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: 1-164 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 | R113. WHOLE MARKED SOURCE, JOINT GENERATOR AND ACTUAL SELECTOR CONTACTS 000002 | 3 November 2026. Continuation of R112 on the existing source, no new model. 000003 | Verdict: CONDITIONAL_JOINT_GENERATOR_DERIVED_BRST_CONTACT_SHORTCUT_EXCLUDED. 000004 | 000005 | The primitive physical S_adm and its joint physical spectrum are not derived. 000006 | The new calculation gives the joint boson/fermion generator of the existing 000007 | whole conditional Clifford-source branch under its explicitly stated 000008 | short-time and source-identification premises. It also gives the exact 000009 | selector-corrected source identity with the original whole P outside the 000010 | boson integral. The original BRST rule cannot supply the extra full linear 000011 | fermion-contact premise by regular gauge directions alone. 000012 | 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. 000097 | 000098 | 3. KEEP THE WHOLE ORIGINAL SELECTOR OUTSIDE THE BOSON INTEGRAL 000099 | 000100 | Let P=P_adm be the original whole physical selector on the SAME common 000101 | space, and A_t[B]=P Q_t[B] P. No fiber decomposition P=P_b is assumed. 000102 | No replacement by Gamma of a one-particle projector or fermion parity. 000103 | In particular a gauge/structural selector may mix boson boundary data. 000104 | 000105 | Exact algebra applied to (3) gives 000106 | A_t[1] c_i^dagger-sum_j c_j^dagger A_t[(R_t)_ji] 000107 | =P Q_t[1][P,c_i^dagger] 000108 | +sum_j[P,c_j^dagger]Q_t[(R_t)_ji]P. (8) 000109 | Proof: expand both commutators. The middle terms are 000110 | -P Q_t[1]c_i^dagger P+sum_j P c_j^dagger Q_t[(R_t)_ji]P=0 000111 | by(3), leaving exactly the left side. No [P,Q_t[B]]=0 is used. 000112 | For distinct actual incoming/outgoing spaces retain P_in and P_out, 000113 | and their corresponding field operators, on their respective sides. 000114 | 000115 | For ordered creation words C_I^dagger of length k the same exact identity 000116 | uses det(R_t)_(J,I) as its original scalar marks and commutators 000117 | [P,C_I^dagger],[P,C_J^dagger]. The formula follows from the already given 000118 | whole exterior source and the same two-line expansion. Adjoint and mixed 000119 | words retain the actual adjunction, ordering and field domains. 000120 | There is NO extra P between each charged letter of an original word. 000121 | Neutral composite words can contain charged intermediate propagation. 000122 | 000123 | The original local net requires BOTH A and A^dagger to preserve Ran(P). 000124 | For bounded A this is equivalent to [A,P]=0. For these actual allowed 000125 | field combinations the corresponding selector contacts vanish. 000126 | This is not a requirement that every charged CAR coordinate be a physical 000127 | observable. No universal ambient-CAR prerequisite is introduced. 000128 | 000129 | For the original color center let 000130 | P_r=(1/3)sum_(k=0)^2 omega^(-kr) Z_c^k, 000131 | Z_c c_q^dagger Z_c^-1=omega^q c_q^dagger. 000132 | Then EXACTLY 000133 | P_r c_q^dagger=c_q^dagger P_(r-q), 000134 | [P_0,c_q^dagger]=c_q^dagger(P_(-q)-P_0), 000135 | P_0 c_q^dagger P_0=0 for q!=0. (9) 000136 | For the full original P=P_rest P_0 retain also 000137 | [P,c]=P_rest[P_0,c]+[P_rest,c]P_0. 000138 | No field action of the gap, seam or determinant selector is invented. 000139 | The color example computes an ORIGINAL selector contact, not a new model. 000140 | 000141 | 4. PHYSICAL RESTRICTION REQUIRES THE ACTUAL TIME SEWING 000142 | 000143 | If the SAME original P reduces H, then the physical generator is the 000144 | restriction H|Ran(P), with (6) on that range. If it does not reduce H, 000145 | P exp(-tH)P is not automatically a closed physical semigroup. At a 000146 | compatible finite cutoff its exact sewing defect is 000147 | P exp(-2tH)P-(P exp(-tH)P)^2 000148 | =P exp(-tH)(I-P)exp(-tH)P >=0, 000149 | and its first nonzero short-time coefficient is 000150 | t^2 P H(I-P)H P. (10) 000151 | Vanishing of the defect for all t implies P commutes with exp(-tH) 000152 | and hence reduces the selfadjoint H. This is an exact condition on the 000153 | original caller, not a new projection inserted after every time step. 000154 | At a finite cutoff, elimination of the unchanged complementary sector 000155 | for initial data in Ran(P) gives the EXACT memory equation 000156 | dot psi_P(t)=-H_PP psi_P(t) 000157 | +integral_0^t H_PQ exp(-(t-s)H_QQ) H_QP psi_P(s) ds. (10a) 000158 | Here Q=I-P in this block notation, not the source symbol Q_t[B]. Replacing 000159 | it by PHP evolution would discard this same-source return term. 000160 | If t is only a collar width, (3),(8) remain collar identities and (6) 000161 | is not declared a physical time generator without the actual time map. 000162 | 000163 | 5. WHY THE ORIGINAL BRST RULE DOES NOT SUPPLY A LINEAR FERMION CONTACT LAW 000164 |