TFPT scientific source excerpt Original repository path: experiments/theory-contracts/primitive-seam-operator-reconstruction-20261025/SPECTRAL-SEAM-RECONSTRUCTION.txt Original file SHA-256: 45c472819667e7a981f7da4246d74d31e15104335e941cde0e0893808d674654 Inclusive original lines: 201-278 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 | circle, line bundle, source, and clock first. If the original full B_Sigma 000202 | fails this relation, the theorem cannot be applied to it. Restriction to a 000203 | sector is allowed only after the original construction supplies that sector 000204 | and its reducing/source/time properties; no projector is added by fiat. 000205 | 000206 | 8. Locality, fields, and the full TFPT completion boundary 000207 | ---------------------------------------------------------- 000208 | The unitary W identifies C*(u) with continuous functions on S1, including 000209 | continuous functions supported in proper arcs. The Borel arc projections 000210 | 1_I(u) belong to W*(u), not to C*(u): the connected circle has no nontrivial 000211 | continuous projection-valued scalar functions. This reconstructs a circle 000212 | coordinate algebra. It does not identify those arcs with the original 000213 | spacetime regions or prove that the original locality/reflection maps agree. 000214 | 000215 | Ordinary complex CAR quantization of H is mathematically available for any 000216 | complex H. Its physical use here must still be tied to the original graded 000217 | field algebra, full source functional and the original positive time. If that 000218 | same-time full CAR ground-state interface is supplied, the existing 000219 | GROUND-STATE-SOURCE-REDUCTION theorem forces the negative-B polarization and 000220 | all Wick moments. No independently tuned experimental pulse is required. 000221 | 000222 | The odd CAR fields may belong to a graded charged field algebra F; they need 000223 | not preserve the physical admissible subspace. Physical observables are 000224 | obtained by the actual invariant/admissible restriction. Requiring the odd 000225 | field algebra itself to lie inside A_adm would impose an unjustified extra 000226 | condition. 000227 | 000228 | The Calderon Cauchy-data projection P_C is not used anywhere in this proof. 000229 | No equality P_C=P_B, or antiunitary complement relation for P_C, is required 000230 | for ordinary complex CAR. Those identities would matter only for an 000231 | additional, explicitly proposed identification of classical Cauchy data with 000232 | a particular self-dual quantum polarization. The original B,u pairing is the 000233 | smallest sufficient operator test pursued here; it is not a no-go theorem for 000234 | all alternative TFPT quantizations. 000235 | 000236 | The operator theorem and the full-moment OS repair do not close the coupled 000237 | four-dimensional source, interactions, generation/mirror selection, physical 000238 | state, or infrared universality. In the original research contracts the 000239 | consistent local lattice route is allowed; an exact continuum OS limit is an 000240 | additional strengthening, not a universal prerequisite for that route. 000241 | 000242 | Decision-relevant outcome 000243 | -------------------------- 000244 | The rank-one seam mode is no longer a vague free choice within the stated 000245 | operator class: Pi=1_(0,1](B), and the complete channel is generated by u^n. 000246 | The first decisive original-source test is (1) together with the SAME-pair 000247 | index (H3). Quarter phase, spacetime localization and the full graded source 000248 | functional must be identified in their proper types. None is replaced by a 000249 | matching dimension, topological number, or finite numerical example. 000250 | 000251 | 9. Why winding and the existing offset cannot supply the covariance 000252 | ------------------------------------------------------------------- 000253 | The distinction persists inside a fixed circle Hilbert space with a fixed 000254 | quarter-shifted B, without changing its CAR state. Put 000255 | 000256 | B=-i partial_theta+1/4, 000257 | b_a(z)=(z-a)/(1-a z), u_a(z)=z^2/b_a(z), 0 0, 000262 | 000263 | because the maximum of the subtracted Poisson kernel is (1+a)/(1-a)<2. 000264 | Nevertheless 000265 | 000266 | u_a* B u_a-B = d arg(u_a)/dtheta 000267 | 000268 | is nonconstant. Hence (H1) fails while the winding and quarter offset agree. 000269 | With the same Hardy cut, the negative Fourier coefficients of u_a are 000270 | (1-a^2)a^(m+1) at z^(-m), m>=1. Thus 000271 | 000272 | ||(1-P_B)u_a e_0||^2 = a^4(1-a^2) > 0. 000273 | 000274 | At a=1/4 this equals 15/4096. The Toeplitz index remains -1 by its actual 000275 | circle-symbol pairing, but the compression is not an isometry. This is a 000276 | counterexample to inferring (H1) or isometry from that index/winding/offset 000277 | alone. It is not a pair of complete TFPT models and does not exclude an 000278 | additional original source rule which identifies the correct representative.