TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/JOINT-HIGGS-TWOFACE-SOURCE-CONSTRAINT-20261102.txt
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000015 | 1. Quantitative bound on the same two-face carrier
000016 | -------------------------------------------------
000017 | Let mu be a positive normalized Higgs weight. On ONE common two-face CAR
000018 | space, suppose each conditional one-particle covariance P_H is selfadjoint
000019 | with 0 <= P_H <= 1. Set Pbar = integral P_H dmu and let L be the fixed rank 6
000020 | projector on one color and the six left-quark modes of the second face.
000021 | Let m(H)=Tr(L P_H L). Direct expansion gives the operator identity
000022 | 
000023 |  Pbar-Pbar^2 = integral (P_H-P_H^2) dmu
000024 |                + integral (P_H-Pbar)^2 dmu.
000025 | 
000026 | The first term is nonnegative by 0 <= P_H <= 1, the second is an average of
000027 | operator squares. For X=P_H-Pbar, Hilbert-Schmidt Cauchy-Schwarz gives
000028 | 
000029 |  Tr(L X^2 L) >= (Tr L X L)^2 / 6,
000030 |  Tr L(Pbar-Pbar^2)L >= Var_mu(m)/6.                         (1)
000031 | 
000032 | Using the existing R054 result, additionally assuming its actual constant-H
000033 | color-symmetric, conditional quasifree fermion fibers and the same weights,
000034 | the color-singlet four-word Wick defect is Delta_O=24 Var_mu(m). Hence
000035 | 
000036 |  Tr L(Pbar-Pbar^2)L >= Delta_O/144.                        (2)
000037 | 
000038 | Writing Pbar=[[A,X],[X*,C]], the left side is
000039 | Tr L(C-C^2-X*X)L, with L now on the second face. Thus (2) connects the full
000040 | two-face crossing, the single-face response, and the four-word defect on the
000041 | same carrier. If Var(m)>0, Pbar cannot be a projection. This was already
000042 | known qualitatively; the rank-six lower bound makes the relation quantitative.
000043 | It does not rule out a pure primitive state before H is integrated out.
000044 |