TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/WEIGHT.txt
Original file SHA-256: 7857f9185fc245b3de23789bcf1020b0b88e64dfd814d6ba71004e7d185fd345
Inclusive original lines: 62-132
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000062 | Metrology001:30–67 puts D_geo and the DECLARED D_ref on the given graded
000063 | relative bundle with APS boundary conditions. D_geo is the Dirac-type
000064 | metric/spin/connection anchor; inner fluctuations of the same anchored
000065 | spectral datum generate the gauge sector. D_rel is comparison bookkeeping,
000066 | not a replacement local principal symbol.
000067 | 
000068 | Metrology001:103–130 defines
000069 |  Zeta_rel(s;sigma)
000070 |  =Tr_rel[|D_sigma,geo+B_rel|^(-2s)-|D_ref|^(-2s)]
000071 |    +i pi Deltaeta/2,
000072 |  E_D=Res_(s=1)Zeta_rel,
000073 |  Gamma_D4=FP_(s=0)Zeta_rel.
000074 | This is an actual SAME-FUNCTIONAL relative constructor, not a generic
000075 | proposal to build one. Its first operator is the printed anchored/fluctuated
000076 | D_sigma,geo+B_rel, and its second is the printed bound D_ref.
000077 | 
000078 | In the explicitly stated constant-scale reduction sigma=sigma0,
000079 | chi=chi0, Metrology001:132–149 gives
000080 |  Gamma_grav=-6chi0² E_D+Gamma_D4.
000081 | On the corresponding compact compatible APS/regulator branch, introduce
000082 | only the already displayed absolute spectral-residue functional
000083 |  A_geo,chi0(T)
000084 |   =-6chi0² Res_(s=1)Tr_rel|T|^(-2s)
000085 |     +FP_(s=0)Tr_rel|T|^(-2s)+i pi eta_Sigma(T)/2.
000086 | Linearity of Res and FP gives EXACTLY
000087 |  Gamma_grav
000088 |  =A_geo,chi0(D_sigma0,geo+B_rel)-A_geo,chi0(D_ref).       (3)
000089 | The constant eta term has no pole at1 and enters FP0 once, with its original
000090 | coefficient. It is not duplicated by this separation.
000091 | 
000092 | Thus the geometric reference weight is forced:
000093 |  R_geo=exp(-Gamma_grav)=exp(6chi0² E_D-Gamma_D4).        (4)
000094 | There is no fitted counterphase, independent Higgs normalization or second
000095 | spectral reference subtraction in(3)–(4). This is the concrete constructive
000096 | answer in the older ORIGINAL geometric action branch.
000097 | 
000098 | The local-sigma zeta DIFFERENCE is also explicitly given and its residues
000099 | remain linear. However the simple factor -6chi0² is printed in the
000100 | constant-scale corollary. No new unprinted position-dependent weighted
000101 | residue prescription is inserted here to extend that corollary by notation.
000102 | For a relative-only/noncompact operator class keep its supplied relative
000103 | functional; separate absolute traces require the actual stated realization.
000104 | 
000105 | 3. WHICH TERMS ARE REALLY SHARED
000106 | 
000107 | Metrology002:39–170 supplies the common relative endomorphism
000108 |  E_rel=-R I48/4-(Phi*Phi)P2+(gamma^{mu nu} F_mu nu)/2,
000109 | and the actual common residue expansion:
000110 |  E_D=-int sqrt(g)[R/(24pi²)+(Phi*Phi)/(48pi²)]+E_Sigma,
000111 |  Gamma_D4=int sqrt(g)[|D Phi|²/(16pi²)+(Phi*Phi)²/(16pi²)
000112 |      +R Phi*Phi/(96pi²)+tr F²/(192pi²)+R²/(6M²)+...]+A_Sigma.
000113 | These are terms of the ONE relative constructor, not separate optional
000114 | weights. The reference follows that same constructor and trace. The source
000115 | explicitly calls a0 vacuum-energy bookkeeping, a2 the Einstein/Higgs scale
000116 | part, a4 the movement/quartic/gauge/curvature part, with boundary/eta data
000117 | kept separately in the same table.
000118 | 
000119 | The printed homogeneous Gamma_grav therefore has exactly the resulting
000120 | signs and coefficients, including the -6chi0² multiplication of E_D.
000121 | This review does not change the Higgs sign to force symmetry breaking or
000122 | revive an old convention. The earlier action/argument/sign audits remain.
000123 | Nor does the residue pair include an independently chosen a0 profile or
000124 | turn FP0 into logdet. This is the OLD RESIDUE branch, not automatically
000125 | the current heat-profile/v259 branch.
000126 | 
000127 | The two branches' differences are not a new global failure: they must be
000128 | transported with their own actual operator/argument/trace conventions.
000129 | Neither 'same Gamma' nor 'same KMS' identifies them automatically.
000130 | 
000131 | 4. LATER FIELD TERMS: WHAT IS AND IS NOT TRANSPORTED BY THESE FORMULAS
000132 |