TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/session_coverage/supplements/PRIMITIVE-DIRECT-RECONSTRUCTION.txt Original file SHA-256: 308e8e741e1999dcedf422ca1b063150b1da00f83ced7221851565c4f49653f9 Inclusive original lines: 201-366 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 | 000203 | The fixed Ext_H(U) theorem says that the extension is generated over 000204 | 000205 | U = V_D5 tensor V_A3 000206 | 000207 | by the marked weight one sector M_h with h=(s,f). Therefore it is not 000208 | necessary to start by mapping all 248 currents or a complete free fermion 000209 | algebra. The first test is: 000210 | 000211 | construct a local P1 map iota_U for the base net U and one nonzero 000212 | localized operator X_h(f) in A_loc carrying the marked h sector; verify 000213 | its normalized Q system product with U, its adjoint sector, and the three 000214 | OS relations (3.3) to (3.5). (4.1) 000215 | 000216 | If (4.1) holds, the unique marked extension generates the rest, and (3.6) to 000217 | (3.8) reconstruct its cyclic representation and time without a global Fock 000218 | identification. If it fails, the proposed P1 origin of that exact local E8 000219 | class fails at its first charged generator. 000220 | 000221 | The source test is immediate and decisive. The original operational seed 000222 | names A_loc but does not exhibit iota_U or X_h. The definition of the local 000223 | source algebra says only "generated by local field insertions." The primitive 000224 | action is 000225 | 000226 | S_prim[chi_seed] 000227 | = Tr f(D_rel/chi_seed) - Tr f(D_ref/chi_seed) 000228 | + (i pi/2) Delta eta_Sigma, (4.2) 000229 | 000230 | for a declared profile f. It does not give D_rel[Phi], D_ref[Phi], or a 000231 | supported perturbation carrying sector h. Therefore differentiation of 000232 | (4.2) cannot yet manufacture X_h. This is concrete source counterevidence to 000233 | the claim that the primitive spectral kernel already supplies the local E8 000234 | field. It does not refute the stronger operational seed, which assumes a 000235 | local algebra and state from the start. 000236 | 000237 | 000238 | 5. P2 already gives the internal Clifford packet without a Cartan choice 000239 | --------------------------------------------------------------------------- 000240 | 000241 | The carrier source states 000242 | 000243 | E = E3 direct sum E2, rank E = 5, 000244 | S+ = Lambda_even(E), dim S+ = 16. 000245 | 000246 | See 000247 | 000248 | documents/archive--tfpt-45--source_extracts--02_carrier_source/sections/ 000249 | 001-subsection-carrier-algebraic-normal-form.tex, lines 17 to 32 and 000250 | 238 to 252, and 000251 | 002-subsection-internal-carrier-stabilizer-and-one-family-pa.tex, 000252 | lines 197 to 330. 000253 | 000254 | This construction is intrinsically Clifford. For v in E and lambda in E*, 000255 | let epsilon(v) be exterior multiplication and iota(lambda) contraction on 000256 | Lambda(E). They obey 000257 | 000258 | {epsilon(v),epsilon(w)}=0, 000259 | {iota(lambda),iota(mu)}=0, 000260 | {iota(lambda),epsilon(v)}=lambda(v) I. (5.1) 000261 | 000262 | Hence 000263 | 000264 | c(v+lambda)=epsilon(v)+iota(lambda) (5.2) 000265 | 000266 | represents the Clifford algebra of E plus E* with its canonical split 000267 | quadratic form. The even Clifford algebra preserves Lambda_even(E), whose 000268 | dimension is 2^(5-1)=16. Equations (5.1) and (5.2) use only the evaluation 000269 | pairing E* tensor E -> C. An adapted basis gives the familiar five CAR or 000270 | Jordan Wigner matrices, but that basis is a coordinate description of the 000271 | module, not an additional origin premise. A Hermitian metric is useful for a 000272 | unitary real form; it is not needed for the algebraic Clifford module itself. 000273 | 000274 | Riemann Roch can determine a five dimensional section space in the declared 000275 | geometric setting, and P2 separately fixes the 3 plus 2 split. Neither fact 000276 | identifies these internal sections with physical boundary fermion modes. The 000277 | correct gain is narrower and useful: no global Cartan choice and no physical 000278 | five mode CAR vacuum are required to construct the internal 16. 