TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-BOUND-REFERENCE-PROOF.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000049 | with actual transported tangent A and a common stable full domain. Let
000050 | theta reverse r, and let the supplied scalar/bundle pullback give H_tau.
000051 | The given pullback intertwines L and transports the supplied Y and MR
000052 | unchanged in this chart; a different internal transport needs its actual map.
000053 | Set p=(H+H_tau)/2, m=(H-H_tau)/2. Thus p is even, m odd, and constant DM
000054 | is even under this BASE pullback. L is real linear, so
000055 |  Phi_even=L(p)+DM, Phi_odd=L(m).
000056 | 
000057 | For the supplied lift tauS=S R_r,
000058 |  tauS D tauS^-1=D0-Z Phi_tau,
000059 |  Dref,S=D0+Z L(m), Drel,S=Z[L(p)+DM].                 (3)
000060 | For the distinct supplied product lift tauT=S gamma_F R_r,
000061 |  tauT D tauT^-1=D0+Z Phi_tau,
000062 |  Dref,T=D0+Z[L(p)+DM], Drel,T=Z L(m).                 (4)
000063 | Neither formula selects the primitive double lift. The product reflection
000064 | in the established RP/state construction is not thereby a geometric deck.
000065 | For nonconstant fields normal derivatives reverse parity: partial_r p is
000066 | odd and partial_r m even; tangential derivative parities remain as before.
000067 | Treat both factors in the derivative contractions, not just field values.
000068 | 
000069 | 4. Whole relative bulk Higgs response with both references included
000070 | Use the standard local geometric PAIR heat response on a theta-invariant
000071 | double, invariant metric/trace, the same Dirac-type connection and covariantly
000072 | constant original DM. Variations vanish in the boundary/seam collar, or use
000073 | a supplied closed double. This fixes boundary variation; it does not select
000074 | an APS extension. No arbitrary nonlocal Padm insertion into a heat trace is
000075 | assumed. Subtract the H=0 background, preserving its absolute Z[0] elsewhere.
000076 | All displayed a2/a4 below are these H-dependent integrated differences.
000077 | 
000078 | Put P=|p|², M=|m|², v=Re(p*m), and
000079 |  QS=P²+2PM+4v², QT=M²+2PM+4v².
000080 | The original96 trace identities, together with reflection integration, give
000081 |  a2,S=-A/(4pi²) int P,
000082 |  a4,S=1/(8pi²) int[A|Dp|²+B QS+4C(P+M)]
000083 |                          +A/(48pi²) int R P;          (5)
000084 |  a2,T=-A/(4pi²) int M,
000085 |  a4,T=1/(8pi²) int[A|Dm|²+B QT+4C M]
000086 |                          +A/(48pi²) int R M.          (6)
000087 | Every formula retains all colors/families/Y/MR and the moving reference.
000088 | In (5) the reference has NO DM, so the Majorana contribution to M remains
000089 | even though M's pure kinetic and pure quartic contributions cancel.
000090 | In (6) the reference DOES have DM, so its Majorana term leaves only M.
000091 | 
000092 | Proof: TrPhi²=Au+TrDM² and TrPhi⁴=Bu²+4Cu+TrDM⁴, valid for all H by
000093 | R028/R031. Tr(dPhi)^2=A|dH|². Subtract the correct reference moments, then
000094 | pair r and-r. All terms proportional to v, Rv, (P+M)v and Re(Dp*Dm)
000095 | are odd and integrate to zero. The remaining quartics are QS and QT.
000096 | This is a reflection-pair identity. The ORIGINAL pointwise difference
000097 | |p+m|⁴-|m|⁴ can be negative (p=-m!=0). Only its even representative QS
000098 | is pointwise nonnegative. No pointwise positivity of the raw density is
000099 | claimed, and positivity of this integrated action is NOT an RP proof.
000100 | 
000101 | At R=0 and fixed positive chi, -6chi²a2,S+a4,S is nonnegative for every
000102 | p,m, with positive even Higgs quadratic and nonnegative C mass on both.
000103 | It cannot repair R029's mass sign or R030's same-geometry ratio: on the
000104 | surviving even field, a2/a4_R remains-12. In the T branch a PURELY EVEN
000105 | H has Dfull=Dref pointwise as operators, so its entire unscaled geometric
000106 | pair response vanishes, not just one heat coefficient. It cannot be
000107 | identified with the displayed nonzero even-H Higgs kinetic/quartic producer.
000108 | This excludes those specific source identifications, not full TFPT or
000109 | effects from the original separate fermion functional or other vertices.
000110 | 
000111 | 5. Execute the original symmetric conformal sandwich, not a new field choice
000112 | Let f=exp(-sigma/2), Dsigma=f D f, with original smooth real sigma. Its
000113 | principal symbol squares to exp(-2sigma)g^-1, hence ghat=exp(2sigma)g.
000114 | For the canonical conformal spinor isometry F:S_g->S_ghat,
000115 |  Dirac_ghat=F exp(-5sigma/2) Dirac_g exp(3sigma/2) F^-1.
000116 | The UNITARY L2 map U:L2_ghat->L2_g is exp(2sigma)F^-1. Consequently
000117 |  U^-1 Dsigma U=Dirac_ghat+gamma5 Phihat,
000118 |  Phihat=exp(-sigma)Phi,
000119 |  Hhat=exp(-sigma)H, DMhat=exp(-sigma)DM.              (7)
000120 | Metric, volume, Higgs and Majorana scale TOGETHER. In particular DMhat
000121 | has derivative -exp(-sigma)DM d sigma. The spin/twisted connection is
000122 | transported as well. This is an actual derived dictionary within the
000123 | supplied spin-Dirac realization, not a derivation of that realization.
000124 | 
000125 | All H variations here hold sigma fixed. Joint sigma backreaction is not
000126 | computed. The unitary operator identity uses the transported common domain;
000127 | local interior coefficient claims do not assume an arbitrary APS domain is
000128 | conformally invariant. For compact interior H variations the coefficient satisfies