TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/HIGGS-MIXTURE-WICK-PROOF.txt
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000001 | R054. SHARED ORIGINAL HIGGS FLUCTUATIONS HAVE AN INVARIANT FOUR-FIELD WITNESS
000002 | 31 October 2026. Established scoped theorem, not a physical source selection.
000003 | 
000004 | 1. Decision and actual prerequisites
000005 | Compare the existing whole H source with a reconstruction retaining ONLY
000006 | its averaged two-point covariance. No Yukawa or Majorana is refitted.
000007 | Use the original v252 coefficient Phi(H)=L(H)+DM, its fixed left grading
000008 | gamma=+1, and the already supplied common-quarter constant product chart
000009 |   h_H=p gamma-i gamma Phi(H), p=1/4, ell>0,
000010 |   C_H=(I+exp(2ell h_H))^-1.
000011 | This is the SECOND FACE contraction, not the whole Cauchy projection G.
000012 | The finite/regulator CAR source is the existing charged source chart.
000013 | We use its quark particle subalgebra, not a new integration over96 physical
000014 | fermions. The independent64-state Fock control below is only algebra QA.
000015 | 
000016 | Hypothesis on the H integration: it actually represents a positive
000017 | normalized mixture of these CONDITIONAL quasifree states. Write
000018 |   omega(X)=int omega_H(X) dmu(H), dmu>=0, int dmu=1.
000019 | For an unnormalized original fiber, mu must include its actual Z_F(H),
000020 | bosonic weight, reference, eta and every other factor before normalization
000021 | by Z[0]. Positivity concerns the complete weight, not each named factor.
000022 | This is a hypothesis to verify at the original source, not a chosen measure.
000023 | Quadratic dependence on FERMIONS gives conditional quasifreeness;
000024 | nonlinear H dependence alone does not destroy it. Additional fermion
000025 | vertices can destroy it and require the actual conditional four-point term.
000026 | 
000027 | 2. Whole four-real H fiber, original three colors and arbitrary given Y
000028 | Put u=H*H and q(H)=[[bar h2,h1],[-bar h1,h2]]. Then q*q=uI2.
000029 | At u>0, U=q/sqrt(u) is unitary; at u=0 use continuous extension.
000030 | The original quark LR block in each color is
000031 |   M(H)=(q(H) tensor I3fam) diag(Yu,Yd).
000032 | Thus Phi_quark=[[0,M],[M*,0]], h_H^2=p^2+Phi(H)^2, and
000033 |   (h_H^2)_LL=p^2+MM*.
000034 | The six eigenvalues of MM* are u times the squared singular values y_aj^2
000035 | of the actual Yu,Yd, regardless of their noncommutation or weak H direction.
000036 | The Majorana lies in the original neutrino block. It is retained in the
000037 | whole coefficient and the H weight, not discarded because this witness
000038 | is read in the quark subalgebra.
000039 | 
000040 | Set g(E)=tanh(ell E)/E. Functional calculus gives
000041 |   C_H=I/2-h_H g(sqrt(h_H^2))/2.
000042 | For A_H the6x6 left-quark covariance in any one color,
000043 |   m(u)=Tr A_H
000044 |       =3-(p/2) sum_(a=u,d;j=1..3) g(sqrt(p^2+y_aj^2 u)).       (1)
000045 | All three color blocks are identical A_H and cross-color two-point blocks
000046 | vanish. This uses the ORIGINAL scalar pI background; an arbitrary nonzero
000047 | weak connection is not covered by(1). Full H-direction dependence of A_H
000048 | is retained even though its trace is radial.
000049 | 
000050 | For x>0, b(x)=tanh x-x sech^2 x has b(0)=0 and
000051 |   b'(x)=2x sech^2 x tanh x>0.
000052 | Therefore g'(E)=-b(ell E)/E^2<0. Since p>0 and at least one quark y is
000053 | nonzero, m(u) is strictly increasing for u>=0. It is bounded, so its
000054 | variance exists for ANY positive probability mu on finite H values,
000055 | even when an unbounded H moment has not been assumed integrable.
000056 | Consequently Var_mu(m)>0 iff u is not mu-almost-surely constant.
000057 | No discretization, radial potential, preferred H direction or Y scan.
000058 | 
000059 | 3. A pure quartic witness independent of the constant internal frame
000060 | Let a_ic be the six left modes i per color c, with the actual CAR adjoint.
000061 | Use C_ij=omega(a_j* a_i); the anomalous pairs vanish conditional on H and
000062 | hence in their mixture. Define bounded operators on this finite subalgebra
000063 |   E_cd=sum_i a_ic* a_id, N=sum_c E_cc,
000064 |   Q_col=sum_cd E_cd E_dc, O=3N^2-Q_col.                      (2)
000065 | The E matrix changes by conjugation under a CONSTANT color U3 frame.
000066 | N and Q_col are invariant; a common unitary on the six internal modes
000067 | also preserves every E_cd. In particular O does not require an H-dependent
000068 | choice of weak direction. This is internal-frame invariance at the fixed
000069 | circle mode, not a proof of a local3+1D gauge observable or spacetime locality.
000070 | 
000071 | Normal ordering with the CAR gives
000072 |   N^2=N+sum_ab a_a* a_b* a_b a_a,
000073 |   Q_col=3N-sum_cdij a_ic* a_jd* a_id a_jc.
