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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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.