TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/WHOLE-SCALE-ENERGY-RECONSTRUCTION-PROOF.txt
Original file SHA-256: 9693f9051ae803cb2e750035b934cd99d22044efbd5430cb592eb7ad47b2a96f
Inclusive original lines: 131-191
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000131 | Then deltaL=-(hL+Lh)/2-KhK. Duhamel in a trace-class convergence region,
000132 | followed by the SAME justified meromorphic continuation, gives
000133 |  delta zeta_L(s)=2s Z_h(s), Z_h(s)=Tr(h L^-s).       (8)
000134 | A zero-excluded convention inserts I-P0 throughout; constant nullity and
000135 | its varying projector are retained. No zero is lifted. For simple poles,
000136 |  delta Res1 zeta=2 Res1 Z_h,
000137 |  delta FP0 zeta=2 Res0 Z_h,
000138 |  delta logdet L=-2 FP0 Z_h                          (9)
000139 | when the usual zeta determinant is defined. More generally keep Res[2sZ_h]
000140 | if higher poles are present. For smooth elliptic local inner data Z_h is
000141 | regular at0, so delta FP0=0, while delta logdet=-2[a4(h,L)-Tr(hP0)].
000142 | An anomaly in logdet is not a correction to the original FP0 functional.
000143 | This is consistent with the conformal invariance already proved in R033.
000144 | 
000145 | At sigma0, the full existing Spin4x96 product has the LOCAL Higgs density
000146 |  a4H(x)=(16pi^2)^-1[2A|DH|^2+2Bu^2+8Cu+(A/3)Ru-(2A/3)Delta u],
000147 |  u=|H|^2, Delta=nabla^mu nabla_mu.                  (10)
000148 | The last divergence term follows from60 DeltaE/360 with tr_spin E_H=-4Au.
000149 | It vanishes in the integrated h=1 expression R031, but MUST be retained
000150 | with local h. Thus a4H(h)=int h a4H(x) is not obtained by multiplying h
000151 | into a formula after dropping divergences. a2H(h)=-A int h u/(4pi^2).
000152 | These are the existing A,B,C,baselineH0=0; another H0 uses R034 transport.
000153 | Local gauge/gravity terms and all H-independent Majorana data remain.
000154 | 
000155 | 6. Actual exterior Brel and actual relative trace are not omitted
000156 | For the literal K=fDgeo f+Bout define K0sigma=Dgeo+f^-1 Bout f^-1.
000157 | Equations(1)-(2) apply with this K0sigma. Section2's fixed Cauchy identity
000158 | requires K0sigma to be independent of sigma; it is NOT automatic here.
000159 | At fixed Dgeo, put Wext=deltaBout+{h,Bout}/2. Then exactly
000160 |  deltaK=-{h,K}/2+Wext,
000161 |  delta zeta=2s[Z_h(s)-Xi_W(s)],
000162 |  Xi_W(s)=Tr(Wext K L^(-s-1)).                       (11)
000163 | If Dgeo also changes, add f(deltaDgeo)f to Wext. This is the original full
000164 | response, not a newly chosen counterterm. Wext=0 is SUFFICIENT for the
000165 | pure flow, not necessary for a vanishing particular residue. A nonzero
000166 | Wext can have a zero residue. Do not infer an if-and-only-if assertion.
000167 | For the bound reference subtract its OWN variation; add i pi deltaEta/2
000168 | to the FP0 response and keep moving domains/trace weights if present.
000169 | 
000170 | Cyclicity cannot be assumed for an arbitrary Tr(W dot). For a fixed weight
000171 | commuting with K and the variation, the corresponding trace proof works.
000172 | TOTAL chiral Pc does NOT commute with K: PcK=KQc. If K and deltaK remain
000173 | odd in the same grading-stable domain, R039's full positive pairing instead
000174 | gives delta zeta_Pc=(1/2)delta zeta, hence s[Z_h-Xi_W]. It is generally NOT
000175 | 2sTr(Pc hL^-s). Local chiral index density cannot be deleted. The original
000176 | Tr_rel module, weights, reference and continuation still decide which call
000177 | is physical. Bulk eta symmetry is not a proof about boundary eta.
000178 | 
000179 | 7. Consequence for the original task
000180 | The whole original scale is now transported into an explicit weighted
000181 | energy problem, with the correct full spectral reader. A complete normal
000182 | profile is uniquely recoverable from the specified existing source's first
000183 | energy response. Mere normalized zero-energy boundary words do not provide
000184 | that datum. Absolute source factors and the ORIGINAL geometric measure
000185 | must stay in the common process; they are not fixed by the Cauchy equality.
000186 | Next use the original relative/B_Ea insertion and its actual scale response
000187 | to evaluate Wext, along with the declared reference/Tr_rel and full energy
000188 | response. Do not start another free scale/Y/lift/projector search. Preserve
000189 | whole words/memory,Z[0],state/time/RP,E8/clocks and solved Flavor/Alpha.
000190 | F1 remains open. No physical normal metric,energy-data owner,3+1D origin,
000191 | Yukawa values or equality Ladm=Ksigma^2 was produced by these identities.