TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-HEAT-PROOF.txt
Original file SHA-256: 5904d71d3b9cc978ef5025b86a43931db227d7ee9e87d894384f844b10f1d805
Inclusive original lines: 1-110
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.

000001 | ORIGINAL HIGGS HEAT CONSUMER AND SAME-SOURCE COEFFICIENT COMPATIBILITY
000002 | 31 October2026. R030/R031. Scoped source audit and conditional consumer.
000003 | No new physical parameters, new measure, paper edits or ledger promotion.
000004 | 
000005 | 1. Actual question, original sources, and the first extra identification
000006 | Continue R029 at the printed producer, not at a new Yukawa search. The v45
000007 | metrology001 definition108-146 states ED=Res_1 Zrel and G4=FP_0 Zrel, with
000008 | Zrel=Tr_rel(|Dsigma,geo+Brel|^-2s-|Dref|^-2s)+i pi DeltaetaSigma/2.
000009 | Metrology002:64-74 displays Erel=-R I48/4-u P2+gammaF/2, u=Phi*Phi.
000010 | Metrology002:143-224 prints ED_H=-u/(48pi²) and G4_RH=Ru/(96pi²), as
000011 | well as kinetic,quartic,Einstein and -6chi² ED+G4 formulas already in R029.
000012 | The precise compatibility question is whether these two coefficients can
000013 | come from this displayed endomorphism through one ordinary local heat trace.
000014 | 
000015 | For this test state the convention explicitly: positive scalar curvature R
000016 | on spheres, Laplace-type P=-(nabla²+E), same local linear trace for both
000017 | residues, fixed common smooth geometry, no extra H-dependent connection or
000018 | curvature endomorphism hidden in E or Omega. The bulk heat formula is
000019 |  a2=(16pi²)^-1 Tr(E+R/6),
000020 |  a4=(16pi²)^-1 Tr(E²/2+RE/6+Omega²/12+geometric terms
000021 |                   +total derivatives).                              (1)
000022 | This is the specific local heat identification made for the displayed Erel.
000023 | If Tr_rel includes a nonlocal compression, moving projector, different
000024 | principal symbol or differently normalized residue definition, its actual
000025 | local asymptotics must be supplied before making the same identification.
000026 | The test does not prove those additional maps absent from the whole theory.
000027 | 
000028 | 2. Universal linear mass/curvature compatibility, independent of rank
000029 | Set E=-R I/4-X(H) and retain the terms linear in X. Then
000030 |  a2_H=-(16pi²)^-1 Tr X,
000031 |  a4_RH=+(192pi²)^-1 R Tr X.                           (2)
000032 | The second identity follows from (R/4)X in E²/2 minus RX/6.
000033 | Thus, for X=uP2, or for ANY X with H² coefficient X2,
000034 |  coeff_a2(u)=-12 coeff_a4(Ru).                         (3)
000035 | Idempotence, rank2, positivity and simultaneous diagonalization are not
000036 | needed. A common linear weighted trace obeys (3), even a signed trace.
000037 | Subtracting two same-geometry Dirac-type producers with the same -R/4
000038 | curvature term also preserves (3), including both H-dependent reference
000039 | jets. This last statement requires BOTH actual operators to be of that
000040 | type; the primitive even part is not automatically such a Dirac operator.
000041 | Positive common field rescaling and common sheet/trace weights cancel in
000042 | the ratio. Reversing the curvature convention reverses its sign, not the
000043 | factor6 mismatch when source formulas are transformed consistently.
000044 | 
000045 | The PRINTED ratio is (-1/48)/(1/96)=-2, rather than -12. Keeping the printed
000046 | G4_RH forces ED_H=-u/(8pi²); keeping ED_H forces G4_RH=Ru/(576pi²).
000047 | These are ALTERNATIVE necessary repairs of the given local identification,
000048 | not permission to change one original formula or to choose either repair.
000049 | An extra original Ru contribution or a different trace/field/operator map
000050 | could change the comparison, but has to be calculated from that source.
000051 | Changing only a common normalization cannot do so.
