TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-INNER-SCALE-PROOF.txt
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Inclusive original lines: 171-198
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000171 | constant-dimensional kernel, and its stated scale operation really is fBf
000172 | with the transported domain. This assumption is not inferred from its index.
000173 | Let S:C^k -> kerB be the given isometric basis map. Then
000174 |  ker(fBf)=f^-1 kerB,
000175 |  G=S* f^-2 S>0, T=f^-1 S G^-1/2,
000176 |  T*T=I, rangeT=ker(fBf).                            (8)
000177 | For an ACTUAL vertex X whose relevant mode vectors are in its domain,
000178 |  T*(fXf)T=G^-1/2 S*XS G^-1/2,
000179 |  T*X T=G^-1/2 S*f^-1Xf^-1 S G^-1/2.                (9)
000180 | No scalar exponent is assumed: G need not be diagonal or scalar if f varies
000181 | across the normal fiber. These are full matrix identities. For q=f² constant
000182 | ON that fiber, G=q^-1I and T=S in the same original Hilbert metric.
000183 | The overlaps then scale by q for fXf, and by1 for X. Mode normalization
000184 | alone does not choose which vertex the original inserts.
000185 | 
000186 | For the given k=2 COMPLEX mode coefficients, h_norm=G^1/2 H_raw, hence
000187 |  d4H_raw=(det_C G)^-1 d4h_norm                     (10)
000188 | when that coefficient measure is flat. This is a DIFFERENT change from(5),
000189 | whose Jacobian is q^4. Do not confuse normalized mode coordinates, the affine
000190 | seed coordinate, and the conformally transformed physical Higgs field qH.
000191 | Use the pushforward for a non-flat original measure. For x-dependent modes
000192 | retain the induced connection T*nabla T and exterior derivative couplings;
000193 | the exact Schur/memory response is not discarded by retaining only zero modes.
000194 | No actual normal metric, profile or overlap insertion is chosen here.
000195 | Isolation of the zero eigenvalue is required for the usual smooth spectral
000196 | projection perturbation theorem, not for the pointwise algebraic kernel
000197 | identity(8). No extra isolation gate is imposed on that identity.
000198 |