TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-BOUND-REFERENCE-PROOF.txt
Original file SHA-256: 13e79f2be052ca918fd2be8faf8cf6f1669fc3ad535ccf9a3fdff5b7c0b1b3c5
Inclusive original lines: 112-191
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

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
000129 |  a2,H[Dsigma]=-A/(4pi²) int_g exp(2sigma)u,
000130 |  a4,H[Dsigma]=a4,H[D] after integration.              (8)
000131 | To verify the second identity use
000132 |  Rhat=exp(-2sigma)[R-6Delta sigma-6|d sigma|²],
000133 |  dvolhat=exp(4sigma)dvolg.
000134 | The kinetic transforms to A|DH-H d sigma|²/(8pi²). Combining it with
000135 | the transformed Rhat|Hhat|² term gives the OLD kinetic/curvature terms
000136 | minus A div(u grad sigma)/(8pi²). This divergence integrates to zero in
000137 | the stated chart. The quartic and mixed Majorana terms have weights
000138 | exp(4sigma)exp(-4sigma)=1. Mixed derivative trace Tr(dL(Hhat)dDMhat)=0
000139 | by original slot orthogonality. DMhat-only derivative terms are H-independent
000140 | and are retained in the baseline, not declared absent from the whole action.
000141 | 
000142 | If sigma is deck-even, f commutes with each given tau, so the even/odd
000143 | operator split of fDf is f Dref f / f Drel f. But a declared UNSCALED
000144 | Dref in the geometric Z expression is a DIFFERENT subtraction from fDref f.
000145 | Do not move f onto it merely to preserve a preferred answer.
000146 | 
000147 | 6. Unscaled-reference response and simultaneous Einstein/Higgs test
000148 | For that explicit unscaled reference convention set t(x)=exp(2sigma(x)),
000149 | with sigma deck-even. Equations(5)-(6) change at a2 to
000150 |  a2,S,rel,H=-A/(4pi²) int[t P+(t-1)M],
000151 |  a2,T,rel,H=-A/(4pi²) int[t M+(t-1)P].               (9)
000152 | Their integrated a4,H stay(5)-(6), by(8). In particular the odd coefficient
000153 | t-1 need not be positive. The unscaled result is not the previously matched
000154 | reference result. For pure even H, branch S keeps tP; branch T has (t-1)P
000155 | but no integrated H kinetic/quartic. In the T branch it is WRONG to extend
000156 | the unscaled sigma0 whole-pair cancellation to arbitrary unequal metrics.
000157 | 
000158 | For constant sigma, pure even H and the S branch, in the ORIGINAL g/H
000159 | frame the a2_H/a4_RH coefficient ratio becomes -12t. The printed-2 could
000160 | alone suggest t=1/6. Test the SAME pair's Einstein coefficient before using
000161 | that suggestion. With N=Tr_total I>0 on the same spin bundle and ordinary
000162 | positive trace, the two geometric curvature terms give
000163 |  a2,rel,R=-N/(192pi²)(t-1) int_g R.                  (10)
000164 | At t=1/6 this is +5N/(1152pi²) int R, POSITIVE, whereas the named printed
000165 | ED_R is negative. Thus this scale/frame explanation cannot simultaneously
000166 | match the two printed coefficients for ANY positive N. No rank48 or rank2
000167 | identification is needed. At t=1 the same-bundle Einstein coefficient
000168 | cancels entirely. If both reference and full operator are conformally
000169 | transported together, that same pure geometric a2_R cancels for every sigma.
000170 | In the ghat/Hhat frame the unsmeared local Dirac ratio is instead-12, with
000171 | reference and curvature terms explicitly transformed. Never compare only
000172 | the Higgs coefficient while silently changing the gravity frame.
000173 | 
000174 | For variable sigma the pure geometric difference also contains, after
000175 | integration by parts, -N/(192pi²) int[ (t-1)R+6t|d sigma|² ] in this chart.
000176 | It is not obtained by treating a global ED functional as a pointwise scalar.
000177 | A local chi² insertion requires its declared smearing/density prescription.
000178 | 
000179 | 7. Keep the actual Brel and source contract; no blind replacement
000180 | The original operator is Ksigma=f Dgeo f+Brel. Equations(7)-(10) apply to
000181 | the explicitly tested Brel0/matched mass chart, not an unspecified sum.
000182 | Generally U^-1 Ksigma U equals the RHS of(7)+U^-1 Brel U. If the ACTUAL
000183 | zero-order Brel also has the forced form gamma5 Psi, the total coefficient
000184 | is exp(-sigma)Phi+Psi, with derivatives of Psi retained. It cannot be
000185 | counted once in Dgeo and then inserted a second time as Brel by notation.
000186 | Different first-order Brel, symbol, trace or moving domain needs its actual
000187 | producer. A new Psi chosen to repair the coefficients is not authorized.
000188 | The retrieved metrology owner names Brel, its package and its occurrence
000189 | in Ksigma but does not evaluate its H-dependent component there. This is
000190 | a precise source component still to resolve, not a claim the corpus lacks it.
000191 |