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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000001 | ORIGINAL VARIABLE HIGGS, BOUND REFERENCE AND CONFORMAL SOURCE MAP
000002 | 31 October2026. R032/R033. Conditional original-source calculations.
000003 | No new Y, lift, spacetime, action, reference prescription or ledger promotion.
000004 | 
000005 | 1. Reused originals and the genuinely new question
000006 | Keep R026-R031, actual full96 v252 finite coefficients and entire source.
000007 | The old reviewed DECK-REFERENCE-AND-ACTION.txt already proves the two full
000008 | lift identities below, including the BASE pullback. Do not repeat their
000009 | constant-field classification or the old bounded-Drel trace obstruction.
000010 | The new step is the whole VARIABLE Higgs/derivative/reference response,
000011 | and the original Dsigma sandwich with the same metric and field map.
000012 | The geometric source uses Dgeo=Dref+Drel=D, and a separate displayed
000013 | Ksigma=Dsigma,geo+Brel. Its local anchor is Dgeo, NOT Padm Drel Padm.
000014 | Primitive f(Drel)-f(Dref) and geometric f(Dfull)-f(Dref) are different
000015 | operator contracts. No claim here replaces one by the other.
000016 | 
000017 | 2. A fixed Clifford connection forces the Higgs Clifford factor
000018 | On the given local complex spin4 module S tensorF, Clifford acts by
000019 | gamma^mu tensorI. Assume D(H)=Dconn+V0+W(H), with fixed Clifford connection,
000020 | H only in order zero, and no change of first-order coefficients in D(H)^2.
000021 | The first-order change is -i{gamma^mu,W(H)}nabla_mu. It vanishes iff
000022 |  {gamma^mu,W}=0 for every mu.                         (1)
000023 | Since gamma5 anticommutes with every gamma, gamma5 W then COMMUTES with
000024 | the irreducible complex Clifford action. Its commutant is Ispin tensorEndF.
000025 | Thus W=gamma5 tensorL(H), uniquely. An H-independent V0 is not constrained
000026 | by this derivative statement; use the already given V0=gamma5 tensorDM
000027 | only when that is the actual original baseline. An H-dependent Bochner
000028 | connection, different Clifford action, or different anchor changes (1).
000029 | 
000030 | This removes an independent gamma-factor choice WITHIN this fixed-connection
000031 | contract; it does not derive the original4D manifold, module or connection.
000032 | In the supplied collar choose gamma^r=PauliY tensorI2 and the three tangent
000033 | gammas=PauliX tensorPauli_i. Their volume is PauliZ tensorI2. Hence the
000034 | existing normal Z is precisely gamma5 in that chart, not a newly fitted
000035 | mass matrix. S=PauliX tensorI preserves the tangent gammas and reverses
000036 | normal gamma and gamma5. Actual supplied internal gamma_F and J_F stay.
000037 | 
000038 | For the whole supplied Phi(H)=L(H)+DM one therefore MUST retain
000039 |  E=-R/4-Phi²+i gamma^mu gamma5 nabla_muPhi.             (2)
000040 | The derivative term is forced by squaring this D, not an extra interaction.
000041 | R031 gives the full moments A=Tr96 DY², B=Tr96 DY⁴,
000042 | C=Tr96 DY² DM²>=0 and Tr(L(X)L(Y))=A Re(X*Y).
000043 | Only TRACE identities are invariant. In general L(X)^2 is NOT |X|²DY²
000044 | as a matrix when up/down Yukawas differ. The actual original constructor
000045 | supplies a strict witness against that mistaken Gemini matrix identity.
000046 | 
000047 | 3. Both actual reference jets for all Higgs fields
000048 | Use the already supplied collar expression D=D0+Z Phi, D0=N partial_r+SA,
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