TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/HIGGS-COLOR-HOLONOMY-PROOF.txt Original file SHA-256: b018cd619e98f9cc05b1994cf18b77427af61932293070036dc633bad05adc2d Inclusive original lines: 1-185 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. 000001 | R067. Original Higgs/color response tests the free geometric source jointly. 000002 | 1 November2026, Sitzung035. Scoped theory result, no primitive F1 closure. 000003 | 000004 | 1. Actual question, source and first identification 000005 | R066 relates the GIVEN free quarter source to the real unit-charge elliptic 000006 | Quillen norm. Does that same response survive the ALREADY GIVEN original 000007 | Higgs coupling, without changing Y, choosing an H distribution or a profile? 000008 | Use the actual quark-PARTICLE CAR subalgebra of the existing second-face 000009 | source,36 one-particle modes per p=n+a, a=1/4 or3/4, b=4pi. 000010 | The other slots,Majorana and full two-face graph stay in the whole source 000011 | and its weight. We do not quantize96 independent physical fermions. 000012 | The two quarters are independent blocks conditional on the specified H, 000013 | not the two correlated annulus faces. Existing full-H statements are reused. 000014 | 000015 | On each of the three colors, the original6x6 LR coefficient is 000016 | M(H)=q(H)diag(Yu,Yd), q*q=uI2, u=H*H, with family identities understood. 000017 | Phi=[[0,M],[M*,0]], gamma=diag(I6,-I6), 000018 | h_H=p gamma-i gamma Phi, C_H=(1+exp(bh_H))^-1. 000019 | The original weak-left gauge orbit reduces the six singular values of M 000020 | to sqrt(u)y_j, j=1..6, without rotating or fitting either Yukawa matrix. 000021 | Hence h_H has pairs +-E_j, E_j=sqrt(p²+y_j²u), each color-degenerate. 000022 | 000023 | The original marked COLOR group G in SU3 acts equally on these L/R slots 000024 | and trivially on the weak/family labels. It commutes with h_H. This is 000025 | not a guessed scalar coordinate form or a new current-to-Higgs map. 000026 | Use the actual real gauge lift U(u)=pi(u) J_F pi(u) J_F^-1 with 000027 | u=(1,I2,bar(G)). Its particle quark restriction is precisely G on color; 000028 | the antiparticle block is conjugate. Raw left pi alone would be wrong. 000029 | Our insertion is its bounded existing CAR implementer U_col(G), and 000030 | Xi_H(G)=omega_H(U_col(G)). It counts a GLOBAL internal charge/holonomy 000031 | probe in this specified state, not a local3+1D observable or an event rule. 000032 | Success: evaluate that actual mixed response, INCLUDING all modes and 000033 | normalization, and compare it to the free R066/Quillen response. Abort 000034 | any inference equating this insertion with primitive EM or metric variation 000035 | without its original source/reference dictionary. 000036 | 000037 | 2. Exact whole nonabelian group response, not only a fitted Cartan jet 000038 | Write q_j=exp(-bE_j), s(E)=q/(1+q)²=1/[4cosh²(bE/2)]. 000039 | The occupied negative-energy color triplet has group determinant detG=1. 000040 | Thus its sea/reference phase is exactly1 for EVERY G in SU3. 000041 | Particle and hole excitations have weights q and color G,G*. The original 000042 | conditional Fock trace gives for one p,j 000043 | F_E(G)=det3(1+qG)det3(1+qG*)/(1+q)^6 000044 | =det3[I-s(E)(2I-G-G*)]. 000045 | Equivalently if eigenvalues of G are exp(i theta_c), 000046 | F_E(G)=product_c[1-4s(E)sin²(theta_c/2)]. 000047 | No charged branch,extra determinant multiplicity or Nambu square root has 000048 | been inserted: this is exactly the given quark-particle CAR restriction. 000049 | The finite Fock determinant identity supplies each finite block; the limit 000050 | below is proved separately, not asserted from a finite trace theorem. 000051 | 000052 | For the entire original mode tower, 000053 | Xi_a,u(G)=product_(n in Z;j=1..6) F_(sqrt((n+a)²+y_j²u))(G). 000054 | For both quarters of the SAME constant H use their product, not two 000055 | independent H draws. It is a real positive conjugacy-class function. 000056 | Because |n+a|>=1/4, every factor has the uniform lower bound 000057 | 1-4s(1/4)>0. Also 0<=1-F_E(G)<=12exp(-b|n+a|). 000058 | The summable bound proves convergence,strict positivity and a uniform 000059 | tail bound for ALL u>=0 and ALL G in SU3. The whole group class is covered. 000060 | For finite H,dominated summation also justifies derivatives near identity. 