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: 120-145
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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000120 | At a pole zeta=r/s+f+..., FP becomes f+2sigma r. If the reference is fixed,
000121 | its FP is subtracted linearly, without this scale shift. If B is fixed,
000122 | Dsigma+B is NOT exp(-sigma)(D+B). Neither dimensional correctness of the
000123 | extra chi² factor nor physical double counting follows from notation.
000124 | Retain the actual scale/coordinate/reference contract before deciding it.
000125 | 
000126 | 4. Full supplied96 Higgs family through the conditional product consumer
000127 | Use EXACT original v252 full_D and the R026/R028 one-form map, all supplied
000128 | complex Yu,Yd,Ye,Ynu and symmetric complex MR. Let DY=full_D(Y,0),
000129 | DM=full_D(0,MR), u=|H|². Define internal traces
000130 |  A=Tr96 DY²=4Y2, B=Tr96 DY⁴=4Y4, C=Tr96(DY² DM²)=2T,
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