TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/FOCK-DEFECT-REVIEW.txt Original file SHA-256: bedb724c59b146200ea0fe9402b8981928be66b8c0ebdcf6e18cb285fc2798ab Inclusive original lines: 201-306 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 | A(z) A_u(z)^(-1) 000203 | =I+[z/(1-z)]Pi. (16) 000204 | 000205 | The right side differs from the identity by rank one. With the explicit 000206 | orientation 000207 | 000208 | det_rel(A,A_u):=det_F(A A_u^(-1)), 000209 | 000210 | one obtains 000211 | 000212 | det_rel(A,A_u)=(1-z)^(-1). (17) 000213 | 000214 | Reversing the ordered pair gives 000215 | 000216 | det_rel(A_u,A)=1-z. (18) 000217 | 000218 | For |z|<1 both are analytic at z=0, and (17)--(18) require no regularization 000219 | beyond the ordinary rank-one Fredholm determinant. At z=1 the inverse in 000220 | (16) is singular. 000221 | 000222 | This exact factor is not by itself a CAR state sum. For one ordinary fermion 000223 | mode, the number-graded trace is 000224 | 000225 | Tr(z^N)=1+z, 000226 | 000227 | while the supertrace is 1-z. A Berezin Gaussian with an inverse-kernel 000228 | deformation can also produce 1-z, with z acting as a source or kernel 000229 | coupling rather than a fugacity. The inverse factor in (17) is fixed by the 000230 | chosen direction of the relative covariance quotient. Therefore none of 000231 | (17), (18), 1+z or their powers may be substituted for another without the 000232 | original response, trace or Berezin contract. In particular, (17) does not 000233 | derive the later electromagnetic factor merely because it equals a familiar 000234 | rank-one expression. 000235 | 000236 | 5. Observable and superselection scope 000237 | --------------------------------------- 000238 | 000239 | If the physical Hilbert space is restricted to total charge zero, U Omega 000240 | lies in a different charge sector. A neutral observable cannot prepare it 000241 | from Omega. The seam implementer U is then a charged field or sector 000242 | intertwiner, not automatically an observable intervention. 000243 | 000244 | The covariance difference remains visible to neutral observables. For every 000245 | f,g, the bilinear c(f)*c(g) is gauge invariant, and (10) gives the exact 000246 | neutral response 000247 | 000248 | omega_U(c(f)*c(g))-omega_C(c(f)*c(g))=. (19) 000249 | 000250 | Thus the rank-one defect survives as a gauge-invariant two-point response 000251 | even when the vector U Omega itself is not in the vacuum charge sector. This 000252 | does not show that the source permits U as a physical operation. 000253 | 000254 | 6. What this closes and what remains typed separately 000255 | ------------------------------------------------------ 000256 | 000257 | PROVED from the recovered circle pair and the conditional original CAR/time 000258 | interface: 000259 | 000260 | * u shifts the sea covariance by exactly +Pi; 000261 | * [u,C] is rank one and hence CAR-implementable; 000262 | * the stated implementer sends the vacuum to the unique charge +1 state 000263 | c_0*Omega up to phase; 000264 | * that state has exact positive energy a; 000265 | * the reversed shift creates the charge -1 hole of energy 1-a; 000266 | * the directed relative covariance determinants are (1-z)^(-1) and 1-z. 000267 | 000268 | The first source condition is still the same one already isolated in 000269 | GROUND-STATE-SOURCE-REDUCTION.txt: the full graded CAR algebra and its 000270 | adjoints must be embedded in the original positive-energy representation, 000271 | and its time must be the Bogoliubov time of this same B_a. Once that condition 000272 | is met, C=P_- follows and is not an independently chosen sea. 000273 | 000274 | NOT PROVED by this calculation: 000275 | 000276 | * P_adm=P_+ or any identification of their Hilbert spaces; 000277 | * that U is an admissible neutral observable; 000278 | * that z is an electromagnetic fugacity, source coupling or kernel 000279 | deformation; 000280 | * equality of the relative determinant with the original electromagnetic 000281 | answer functional; 000282 | * a complete P1 source reconstruction. 000283 | 000284 | The new direct connection is narrower and exact: the recovered old 000285 | circle representative has the same displayed one-particle mode shift that the 000286 | previously constructed source W1 implements, and its old rank-one line is the 000287 | excess covariance of that shifted sea rather than a freely selected 000288 | one-particle ray. The physical P1 source map and answer map remain the next 000289 | typed interfaces. 000290 | 000291 | Sources used 000292 | ------------ 000293 | 000294 | * experiments/theory-contracts/older-paper-source-recovery-20261025/ 000295 | ORIGINAL-CIRCLE-REVIEW.txt 000296 | * experiments/theory-contracts/primitive-seam-operator-reconstruction-20261025/ 000297 | SPECTRAL-SEAM-RECONSTRUCTION.txt 000298 | * experiments/theory-contracts/paper-source-reconstruction-20261025/ 000299 | GROUND-STATE-SOURCE-REDUCTION.txt 000300 | * experiments/theory-contracts/mixed-current-origin-20261014/ 000301 | ATLAS-KMS-LOCALITY-AND-SOURCE-WINDING.txt:75-164 000302 | * experiments/theory-contracts/seam-joint-ward-state-20261014/ 000303 | PROOF.txt:742-850,1003-1087 000304 | * documents/archive--paper-latex--tfpt-theory-fullv21/sections/ 000305 | 001-section-introduction-the-problem-of-parameters.tex:230-251 000306 |