Listing: Listing

# time_only_helmholtz.py output  (python 3.13.5, sympy 1.14.0; exact QQ; CONDITIONAL on (S-bv) via M.2 basis)
# XVI-M work package C: universal time-only Helmholtz classification (reading time_only)
CHECK dim C_coh = 3321, graph table 3282 lines: True
# input sha256: {'graph_general_table.txt': '4cd29d3c9f691340b314aecbfe7d6aef3590b9cee21dcdb486fe9fbcdd679e49', 'kernel_basis.json': 'fcbe2876abd2a6b410813f732356f829c970c22c3a333f7e4ec34909c0e0793b', 'm4_common.py': '932b716db827efc0cbb3d9eca91c633d9eecf932c50c1988008f98795337bf1e'}
CHECK dim C_reg = 3150 (N>=3), dim K block = 171 (N<=2): True
# === C.2 specialization tests ===
CHECK S_t(D_mu F) = D_mu^(t) S_t(F) for 400 random exact polynomials x 4 directions, and S_t(D^beta F) = D^(t)beta S_t(F) for random |beta|<=4: True
CHECK negative control: a D_1^(t) that differentiates sigma (sigma'(t) for d_1) violates the intertwining: True
CHECK sympy Function route: d_mu[F(sigma(t), chi(x), h(x))] = D_mu^(t) S_t(F) on 25 random polynomials (sigma a function of x0 only): True
CHECK S_t(D_mu F) = D_mu^(t) S_t(F) on 960 actual collapse / adjoint-defect expressions (120 random contact variables x 4 directions): True
# ######## reading time_only ########
# === C.3/C.4 blockwise: route 1 vs route 2, exact kernels ===
# blockwise pass 64.9s
CHECK [time_only] ROUTE 1 = ROUTE 2 for Coll_t: all 3321 columns equal exactly (nnz 446292): True
CHECK [time_only] ROUTE 1 = ROUTE 2 for H_t: all 3321 adjoint-defect columns equal exactly (nnz 573582): True
CHECK [time_only] grading: every time-only key in block (D, parity) has total derivative count D and parity of the block (blocks have disjoint rows): True
# supplementary mod-p ranks (all rows, p=1e9+7, 998244353) agree with exact ranks in every block: True
CHECK [time_only] ker Coll_t is contained in ker H_t (each ker Coll_t vector has zero time-only adjoint defect): True
CHECK [time_only] the 26 formal SA vectors (helmholtz_certificates.txt) lie in ker H_t (specialization: ker H_formal <= ker H_t): True
CHECK [time_only] D=0: dim Coll_t(ker H_t) = 26 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only] D=0: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 26: True
CHECK [time_only] D=1: dim Coll_t(ker H_t) = 0 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only] D=1: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 0: True
CHECK [time_only] D=2: dim Coll_t(ker H_t) = 0 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only] D=2: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 0: True
CHECK [time_only] D=3: dim Coll_t(ker H_t) = 0 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only] D=3: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 0: True
CHECK [time_only] D=4: dim Coll_t(ker H_t) = 0 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only] D=4: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 0: True
