TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/OLD-SOURCE-MAP-AUDIT.txt Original file SHA-256: 6f93a94456b33f286ae54233928afcc421d6d1971963c3349eb1b77abbd49aa4 Inclusive original lines: 191-378 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 | The same source defines SEAM.EQUIV.01 as the claim that the raw 000193 | reflection-positive seam state is the holomorphic (E8)_1 net at tau=i and 000194 | keeps the unconditional parent open; see lines 138-147. It also states at 000195 | lines 305-316 that the common parent, microscopic MMST source, and global 000196 | state are not supplied. 000197 | 000198 | The MMST branch does not secretly provide the P1 map: 000199 | 000200 | * verification/v366_mmst_seam_collar.py:75-90 sets 000201 | inclass=True after arithmetic checks. It does not construct the local 000202 | source or an intertwiner from P1. 000203 | 000204 | * verification/v994_mmst_criteria_battery.py:72-85 defines a QWZ/p+ip strip 000205 | Hamiltonian, and lines 88 onward extracts its edge covariance. This is a 000206 | separately specified lattice collar. 000207 | 000208 | * v994's own header, lines 14-46, says Osborne-Stottmeister covers the free 000209 | fermion/Virasoro and bilinear limits, while the mu4/GSO spin-field 000210 | extension and net-level modular step remain outside the executed result. 000211 | 000212 | * verification/v998_seam_modular_closure.py:24-28 fences its scope to 000213 | finite QWZ cylinders and finite q-expansions and leaves conformal-net 000214 | modular covariance open. 000215 | 000216 | Hence SEAM.EQUIV supplies a conditional or separately modeled downstream 000217 | scaling route. It does not identify that QWZ covariance, state, or local net 000218 | with the original P1 boundary kernel. 000219 | 000220 | The similarly numbered software checks are not the bridge either. 000221 | verification/v31_R_dictionary.py searches a D4-fixed monodromy dictionary and 000222 | leaves a Hitchin/Mehta-Seshadri residual. verification/v42_exterior_leg.py 000223 | computes an exterior-area ratio and retains a physical-identification premise. 000224 | Neither acts on the P1 collar or its Cauchy data. 000225 | 000226 | 2. Smallest decisive sewing theorem 000227 | ==================================== 000228 | 000229 | There is one useful compression. The analytic boundary entries B and P_C do 000230 | not need independent guesses once a genuine marked collared spin isomorphism 000231 | has been supplied. 000232 | 000233 | Let 000234 | 000235 | F:(M_1,S_1,D_1) -> (M_2,S_2,D_2) 000236 | 000237 | be a unitary spin-bundle isomorphism of manifolds with boundary that preserves 000238 | the oriented normal, collar coordinate, boundary trace, and the full Dirac 000239 | operator: 000240 | 000241 | F D_1 F^(-1)=D_2, 000242 | Tr_2 F=W Tr_1. 000243 | 000244 | Writing D_j=gamma_n(partial_n+B_j) in the transported product trivialization 000245 | gives 000246 | 000247 | W B_1 W^(-1)=B_2. (A) 000248 | 000249 | Moreover F maps harmonic interior solutions bijectively, so it maps their 000250 | boundary traces. The orthogonal projections onto those Cauchy-data spaces 000251 | therefore satisfy 000252 | 000253 | W P_C,1 W^(-1)=P_C,2. (B) 000254 | 000255 | Equations (A) and (B) are direct consequences of intertwining and require no 000256 | new model. They agree with the standard construction of Calderon projectors 000257 | from the product collar, invertible double, and Poisson operator in 000258 | Booss-Bavnbek, Lesch and Zhu, "The Calderon Projection: New Definition and 000259 | Applications", J. Geom. Phys. 59 (2009), arXiv:0803.4160. That paper gives a 000260 | canonical Calderon construction for a supplied elliptic first-order operator; 000261 | it does not identify two previously unrelated physical sources. Guillarmou, 000262 | Moroianu and Park, arXiv:1009.3179, likewise construct the natural orthogonal 000263 | Calderon projector for a supplied spin Dirac operator. Neither theorem creates 000264 | the missing F. 000265 | 000266 | The scope is exact: 000267 | 000268 | * B and P_C follow from F. 000269 | 000270 | * The seam multiplier requires the extra marked relation 000271 | 000272 | W u_1 W^(-1)=u_2. (C) 000273 | 000274 | Winding one alone does not imply (C). Many inequivalent unitary 000275 | representatives have the same K1 class. 000276 | 000277 | * Reflection follows only when F is equivariant for the relevant reflected 000278 | half and its Pin/Real lift. The Schottky reflection rho, free Moebius deck 000279 | delta_A, and quarter transport Q are distinct operations. 000280 | 000281 | * A quasifree covariance or OS state follows only if it is defined from the 000282 | same transported full Cauchy graph or is separately required to satisfy 000283 | 000284 | omega_2 = omega_1 o Ad(W^(-1)). (D) 000285 | 000286 | The Calderon projector alone is not a general interacting Schwinger 000287 | functional. 