TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PROJECTED-JOINT-SOURCE-GENERATOR-20261103.txt Original file SHA-256: bac12849d64dff9abe77e0f02c6d47bd33359501ccaa0f5730fa291558c90796 Inclusive original lines: 13-96 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. 000013 | 1. ACTUAL SOURCE INPUT AND THE CHANGE FROM R112 000014 | 000015 | The existing original field chart has its entire conditional exterior 000016 | transfer Q_t[b]=s_t[b] Gamma(R_t[b]), including the original scalar s. 000017 | Its full-field relation is 000018 | Q_t[b] c_i^dagger = sum_j c_j^dagger (R_t[b])_ji Q_t[b]. (1) 000019 | This is reused from the existing whole Clifford source, not asserted for 000020 | an otherwise unidentified interacting primitive source. The c_i are its 000021 | actual coordinates, not automatically physical particle annihilators. 000022 | Retain all original reference, eta, orientation, boundary and rest legs. 000023 | 000024 | Let the ACTUAL original ordered unnormalized boson integration define 000025 | Q_t[B]=I_(B,t)(w_(B,t)[b] B[b] Q_t[b]). (2) 000026 | No new measure is chosen. A bosonic history is not frozen. Its scalar 000027 | marks (R_t)_ji stay inside the same integral. No B-dependent normalization 000028 | or product of separately averaged quantities is introduced. 000029 | 000030 | All fields must act in the same actual transported field frame, independent 000031 | of b; the original integral must allow constant left/right field factors 000032 | to pass through it. Scalar marks are even. Keep the given frame connection 000033 | if a field-dependent chart is used instead. On a compatible finite 000034 | regulator these are ordinary operator equalities. Unbounded transfers 000035 | require their actual common invariant product domain and convergence. 000036 | 000037 | Then (1) integrates exactly to 000038 | Q_t[1] c_i^dagger = sum_j c_j^dagger Q_t[(R_t)_ji]. (3) 000039 | In particular the right side is NOT c^dagger Q_t[1]. 000040 | 000041 | 2. EXPLICIT JOINT GENERATOR FROM THE SAME SHORT-TIME SOURCE 000042 | 000043 | Suppose that the SAME full caller in (2), in the ORIGINAL time parameter, 000044 | is a strongly continuous selfadjoint semigroup 000045 | Q_t[1]=exp(-t H), Q_0[1]=I, 000046 | and the actual short-time marks obey R_0=I, dot R_0=-h[b]. 000047 | The product-rule passage under the original integral, including its 000048 | varying density/boundary kernel, must exist on the actual common core. 000049 | It then gives 000050 | -partial_t Q_t[(R_t)_ji]|0 = delta_ji H+V_ji, 000051 | V_ji:=Q_0[h_ji]. (4) 000052 | Here Q_0[h_ji] is the actual ZERO-TIME MARKED OPERATOR. It is not a 000053 | normalized mean of h. In the existing Clifford chart 000054 | h_b=A_b gamma_F-i gamma_F Phi_b; 000055 | the circle derivative, full original Phi and quarter parameter remain. 000056 | The right side of (4) includes every density/endpoint derivative in H; 000057 | only the additional mark derivative produces V. Without this justified 000058 | product rule retain the original marked source derivative itself. 000059 | 000060 | Differentiating (3) now gives the actual commutator 000061 | [H,c_i^dagger]=sum_j c_j^dagger V_ji. (5) 000062 | If the resulting V_ji act only on the remaining boson/rest factor and 000063 | V_ji^dagger=V_ij, put 000064 | H_F=sum_(j,i) V_ji tensor c_j^dagger c_i. 000065 | CAR gives [H_F,c_i^dagger]=sum_j c_j^dagger V_ji. The adjoint relation 000066 | also holds. Hence H-H_F commutes with the ENTIRE original finite CAR 000067 | algebra, so its commutant gives 000068 | 000069 | H=H_B tensor I+sum_(j,i) V_ji tensor c_j^dagger c_i. (6) 000070 | 000071 | No commutativity of the different boson operators V_ji is required. 000072 | At an unbounded cutoff actual strong reduction by the CAR matrix units 000073 | and the closed operator realizations must be justified. A formal core 000074 | commutator alone is insufficient. That infinite-dimensional extension is 000075 | not proved here merely by the finite-regulator algebra. 000076 | Higher explicit fermion-degree terms are absent in this EXISTING 000077 | conditionally exterior source class. Integrating its dynamic bosons can 000078 | still produce a non-Gaussian fermion source, as R112 already derives. 000079 | Equation(6) is not imposed on every possible physical TFPT realization. 000080 | 000081 | The original Fock vacuum matrix leg fixes the remaining operator: 000082 | H_B=(I tensor ), 000083 | C_t^00=(I tensor ) 000084 | =exp(-t H_B). (7) 000085 | Omega_F is the existing coordinate leg of this source, not a new choice 000086 | of physical OS vacuum. It retains the original scalar s and absolute 000087 | reference. A fully traced partition Z_t[0] is not C_t^00. Neither the 000088 | normalized Cauchy graph nor an independently chosen boson action supplies 000089 | the missing scalar/operator leg. No extra determinant is multiplied in. 000090 | 000091 | The exact joint spectral problem in this source class has number-sector 000092 | operators H^(n)=H_B tensor I+sum V_ji tensor dGamma_n(E_ji). 000093 | These act on BOSON SPACE tensor Lambda^n(C^m), not on a numerical m x m 000094 | mass matrix. Their eigenvalues are not sums of eigenvalues of . 000095 | The coefficient operators and bosonic body caller must actually be 000096 | evaluated before any spectrum is numerically claimed.