TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE33-FACE-GRAPH-AND-TIME-REVIEW.txt
Original file SHA-256: c106777f365804a4f3a20d880eb88560f4f5e567b2713040b0fa7aff88b77a87
Inclusive original lines: 201-272
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 | The tangential pullback sends the periodic quarter data as
000202 | 
000203 |   alpha_p=-alpha_q mod 1,  U_p=U_q^-1.                      (20)
000204 | 
000205 | Thus it exchanges the 1/4 and 3/4 blocks. This is transport, not a fixed
000206 | quarter claim.
000207 | 
000208 | For a smooth constant-frame annulus domain f(ell)=u_q f(0), radial exchange
000209 | alone inverts the gluing orientation. If in addition
000210 | 
000211 |   R_x u_q R_x^-1=u_q^-1,                                   (21)
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000217 | 
000218 | For a variable frame, the original endpoint factors H(0),H(ell) must be
000219 | inserted in (21). At the Fuchs corners z=plus/minus1 the smooth pullback does
000220 | not by itself prove preservation of the closed corner domain or its selected
000221 | reference subspace. The reference is transported to its Theta image; it is
000222 | fixed only if the original reference data satisfy that extra equivariance.
000223 | These are the remaining domain checks, not reasons to introduce another p
000224 | background.
000225 | 
000226 | Finally, (8) and (18) give
000227 | 
000228 |   Theta_faces (Ical B_q) Theta_faces^-1=Ical B_p.            (22)
000229 | 
000230 | Therefore
000231 | 
000232 |   Theta_faces exp(t Ical B_q) Theta_faces^-1
000233 |     =exp(t Ical B_p).                                        (23)
000234 | 
000235 | The actual circle parameter is +t. A replay of (22)-(23) gave zero generator
000236 | residual and time residual 2.25e-16. This statement concerns the existing
000237 | circle evolution; it does not identify it with positive radial OS time.
000238 | 
000239 | FINAL SCOPE
000240 | 
000241 | Theta_- gives a complete conditional whole-source map on the smooth annulus:
000242 | operator, q-to-p image coefficient, selfdual field/adjoint chart, both faces,
000243 | full graph state and circle time all transport together. The exact remaining
000244 | source question is narrower than a second-background problem: prove that the
000245 | original geometric spin inversion uses the same field lift and old-i phase,
000246 | and verify the variable-frame Fuchs corner/reference domain.
000247 | 
000248 | The spin equality r^6=-I is relevant only after that intertwiner is supplied.
000249 | Until then it cannot be used to upgrade equality of operator conjugations to
000250 | equality of whole CAR field maps or to declare a physical P1 execution.
000251 | 
000252 | Sources reviewed
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