TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/REFERENCE-SOURCE-COMPLETION.txt
Original file SHA-256: 5ddc9c63c5f20f42befba6ae32fc57fdecb61c4b70dd9ab2d7b26fe667380cbb
Inclusive original lines: 82-140
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000082 |  Z_adm^U[J_new]=Z_adm[J],
000083 |  Z_ref^U[0]=Z_ref[0],
000084 |  Z_rel^U[J_new]=Z_rel[J].
000085 | This equality includes the original geometric, Higgs, gauge, fermionic,
000086 | ghost and eta factors whenever they belong to that given call.
000087 | It preserves the null-source factor, rather than resetting it to1.
000088 | Higgs/source jets commute with the fixed U and retain actual reference
000089 | variations. Exact equality at every compatible regulator passes to any
000090 | existing limit of that same family, with no extra limiting estimate.
000091 | If a source is only formal, this is the same formal coefficient identity,
000092 | not a proof of its analytic existence.
000093 | 
000094 | In particular it is unnecessary to first represent Z_rel as an expectation
000095 | in a separately supplied Gibbs reference state. That optional rewrite has
000096 | its own same-basis identity, as correctly stated in WEIGHT.txt. Its extra
000097 | matching condition is not a prerequisite for the direct full-source
000098 | transport just established. There is no global new omega_ref requirement.
000099 | Gauge fixing and ghosts also need not select new physical dynamics when
000100 | they are the original representation of the same gauge quotient.
000101 | 
000102 | The actual full state transports by its word algebra. With original
000103 | reflection, time and operations transported by U, every complete reflected
000104 | word and finite history has exactly the same expectation. The existing
000105 | full-moment OS theorem then gives an isometry of the reconstructed spaces
000106 | and their time evolutions. This uses full moments of Z/Z[0], not connected
000107 | cumulants as moments. It does not identify an independently chosen current
000108 | clock or state with this transported one.
000109 | 
000110 | 5. THE SPECTRAL WEIGHTS ALREADY FIXED BY THE ORIGINALS
000111 | 
000112 | The primitive spectral seed literally reads
000113 |  S_prim=Tr f(D_rel/chi)-Tr f(D_ref/chi)+i pi Delta_eta_Sigma/2.
000114 | Within its actual trace/regulator and declared relative eta contract,
000115 | its spectral ratio is exp(-S_prim). Writing a difference A(D_rel)-A(D_ref)
000116 | additionally requires that this Delta_eta is the difference of those same
000117 | actual eta functionals; it is not inferred merely from the seed notation.
000118 | The later geometric original uses the full pair
000119 |  (D_sigma,geo+B_rel,D_ref).
000120 | In its stated constant-scale residue branch,
000121 |  Gamma_grav=-6chi0^2 E_D+Gamma_D4,
000122 |  R_geo=exp(6chi0^2 E_D-Gamma_D4).
000123 | This common constructor already includes its Einstein, Higgs, gauge and
000124 | curvature responses with their original coefficients. The reference must
000125 | not be subtracted a second time. A relative-only trace stays relative;
000126 | no arbitrary absolute infinite factors are assigned to its members.
000127 | These are exact existing constructors, not an automatic equality between
000128 | primitive action, bare S_adm, full integrated source, and Gamma_ren.
000129 | 
000130 | 6. DECISIVE LIMIT OF THE COMPOSITION
000131 | 
000132 | The full U maps D to mathfrakD_h. It does not map D to the finite
000133 | coefficient Phi alone. The latter loses the original derivatives and
000134 | boundary domains. On an actual constant original coefficient and circle
000135 | mode p=n+alpha, the interior principal operator has
000136 |  mathfrakD_h^2=-partial_r^2+p^2+Phi^2,
000137 | whereas the reconstructed finite coefficient has no p or normal derivative.
000138 | The existing current v259 computes profile moments, and v255 computes
000139 | finite heat coefficients. Their valid geometric expansion contracts remain
000140 | intact. They are not themselves whole operator-trace equalities.