# === CLASSIFICATION TABLE (time_only reading; conditional on (S-bv)) ===
# D | size | kerColl | imColl | kerH | Coll(kerH) | kerColl^Creg | kerColl^K | kerH^Creg | kerH^K | Coll(kerH^Creg)
# 0 | 35 | 0 | 35 | 26 | 26 | 0 | 0 | 26 | 0 | 26
# 1 | 160 | 0 | 160 | 0 | 0 | 0 | 0 | 0 | 0 | 0
# 2 | 440 | 0 | 440 | 0 | 0 | 0 | 0 | 0 | 0 | 0
# 3 | 944 | 0 | 944 | 0 | 0 | 0 | 0 | 0 | 0 | 0
# 4 | 1742 | 0 | 1742 | 0 | 0 | 0 | 0 | 0 | 0 | 0
# totals: {'size': 3321, 'size_Creg': 3150, 'size_K': 171, 'dim_ker_Coll_t': 0, 'dim_im_Coll_t': 3321, 'dim_ker_H_t': 26, 'dim_Coll_t_of_ker_H_t': 26, 'dim_ker_Coll_t_cap_Creg': 0, 'dim_ker_Coll_t_cap_K': 0, 'dim_ker_H_t_cap_Creg': 26, 'dim_ker_H_t_cap_K': 0, 'dim_regression_preserving_variational_image': 26}
# variational-image monomial structure per D: {"0": {"n_vectors": 26, "monomial_type_counts": {"sigma^(0) chi_0 h_0": 198}, "sigma_orders_present": [0], "has_chi_derivatives": false, "max_h_order": 0}}
# === C.5 functional probes (weight w = chi_I(x) sigma_eps(t); all preparation/switch derivatives retained) ===
CHECK [time_only] weighted P1 target: total derivative degree 0, parity 0: True
CHECK [time_only] weighted P1 target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only] weighted P2 target: total derivative degree 0, parity 0: True
CHECK [time_only] weighted P2 target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only] weighted EH target: total derivative degree 2, parity 0: True
CHECK [time_only] weighted EH target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only] weighted R2 target: total derivative degree 4, parity 0: True
CHECK [time_only] weighted R2 target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only] weighted RC2 target: total derivative degree 4, parity 0: True
CHECK [time_only] weighted RC2 target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only] divergence identity: d_mu E^{mu nu} = 0 for weighted R2 with w = chi sigma(t) (time-only jets): True
CHECK [time_only] divergence identity: d_mu E^{mu nu} = 0 for weighted RC2 with w = chi sigma(t) (time-only jets): True
CHECK [time_only] divergence identity: d_mu E_EH^{mu nu} = (1) (d_mu w) G_lin^{mu nu} for weighted EH (w = chi sigma(t)): True
CHECK [time_only] ultralocal weighted P1: divergence nonzero (not gauge invariant): True
CHECK [time_only] P1+P2/C_coh: explicit preimage of direction ['1', '-89/140'] reproduces the weighted target exactly: True
CHECK [time_only] P1+P2/C_coh: route-2 collapse of the preimage equals the target: True
CHECK [time_only] P1+P2/C_coh: preimage lies in ker H_t (variational): True
CHECK [time_only] P1+P2/C_coh: EL(1/2 <h, E_C[h]>) = E_C (independent sympy time-only jet route): True
CHECK [time_only] P1+P2/C_coh: EL(1/2 <h, E_C[h]>) = E_C via sympy euler_equations with sigma(x0), chi(x), h(x) Functions: True
# probe P1+P2   C_coh D=0 block=10 : dim(span cap im Coll_t)=1 ; members={'P1': False, 'P2': False} ; directions=[['1', '-89/140']] (0.1s)
CHECK [time_only] P1+P2/C_reg: explicit preimage of direction ['1', '-89/140'] reproduces the weighted target exactly: True
CHECK [time_only] P1+P2/C_reg: route-2 collapse of the preimage equals the target: True
CHECK [time_only] P1+P2/C_reg: preimage lies in ker H_t (variational): True
CHECK [time_only] P1+P2/C_reg: EL(1/2 <h, E_C[h]>) = E_C (independent sympy time-only jet route): True
CHECK [time_only] P1+P2/C_reg: EL(1/2 <h, E_C[h]>) = E_C via sympy euler_equations with sigma(x0), chi(x), h(x) Functions: True
