ambient trace core. It is not a no-go theorem for every 000211 | possible primitive boundary operator. 000212 | 000213 | There is also a framing distinction. In the holomorphic frame e^2=dz, the 000214 | coefficient number operator is N=-i partial_theta with integer modes. In an 000215 | adapted bounding-spin frame the geometric tangential Dirac can be N+1/2, up 000216 | to the boundary orientation, with Neveu-Schwarz spectrum Z+1/2. Both have the 000217 | same Hardy sign range after the corresponding convention is fixed, and both 000218 | satisfy the same one-step commutator with M_z. The original source must say 000219 | which framed operator and domain its B_Sigma denotes. 000220 | 000221 | 000222 | 5. THE CONDITIONAL MARKED OPERATOR DICTIONARY 000223 | ============================================== 000224 | 000225 | On the normalized three-disk space, the natural candidate dictionary is 000226 | 000227 | boundary Hilbert space -> L2(S1)^3, 000228 | positive Cauchy range -> H2(D)^3, 000229 | seam multiplier U_Gamma -> M_z, 000230 | cubic symmetry c -> cyclic sheet permutation, 000231 | quarter map Q -> z->i z with its S_0 spin lift. (19) 000232 | 000233 | In the holomorphic coefficient convention, 000234 | 000235 | [N,M_z]=M_z, 000236 | c M_z c^(-1)=M_z, 000237 | Q^* M_z (Q^*)^(-1)=i M_z, 000238 | Q^* c (Q^*)^(-1)=c^(-1). (20) 000239 | 000240 | The Q lift on S_0 contributes the already proved square-root phase. Equations 000241 | (19) and (20) are exact on the named normalized candidate. They become facts 000242 | about the primitive source only after a unitary domain intertwiner is given. 000243 | 000244 | 000245 | 6. WHAT THE ORIGINAL SOURCES ACTUALLY PROVIDE 000246 | ============================================== 000247 | 000248 | TFPT45 starts from 000249 | 000250 | T_bound=(A_+,H_+,D_+,J,Gamma,B_Sigma), 000251 | D_+=gamma_n(partial_n+B_Sigma), (21) 000252 | 000253 | where B_Sigma is assumed self-adjoint on the smooth compact boundary of a 000254 | reflection-regular product collar. It then constructs the Calderon projector 000255 | from that supplied datum. The source gives no explicit B_Sigma formula or 000256 | domain on the cubic curve. 000257 | 000258 | The older fullv21 source separately takes 000259 | 000260 | D_Gamma=i d/dtheta, 000261 | U_Gamma(theta)=exp(i theta), (22) 000262 | 000263 | and computes spectral flow one. This fixes a circle representative and a 000264 | one-step winding. It does not identify D_Gamma with B_Sigma, identify its 000265 | circle with the normalization of pi^(-1)D, or transport the six-edge domain. 000266 | 000267 | The distinction is decisive because the hypotheses of (21) are a smooth 000268 | boundary and a product collar. The ambient cubic boundary in Section 4 is a 000269 | singular graph. The normalized boundary is smooth, but passing to it discards 000270 | the H1 vertex matching. Neither interpretation follows merely from (21). 000271 | 000272 | Standard results on Calderon projectors are functorial for a supplied smooth 000273 | Dirac operator and its boundary domain. They do not manufacture an 000274 | identification between two source problems. See: 000275 | 000276 | C. Baer and W. Ballmann, Boundary value problems for elliptic differential 000277 | operators of first order, arXiv:1101.1196; 000278 | 000279 | C. Guillarmou, S. Moroianu and J. Park, Bergman and Calderon projectors for 000280 | Dirac operators, arXiv:1009.3179, for the natural spin Calderon 000281 | construction used by the earlier source audit. 000282 | 000283 | 000284 | 7. THE DIRECT PUSHFORWARD IS NOT THE OLD RANK THREE ANCHOR 000285 | ========================================================== 000286 | 000287 | The direct pushforward supplies a useful exclusion without choosing a new 000288 | bundle. Let E=pi_*S_0. Since pi is finite, 000289 | 000290 | rank E=3, 000291 | chi(E)=chi(S_0)=deg(S_0)+1-g(C)=1+1-2=0. (23) 000292 | 000293 | On P1 this gives deg E=-3. The selected even theta characteristic is 000294 | non-effective, so 000295 | 000296 | H0(P1,E)=H0(C,S_0)=0. (24) 000297 | 000298 | Every Birkhoff-Grothendieck summand therefore has degree at most -1. Their 000299 | sum is -3, hence 000300 | 000301 | pi_*S_0 isomorphic O(-1) direct-sum O(-1) direct-sum O(-1). (25) 000302 | 000303 | The original v117/P2 anchor O(-2) direct-sum O(-1)^2 is therefore not the 000304 | direct pushforward of this C3 spin source. An elementary modification at 000305 | z=0 could change (25) to that splitting without choosing one q point, but it 000306 | would be an additional modification at a regular point. No checked original 000307 | source requires it, and it may change the connection by an apparent