TFPT scientific source excerpt Original repository path: experiments/theory-contracts/primitive-seam-operator-reconstruction-20261025/OS-SOURCE-REPAIR.txt Original file SHA-256: 50bc771398e8a363a617c677f503f556c9f2adccf6e40cfcb460ca4eb9f55ba6 Inclusive original lines: 201-375 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 | 000202 | (R4) if a full relativistic reconstruction is claimed, the full moments 000203 | reconstructed from Z satisfy the required graded symmetry, Euclidean 000204 | covariance, and the uniform growth/regularity hypotheses of that 000205 | reconstruction theorem. Temperedness of each connected distribution 000206 | separately is not a substitute for these full-family hypotheses. 000207 | 000208 | Then 000209 | 000210 | _OS = omega_Z((Theta F)G) 000211 | 000212 | is positive semidefinite. Quotienting P_+ by its null space and completing 000213 | gives the OS Hilbert space. Under (R3), positive-time translation descends to 000214 | a strongly continuous symmetric contraction semigroup exp(-tH), H>=0. The 000215 | self-adjointness conclusion uses symmetry, not merely the general 000216 | Hille--Yosida theorem for an arbitrary contraction semigroup. In an application 000217 | one may prove (R3) from the complete standard OS assumptions instead of 000218 | assuming it separately. Truncated 000219 | functions may be used to encode Z through exp W, but never as replacements for 000220 | the entries of this Gram form. 000221 | 000222 | 000223 | 5. State-preserving primitive boundary transport is sufficient 000224 | --------------------------------------------------------------- 000225 | 000226 | Let P_boundary,+ be a graded boundary field-word algebra built on the primitive 000227 | boundary data, and let P_adm,+ be the positive-time field-word algebra whose 000228 | full functional is the original Z_adm. Assume that this full original 000229 | functional is reflection positive. Suppose there is a unital graded word map 000230 | 000231 | iota : P_boundary,+ -> P_adm,+ 000232 | 000233 | such that 000234 | 000235 | (B1) iota intertwines reflection and the declared grading; 000236 | 000237 | (B2) it preserves the complete reflected word functional, 000238 | 000239 | omega_boundary((Theta F)G) 000240 | = omega_adm((Theta iota(F)) iota(G)) 000241 | 000242 | for all F,G in P_boundary,+; 000243 | 000244 | (B3) it intertwines positive Euclidean translation on a common invariant 000245 | dense word domain. 000246 | 000247 | Then the boundary OS Gram matrix is the pullback of the original admissible OS 000248 | Gram matrix. In particular it is positive, and 000249 | 000250 | [F]_boundary -> [iota(F)]_adm 000251 | 000252 | is a well-defined isometry of OS quotient spaces. By (B3) it also intertwines 000253 | the reconstructed positive-time semigroups. This is the smallest positive 000254 | repair path: no independently chosen boundary state is needed if the complete 000255 | state-preserving map into the already defined original Z_adm is proved. 000256 | 000257 | Preservation of the two-point function alone is insufficient outside a proven 000258 | quasi-free sector. After a fluctuating bosonic background has been integrated, 000259 | a mixture of Gaussian fermion states need not remain Gaussian, so its full 000260 | four-point function is not fixed by the averaged two-point kernel. 000261 | 000262 | 000263 | 6. Field algebra and observable algebra 000264 | --------------------------------------- 000265 | 000266 | The repaired theorem is first a statement about the graded field-word algebra. 000267 | Odd CAR generators may live in a charged field algebra F. The physical 000268 | observable algebra is obtained only afterward by the actual gauge-invariant or 000269 | admissible restriction used by the theory. In the cited primitive source, 000270 | 000271 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000272 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex:319-343 000273 | 000274 | defines A_adm by restriction of operators which preserve Ran(P_adm), together 000275 | with their adjoints. It does not require each odd charged field to be an 000276 | element of A_adm. The boundary map above must therefore map field words into 000277 | the graded source algebra before this observable restriction; the restriction 000278 | can then be checked on invariant/even words. No identification of Q_Theta with 000279 | fermion parity is used here. 000280 | 000281 | 000282 | 7. Source audit: what the spectral and Chern--Simons data do not supply 000283 | ------------------------------------------------------------------------ 000284 | 000285 | The following statements are source-level facts, not a general impossibility 000286 | claim. 