TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PAIRED-REVIEW.txt
Original file SHA-256: f51a4d21315fac64cf42e2edceea20073b78f86b74e5e04e5bf66a5236280867
Inclusive original lines: 201-239
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 | requires its actual graph/polar transport convergence; it is not obtained
000202 | just by applying the finite formula at growing sizes.
000203 | 
000204 | 7. Moving polar connection must be retained
000205 | 
000206 | When V_K depends differentiably on an original background b, the exact
000207 | variation of the companion is
000208 |  d(V_K W V_K^dagger)
000209 |   =V_K(dW)V_K^dagger+[(dV_K)V_K^dagger,V_K W V_K^dagger]. (10)
000210 | The differentiability/domain conditions for this polar family are actual
000211 | hypotheses; in finite invertible families they follow from the usual smooth
000212 | polar calculus. They do not automatically extend uniformly when singular
000213 | values approach0 in an infinite family.
000214 | For a genuinely moving time frame, use the endpoint-correct transport
000215 |  W2(t,s)=V_K(t)W(t,s)V_K(s)^dagger.
000216 | If W'=-i F_I W, its generator is
000217 |  V_K F_I V_K^dagger+i dotV_K V_K^dagger.
000218 | Dropping this connection term would change the original field variation.
000219 | 
000220 | 8. Scope of the positive resolution
000221 | 
000222 | The true Dyson one-particle evolution is now controlled in S2, and the
000223 | whole graph supplies a canonical positive paired coefficient. Its phase
000224 | has not been selected by an arbitrary quadratic subtraction. In the
000225 | nonsymmetric case the polar transport supplies the missing second-face
000226 | operation. A failure for a literal untransported diagonal word cannot be
000227 | carried over to this correctly transported paired word.
000228 | 
000229 | The proven object is an adjoint-paired/full-graph response with the same
000230 | given initial source. It is not the old UNPAIRED Dyson amplitude, whose
000231 | quadratic common phase was analyzed separately. A proof that the primitive
000232 | S_adm/Z_rel source uses this pair still needs its ACTUAL reflected field
000233 | map and, under background changes, connection(10). No extra global CAR
000234 | route is required for TFPT. No full physical OS/time,3+1, absolute Z_rel[0]
000235 | or fundamental-origin completion is promoted by the present theorem.
000236 | 
000237 | This review is analytic. Root owns native smooth-pulse controls; none are
000238 | duplicated here. No canonical/Atlas mutation. Existing R032/R080/R101 and
000239 | the variable-background source theorem are preserved.