TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/original_insertions/PROOF.txt
Original file SHA-256: 637007e39986ac8f2ec07a6c2ce96f0d5686171583b226699ea9dfe9ca3bb4fc
Inclusive original lines: 201-255
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 | On the ALREADY reducing finite normal kernel K0, the old scalar-current
000203 | calculation and direct sums immediately give
000204 |   c_kernel=(dim ker D_N^+ - dim ker D_N^-)c=(index D_N^+)c.
000205 | This only uses the finite kernel and its existing product embedding. It does
000206 | not apply an unrenormalized circle trace formula to the full higher-dimensional
000207 | bulk. The determinant-line spin case has one positive mode and no negative
000208 | one; it transports both level1 and the same source-time Ward on that image.
000209 | This closes the conditional current-algebra part of that OLD product result.
000210 | 
000211 | For the full E2 spin twist, the old example E2=O(k) direct_sum O(1-k)
000212 | has r_plus=k,r_minus=k-1. Its NET scalar cocycle is still level1. Thus kernel
000213 | rank1 is unnecessary if the requested observable is ONLY this net phase.
000214 | It remains necessary to check the stronger one-current/time identification:
000215 |   [B_normal,J_kernel,n]=-n(J_plus,n-J_minus,n),
000216 | whereas J_kernel,n=J_plus,n+J_minus,n. If r_minus>0 this is not the old single
000217 | right-current Ward. In particular net index1 alone does not provide the
000218 | all-word/time recognition theorem or a canonical actual rank1 subchannel.
000219 | No arbitrary positive mode or mass profile is chosen to remove the extra pairs.
000220 | 
000221 | The remaining original question is the already named product-source map and
000222 | horizontal transport, now with explicit mixed current operators to test.
000223 | Neither a different Spin-c operator (index2) nor the bosonic Callias witness
000224 | can be substituted merely because an index is small or equals1.
000225 | 
000226 | 8. What this actually resolves and what remains the single source step
000227 | 
000228 | Resolved in the given full CAR collar: the mixed vector/axial route provides
000229 | an explicit chiral current, and its original geometric source time is linear.
000230 | No additional chirally quantized measure is required for this restriction.
000231 | Resolved for the constant finite product coefficient: the same bare current
000232 | fails time invariance; ALL constant reducing channels and their net scalar
000233 | levels are classified. Correct total grading and an internal reflection lift
000234 | survive; we do not declare the product construction inconsistent.
000235 | 
000236 | The next original-source question remains the missing actual embedding in
000237 | the old normal result: does its coupled normal/admissibility operator really
000238 | have that product/family action and horizontal transport, with the required
000239 | current image, seam/quarter action and joint reflection? Its actual state and
000240 | OS time must be carried through that embedding. A supplied invertible finite
000241 | Yukawa block does not answer this normal index problem.
000242 | No arbitrary zero Yukawa, normal kink or chirality filter is inserted here.
000243 | The existing finite Higgs insertion is not itself that source derivation.
000244 | Full physical P1 pushforward, electromagnetic readout and T1-T8 remain open.
000245 | 
000246 | 9. Verification and source protection
000247 | 
000248 | check_original_insertions.py imports the actual v252 constructor and rep,
000249 | with exact integer controls, and converts entries to exact Gaussian integers
000250 | before testing. It checks all four quaternion witnesses, source Clifford/time
000251 | and grading relations, all linear commutants for four decisive signed/kernel
000252 | cases, non-diagonal reducing and unpaired kernel projections, and VA phases.
000253 | All finite checks use explicit exceptions, so python -OO retains them.
000254 | Finite controls support but do not replace the all-parameter proof above.
000255 | No original paper, verification script, ledger, manifest or website is edited.