000287 | 000288 | 1. Primitive relative spectral action. 000289 | 000290 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000291 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex:302-317 000292 | defines 000293 | 000294 | S_prim = tr f(D_rel/chi) - tr f(D_ref/chi) + (i pi/2) Delta eta_Sigma. 000295 | 000296 | This is a spectral functional. It contains no Grassmann integration, 000297 | fermionic source coupling, inverse Dirac kernel, or definition of Z_adm. 000298 | 000299 | 2. Relative generating functional. 000300 | 000301 | documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000302 | 002-section-closure-theorems-from-the-admissibility-complex.tex:256-316 000303 | defines only 000304 | 000305 | Z_rel[J,eta,bar eta] 000306 | = Z_adm[J,eta,bar eta] / Z_ref[0,0,0]. 000307 | 000308 | Its reflection-positivity lemma explicitly assumes that the admissible 000309 | fermionic CAR kernel is reflection positive. It does not derive that kernel 000310 | or Z_adm from D_+ and B_Sigma. 000311 | 000312 | 3. Geometric measure. 000313 | 000314 | documents/archive--tfpt-45--source_extracts--05_metrology_source/sections/ 000315 | 005-section-absolute-dimensionless-metrology-from-the-bounda.tex:136-178 000316 | constructs a measure weighted by exp[-tr f_*(D_V^2/Lambda^2)] on the 000317 | BRST-reduced geometric configuration space. It does not include D psi, 000318 | define a Berezin measure, or provide the missing mixed source functional. 000319 | 000320 | 4. Chern--Simons material. 000321 | 000322 | documents/tfpt_research_contracts/sections/005-source.tex:97-108 uses an 000323 | abelian bosonic 2+1-dimensional Chern--Simons/K-matrix theory to type the 000324 | anyon-condensation and holomorphic-edge step. It is not a definition of the 000325 | local fermionic Z_adm. The Maxwell/Chern--Simons reading in 000326 | documents/tfpt_1_architecture_e8/sections/007-source.tex:140-150 is itself 000327 | conditional and likewise does not provide Grassmann sources or the full 000328 | Schwinger family. 000329 | 000330 | 5. Explicit coupled source remains a demand. 000331 | 000332 | documents/tfpt_research_contracts/sections/017-source.tex:153-191 asks for 000333 | 000334 | Z_a[J] = integral DU Dpsi DPhi exp(-S_a + J O), 000335 | S_a = S_Wilson + bar psi D_GW[U,Phi] psi 000336 | + S_Phi + S_seam/top, 000337 | 000338 | and marks the complete T1--T7 package open. This confirms where an actual 000339 | fermion-plus-boson source would enter; it is not supplied by the primitive 000340 | spectral action. 000341 | 000342 | 000343 | Final boundary 000344 | -------------- 000345 | 000346 | The exact repair is therefore: 000347 | 000348 | derivatives of log Z -> connected cumulants 000349 | exponentiation -> full graded moments 000350 | full Z moments -> OS Gram form 000351 | state-preserving map -> boundary OS isometry. 000352 | 000353 | The first three arrows are mathematical identities or the standard OS input. 000354 | The fourth is a precise sufficient interface. The original TFPT sources cited 000355 | above do not yet construct Z_adm from the primitive D_+, B_Sigma data and do not 000356 | prove that interface. No claim of complete primitive seam reconstruction is 000357 | made. 000358 | 000359 | Primary mathematical references 000360 | -------------------------------- 000361 | 000362 | K. Osterwalder and R. Schrader, Axioms for Euclidean Green's Functions II, 000363 | Communications in Mathematical Physics 42 (1975), 281-305, 000364 | doi:10.1007/BF01608978. The corrected theorem and its full-family growth 000365 | hypotheses are in Sections III-IV, especially IV.1. This reference supplies 000366 | the standard reconstruction framework, not the missing TFPT functional. 000367 | 000368 | P. F. Baum and E. van Erp, The Index Theorem for Toeplitz Operators as a 000369 | Corollary of Bott Periodicity, arXiv:2010.02906v2 (2020), Section 6, 000370 | Proposition 13; Quarterly Journal of Mathematics 72 (2021), 547-569, 000371 | doi:10.1093/qmath/haab008. The Calderon and nonnegative boundary spectral 000372 | projections there differ by a compact operator and give the same index. 000373 | Equality of that index does not give an exact equality of projections, 000374 | an isometric shift, a CAR state, or the time covariance used in the companion 000375 | operator theorem.