000288 | 000289 | * Local support follows only when F maps regions and intertwines the local 000290 | algebras. A Hilbert-space unitary can satisfy (A) while being nonlocal. 000291 | 000292 | Thus there are not six unrelated gaps, but there is also not a bare 000293 | one-particle shortcut. The minimal sufficient datum is one marked local 000294 | collared spin/source isomorphism 000295 | 000296 | I_source=(F,W; seam faces, deck, Pin lift, Q, u, local net, state), 000297 | 000298 | whose Dirac part automatically yields (A) and (B), while its marks impose (C) 000299 | and (D). 000300 | 000301 | 3. Exact gap dictionary 000302 | ======================= 000303 | 000304 | Object Present in older sources Missing equality 000305 | ------ ------------------------ ---------------- 000306 | B fullv21 B_alpha; later B_Sigma W B_alpha W^-1=B_Sigma 000307 | domain alpha-twisted H1; later collar domain W Dom(B_alpha)=Dom(B_Sigma) 000308 | u fullv21 M_z; later only [u_Sigma]=1 W M_z W^-1=u_Sigma 000309 | P_C TFPT45 Cauchy projector full two-face relation (B) 000310 | reflection old Moebius/Pin data; later Theta/R_tau equivariant lift under F 000311 | state supplied omega in Seed_op; QWZ sea transported equality (D) 000312 | locality supplied A_loc; separate QWZ support region-preserving net map 000313 | cover old annulus; elliptic paired annuli I_cover with seam traces 000314 | metric old flat source modulus 1; elliptic 1/2 full operator/conformal transport 000315 | 000316 | The last row is a genuine full two-dimensional obstruction. Boundary spectra 000317 | can be rescaled into the same angular units, but a uniform rescaling cannot 000318 | change the annulus modulus. Therefore the exact boundary tuple found on the 000319 | elliptic annulus does not by itself prove equality of the full old flat source. 000320 | 000321 | 4. Relation to the recovered elliptic annulus 000322 | ============================================== 000323 | 000324 | The current elliptic work already supplies the target side of I_source: 000325 | 000326 | E:y^2=x^4-1, 000327 | {rho_+,rho_-}={hQkappa,hQ^3kappa}, 000328 | delta=hQ^2, 000329 | chi(A)=chi(B)=-1, 000330 | 000331 | and the exact NS boundary tuple. It also proves that Q exchanges the two 000332 | dividing annulus charts, while rho, delta_A, and Q occupy different operator 000333 | slots. See 000334 | 000335 | experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/ 000336 | ORIGINAL-RELEVANCE.txt:23-79,113-152,293-307,381-418. 000337 | 000338 | The same audit already isolates I_cover as the first additional premise. The 000339 | present historical continuation adds the negative result that neither the 000340 | v31 closure paper, the v42 master paper, TFPT45, nor the MMST branch supplies 000341 | that premise under another name. 000342 | 000343 | 5. Smallest decisive original-source test 000344 | ========================================== 000345 | 000346 | No further finite spectrum or character calculation is needed. The first 000347 | decisive test is whether the original P1 construction supplies a marked local 000348 | collar map W satisfying simultaneously 000349 | 000350 | W Dom(B_P1)=H1_NS, 000351 | W B_P1 W^-1=2 pi B_cov, 000352 | (W direct-sum W) P_C,P1 (W direct-sum W)^-1=P_C,ann, 000353 | W u_P1 W^-1=e^(2 pi i s), 000354 | 000355 | with the original reflected local algebra and state transported as in (D). 000356 | 000357 | Success criterion: these are consequences of the original P1 circle kernel 000358 | and its canonical double, with elliptic annulus modulus 1/2 or an explicit 000359 | operator/conformal transport from the old modulus. 000360 | 000361 | Abort criterion: the source yields only [u]=1, a matching group order, a 000362 | boundary spectral resemblance, or a separately chosen QWZ/Hardy state. Any 000363 | one of those leaves I_source open. 000364 | 000365 | Final classification 000366 | ==================== 000367 | 000368 | PROVED IN THE ORIGINALS: an explicit old circle B_alpha,u pair; a later 000369 | abstract operational seed; canonical Calderon reconstruction from a supplied 000370 | collared Dirac problem; downstream heat and MMST/QWZ constructions. 000371 | 000372 | PROVED BY THIS AUDIT: the analytic B and P_C entries collapse functorially to 000373 | one supplied marked collared spin isomorphism. The old paper chain contains no 000374 | such isomorphism and does not identify the state or local net. 000375 | 000376 | OPEN: the actual P1-to-elliptic marked local source map I_source. This is the 000377 | single load-bearing original relation. It is more precise than a request for 000378 | a generic "field dictionary" and smaller than adding a new CAR model.