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: 131-158
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000131 |  Y2=Tr3(3Yu*Yu+3Yd*Yd+Ye*Ye+Ynu*Ynu),
000132 |  Y4=Tr3(3(Yu*Yu)²+3(Yd*Yd)²+(Ye*Ye)²+(Ynu*Ynu)²),
000133 |  T=Tr3(Ynu*Ynu MR MR*)=||Ynu MR||_HS²>=0.             (4)
000134 | Here * is adjoint. No family matrices are assumed commuting. A,B>0 for
000135 | nonzero Y, C>=0 because C=||DY DM||_HS².
000136 | 
000137 | The whole original block proof, using the left rotation R028, gives
000138 |  Tr96(Phi²-DM²)=A u,
000139 |  Tr96(Phi⁴-DM⁴)=B u²+4C u,
000140 |  Tr96(dPhi dPhi)=A |dH|².                             (5)
000141 | In radial coordinates Phi=rDY+DM. Every word with an odd count of DY has
000142 | zero trace by the original chirality/slot paths. DM DY DM=0, so the six
000143 | quartic mixed words reduce cyclically to four DY² DM² words, giving4C.
000144 | Polarizing the exact quadratic identity for arbitrary four real H
000145 | coordinates gives Tr(L_a L_b)=A delta_ab, proving the derivative identity
000146 | also for nonconstant H with the supplied fixed Y and constant MR.
000147 | For gauge-covariant derivatives use the actual gauge intertwiner and
000148 | covariantly constant invariant MR. Varying MR would add its own terms.
000149 | 
000150 | The first extra spacetime realization is explicitly the STANDARD product
000151 |  D=-i gamma^mu nabla_mu tensorI96+gamma5 tensorPhi(H),
000152 | with spin4, Euclidean Clifford algebra and a fixed local spin connection.
000153 | It is used as a conditional consumer of the given original finite map,
000154 | not claimed to be a derivation of the primitive-to-geometric spacetime map.
000155 | In a flat chart D²=-partial²+Phi²-i gamma^mu gamma5 partial_muPhi.
000156 | The same curvature convention yields
000157 |  E=-R/4-Phi²+i gamma^mu gamma5 nabla_muPhi.
000158 | The derivative term has spin trace zero, and its square has spin trace