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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000001 | R096. Joint Higgs, original quark and two-face source: conditional identities
000002 | 
000003 | Scope and provenance
000004 | --------------------
000005 | This is a strengthening and connection of the existing R054 color-singlet
000006 | four-word result and the previously observed failure of a pure averaged
000007 | projector (REGIONAL-SOURCE-PARTNER-PROOF.txt, section 8). It is not a second,
000008 | independent physical observation. Use the original v252 full 96 constructor,
000009 | the fixed quarter mode, the given Yukawas and the actual two-face covariance
000010 | of ACTUAL-ONEFORM-COUPLED-SOURCE-RESPONSE.txt. The latter is a fixed-H source.
000011 | The original Z_adm and Z_rel are formal source contracts; the complete
000012 | interacting Higgs pushforward, conditional fermion state, face transport and
000013 | its positivity are not identified by this calculation.
000014 | 
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 | 
000045 | 2. Linear mixed-source identity without positivity
000046 | --------------------------------------------------
000047 | For the same R054 conditional fibers, O=3N^2-sum_cd E_cd E_dc obeys
000048 | <O>_H=24m(H)^2 and <N>_H=3m(H), with the stated color symmetry. Therefore,
000049 | for the same integrable unnormalized weight on both sides,
000050 | 
000051 |  Z[O] = 8 Z[m(H^dagger H) N].                              (3)
000052 | 
000053 | The identity is linear and does not require positivity of the weight.
000054 | It requires the mixed insertion m(H^dagger H)N and the stated conditional
000055 | fiber identities. It cannot be added to the original solver alphabet until
000056 | the original H carrier, source insertion, and conditional fibers are actually
000057 | connected. With normalized positive mu, (3) recovers Delta_O=8 Cov(m,N),
000058 | so the mixed word and the quartic defect express the same relation, not two
000059 | independent measurements. Finite or complex normalization is not presumed.
000060 | 
000061 | 3. A second mixed source word reading the actual crossing block
000062 | ---------------------------------------------------------------
000063 | On the original 36-dimensional quark particle module Q, use the two-face
000064 | cross block X_H=(2 cosh(ell h_H))^-1 at fixed constant H, h_H^2=p^2+Phi_H^2.
000065 | With a shared real face frame, define
000066 | 
000067 |  B_Q = (1/2) sum_{a in Q}(a^dagger_{left,a} a_{right,a}+h.c.).
000068 | 
000069 | The two face labels are not the internal L/R chiralities. Put u=H^dagger H,
000070 | and y_j the six given quark Yukawa singular values. Then
000071 | 
000072 |  g(u)=<B_Q>_H=Tr_Q X_H
000073 |      =6 sum_{j=1}^6 [2 cosh(ell sqrt(p^2+y_j^2 u))]^-1. (4)
000074 | 
000075 | Each of six singular values has multiplicity 3 colors times 2 chiralities.
000076 | For ell>0, p>0 and at least one y_j nonzero, g is strictly decreasing on
000077 | u>=0. For f(u)=u/(1+u), any positive normalized mu with nonconstant u gives
000078 | 
000079 |  Cov_mu(f,B_Q) = (1/2) integral integral (f(u)-f(v))
000080 |                             (g(u)-g(v)) dmu(u)dmu(v) < 0.  (5)
000081 | 
000082 | If mu assigns masses m_A,m_B to [0,a] and [b,infinity), 0<=a<b, then
000083 | Cov_mu(f,B_Q) <= -m_A m_B [f(b)-f(a)][g(a)-g(b)]. Both f and B_Q are
000084 | bounded for this finite carrier. A common unitary frame transformation on
000085 | both faces leaves B_Q invariant. Independent face gauge transformations
000086 | require the actual transported identification; no Wilson bridge is assumed.
000087 | 
000088 | The real face chart, inclusion of the conditional H fibers in the same
000089 | original source, all further vertices, reference/eta/Z0, H-dependent weights,
000090 | and comparison with the primitive P1 source remain to be established. In
000091 | particular no actual Higgs probability or nonzero H fluctuation follows here.
000092 | The finite constant-H slice alone says nothing about generic H(x) measures.
000093 | 
000094 | 4. Reproducible controls
000095 | ------------------------
000096 | Run python3 -B check_joint_higgs_twoface_response_20261102.py. It evaluates
000097 | the full original 96 matrix and the 192-dimensional two-face projector for
000098 | four diagnostic Higgs values, including two weak directions at equal u;
000099 | checks (4), monotonicity and (1); and independently checks equality of the
000100 | rank-six bound in a rational abstract covariance example. The seed252 Yukawas,
000101 | ell=1 and two-point weights are diagnostic inputs, not a physical fit or
000102 | derived TFPT measure. The old exact Fock proof remains the evidence for the
000103 | R054 four-word formula; the new script does not re-prove it from CAR matrices.