TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/SPIN-PRODUCT-CAR-U6-TRANSPORT-REVIEW.txt Original file SHA-256: 6a107b94df366affec36a50d940a74420e192010d371d34a6aeeccfe074a1e5b Inclusive original lines: 191-307 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 | h_Phi=A gamma_F-i gamma_F Phi, (12) 000193 | 000194 | and the finite-collar graph/HS state has 000195 | 000196 | C_Phi=(1+exp(2 ell h_Phi))^-1, 000197 | X_Phi=(2 cosh(ell h_Phi))^-1. (13) 000198 | 000199 | The real-time source automorphism is implemented by exp(i t h_Phi), and the 000200 | graph state is invariant under that actual group. If a unitary r on the same 000201 | charged Hilbert space is to act on the same source object, it must satisfy 000202 | 000203 | r C_Phi r^*=C_Phi. (14) 000204 | 000205 | For finite ell>0 the scalar function x->(1+exp(2 ell x))^-1 is injective on 000206 | the real line. Functional calculus therefore makes (14) equivalent to 000207 | 000208 | r h_Phi r^*=h_Phi, or [r,h_Phi]=0. (15) 000209 | 000210 | Then the cross covariance X_Phi is preserved as well and the CAR 000211 | automorphism fixes the graph state. If instead 000212 | 000213 | r h_Phi r^*=h_(Phi^R), (16) 000214 | 000215 | it transports the whole state and time to another member of the source 000216 | family. This is a valid covariance, but not a symmetry of one fixed source. 000217 | 000218 | The exact E0 review provides an example of (16): it transports the complete 000219 | F5 graph, T_faces chart and circle group to explicitly transformed 000220 | coefficients. It becomes a same-source operation only for appropriately 000221 | fixed controls or after a separately allowed commutant equivalence. No 000222 | analogue of (15) or (16) has been shown for Rhat. The direct embedding (11) 000223 | is absent, and the alternative q-to-p source-family pullback has not yet been 000224 | checked on h_Phi and the whole F5 chart. 000225 | 000226 | The Hodge inner product on the degree-sixteen Kummer cover proves that R_V 000227 | is unitary on Voff. It is not the covariance (13), does not define h_Phi, 000228 | and does not select a Fock sea. Extending the six-mode Rhat by the identity on 000229 | an arbitrarily chosen orthogonal complement would create a new source 000230 | operation rather than derive the original one. 000231 | 000232 | There is also a sharp consequence if a future same-source map satisfies 000233 | (14). For a nontrivial eigenoperator O with alpha_R(O)=rho O, invariance gives 000234 | 000235 | omega_Phi(O)=omega_Phi(alpha_R(O))=rho omega_Phi(O)=0. (17) 000236 | 000237 | Thus source symmetry alone does not prepare a nonzero rho one-point 000238 | amplitude. Relative insertions, two-point functions or a covariant source 000239 | family can still see the character. Their consumer must be the existing 000240 | source operation, not a state chosen from the Hodge metric. 000241 | 000242 | 6. Smallest missing whole-field alternatives 000243 | 000244 | A sufficient next theorem for a direct six-mode embedding is the explicit 000245 | isometric map (11), not another U6 spectrum calculation. It would make the 000246 | following data commute together: 000247 | 000248 | q+p spin zero modes --Rhat--> q+p spin zero modes 000249 | | iota_6 | iota_6 000250 | v v 000251 | actual charged H ------r------> actual charged H or H_(Phi^R), 000252 | 000253 | and it must additionally verify: 000254 | 000255 | 1. its q restriction agrees with the already declared q-to-generation 000256 | comparison, while its p restriction has an original charge/dual label; 000257 | 2. r respects gamma_F, the represented finite algebra and J_F in their 000258 | actual types; 000259 | 3. r extends selfdually through C0 and all four T_faces slots; 000260 | 4. the pair-product square commutes with mu_ab and the CAR pair operation 000261 | through the forced alternating intertwiner (8), including (9)-(10); 000262 | 5. either the fixed-source equation (15) or the explicit source-family 000263 | covariance (16), with the already given graph state and circle time; 000264 | 6. the original reflection and reference data, rather than only the Hodge 000265 | target norm. 000266 | 000267 | This diagram is not a universal necessity. The other concrete route is a 000268 | q-to-p pullback between two source-family members. Its first decisive test is 000269 | the actual action of the original spinframe pullback, including its basis 000270 | inversion sign, together with the represented linear chiral/dual swap on 000271 | 000272 | h_Phi=A gamma_F-i gamma_F Phi, 000273 | 000274 | followed by C_Phi, circle time and all T_faces slots. Success would establish 000275 | (16) without embedding K6 into one fixed charged space; failure at that 000276 | operator identity would rule out this source-family route in the tested 000277 | chart. 000278 | 000279 | Until either source map is supplied, the strongest exact statement is: 000280 | 000281 | * the original curve gives a canonical geometric mixed-product U6; 000282 | * the stipulated six one-particle modes admit a canonical CAR star 000283 | automorphism and an equivalent signed pair U6 module; 000284 | * the full star closed pair module is Voff plus its conjugate; 000285 | * no reviewed source identifies this module with the existing F5/96 000286 | state/time object. 000287 | 000288 | These are two precise missing source maps, not a proposal for a new six-mode 000289 | Gaussian source. The direct iota_6 diagram is the stronger embedding claim; 000290 | the q-to-p pullback alternative asks only for whole-family covariance. 000291 | 000292 | Sources reviewed 000293 | 000294 | [OMITTED: non-scientific conversation or execution metadata] 000295 | [OMITTED: non-scientific conversation or execution metadata] 000296 | [OMITTED: non-scientific conversation or execution metadata] 000297 | [OMITTED: non-scientific conversation or execution metadata] 000298 | [OMITTED: non-scientific conversation or execution metadata] 000299 | [OMITTED: non-scientific conversation or execution metadata] 000300 | [OMITTED: non-scientific conversation or execution metadata] 000301 | [OMITTED: non-scientific conversation or execution metadata] 000302 | [OMITTED: non-scientific conversation or execution metadata] 000303 | [OMITTED: non-scientific conversation or execution metadata] 000304 | [OMITTED: non-scientific conversation or execution metadata] 000305 | [OMITTED: non-scientific conversation or execution metadata] 000306 | [OMITTED: non-scientific conversation or execution metadata] 000307 | [OMITTED: non-scientific conversation or execution metadata]