TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DYNAMICAL-HIGGS-SOURCE-PROOF.txt
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000109 | truncation or a substitute boson measure. The complex H2 coefficient is also
000110 | checked directly in the existing joint_source exterior algebra.
000111 | 
000112 | 4. Fermion-first evaluation, determinant and mixed correlations
000113 | 
000114 | The full source can also be evaluated by integrating fermions first. For a
000115 | fixed background H in an actually specified bilinear charged chart it gives
000116 | the unnormalized Berezin polynomial P_H[eta,bar eta], including det K[H].
000117 | On the invertible chart only,
000118 |  P_H=det K[H] exp(bar eta K[H]^-1 eta),
000119 | with the original Berezin orientation. Coefficients are complementary minors
000120 | and extend polynomially through singular K; do not discard zero-determinant
000121 | fibers that still carry nonzero sourced coefficients. Actual Majorana/chiral
000122 | blocks require the actual skew pairing and Pfaffian instead.
000123 | 
000124 | Reuse the established same-collar skew pairing
000125 |  [[0,-Lplus^t],[Lplus,0]]
000126 | whose Pfaffian is (-1)^(n(n+1)/2)det Lplus at finite n. The orientation fixes
000127 | this sign. One must not integrate all96 doubled internal labels again as
000128 | independent physical complex fermions, square the determinant, replace a
000129 | Pfaffian by abs(det) or infer the physical chiral restriction from96 alone.
000130 | This kinetic/KO distinction was already established and is reused. Primary
000131 | context: Chamseddine,Connes,Marcolli hep-th/0610241, section4,especially4.4.1.
000132 | 
000133 | When the actual fixed-H fibers are Gaussian, the two- and four-point source
000134 | coefficients of their common integral are weighted averages of C_H and of
000135 | C_H,ij C_H,kl-C_H,il C_H,kj, respectively. Subtracting the Wick expression
000136 | of the averaged C gives the difference of two weighted covariances. Use the
000137 | actual det/Pf-weighted functional, not the bare boson measure, and only divide
000138 | where its scalar is nonzero. Signed/complex weights are not probabilities.
000139 | Variation of C_H generically yields a non-Gaussian marginal, but neither H