000061 | 000062 | 3. Genuine mixed Higgs/charge response 000063 | For G(theta)=diag(exp(i theta),exp(-i theta),1), 000064 | Xi_a,u(theta)=product_(n,j)[1-4s(E_j)sin²(theta/2)]². 000065 | The original Hermitian generator t=diag(1,-1,0) has tr3(t²)=2. 000066 | Its quark-source mean is0 at every H. Its variance is exactly 000067 | K_a(u)=-partial_theta² logXi_a,u(0)=4 sum_(n,j) s(E_j). 000068 | This also follows independently from Tr36[t² C_H(1-C_H)]. 000069 | On the original36 subalgebra Tr(t²)=24, not48 or96. 000070 | Since s'(E)=-b s(E)tanh(bE/2)<0 for E>0, 000071 | K_a'(u)=-2b sum_(n,j) y_j² s(E_j)tanh(bE_j/2)/E_j <0 000072 | when at least one given quark singular value is nonzero. 000073 | This is a genuine response of the supplied joint operator, not an internal 000074 | frame change. It cannot be removed by a simultaneous weak/color basis 000075 | transport, since u and the given singular values are invariant. 000076 | 000077 | At the unmodified H=0 source, R066's exact Var(Q_a)=1/(8pi) gives 000078 | K_a(0)=24/(8pi)=3/pi, K_pair(0)=6/pi. 000079 | For a nonzero H with a nonzero given quark Y, these values strictly decrease. 000080 | The normalized free value K_a(0)/24 equals the original numerical c3, 000081 | but K_a(u)/24 is NOT a state-independent definition of primitive c3. 000082 | 000083 | The stronger all-group statement follows directly from the factors: 000084 | 00, G!=I, some y_j!=0. 000085 | There is a strict factor for any nonidentity G and a changed massive mode; 000086 | the positive convergent remaining product preserves the strict inequality. 000087 | No small-H truncation and no choice of preferred weak direction is used. 000088 | Rank-deficient Y merely retains its massless factors; all Y=0 is the 000089 | explicit non-strict exceptional case. At G=I all responses equal1. 000090 | 000091 | 4. Exact comparison to the existing free Quillen norm 000092 | Let phi_a(theta)=Theta_a(theta)/Theta_a(0) be the original full single-quarter 000093 | charge characteristic, with its nonzero quarter mean retained. Its sea 000094 | phase cancels in the color trace. The free source gives 000095 | Xi_a,0(theta)=|phi_a(theta)|^24. 000096 | For the pair, the already proved square-theta identity gives 000097 | Xi_pair,0(theta)=N(theta/(2pi),0)^12, 000098 | where N is the EXISTING real unit-charge elliptic Quillen norm, normalized 000099 | at zero. The exponent12 is fixed by this36-mode color restriction and 000100 | both original quarters, not a free EM exponent or twelve new species. 000101 | Its real curvature is exactly -24pi with respect to v=theta/(2pi), 000102 | consistent with the original N curvature -2pi and K_pair(0)=6/pi. 000103 | 000104 | There is an especially simple EXACT bounded discriminator: 000105 | G_star=diag(-1,-1,1), theta=pi. At the already given square parameter, 000106 | theta2(0|i)=theta4(0|i)>0 by the modular transformation. Jacobi's identity 000107 | theta3(0)^4=theta2(0)^4+theta4(0)^4 therefore gives N(1/2,0)=1/sqrt2. 000108 | Consequently Xi_pair,0(G_star)=1/64 exactly (each quarter is1/8). 000109 | This is a parity EXPECTATION of the existing color operation, not a 000110 | probability1/64, a record count, an E8 family event or a new constant. 000111 | 000112 | 5. Entire positive H integration class: one bounded probe suffices 000113 | Retain the R054 conditional premise 000114 | omega(X)=integral omega_H(X) dmu(H), mu>=0, mu(total)=1, 000115 | where the actual complete weight includes Z_F(H),bosons,Majorana,reference, 000116 | eta and Z[0]. We do NOT choose mu or assert the actual primitive weight 000117 | is positive. Conditional source quasifreeness and fixed scalar quarter 000118 | kinetics are explicit. Other conditional fermion/gauge interactions are 000119 | not silently Gaussianized. Both quarters see the same H. 000120 | For any fixed G!=I and any such positive weight with mu(u>0)>0, 000121 | Xi_mixture(G)=integral Xi_H(G)dmu > Xi_free(G). 000122 | In particular along the Cartan path, for theta not in2piZ, 000123 | Xi_pair,mixture(theta)>N(theta/(2pi),0)^12. 000124 | Every Xi_H is a bounded observable expectation, so no H moment or new 000125 | integrability condition is required. It is enough to test ONE nonidentity 000126 | group element, e.g.diag(-1,-1,1), rather than an infinite list of moments. 000127 | At G_star the strict inequality is simply Xi_pair,mixture>1/64. 000128 | Likewise K_mix=integral K_H dmu