TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/REFERENCE-SOURCE-COMPLETION.txt
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000001 | ORIGINAL REFERENCE AND WHOLE SOURCE TRANSPORT
000002 | L20261102G. 2 November 2026. Source-reviewed continuation.
000003 | 
000004 | RESULT AND QUESTION ANSWERED
000005 | 
000006 | The old bound reference is not an extra missing input to the complete
000007 | existing carrier chart. The same full chart carries D and D_ref, the actual
000008 | reflection and charge/adjoint data, and every original unnormalized source
000009 | call on their transported domains. The current complete odd reader then
000010 | reconstructs the reference coefficient in its established product class.
000011 | No second reference state or new measure is needed for this direct route.
000012 | This closes that particular source-representation question. It does not
000013 | choose the physical source from the original principles or prove an
000014 | identity between different historical bare/effective callers.
000015 | 
000016 | 1. USE THE ENTIRE EXISTING FRAME
000017 | 
000018 | On the old product collar write
000019 |  D=N(partial_r+B), B=Z A-S Phi, S=NZ,
000020 |  Phi=L(H)+D_M, {gamma,Phi}=0, [A,gamma]=0.
000021 | For E v=(Pplus v,i Pminus v), the full U=(E,NE) is unitary.
000022 | It retains both original normal summands and the entire96 coefficient.
000023 | It is independent of H and of the base coordinates in this fixed frame.
000024 | Direct action, without [A,Phi]=0, gives
000025 |  U*D U=mathfrakD_h=[[0,-partial_r+h],[partial_r+h,0]],
000026 |  h=A gamma-i gamma Phi,
000027 |  Dom(mathfrakD_h)=U*Dom(D).
000028 | Closed original boundary conditions, including any endpoint coupling, are
000029 | transported. Independent new normal conditions are not imposed.
000030 | 
000031 | 2. THE ACTUAL REFERENCE OPERATION COMMUTES WITH THIS FRAME
000032 | 
000033 | For the original D_ref=(D+tau D tau^-1)/2 and tau_U=U*tau U,
000034 |  U*D_ref U=(mathfrakD_h+tau_U mathfrakD_h tau_U^-1)/2.
000035 | For the actual R032 S and T lifts this is mathfrakD_href with
000036 |  h_ref,S=A gamma-i gamma L(m),
000037 |  h_ref,T=A gamma-i gamma[L(p)+D_M],
000038 |  p=(H+H_tau)/2, m=(H-H_tau)/2.
000039 | The base reflection is retained:
000040 |  tau_S,U=offdiag(gamma,gamma)R_r,
000041 |  tau_T,U=offdiag(I,I)R_r.
000042 | Neither lift has been newly selected as the physical one.
000043 | 
000044 | The existing same-domain functional calculus consequently preserves the
000045 | full pair and its actual eta, including their regulators and zero modes.
000046 | It also preserves the earlier bounded difference D-D_ref as an orderzero
000047 | operator; a frame change cannot give that difference a principal symbol.
000048 | The primitive profile of that difference and the later full geometric pair
000049 | remain distinct actual original calls.
000050 | 
000051 | For field-dependent tau the correct first derivative contains
000052 |  [ (delta tau)tau^-1, tau D tau^-1 ]/2 in addition to the two deltaD terms.
000053 | A fixed U carries this term too. A moving geometric frame would require
000054 | its actual connection derivatives; this result uses the supplied fixed
000055 | product frame, not a new assertion about arbitrary variable metrics.
000056 | 
000057 | 3. APPLY THE EXISTING COMPLETE READER TO THE REFERENCE
000058 | 
000059 | In the established normal-independent product class,
000060 |  C_ref=(I+exp(2ell h_ref))^-1,
000061 |  R_ref=mu_geo Pi_odd^gamma log((I-C_ref)C_ref^-1)
000062 |       =-2ell mu_geo i gamma Phi_ref
000063 |       =2ell mu_geo U_gamma Phi_ref U_gamma*,
000064 |  U_gamma=(I-i gamma)/sqrt2.
000065 | Thus
000066 |  Phi_ref=U_gamma*R_ref U_gamma/(2ell mu_geo),
000067 |  h_ref=A gamma+R_ref/(2ell mu_geo).
000068 | This is an application of R003, with its retained A, ell and mu_geo.
000069 | It neither chooses a value of mu_geo nor equates the raw modular operator
000070 | with Phi. In the normal-variable case retain the original ordered K_ref,
000071 | logodds(C_ref)=-log(K_ref K_ref*), and its polar crossing blocks.
000072 | 
000073 | 4. THE COMPLETE GIVEN SOURCE MOVES WITH ITS REFERENCE
000074 | 
000075 | Extend U by the identity on every other original field coordinate.
000076 | On a compatible finite regulator use psi=U psi_new and its dual source
000077 | map, carrying the actual even field weight and reference factors.
000078 | The paired complex Berezin Jacobian is1. For a genuine Majorana pairing,
000079 | Pf(U^T M U)=det(U)Pf(M), compensated by the transformed Berezin measure
000080 | with its actual orientation. Do not use a one-slot symmetric pairing.
000081 | Therefore, for EACH actually supplied complete original source,
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.
000141 | The actual later4D/IR source map must therefore be retained when invoking
000142 | those readouts. It is not supplied merely by changing coordinates here.
000143 | 
000144 | VERDICT
000145 | 
000146 | Reference operator and direct full-source representation transport are
000147 | closed in this original product/source class. Our earlier limited script
000148 | absence is not a global missing-reference theorem. The original joint
000149 | source needs no extra freely chosen density to be copied to this chart.
000150 | This is not a complete fundamental TFPT solution. It has not constructed
000151 | a previously absent physical joint source or proved its full continuum
000152 | and source-selection hypotheses. The existing original conditional
000153 | S/Z/OS/Gamma architecture, all earlier exclusions in their scope, and
000154 | the concurrent R101-R103 results remain in force. F1 stays in progress.
000155 | 
000156 | EVIDENCE
000157 | 
000158 | PAIR-TYPING.txt: original pair, actual lifts, whole operator derivation.
000159 | FUNCTOR.txt: independent real/adjoint, domains, Berezin, source jets review.
000160 | WEIGHT.txt and WEIGHT-FULL.txt: actual primitive/residue and gauge quotient.
000161 | FULL-FUNCTOR-REVIEW.txt: independent field and current-action comparison.
000162 | Pair control19 and independent full96 control20 are exact coefficient
000163 | controls of the analytic derivation, not full physical validation.
000164 | Original source handles and hashes are attached in those reviews.
000165 | No original paper, Lean theorem or ledger status was changed.