TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
Original file SHA-256: 3537a2b080df0bb8b5d30c73da7dd83dc8e1ae5a6cfea4bb48a8b99f0080dfce
Inclusive original lines: 271-326
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.

000271 | 10. Joint unnormalized source integration: the original identification test
000272 | 
000273 | 10.1 What the complete original source contains
000274 | 
000275 | The full source is more than a normalized Cauchy graph or the number c3.
000276 | It supplies one-sided operator data, domains, reference/comparison operators and
000277 | an eta contribution. In 01_boundary_kernel_source.tex:122-130,
000278 | D_ref=(D+tau*D tau*)/2 and D_rel=(D-tau*D tau*)/2 are EVEN/ODD PARTS of D.
000279 | Thus D_rel is not automatically the full physical fermionic kinetic operator K.
000280 | It must not be inserted in a Gaussian determinant without that identification.
000281 | The canonical P1/P2 statement in tfpt_1_architecture_e8.tex:163-189 gives a
000282 | primitive reflection-positive unit-winding seam kernel and the five-slot 3+2
000283 | interface. Neither postulate states a Berezin measure or its source coupling.
000284 | 
000285 | Suppose a complex fermion kinetic operator K, reference, local source coupling
000286 | and ordered Berezin measure really ARE given by that same source. On an
000287 | invertible finite regulator, direct nilpotent integration gives, with orientation
000288 | integral exp(-sum_j barpsi_j psi_j)=1,
000289 |   Z_F[bareta,eta] = exp(-S0) det K exp(bareta K^(-1) eta).
000290 | This includes its scalar rather than fitting it after a normalized correlation.
000291 | For retained modes b and integrated modes i with invertible K_ii,
000292 |   S_b=K_bb-K_bi K_ii^(-1)K_ib,
000293 |   det K=det K_ii det S_b,  (K^(-1))_bb=S_b^(-1).
000294 | Thus the source response and determinant factor come from the SAME integration.
000295 | In a genuinely geometric cut this supplies a boundary reduction; a generic mode
000296 | partition alone does not derive the physical geometric boundary pairing.
000297 | Joint_source uses direct Grassmann integration independent of the determinant
000298 | formula and checks both formulas and their signs. It keeps det K_ii explicitly.
000299 | 
000300 | For the actual lowest particle/hole pair of section6, close two copies of the
000301 | given collar under the DECLARED ordinary fermionic trace. Then
000302 |   K=I+R_ell^2=diag(17/16,4097/4096),
000303 |   det K=69649/65536,
000304 |   K^(-1)=diag(16/17,4096/4097),
000305 |   n=I-K^(-1)=diag(1/17,1/4097).
000306 | K^(-1) is the coherent source contraction, whereas n is the normal-ordered CAR
000307 | occupation. They are not interchangeable. Eliminating the hole MODE produces
000308 | the retained kinetic17/16 AND scalar4097/4096. This example is not an integration
000309 | of a geometric collar interior. Its CAR/Berezin and trace-closure assumptions
000310 | remain explicit; it does not supply a new physical field assignment from P1.
000311 | 
000312 | 10.2 A decisive test against the original primitive spectral action
000313 | 
000314 | For a direct identification of the real primitive action with the integrated
000315 | complex-fermion action -log det K, vary a positive commuting kinetic eigenvalue
000316 | x=K/chi. The necessary identity is x f'(x)+1=0. Hence on that spectral interval
000317 |   f(x)=-log x+constant.
000318 | A generic declared spectral test profile does not satisfy it: f(x)=exp(-x^2)
000319 | gives residual1-2x^2 exp(-x^2). Rewriting exp(-Tr f(K)) as det(exp(-f(K)))
000320 | changes the kinetic operator and therefore does not prove covariance K^(-1).
000321 | This only tests the proposed DIRECT identification. It is not a no-go against a
000322 | theory containing a separate bosonic spectral action AND fermionic determinant.
000323 | The original04:463-471 normalizes Z_adm by a reference; 04:2160-2173 introduces
000324 | a formal measure and S_adm; 04:2270-2301 retains higher interaction vertices.
000325 | Those definitions do not prove that the full source is this quadratic model.
000326 |