TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DYNAMICAL-HIGGS-SOURCE-PROOF.txt Original file SHA-256: eda5bb664b1fb20370e59b39dbd5e3cb04d79864a8d1d0a0eb6c791cdaf98f49 Inclusive original lines: 34-139 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000034 | Work first at a finite regulated field/source algebra. Keep all remaining 000035 | original fields A fixed only as CONDITIONAL arguments of the SAME Z_adm. 000036 | Let Z_H[J|A] denote its actual H fiber integral, with its original measure, 000037 | regulator, support, reference, phases, eta and Jacobian information. It is not 000038 | an extra Gamma_H added to a master that already contains this contribution. 000039 | 000040 | On the declared original affine Yukawa sector, write 000041 | S_Y=integral[ h_a j_a + bar h_a jbar_a ], 000042 | where j are Grassmann-even bilinears from the above whole q(H)M. More 000043 | generally use real component labels r=1,...,4 and S_Y=h_r j_r. Since the 000044 | currents are even, they commute in the Euclidean Grassmann algebra. Exactly 000045 | int DH exp(-S_H+J.H-S_Y)=Z_H[J-j|A]. 000046 | If Z_H[J] is nonzero this equals 000047 | Z_H[J|A] exp[ sum_(n>=1) (-1)^n/n! kappa_H,n[J|A](j,...,j) ]. 000048 | Finite Grassmann nilpotency terminates the sum, requiring only finitely many 000049 | original H moments for any specified finite word. No Gaussianization, small 000050 | Y expansion or inverse-heat convergence claim is involved. Analytic infinite 000051 | field limits need their actual moment/domain/limit controls separately. 000052 | 000053 | In induced action notation, with no additional H-dependent fermionic vertices, 000054 | S_eff=S_F,0-log Z_H[J|A] 000055 | +kappa_1 j -(1/2)kappa_2 jj +(1/6)kappa_3 jjj 000056 | -(1/24)kappa_4 jjjj + ... . 000057 | These are2,4,6,8-fermion terms respectively. The cubic and quartic boson 000058 | cumulants are retained; the boson density need not be Gaussian. All original 000059 | external fermion sources and subsequent Grassmann integrations remain. 000060 | The conditional log Z_H[J|A] depends on A and must not be dropped as a global 000061 | constant; it contributes to the total Z[0] and its geometry/reference scores. 000062 | 000063 | IMPORTANT: the original full S_adm allows higher vertices. If its actual 000064 | interaction has an additional term S_extra(H,psi), the affine rule alone is 000065 | not the full answer. Retain it explicitly: for a polynomial admissible finite 000066 | interaction the exact replacement is 000067 | exp[-S_extra(delta_J,psi)] Z_H[J-j|A], 000068 | with the actual operator/source order and the corresponding moments. Extra 000069 | H-dependent four-fermion terms, for example, can already modify the induced 000070 | four-fermion coefficient. No such term is silently assumed absent in the full 000071 | primitive theory. Without a declared polynomial/analytic representation, use 000072 | the original unnormalized fiber integration directly. This is the R024 rule. 000073 | 000074 | 3. First explicit common-sector coefficient and correct meaning 000075 | 000076 | For a pair U,D=(nu,e) or (u,d), suppress color/family sums and use the usual 000077 | already oriented charged Dirac chart. The above Yukawa bilinear plus its 000078 | specified adjoint gives 000079 | j1=bar U_L Yd D_R - bar U_R Yup^dagger D_L, 000080 | j2=bar D_L Yd D_R + bar U_R Yup^dagger U_L, 000081 | jbar1=-bar D_L Yup U_R + bar D_R Yd^dagger U_L, 000082 | jbar2=bar U_L Yup U_R + bar D_R Yd^dagger D_L. 000083 | For leptons replace Yup,Yd by the actual Ynu,Ye. Each current is summed over 000084 | all original families, and over three colors only for quarks. The quaternion 000085 | signs and the original adjoint determine these currents. They are not a new 000086 | choice of physical reflection or a Majorana-independent Dirac doubling. 000087 | 000088 | Write G_ab(x,y)=_c in the ORIGINAL conditional boson fiber. 000089 | The induced action has the term -integral j_a(x)G_ab(x,y)jbar_b(y), plus 000090 | -(1/2)_c jj and its conjugate. No symmetry that kills anomalous 000091 | is imposed here, and no translation invariance or local limit is assumed. 000092 | 000093 | In particular the original common H2 gives, in the displayed ordered word, 000094 | -G_2bar2(x,y) Ynu[a,b] conjugate(Yu[d,c]) 000095 | * [bar nu_L,a(y) nu_R,b(y)] [bar u_R,c(x) u_L,d(x)]. 000096 | The two bilinears commute, so their interchanged terms cancel the1/2 rather 000097 | than producing an extra factor2. The complete even source fixes ordering 000098 | inside each bilinear. No color multiplicity is inserted for a fixed color; 000099 | the actual three-color sum is explicit. Nonzero coefficient needs the actual 000100 | nonzero boson correlator and Yukawa entries; it is not universal in every 000101 | degenerate sector. This is an EXACT induced four-fermion ACTION coefficient 000102 | of the affine sector, not the full connected/1PI scattering amplitude after 000103 | the remaining fermion/gauge integrals. No blanket attractive-force assertion. 000104 | 000105 | Four disjoint original directed bilinears give an exact polynomial test up to 000106 | eight fermions, with symbolic moments mu1,...,mu4. The logarithm reproduces 000107 | all four cumulants, including the required6/8-fermion terms. This selects 000108 | coefficients of the original field polynomial; it is not a four-mode physical 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