Listing: Listing

# helmholtz_check.py output   (exact QQ; conditional on (S-bv) through M.2)
# M.4 formal self-adjoint (variational) subspace under Gate V; Helmholtz certificates
CHECK dim C_coh = 3321: True
CHECK unit test: Euler derivatives of 30 random quadratic Lagrangians (independent sympy route) are self-adjoint under m4_common.adjoint, all readings, D=0..4: True
CHECK negative control: E^{00}=d_0 h_{01} is NOT self-adjoint in every reading: True
CHECK formal: every exported SA vector satisfies E_C = E_C^dag exactly: True
# formal           dim SA = 26 ; per D = {0: 26, 1: 0, 2: 0, 3: 0, 4: 0} ; defect-map nnz = 1482596 (37.2s)
CHECK chi_plateau: every exported SA vector satisfies E_C = E_C^dag exactly: True
# chi_plateau      dim SA = 26 ; per D = {0: 26, 1: 0, 2: 0, 3: 0, 4: 0} ; defect-map nnz = 645026 (16.6s)
CHECK double_plateau: every exported SA vector satisfies E_C = E_C^dag exactly: True
# double_plateau   dim SA = 1347 ; per D = {0: 26, 1: 0, 2: 271, 3: 12, 4: 1038} ; defect-map nnz = 60016 (28.2s)
CHECK formal: dim Collapse(SA) = dim SA - dim ker Collapse = 26 - 0 = 26 (mod-p rank lower bound meets the upper bound): True
# formal           dim (T_variational cap im Collapse) = 26 ; per D = {0: 26, 1: 0, 2: 0, 3: 0, 4: 0}
CHECK chi_plateau: dim Collapse(SA) = dim SA - dim ker Collapse = 26 - 0 = 26 (mod-p rank lower bound meets the upper bound): True
# chi_plateau      dim (T_variational cap im Collapse) = 26 ; per D = {0: 26, 1: 0, 2: 0, 3: 0, 4: 0}
CHECK double_plateau: dim Collapse(SA) = dim SA - dim ker Collapse = 1347 - 22 = 1325 (mod-p rank lower bound meets the upper bound): True
# double_plateau   dim (T_variational cap im Collapse) = 1325 ; per D = {0: 26, 1: 0, 2: 271, 3: 8, 4: 1020}
CHECK FORMAL: SA = 26-dimensional, supported only at D=0 (no variational collapse with any derivative): True
CHECK chi_plateau: same as formal: True
# formal           dim (SA cap C_reg) = 26
# chi_plateau      dim (SA cap C_reg) = 26
# double_plateau   dim (SA cap C_reg) = 1208
CHECK Helmholtz certificate A (formal, D=0): Euler derivative of I_C = (1/2) int <h,E_C[h]> equals E_C: True
CHECK Certificate A K-matrix satisfies CORRECT Gate-V identity w_mn K^{mn}_ab = w_ab K^{ab}_mn: True
CHECK Certificate A K-matrix FAILS the reversed identity w_ab K^{mn}_ab = w_mn K^{ab}_mn (as required): True
CHECK Certificate A numeric check: w00 K00_02 = -56/3 = w02 K02_00: True
CHECK Helmholtz certificate B (double plateau, D=4, fourth-order operator): Euler(I_C) = E_C: True
CHECK Helmholtz certificate B2 (double plateau, D=2, second-order operator): Euler(I_C) = E_C: True
CHECK negative control: for a non-self-adjoint collapse E_C, Euler(I_C) != E_C: True
# wrote helmholtz_certificates.txt
# --- certificate A (formal, D=0): explicit matched quadratic functional ---
#   C[N=4, b={(),0222}, gamma=()] = 1
#   E^{00} = -56*h_02_*s_*x_/3
#   E^{02} = -28*h_00_*s_*x_/3 - 28*h_22_*s_*x_/3
#   E^{22} = -56*h_02_*s_*x_/3
#   I_C = int d^4y L ,  L = -56*h_00_*h_02_*s_*x_/3 - 56*h_02_*h_22_*s_*x_/3
# --- certificate B (double plateau, D=4): 4 E-terms ; B2 (D=2): 4 E-terms (see certificates file)
# === SUMMARY ===
# formal           {'dim_SA': 26, 'perD': {0: 26, 1: 0, 2: 0, 3: 0, 4: 0}, 'dim_SA_cap_Creg': 26, 'dim_variational_image': 26, 'variational_perD': {0: 26, 1: 0, 2: 0, 3: 0, 4: 0}}
# chi_plateau      {'dim_SA': 26, 'perD': {0: 26, 1: 0, 2: 0, 3: 0, 4: 0}, 'dim_SA_cap_Creg': 26, 'dim_variational_image': 26, 'variational_perD': {0: 26, 1: 0, 2: 0, 3: 0, 4: 0}}
# double_plateau   {'dim_SA': 1347, 'perD': {0: 26, 1: 0, 2: 271, 3: 12, 4: 1038}, 'dim_SA_cap_Creg': 1208, 'dim_variational_image': 1325, 'variational_perD': {0: 26, 1: 0, 2: 271, 3: 8, 4: 1020}}
# total runtime 124.6s
ALL CHECKS PASSED