000074 | The quadratic parts cancel in(2); O is a genuine quartic polynomial.
000075 | It is not an extra term added to the action.
000076 | 
000077 | For ANY conditional gauge-invariant quasifree A_H tensor I3, put
000078 | tau_H=Tr(A_H-A_H^2). Wick gives the general quadratic-word identity
000079 |   omega_H(E_cd E_ef)=delta_cd delta_ef m_H^2
000080 |                      +delta_cf delta_de tau_H.
000081 | Summing actual indices, with all fermionic exchange signs retained,
000082 |   omega_H(N^2)=9m_H^2+3tau_H,
000083 |   omega_H(Q_col)=3m_H^2+9tau_H,
000084 |   omega_H(O)=24m_H^2.                                    (3)
000085 | 
000086 | 4. Exact whole-class discriminator
000087 | Let Cbar=int C_H dmu, and let omega_G be the quasifree state with that SAME
000088 | full two-point data, including zero anomalous pairs on this subalgebra.
000089 | Its left covariance is Abar tensor I3 and Tr Abar=int m_H dmu.
000090 | Equations(2)-(3), without any simultaneous diagonalization in H, imply
000091 |   omega(O)-omega_G(O)=24 Var_mu(m(u)).                      (4)
000092 | Thus if the actual positive H mixture has nonconstant u and a nonzero
000093 | quark Yukawa, this invariant four-field difference is STRICTLY positive.
000094 | The entire positive-weight class is decided by the proof, not by40 checks.
000095 | 
000096 | The covariance logarithm log((I-Cbar)Cbar^-1) remains an exact two-point
000097 | reader; its associated quasifree state fails(4). It cannot be the full
000098 | H-averaged source. In finite faithful restrictions, -log rho of the actual
000099 | mixed state contains terms beyond quadratic in the fermion operators.
000100 | This does NOT assert that its only missing term is quartic or determine a
000101 | physical coupling coefficient. Allowing arbitrary anomalous pairs in a
000102 | replacement changes other two-point data; it does not preserve this source.
000103 | 
000104 | 5. Limits and connection to the existing whole theory
000105 | At constant u, p=0, ell=0, or all quarkY=0 the strict witness can vanish;
000106 | vanishing is not a general proof of quasifreeness. A complex or signed
000107 | weight does not support the positive-variance conclusion. Additional
000108 | conditional interactions require their conditional four-point response,
000109 | which can contribute to or cancel the displayed defect. Arbitrary local
000110 | H(x), gauge fields and continuum measures are not derived by this theorem.
000111 | 
000112 | In particular a primitive quasifree C_Sigma BEFORE H integration remains
000113 | possible. The current original modular route must specify the order:
000114 | conditional source and carrier map first, whole H integration after, or
000115 | a genuine many-body reader preserving(4). Nothing here identifies the
000116 | given conditional H family with that primitive C_Sigma or derives mu/P_F.
000117 | 
000118 | R026's exact full-H cumulant functional is reused. R054 adds the explicit
000119 | invariant quark four-field witness and its strict all-positive-mixture
000120 | test against the covariance-only current reader, not another discovery
000121 | that H integration has cumulants. Earlier conformal-source averaging
000122 | preserved every transported conditional word because its covariance was
000123 | unchanged. Here the original Higgs radial dependence changes(1), so that
000124 | old zero-defect argument does not apply. The changed premise is explicit.
000125 | 
000126 | Whole96, reference/Eta/Z[0]/H0, original source/state/time/RP, word memory,
000127 | E8/clocks and existing Alpha/Yukawa/Flavor remain in their stated scopes.
000128 | No additional normal metric/gap, profile, counterterm, Y or scale choice.
000129 | No physical scattering amplitude, continuum theorem or ledger promotion.
000130 | 
000131 | 6. Independent checks and primary literature
000132 | check_higgs_mixture_wick.py reuses original v252 seed252 coefficient inputs
000133 | and actual real/selfadjoint Higgs one-form, all96 slots and Majorana intact.
000134 | Direct C_H matches(1) at four H controls including real and complex equal-u
000135 | directions. Its actual left grading is +1, fixing the sign in(1).
000136 | An independent rational6-mode/64-state Fock computation verifies CAR,
000137 | normal ordering, constant-frame invariance and(3) without a Wick evaluator.
000138 | Two exact rational states have identical full averaged two-point data to
000139 | their Gaussian replacement, but O differs by4/27=24*(1/162).
000140 | These are diagnostic inputs, not physical H weights or predictions.
000141 | Forty controls pass. An initial unsimplified complex rational expression
000142 | comparison was replaced by exact residual simplification. No new Lean.
000143 | 
000144 | Primary theory support, not a TFPT origin proof:
000145 | I. Peschel, cond-mat/0212631, equations3 and12 and the qualification below12
000146 | explicitly separate the free covariance reconstruction from general
000147 | higher expectations: https://arxiv.org/pdf/cond-mat/0212631
000148 | J. K. Pachos and C. Vlachou, arXiv:2203.04870, Wick-violation diagnostics:
000149 | https://arxiv.org/pdf/2203.04870
000150 | The calculations(1)-(4) above are our scoped original-consumer application.