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Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/FIXED-CAP-DETERMINANT.txt
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000001 | FINITE FREDHOLM BOUNDARY CORRECTION IN THE EXISTING CONSTANT SOURCE
000002 | 29 October 2026. Independent analytic review. No new model, source choice,
000003 | checker run, original edit or Atlas mutation.
000004 | 
000005 | Result
000006 | The fixed old APS exterior cap and the moving coupled-sea cap differ by an
000007 | explicit positive trace-class determinant. This boundary correction is finite
000008 | and nonzero on the WHOLE original circle at both quarter holonomies. It must
000009 | not be dropped when comparing a fixed-domain source with a pure moving-vacuum
000010 | energy expression. It does not renormalize the remaining bulk energy or
000011 | identify either expression with the original absolute P1 source functional.
000012 | 
000013 | 1. Same actual operator and exterior transfer
000014 | Reuse APS-DOMAIN-AND-STATE.txt:sections5-6 and the already proved original
000015 | constant coupled class:
000016 |   K=C^96, gamma=gamma_F with dimensions48+48,
000017 |   p=n+alpha, alpha=1/4 or3/4, n in Z,
000018 |   h0=p gamma, h=h0+V, V=-i gamma Phi,
000019 |   Phi*=Phi, {gamma,Phi}=0, Phi constant,
000020 |   E=(p^2+Phi^2)^(1/2), h^2=E^2,
000021 |   P=P_-(h0), R_ell=exp(-ell h), ell>0.
000022 | Thus E commutes with gamma,h,V and P. On the actual occupied space RanP,
000023 |   M_p=P R_ell P=cosh(ell E)+|p|E^-1 sinh(ell E).
000024 | The old-sea exterior matrix element is det M_p. The moving negative sea
000025 | has exterior amplitude exp(ell Tr_RanP E), using the same underlying
000026 | transfer representative. Their quotient is det C_p, where
000027 |   C_p=exp(-ell E)M_p
000028 |      =a_p+(I-a_p)exp(-2ell E),
000029 |   a_p=(I+|p|E^-1)/2.
000030 | The reference old-sea amplitude exp(ell |p| dimRanP) cancels from this
000031 | quotient. The identification of the E spectrum on old and moving occupied
000032 | spaces follows from the existing odd singular blocks, not an independent
000033 | fit or an extra internal species.
000034 | 
000035 | 2. Exact positivity and trace-class bound
000036 | All factors are commuting functions of Phi^2 on RanP. Spectral calculus gives
000037 |   I-C_p=(I-a_p)(I-exp(-2ell E)) >=0,
000038 |   C_p>=a_p>=I/2,
000039 |   0<=I-C_p<=I-a_p
000040 |      =Phi^2/[2E(E+|p|)]<=Phi^2/(4p^2).
000041 | These identities include zero singular values by continuity; their C_p is I.
000042 | No small-mass expansion or finite-momentum cutoff is assumed.
000043 | 
000044 | Let C be the direct sum of C_p on directsum_p RanP_-(h0(p)). Since the
000045 | internal space is finite and sum_n(n+alpha)^-2 converges, I-C is positive
000046 | trace class. For the actual odd selfadjoint96 component coefficient,
000047 |   Phi=[[0,F],[F*,0]],
000048 |   Tr_(gamma+) Phi^2=Tr_(gamma-) Phi^2=(1/2)Tr_K Phi^2.
000049 | Both signs of p therefore give the same occupied trace bound. In detail,
000050 |   Tr(I-C)<= (Tr_K Phi^2/8) sum_n 1/(n+alpha)^2
000051 |           =pi^2 Tr_K Phi^2/[8 sin^2(pi alpha)]
000052 |           =pi^2 Tr_K Phi^2/4  at alpha=1/4,3/4.
000053 | There are48 occupied slots per p, including any zero singular slots. The
000054 | conjugate test slots of the existing selfdual chart are not counted again
000055 | as additional physical charged fields.
000056 | 
000057 | Because spectrum(C) is contained in[1/2,1],
000058 |   0<=-log C<=2(I-C).
000059 | Consequently log C is trace class and
000060 |   det_F C=exp(Tr log C)>0,
000061 |   exp(-2Tr(I-C))<=det_F C<=1.
000062 | This proves existence and NONVANISHING of the whole Fredholm determinant.
000063 | Finite Fourier exhaustions converge in trace norm for I-C, hence give this
000064 | same positive determinant; this is not a fit to a cutoff product.
000065 | 
000066 | 3. Exact relation to the already established HS sea result
000067 | Reuse Atlas finding:
000068 |   whole-constant-coupled-source-is-implementable-despite-isolated-insertion-domain,
000069 | and primitive_insertions_review/UV-INSERTION.txt:55-78, read again here.
000070 | For P_h=P_-(h), the old occupied compression is
000071 |   P P_h P|RanP=a_p,
000072 | and the finite internal singular blocks give
000073 |   ||P_h(p)-P(p)||_HS^2=2Tr_RanP(I-a_p).
000074 | Thus
000075 |   Tr_RanP(I-C_p)<=1/2 ||P_h(p)-P(p)||_HS^2.
000076 | The boundary trace-class convergence is the direct finite-time cap version
000077 | of the already proved square-summable sea rotation. It does not contradict
000078 | the nonimplementability of the isolated pure insertion group or put the old
000079 | vacuum into the coupled energy form domain. Those previous domain results
000080 | remain unchanged.
000081 | 
000082 | 4. Boundary factor, bulk factor and original source scalar are distinct
000083 | For a finite set of p, the relative exterior amplitudes satisfy exactly
000084 |   A_fixed/A_moving=product_p det C_p,
000085 |   A_moving=exp(ell sum_p Tr_RanP(E-|p|I)).
000086 | Hence the first ratio has the positive finite whole-circle limit det_F C.
000087 | For nonzero constant Phi the second exponent retains the known logarithmic
000088 | divergence: its summand is asymptotic to Tr_K Phi^2/(4|p|). The boundary
000089 | determinant does not remove that divergence or select a bulk subtraction.
000090 | 
000091 | An unknown scalar multiplying the SAME source transfer at a background
000092 | multiplies its fixed and moving cap matrix elements equally and cancels from
000093 | their quotient. This does not determine that scalar, its reference ratio,
000094 | its determinant-line phase or the original Z_adm. No equality to the original
000095 | spectral action, absolute P1 measure or physical time is asserted. The result
000096 | provides the precise finite boundary correction for the existing source
000097 | comparison, conditional on that stated exterior/CAR quantization contract.