TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-joint-source-origin-audit-20261016/PROOF.txt
Original file SHA-256: f91443b115994302aa3b10d2a6b44131e13c3998b13d47b34aeaa59e25c86315
Inclusive original lines: 154-209
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.

000154 | 6. Continue the same source path: the missing datum is a mixed source derivative
000155 | 
000156 | The retained two-mode quarter-source closure already has
000157 |   K0=diag(17/16,4097/4096), det K0=69649/65536.
000158 | Reuse the existing literal Grassmann integration in
000159 | seam-collar-multifield-20261016/joint_source.py; do not build a replacement solver.
000160 | For a PROVIDED kinetic family K(Phi), it gives
000161 |   Z[bareta,eta;Phi]=det K(Phi) exp(bareta K(Phi)^-1 eta).
000162 | For its normalized two-source coefficient C(Phi)=K(Phi)^-1,
000163 |   C'(0)=-K0^-1 Y K0^-1, Y=K'(0),
000164 | and -K0 C'(0) K0=Y. This is the amputated mixed insertion on this specified
000165 | free complex-fermion chart. Physical chiral Yukawa types still require the
000166 | actual field/charge maps; a generic two-mode insertion is not that derivation.
000167 | 
000168 | An exact zero-jet control compares the SUPPLIED diagnostic families
000169 |   K_a(Phi)=K0,
000170 |   K_b(Phi)=K0+Phi [[0,1],[1,0]].
000171 | Their ENTIRE unnormalized purely fermionic Grassmann generating polynomials
000172 | agree at Phi=0, including the absolute det K0. Their mixed derivatives differ:
000173 | zero versus off-diagonal -65536/69649. Direct nilpotent integration, separate
000174 | inverse differentiation and amputation agree exactly.
000175 | 
000176 | Therefore fixed-background fermionic data do not determine K'(0). This is NOT
000177 | a claim that the full Phi-dependent source functional fails to encode its own
000178 | derivative. It is NOT a proof that both diagnostic families obey P1/P2 or all
000179 | original admissibility conditions. A source rule can constrain or select the
000180 | derivative; supplying it is precisely the missing upstream work.
000181 | 
000182 | S1:313-315 introduces carrier-compatible bosonic fluctuations only after the
000183 | carrier theorem; S1:319-343 separately introduces local field insertions. The
000184 | provided primitive relative spectral trace is not already the above charged,
000185 | Phi-dependent kinetic source. The known free reconstruction consequently does
000186 | not yet produce the independently required nonzero carrier Yukawa vertex.
000187 | 
000188 | 7. Best-supported remaining approach and stopping condition of this audit
000189 | 
000190 | Retain the original complete two-face datum, scales and absolute source factors.
000191 | Before further scans, the original source must provide an actually evaluable
000192 | charged field/boson functional or K[Phi], with its physical measure, field types,
000193 | representations and reference. Compute the mixed two-fermion/one-boson response
000194 | in that SAME functional and test the desired carrier support without inserting
000195 | P3/P2 or the desired contraction equivalence first. For an interacting source,
000196 | use its actual connected/1PI vertex; the free inverse formula above is only a
000197 | control and cannot be assumed universally.
000198 | 
000199 | Success requires that the actual source produce the positive-block identification
000200 | and the specific nonzero negative-block contraction with the required group action,
000201 | while supplying the same state/time data. A primitive invariant or an O(1)
000202 | dimension count does not meet that criterion. A zero or wrong-type actual vertex
000203 | rejects that candidate before downstream optimization.
000204 | 
000205 | The inspected original definitions do not give this pre-carrier field-dependent
000206 | functional or its mixed vertex; they place the field coupling after carrier closure.
000207 | No source-derived replacement is constructed here. Consequently the requested
000208 | joint physical derivation remains OPEN, with a narrower exact missing input.
000209 | No extra arbitrary geometry, Hamiltonian or coupling is introduced as its solution.