TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
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Inclusive original lines: 327-366
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000327 | 10.3 Absolute spectral normalization of the same given angular quarter source
000328 | 
000329 | There is a concrete absolute normalizer for the already GIVEN circle operator,
000330 | once complex-fermion zeta quantization and spectral cut are specified. This is
000331 | angular theta, not the normal collar parameter or a positive thermal state.
000332 | Let D_alpha=-i partial_theta on f(2pi)=exp(2pi i alpha)f(0),0<alpha<1,
000333 | with negative eigenvalue argument +pi. Its spectrum is n+alpha,n in Z. Then
000334 |   zeta_D(s)=zeta_H(s,alpha)+exp(-i pi s) zeta_H(s,1-alpha),
000335 |   eta_D(0)=1-2alpha,  zeta_(D^2)(0)=0,
000336 |   det_zeta D_alpha=2sin(pi alpha) exp(-i pi(1-2alpha)/2)
000337 |                   =1-exp(2pi i alpha).
000338 | The last equality follows from the Hurwitz derivative at0 and Gamma reflection.
000339 | At alpha1/4 the absolute complex scalar is1-i, including modulus sqrt2 and the
000340 | eta phase exp(-i pi/4). The diagnostically conjugate domain alpha_ref3/4 gives
000341 | det1+i and ratio-i. This diagnostic reference is NOT the archived D_ref.
000342 | The conjugate determinant pairing has squared norm2. It is not a proof that the
000343 | physical doubled TFPT source uses exactly this pair and no other contribution.
000344 | 
000345 | The same source operator has the Green kernel, away from its insertion,
000346 |   G_-=i/(h-1), G_+=ih/(h-1), h=exp(2pi i alpha),
000347 |   G_+-G_-=i, G_+=h G_-.
000348 | Hence -i partial_theta G=delta and the actual charged quarter holonomy is kept.
000349 | For smooth finite Grassmann sources the conditional angular functional is
000350 |   Z_alpha[bareta,eta]=det_zeta D_alpha exp(bareta D_alpha^(-1)eta).
000351 | This is a fermionic Green propagator, not a positive CAR occupation covariance.
000352 | For alpha variations use the fixed periodic frame -i partial_theta+alpha;
000353 | the displayed piecewise Green kernel uses the twisted frame. Their coincident
000354 | values agree; the field/source frames must be transported together.
000355 | The alpha variation contains a contact/regularization term:
000356 |   partial_alpha log det_zeta D=pi cot(pi alpha)+i pi,
000357 | whereas symmetric PV Tr D^(-1)=pi cot(pi alpha). The missing i pi would be the
000358 | eta-phase derivative, so discarding it breaks the joint source comparison.
000359 | 
000360 | For a general invertible selfadjoint K with the same spectral cut,
000361 |   log det_zeta K=-zeta_(K^2)'(0)/2
000362 |                 +i pi [zeta_(K^2)(0)-eta_K(0)]/2.
000363 | The primitive real trace plus eta expression therefore matches a relative
000364 | fermionic determinant ONLY after compatible actual K/K_ref, logarithmic profile,
000365 | zeta trace prescription and the relative zeta(K^2;0) term are accounted for.
000366 | The last term vanishes in this circle control, not on every geometry.