TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/WEIGHT.txt
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000024 | itself that state identity.
000025 | 
000026 | 1. THE PRIMITIVE SPECTRAL PART CAN BE RECONSTRUCTED EXACTLY
000027 | 
000028 | Boundary001:302–315 gives, literally,
000029 |  S_prim=Tr f(D_rel/chi)-Tr f(D_ref/chi)+i pi Deltaeta/2.
000030 | At a compatible finite regulator, with the original eta branch, define the
000031 | operator functional already displayed there
000032 |  A_f,chi(T)=Tr f(T/chi)+i pi eta_Sigma(T)/2.
000033 | Where Deltaeta is the declared difference of the same eta functionals,
000034 |  S_prim=A_f,chi(D_rel)-A_f,chi(D_ref).                   (1)
000035 | No new action or profile is selected: this is direct reconstruction from
000036 | the two printed terms. It is unique up to a common additive functional
000037 | that cancels identically between these same arguments.
000038 | 
000039 | The first argument in(1) is the ORIGINAL D_rel. It must not be silently
000040 | changed to an uncompressed whole D or D_geo. The other agents are checking
000041 | the original pair/odd-operator dictionary. Equation(1) does not settle
000042 | that dictionary by renaming the operator.
000043 | 
000044 | Consequently the relative primitive spectral weight is already forced:
000045 |  R_spec=exp[-A_f,chi(D_rel)+A_f,chi(D_ref)]=exp(-S_prim). (2)
000046 | If the actual reference source is the Gibbs/source basis of the second
000047 | functional, (2) is inserted into THAT normalized reference functional.
000048 | There is then no additional division by the same reference partition
000049 | function. In that representation its normalization has already entered
000050 | omega_ref. Conversely, under the unweighted original integration basis,
000051 | one uses the actual absolute numerator weight and the original Z_ref
000052 | quotient. These are two correct representations once their basis is matched.
000053 | 
000054 | This mathematical identity is valid before any assertion about infinite
000055 | trace convergence. On a source that only supplies a relative trace, one
000056 | must retain that relative functional rather than assign arbitrary separate
000057 | absolute infinities to its two terms. The actual regulator/APS/reference
000058 | contract remains attached.
000059 | 
000060 | 2. THE OLDER GEOMETRIC GAMMA SOURCE GIVES A STRONGER COMMON CONSTRUCTOR
000061 | 
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