000279 | 000280 | 000281 | 6. The second idea and why it stops earlier 000282 | ------------------------------------------- 000283 | 000284 | A tempting direct P1 route is to define local insertions as spectral action 000285 | jets. For a supported bounded zeroth order perturbation V one would formally 000286 | obtain 000287 | 000288 | delta S_prim(V) 000289 | = chi_seed^(-1) Tr(f'(D_rel/chi_seed) V_rel) 000290 | - chi_seed^(-1) Tr(f'(D_ref/chi_seed) V_ref) 000291 | + delta eta, (6.1) 000292 | 000293 | with divided differences replacing f' in higher noncommuting derivatives. 000294 | An eighth jet along a correctly typed E8 invariant perturbation would be a 000295 | direct candidate source for a weight eight observable, without first choosing 000296 | free CAR fields. 000297 | 000298 | This route fails its first premise in the current original. No map from a P2 000299 | carrier field or marked h sector to V_rel and V_ref is defined, and the test 000300 | profile f is declared rather than derived. Consequently (6.1) is not yet an 000301 | operator in the actual local source algebra and its eighth derivative cannot 000302 | be identified with W8. Pursuing higher jets before constructing that first 000303 | supported perturbation would rebuild the missing dictionary under another 000304 | name. The cyclic OS route in Sections 2 to 4 is therefore the stronger path. 000305 | 000306 | 000307 | 7. What is now proved, conditional and open 000308 | ------------------------------------------- 000309 | 000310 | PROVED from the stated operational seed hypotheses and standard 000311 | reconstruction theorems: 000312 | 000313 | * the cyclic OS Hilbert space and positive time semigroup do not require a 000314 | CAR presentation; 000315 | * a state, reflection and time preserving local star homomorphism induces 000316 | the needed unitary only on its cyclic image, so a global unitary premise 000317 | is unnecessary; 000318 | * the internal P2 half spin packet Lambda_even(E) is basis free and does not 000319 | require a global Cartan or five physical fermion modes. 000320 | 000321 | PROVED in the existing Atlas, not redone here: 000322 | 000323 | * inside the exact marked Ext_H(U) class, the C4 glue determines one local 000324 | E8 vacuum source, with products, vacuum functional and conformal time. 000325 | 000326 | CONDITIONAL: 000327 | 000328 | * if the operator X_h and the base map iota_U in (4.1) exist in the actual 000329 | P1 operational algebra, the entire marked E8 cyclic source follows 000330 | without a free CAR origin or a global P1 to Fock unitary. 000331 | 000332 | OPEN and now sharply localized: 000333 | 000334 | * construct X_h and iota_U from the original seam/collar source and verify 000335 | the Q system product plus equations (3.3) to (3.5); 000336 | * decide whether the operational seed is itself derived from the smaller 000337 | primitive kernel. The current one way functor C_seed does not prove this; 000338 | * identify the reconstructed marked weight eight observable with the 000339 | already verified W8 vector and with the physical P1 insertion, including 000340 | its normalization and contact terms. 000341 | 000342 | 000343 | Primary references checked 000344 | -------------------------- 000345 | 000346 | K. Osterwalder and R. Schrader, Axioms for Euclidean Green's functions, 000347 | Commun. Math. Phys. 31 (1973) 83 to 112: 000348 | https://projecteuclid.org/journals/communications-in-mathematical-physics/ 000349 | volume-31/issue-2/Axioms-for-Euclidean-Greens-functions/cmp/1103858969.pdf 000350 | 000351 | K. Osterwalder and R. Schrader, Axioms for Euclidean Green's functions II, 000352 | Commun. Math. Phys. 42 (1975) 281 to 305: 000353 | https://projecteuclid.org/journals/communications-in-mathematical-physics/ 000354 | volume-42/issue-3/Axioms-for-Euclidean-Greens-functions-II-with-an-Appendix-by/ 000355 | cmp/1103899050.pdf 000356 | 000357 | D. Schlingemann, From euclidean field theory to quantum field theory, 000358 | arXiv:hep-th/9802035. In particular Lemma 3.1, Theorems 3.4 and 3.7, and the 000359 | fermionic reconstruction in Section 4: 000360 | https://arxiv.org/abs/hep-th/9802035 000361 | 000362 | A. Klein and L. J. Landau, Construction of a unique self-adjoint generator 000363 | for a symmetric local semigroup, J. Funct. Anal. 44 (1981) 121 to 137: 000364 | https://doi.org/10.1016/0022-1236(81)90007-0 000365 | 000366 | No paper, Atlas or ledger file was modified.