TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CANONICAL-CLIFFORD-DOUBLE-REVIEW.txt
Original file SHA-256: 5fef5442dbdbfd035dd4a2eb11d17967ec2bd9335f527376354e1123313cd13d
Inclusive original lines: 201-250
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 | Its AP Green branch between opposite points of the normal circle is
000202 | 
000203 |   S_AP(ell,0)
000204 |     =exp(-ell B_Phi)(I+exp(-2 ell B_Phi))^-1
000205 |     =(2 cosh(ell B_Phi))^-1.                           (20)
000206 | 
000207 | Under the charged chart this is X(h_Phi) on one leg and its conjugate on the
000208 | other.  Therefore, once the original source operator is indeed this
000209 | canonical product double, the AP normal spectrum, the positive Fredholm
000210 | gluing factor (18), and the exact first-order midpoint Green operator (20)
000211 | follow from the same given D.  No separate AP boundary selection or fitted
000212 | cross kernel is needed.  The absolute full determinant still requires the
000213 | common bulk regularization just described.
000214 | 
000215 | 6. Exact remaining source boundary
000216 | 
000217 | The original source extract currently displayed in
000218 | 
000219 |   documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000220 |   sections/001-section-primitive-core-and-carrier-interface-handoff.tex
000221 | 
000222 | writes D=D_plus union D_plus^op.  In the inspected passage it does not print
000223 | the stronger identities
000224 | 
000225 |   D_plus^op=-D_plus,       v=J u.                      (21)
000226 | 
000227 | If D_plus^op there denotes the standard canonical invertible double with the
000228 | J clutch, the present calculation closes the earlier AP-origin condition.
000229 | If it denotes only an oppositely oriented geometric copy, (21) still has to
000230 | be supplied by the original construction.  This is a binding question about
000231 | the existing symbol, not freedom to choose AP from the covariance G.
000232 | 
000233 | The result establishes the normal spin closure of the given constant product
000234 | Dirac operator.  It does not select the physical background Phi, a quantum
000235 | state, a P1 source insertion, a determinant/Pfaffian normalization, or a
000236 | physical time/reflection.  Weak unique continuation establishes global
000237 | invertibility of the canonical closed double; the elementary AP Green
000238 | identity (20) already follows on this product collar.
000239 | 
000240 | Local sources checked:
000241 | 
000242 |   experiments/theory-contracts/tfpt-unified-source-derivation-20261029/
000243 |   graded_boundary_functor/PROOF.txt, lines 20-61.
000244 | 
000245 |   experiments/theory-contracts/tfpt-unified-source-derivation-20261029/
000246 |   graded_boundary_functor/FINITE-COUPLED.txt, lines 10-72.
000247 | 
000248 |   experiments/theory-contracts/tfpt-unified-source-derivation-20261029/
000249 |   graded_boundary_functor/check_finite_typing.py, lines 28-54.
000250 |