000052 | 
000053 | If one ADDITIONALLY reads I48 by the ordinary48-dimensional trace and P2
000054 | as a rank2 projector in that same bundle, the raw result is
000055 |  a2=-R/(4pi²)-u/(8pi²),
000056 |  a4_H4=u²/(16pi²), a4_RH=Ru/(96pi²).
000057 | Both printed ED coefficients are then exactly a2/6, while the two stated
000058 | a4 coefficients are unchanged. This localizes a possible normalization
000059 | translation: ED_print=a2/6. It contradicts the unmodified definition
000060 | ED=Res_1 Zrel=a2 for that same ordinary trace; it does not establish that
000061 | the actual compressed original trace has these ranks. Nor does it derive
000062 | a new residue prescription. With this conditional translation the printed
000063 | -6chi²ED becomes -chi²a2, retaining the separate mass-sign problem.
000064 | 
000065 | The flat projector subtest agrees: exp(-t uP2)=I-P2+exp(-tu)P2 gives
000066 |  a2_H=-k u/(16pi²), a4_H4=k u²/(32pi²), k=Tr P2.
000067 | Their coefficient ratio is1/2; the printed quartic/mass ratio is3, again
000068 | factor6. This flat subtest needs the projector identification, whereas (3)
000069 | does not. In general full96 Phi² is NOT u times an idempotent projector.
000070 | Do not infer a rank8 geometry, author intention, or an unexplained division
000071 | by6 from this numerical pattern. The I48, compact P2 and full96 internal
000072 | spaces are distinct until an explicit intertwining map identifies them.
000073 | 
000074 | There is also a derivative-symbol test. If the displayed Erel is COMPLETE
000075 | and the connection is H-independent, a4 has no surviving integrated Higgs
000076 | kinetic term. E² is algebraic in u; Omega² is then H-independent. The only
000077 | derivative term is Tr Delta E/6, proportional to Delta u. For interior
000078 | changes its integral vanishes. Although Delta u contains2|dH|² pointwise,
000079 | the companion2 Re(H*Delta H) cancels it after integration by parts. Thus
000080 | one cannot read the printed nonzero |DH|² coefficient from this complete
000081 | Erel alone. A true Dirac square has the first-derivative term displayed
000082 | in section4; the original connection/endomorphism map must supply it or
000083 | another actually derived kinetic producer. This is an incompleteness of
000084 | that displayed producer under the stated convention, not proof that the
000085 | whole original theory lacks its already printed Higgs kinetic action.
000086 | 
000087 | 3. Boundary, reference, phase and scale: what can actually repair the jet
000088 | On a fixed elliptic domain with the standard heat/Mellin expansion and
000089 | constant nullity, Res_1 zetaP=a2 and zetaP(0)=a4-dimkerP. This statement at
000090 | zero assumes regularity there. Log heat terms/poles require their own FP
000091 | formula and cannot be discarded. FP_0 zeta is not -zetaPrime(0).
000092 | For a compactly supported interior H change, zero near the boundary and
000093 | seam, with the boundary operator and selfadjoint APS domain fixed, boundary
000094 | symbol data and intrinsic etaSigma are unchanged. Local variations of the
000095 | bulk coefficients are the interior variations in (1). A fixed reference
000096 | has zero H derivatives. On this stated branch none of these fixed terms
000097 | supplies the missing real H²/RH² coefficients. A constant nullity term has
000098 | zero derivatives; crossing a zero mode is outside this derivative chart.
000099 | 
000100 | The s-independent term i pi DeltaetaSigma/2 has ZERO Res_1 even if its
000101 | coefficient depends on H. Its FP is that same imaginary term. It cannot
000102 | change the real local ED coefficient in this real selfadjoint chart.
000103 | This is not a claim about phase effects after the whole path integration,
000104 | nor about a complex field continuation not yet established at the source.
000105 | 
000106 | The original reference is bound, not freely selected:
000107 |  Dref=(D+Rdeck D Rdeck)/2, Drel=(D-Rdeck D Rdeck)/2.
000108 | Consequently delta Dref=(V+Rdeck V Rdeck)/2. It vanishes for a genuinely
000109 | deck-odd operator insertion, but not for every scalar named seam-even or
000110 | odd. The actual H lift must determine the operator parity, including its