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: 33-161 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. 000033 | 2. Exact original CAR state/current intertwiner, not a new local field map 000034 | 000035 | An actual charge0 source occupation has descending occupied positions 000036 | r_i=lambda_i-i, a unique partition lambda. The fixed vacuum has lambda=0. 000037 | Map this original orthonormal wedge basis to the orthonormal Schur basis 000038 | s_lambda(p). Thus the actual five inputs become the explicitly saved 000039 | P_i(p)=sum_(|lambda|=32)c_(lambda,i) s_lambda(p), without discarding any 000040 | coefficient. The five support sizes are694,710,711,713,688 in each quarter. 000041 | Every actual input has charge0 in EACH of the eight residue channels, 000042 | hence empty8core and total8quotient size4. Neither dimension matching nor 000043 | a freely chosen isomorphism is used for this identification. 000044 | 000045 | The Hall metric is =delta_lambda,mu z_lambda. Original 000046 | CAR currents intertwine as J_-m=p_m and J_m=m partial_(p_m), m>0. 000047 | Murnaghan--Nakayama removal of each Young rim ribbon of size m, with sign 000048 | (-1)^height, computes the latter. Independent ribbon calculations match 000049 | the FULL CAR output vectors for every m=1,...,32 and each of five inputs, 000050 | both quarters; no Gram-only comparison. Every m>32 annihilates these 000051 | inputs by original positive energy. Raising uses infinite Maya bead moves; 000052 | its tail is not a finite five-state closure or finite mode truncation. 000053 | Full first-mode CCR is independently checked on all five states. 000054 | 000055 | With delta=alpha-1/2 and original Q, the fixed geometric stress is 000056 | Lcal_m^alpha=sum_(n>0)J_-n J_(m+n) 000057 | +1/2 sum_(r=1..m-1)J_(m-r)J_r+(Q+delta)J_m, m>0. 000058 | Negative modes are adjoints; Lcal0=degree+Q^2/2+delta Q+delta^2/2. 000059 | This has the previously verified original CAR coefficients r+m/2+alpha. 000060 | Full positive stress1/2 vectors match CAR independently, both quarters. 000061 | The ground term is retained. Finite stress-mode words follow from these 000062 | same current operators, not a newly chosen geometric coupling. 000063 | 000064 | IMPORTANT LOCALITY LIMIT: this is an original STATE/current intertwiner. 000065 | It does NOT identify the original folded quartic field with a point-local 000066 | source vertex Y_cover(P_i,z) of conformal weight32. Its true pullback still 000067 | involves eight original preimage arcs and the existing point splitting. 000068 | The actual field-product prescription is given separately in section4. 000069 | 000070 | 3. Complete ordered current response, including all profile tails 000071 | 000072 | Define the EXACT finite normal kernel 000073 | K_ij(u,v)=0)u_m p_m) 000074 | P_j(p_m+m v_m)>_Hall. 000075 | Its weighted u and v degrees are at most32, by input energy, not a chosen 000076 | perturbative cutoff. An explicit finite formula in the saved coefficients is 000077 | K_ij=sum_(lambda,mu)conj(c_(lambda,i))c_(mu,j) 000078 | sum_(nu subset lambda,mu) 000079 | s_(lambda/nu)(p_m=m u_m) s_(mu/nu)(p_m=m v_m). 000080 | Thus all coefficients, including mixed-mode correlations, are specified. 000081 | It is unnecessary to expand the very large polynomial to define it exactly. 000082 | 000083 | For a smooth real original profile phi, phi_-m=conj(phi_m), put 000084 | J(phi)=phi0 Q+sum_(m>0)[phi_m J_-m+conj(phi_m)J_m]. 000085 | On charge0 the actual original Weyl response is 000086 | 000087 | =exp(-1/2 sum_(m>0)m|phi_m|^2) K_ij(i phi,i conj(phi)). 000088 | The Gaussian retains EVERY Fourier tail, including m>32. Only the state 000089 | polynomial terminates. In particular a properly supported smooth local 000090 | profile is not replaced with a finite Fourier polynomial. 000091 | 000092 | For any finite ordered list of such source operations, original CCR gives 000093 | W(phi1)...W(phis)=exp(-1/2 sum_(i0)m[conj(phi_m)psi_m-phi_m conj(psi_m)] I. 000095 | All geometric-free time insertions are retained by the original evolution 000096 | U(t)J_-m U(t)^*=exp(-itm/8)J_-m, U(t)=exp(-itK_alpha). 000097 | No independent time calibration or phase rule is supplied. This formula 000098 | computes all finite smooth density-Weyl histories on the actual five input 000099 | states, preserving their actual Gram and all intermediate source modes. 000100 | 000101 | The complete first-mode slice is additionally saved as33 exact Gram 000102 | coefficients: K_ij(u1,v1)=sum_(k=0..32)(u1 v1)^k 000103 | /(k!)^2. This is a diagnostic slice, NOT a selected 000104 | physical profile. Arbitrary exponentiated geometric diffeomorphisms or 000105 | unsmeared coincident stress products are not inferred from finite-mode 000106 | Sugawara alone; their analytic domains need separate control. 000107 | 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