pole. 000308 | It is not inserted here. 000309 | 000310 | 000311 | 8. THE SMALLEST REMAINING PREMISE 000312 | ================================== 000313 | 000314 | The actual source arrow would be proved by one of the following two precise 000315 | data sets. 000316 | 000317 | NORMALIZED OPTION: 000318 | 000319 | W:H_boundary -> L2(S1)^3, 000320 | W Dom(B_Sigma)=Dom(B_spin)^3, 000321 | W B_Sigma W^(-1)=B_spin direct-sum B_spin direct-sum B_spin, 000322 | W U_Gamma W^(-1)=M_z, (26) 000323 | 000324 | with W equivariant for c and the marked Q spin lift, together with either 000325 | an intertwiner of the full one-sided interior Dirac problems and their 000326 | boundary traces, or an independent proof that W maps the actual harmonic 000327 | Cauchy data onto the normalized analytic range. A local isometric embedding 000328 | onto the relevant reducing sector can suffice. Global surjectivity onto all 000329 | three channels is a stronger sufficient condition, not a necessary one. 000330 | 000331 | SINGULAR GRAPH OPTION: 000332 | 000333 | W maps the actual source to a specified six-edge Dirac problem whose 000334 | vertex relations are the H1 closure of the ambient smooth core in (13), 000335 | and it intertwines the same seam and symmetry marks. That graph problem 000336 | must separately be proved self-adjoint and elliptic, and its harmonic 000337 | Cauchy range must separately be identified. (27) 000338 | 000339 | A tangential relation in (26) alone does not imply 000340 | 000341 | W P_C W^(-1)=P_norm. (28) 000342 | 000343 | The Calderon range depends on the full interior operator, not only on its 000344 | adapted tangential operator. Equation (28) follows when the full one-sided 000345 | Dirac problem and boundary trace are intertwined, because W then maps the 000346 | boundary values of harmonic interior solutions. It may also be established 000347 | directly by proving equality of those Cauchy-data ranges. 000348 | 000349 | Standard smooth Calderon functoriality does not apply automatically to the 000350 | singular graph in (27). The H2(D)^3 density theorem in Sections 2 and 3 is a 000351 | theorem about the particular ambient analytic core. It is not a theorem that 000352 | every self-adjoint six-edge vertex condition has that Cauchy range. 000353 | 000354 | No checked original source supplies the strengthened data following (26) or 000355 | the graph theorem in (27). This is now the first missing datum. It is narrower 000356 | than a new field model and stronger than a dimension or winding match. 000357 | 000358 | 000359 | SOURCE ANCHORS 000360 | ============== 000361 | 000362 | * documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000363 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex, 000364 | lines 65-110. SHA256 000365 | 7d6e673d99c29d950dc601283cd51c1cc0b113567eeb79c8396fec9c82a02c79. 000366 | 000367 | * documents/archive--paper-latex--tfpt-theory-fullv21/sections/ 000368 | 001-section-introduction-the-problem-of-parameters.tex, lines 220-258. 000369 | SHA256 000370 | 85bb11511be05385a7cadf6071dbede216b12befeb9d6767bc4460748cbec5d7. 000371 | 000372 | * experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ 000373 | TRACE-METRIC-AND-DOMAIN.txt. SHA256 000374 | fb696ebe6c3e531c0773eab309083dc4407c68011c6a08221abb433bdf39ad45. 000375 | 000376 | * experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ 000377 | INDEPENDENT-BOUNDARY-REVIEW.txt. SHA256 000378 | 7fac8804f89364125b2970eed9c937e7eb5dad2edf7253ffa8e9cc25f3d429fc. 000379 | 000380 | * experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ 000381 | SOURCE-SELECTED-THETA-SPIN.txt. SHA256 000382 | cf4478191ccd75d62a2f049a54193541009bed24877df9cbe0ed9f942d07d717. 000383 | 000384 | * experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ 000385 | boundary_origin_20261027/originals/ACTUAL-REFERENCE-SPIN-TEST.txt. 000386 | SHA256 000387 | df6b3894ee736e3a0d503eb1d90e7b7e4e462af966eb2d6718e82e24516fac72. 000388 | 000389 | 000390 | FINAL SCOPE 000391 | =========== 000392 | 000393 | The cubic geometry itself now supplies a canonical three-Hardy closed range 000394 | and the exact branch exponent. It also shows why this L2 fact does not settle 000395 | the source domain: normalization and ambient H1 matching are different 000396 | operator problems. The actual TFPT primitive B_Sigma remains an input with no 000397 | formula tying it to either one. The next proof must construct the marked 000398 | full interior and Cauchy-data intertwiner accompanying (26), an independent 000399 | proof of the actual Cauchy range, or the explicit graph theorem (27). 000400 | Replacing these data by a convenient Dolbeault operator would change the