# probe P1+P2   C_reg D=0 block=10 : dim(span cap im Coll_t)=1 ; members={'P1': False, 'P2': False} ; directions=[['1', '-89/140']] (0.0s)
# probe EH      C_coh D=2 block=84 : dim(span cap im Coll_t)=0 ; members={'EH': False} ; directions=[] (0.1s)
# probe EH      C_reg D=2 block=80 : dim(span cap im Coll_t)=0 ; members={'EH': False} ; directions=[] (0.0s)
# probe R2+RC2  C_coh D=4 block=292 : dim(span cap im Coll_t)=0 ; members={'R2': False, 'RC2': False} ; directions=[] (1.1s)
# probe R2+RC2  C_reg D=4 block=265 : dim(span cap im Coll_t)=0 ; members={'R2': False, 'RC2': False} ; directions=[] (0.6s)
# ######## reading time_only_chi_plateau ########
# === C.3/C.4 blockwise: route 1 vs route 2, exact kernels ===
# blockwise pass 36.4s
CHECK [time_only_chi_plateau] ROUTE 1 = ROUTE 2 for Coll_t: all 3321 columns equal exactly (nnz 91568): True
CHECK [time_only_chi_plateau] ROUTE 1 = ROUTE 2 for H_t: all 3321 adjoint-defect columns equal exactly (nnz 130110): True
CHECK [time_only_chi_plateau] grading: every time-only key in block (D, parity) has total derivative count D and parity of the block (blocks have disjoint rows): True
# supplementary mod-p ranks (all rows, p=1e9+7, 998244353) agree with exact ranks in every block: True
CHECK [time_only_chi_plateau] ker Coll_t is contained in ker H_t (each ker Coll_t vector has zero time-only adjoint defect): True
CHECK [time_only_chi_plateau] the 26 formal SA vectors (helmholtz_certificates.txt) lie in ker H_t (specialization: ker H_formal <= ker H_t): True
CHECK [time_only_chi_plateau] D=0: dim Coll_t(ker H_t) = 26 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only_chi_plateau] D=0: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 26: True
CHECK [time_only_chi_plateau] D=1: dim Coll_t(ker H_t) = 0 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only_chi_plateau] D=1: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 0: True
CHECK [time_only_chi_plateau] D=2: dim Coll_t(ker H_t) = 156 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only_chi_plateau] D=2: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 156: True
CHECK [time_only_chi_plateau] D=3: dim Coll_t(ker H_t) = 0 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only_chi_plateau] D=3: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 0: True
CHECK [time_only_chi_plateau] D=4: dim Coll_t(ker H_t) = 390 (mod-p rank lower bound = dim ker H_t - dim ker Coll_t upper bound): True
CHECK [time_only_chi_plateau] D=4: regression-preserving variational image dim Coll_t(ker H_t cap C_reg) = 390: True
# === CLASSIFICATION TABLE (time_only reading; conditional on (S-bv)) ===
# D | size | kerColl | imColl | kerH | Coll(kerH) | kerColl^Creg | kerColl^K | kerH^Creg | kerH^K | Coll(kerH^Creg)
# 0 | 35 | 0 | 35 | 26 | 26 | 0 | 0 | 26 | 0 | 26
# 1 | 160 | 0 | 160 | 0 | 0 | 0 | 0 | 0 | 0 | 0
# 2 | 440 | 0 | 440 | 156 | 156 | 0 | 0 | 156 | 0 | 156
# 3 | 944 | 0 | 944 | 0 | 0 | 0 | 0 | 0 | 0 | 0
# 4 | 1742 | 0 | 1742 | 390 | 390 | 0 | 0 | 390 | 0 | 390
# totals: {'size': 3321, 'size_Creg': 3150, 'size_K': 171, 'dim_ker_Coll_t': 0, 'dim_im_Coll_t': 3321, 'dim_ker_H_t': 572, 'dim_Coll_t_of_ker_H_t': 572, 'dim_ker_Coll_t_cap_Creg': 0, 'dim_ker_Coll_t_cap_K': 0, 'dim_ker_H_t_cap_Creg': 572, 'dim_ker_H_t_cap_K': 0, 'dim_regression_preserving_variational_image': 572}
# variational-image monomial structure per D: {"0": {"n_vectors": 26, "monomial_type_counts": {"sigma^(0) chi_0 h_0": 198}, "sigma_orders_present": [0], "has_chi_derivatives": false, "max_h_order": 