TFPT scientific source excerpt Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt Original file SHA-256: 62a9eeb4519b59d4514445f9d4dd5d2059e29179956a4c97ec94b7bae9ae3094 Inclusive original lines: 108-160 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. 000108 | 4. Actual local folded field products: keep the original arcs and subtraction 000109 | 000110 | Let eta_r(x)=psi((x+2pi r)/8)/sqrt8 and 000111 | U_alpha(r,k)=exp(2pi i(k+alpha)r/8)/sqrt8, chi=U_alpha^* eta. 000112 | Write H_a=sum_k V_ka :chi_k^* chi_k:_sea for the mean-zero positive native 000113 | current, with V=M^T O U_+. The conformal current is h_a^conf=H_a+b_a/z. 000114 | Therefore in the original ARC chart its bilinear matrix 000115 | is A_a=U_alpha diag(V_ka) U_alpha^*. The same point-split subtraction and 000116 | the already derived conformal counterterm b_a are retained. The scalar 000117 | source density need not be diagonal in the chi chart. 000118 | 000119 | For the actual source Weyl operation W(phi), annihilation fields obey 000120 | W(phi)psi(theta)W(phi)^*=exp(-i phi(theta))psi(theta). 000121 | Put G_phi(x)=diag_r exp(-i phi((x+2pi r)/8)). At FINITE point splitting, 000122 | W(phi)[eta(x+)^* A_a eta(x-)]W(phi)^* 000123 | =eta(x+)^* G_phi(x+)^* A_a G_phi(x-) eta(x-). 000124 | This is the exact original operation on the existing current; no channel 000125 | matrix is chosen independently. Conjugating the entire existing subtraction 000126 | is essential: rewritten in the same reference normal order, the c-number 000127 | is c_ref(G+^* A_a G-;x+,x-)-c_ref(A_a;x+,x-). The original state covariance 000128 | C_alpha supplies its expectation, with the existing difference C-C_ref 000129 | giving b. No second shift or new subtraction convention is allowed. 000130 | 000131 | Apply this identity to the existing point-split definition of each quartic 000132 | F_i=:f_i(H_++b/z):=:f_i(h_+^conf):, including all its subtraction terms. 000133 | There is exactly ONE b shift, as in the numerical state construction. 000134 | Keep the full point-split Taylor terms: bosonic current normal order is 000135 | not the same as replacing a current polynomial by a fully fermionically 000136 | normal-ordered monomial. The finite fermion derivative terms (already 000137 | present for :J^2:) follow from the unchanged point-split limits. For any finite 000138 | collection of these transformed fields at separated points, expand the 000139 | finite current polynomials before taking their already specified limits. 000140 | The ordered expectation of each resulting even CAR word X1...X_(2n) is 000141 | Pf(K), K_ij= for i