0}, "2": {"n_vectors": 156, "monomial_type_counts": {"sigma^(0) chi_0 h_2": 1188}, "sigma_orders_present": [0], "has_chi_derivatives": false, "max_h_order": 2}, "4": {"n_vectors": 390, "monomial_type_counts": {"sigma^(0) chi_0 h_4": 2970}, "sigma_orders_present": [0], "has_chi_derivatives": false, "max_h_order": 4}}
# === C.5 functional probes (weight w = sigma_eps(t) [chi_I == 1]; all preparation/switch derivatives retained) ===
CHECK [time_only_chi_plateau] weighted P1 target: total derivative degree 0, parity 0: True
CHECK [time_only_chi_plateau] weighted P1 target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only_chi_plateau] weighted P2 target: total derivative degree 0, parity 0: True
CHECK [time_only_chi_plateau] weighted P2 target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only_chi_plateau] weighted EH target: total derivative degree 2, parity 0: True
CHECK [time_only_chi_plateau] weighted EH target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only_chi_plateau] weighted R2 target: total derivative degree 4, parity 0: True
CHECK [time_only_chi_plateau] weighted R2 target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only_chi_plateau] weighted RC2 target: total derivative degree 4, parity 0: True
CHECK [time_only_chi_plateau] weighted RC2 target (time-only Euler tensor) is self-adjoint under the time-only Gate V adjoint: True
CHECK [time_only_chi_plateau] divergence identity: d_mu E^{mu nu} = 0 for weighted R2 with w = chi sigma(t) (time-only jets): True
CHECK [time_only_chi_plateau] divergence identity: d_mu E^{mu nu} = 0 for weighted RC2 with w = chi sigma(t) (time-only jets): True
CHECK [time_only_chi_plateau] divergence identity: d_mu E_EH^{mu nu} = (1) (d_mu w) G_lin^{mu nu} for weighted EH (w = chi sigma(t)): True
CHECK [time_only_chi_plateau] ultralocal weighted P1: divergence nonzero (not gauge invariant): True
CHECK [time_only_chi_plateau] P1+P2/C_coh: explicit preimage of direction ['1', '-89/140'] reproduces the weighted target exactly: True
CHECK [time_only_chi_plateau] P1+P2/C_coh: route-2 collapse of the preimage equals the target: True
CHECK [time_only_chi_plateau] P1+P2/C_coh: preimage lies in ker H_t (variational): True
CHECK [time_only_chi_plateau] P1+P2/C_coh: EL(1/2 <h, E_C[h]>) = E_C (independent sympy time-only jet route): True
CHECK [time_only_chi_plateau] P1+P2/C_coh: EL(1/2 <h, E_C[h]>) = E_C via sympy euler_equations with sigma(x0), chi(x), h(x) Functions: True
# probe P1+P2   C_coh D=0 block=10 : dim(span cap im Coll_t)=1 ; members={'P1': False, 'P2': False} ; directions=[['1', '-89/140']] (0.1s)
CHECK [time_only_chi_plateau] P1+P2/C_reg: explicit preimage of direction ['1', '-89/140'] reproduces the weighted target exactly: True
CHECK [time_only_chi_plateau] P1+P2/C_reg: route-2 collapse of the preimage equals the target: True
CHECK [time_only_chi_plateau] P1+P2/C_reg: preimage lies in ker H_t (variational): True
CHECK [time_only_chi_plateau] P1+P2/C_reg: EL(1/2 <h, E_C[h]>) = E_C (independent sympy time-only jet route): True
CHECK [time_only_chi_plateau] P1+P2/C_reg: EL(1/2 <h, E_C[h]>) = E_C via sympy euler_equations with sigma(x0), chi(x), h(x) Functions: True
# probe P1+P2   C_reg D=0 block=10 : dim(span cap im Coll_t)=1 ; members={'P1': False, 'P2': False} ; directions=[['1', '-89/140']] (0.0s)
# probe EH      C_coh D=2 block=84 : dim(span cap im Coll_t)=0 ; members={'EH': False} ; directions=[] (0.1s)
# probe EH      C_reg D=2 block=80 : dim(span cap im Coll_t)=0 ; members={'EH': False} ; directions=[] (0.1s)
# probe R2+RC2  C_coh D=4 block=292 : dim(span cap im Coll_t)=0 ; members={'R2': False, 'RC2': False} ; directions=[] (1.5s)
# probe R2+RC2  C_reg D=4 block=265 : dim(span cap im Coll_t)=0 ; members={'R2': False, 'RC2': False} ; directions=[] (0.3s)
# === new time-only variational directions: reading time_only_chi_plateau, D>=1 ===
CHECK [time_only_chi_plateau] D=2: every new variational direction is sigma(t) x (purely SPATIAL derivative operator on h): no sigma derivative, no time derivative of h: True
CHECK [time_only_chi_plateau] D=2: EL(1/2 <h, E_C[h]>) = E_C for all 156 new directions (independent sympy time-only jet route) (2.2s): True
CHECK [time_only_chi_plateau] D=2: EL check via sympy euler_equations with sigma(x0), h(x) Functions on 3 sample directions: True
CHECK [time_only_chi_plateau] D=2: every new direction remains SA in the double-plateau reading (specialization sigma == 1): True
CHECK [time_only_chi_plateau] D=2: no new direction is SA in the (spacetime-formal sigma) chi_plateau reading: the directions are time-only-specific: True
# D=2 new directions: dim 156 ; h-derivative multi-indices used {'23': 198, '11': 198, '22': 198, '33': 198, '13': 198, '12': 198} ; support N-counts {4: 210} ; |gamma| counts {2: 210}
CHECK [time_only_chi_plateau] D=4: every new variational direction is sigma(t) x (purely SPATIAL derivative operator on h): no sigma derivative, no time derivative of h: True
CHECK [time_only_chi_plateau] D=4: EL(1/2 <h, E_C[h]>) = E_C for all 390 new directions (independent sympy time-only jet route) (6.3s): True
CHECK [time_only_chi_plateau] D=4: EL check via sympy euler_equations with sigma(x0), h(x) Functions on 3 sample directions: True
CHECK [time_only_chi_plateau] D=4: every new direction remains SA in the double-plateau reading (specialization sigma == 1): True
CHECK [time_only_chi_plateau] D=4: no new direction is SA in the (spacetime-formal sigma) chi_plateau reading: the directions are time-only-specific: True
# D=4 new directions: dim 390 ; h-derivative multi-indices used {'1123': 198, '2223': 198, '2333': 198, '1111': 198, '1122': 198, '1133': 198, '2222': 198, '2233': 198, '3333': 198, '1113': 198, '1223': 198, '1333': 198, '1112': 198, '1222': 198, '1233': 198} ; support N-counts {4: 525} ; |gamma| counts {4: 525}
CHECK [time_only_chi_plateau] the 26 ultralocal SA vectors are supported on N=4, gamma=0: True
CHECK [time_only_chi_plateau] D=2: span{ultralocal SA (26) x spatial gamma (6)} (156 independent vectors) lies in ker H_t and has dim = dim ker H_t = 156, hence equals it: True
CHECK [time_only_chi_plateau] D=4: span{ultralocal SA (26) x spatial gamma (15)} (390 independent vectors) lies in ker H_t and has dim = dim ker H_t = 390, hence equals it: True
CHECK supplementary (floating, 60 digits): Wronskian of rho, rho', ..., rho'''' (XII-C ramp profile) nonzero at s = 1/2, 1/3, 1/7: True
# Wronskian W(rho,...,rho^(4)) at s: {'1/2': '88060251340800.0', '1/3': '2.575905490314e+15', '1/7': '4.10243726191388e+17'}
# total runtime 154.1s
ALL CHECKS PASSED