Listing: Listing

# covariance_constraints.py output  (AUDITED REVISION)
# Stage 5 / M.7: Lorentz action, equivariant contacts, exact SA intersections, linear divergence identities
# CONDITIONAL on Input (S-bv); Gate S-bv not discharged; M-C/M-D not claimed
# Prescription covariance != interacting Ward/BV; linear divergence identity != Ward identity
CHECK d_N=(1,4,10,20,35) under (S-bv): True
CHECK dim C_coh = 3321: True
CHECK W dims and C_coh: True
CHECK generator J12 is eta-skew: True
CHECK generator J13 is eta-skew: True
CHECK generator J23 is eta-skew: True
CHECK generator K01 is eta-skew: True
CHECK generator K02 is eta-skew: True
CHECK generator K03 is eta-skew: True
CHECK p_0=1: True
CHECK B_N basis of Poly_N, N=0: True
CHECK B_N basis of Poly_N, N=1: True
CHECK B_N basis of Poly_N, N=2: True
CHECK B_N basis of Poly_N, N=3: True
CHECK B_N basis of Poly_N, N=4: True
CHECK N=0: [rho(K01),rho(K02)]=rho(J12) on B_N: True
CHECK N=1: [rho(K01),rho(K02)]=rho(J12) on B_N: True
CHECK N=2: [rho(K01),rho(K02)]=rho(J12) on B_N: True
CHECK N=3: [rho(K01),rho(K02)]=rho(J12) on B_N: True
CHECK N=4: [rho(K01),rho(K02)]=rho(J12) on B_N: True
CHECK N=0 trivial action: True
CHECK N=0 J12: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=0 J13: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=0 J23: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=0 K01: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=0 K02: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=0 K03: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=0: [rhoD(K01),rhoD(K02)]=rhoD(J12) on d_gamma delta: True
CHECK N=1 J12: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=1 J13: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=1 J23: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=1 K01: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=1 K02: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=1 K03: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=1: [rhoD(K01),rhoD(K02)]=rhoD(J12) on d_gamma delta: True
CHECK N=2 J12: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=2 J13: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=2 J23: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=2 K01: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=2 K02: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=2 K03: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=2: [rhoD(K01),rhoD(K02)]=rhoD(J12) on d_gamma delta: True
CHECK N=3 J12: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=3 J13: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=3 J23: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=3 K01: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=3 K02: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=3 K03: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=3: [rhoD(K01),rhoD(K02)]=rhoD(J12) on d_gamma delta: True
CHECK N=4 J12: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=4 J13: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=4 J23: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=4 K01: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=4 K02: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=4 K03: pull-back action on d_gamma delta = P rho^poly P (eta intertwiner): True
CHECK N=4: [rhoD(K01),rhoD(K02)]=rhoD(J12) on d_gamma delta: True
CHECK first-order basis order d_0..d_3: True
CHECK first order J12: X.(d_nu delta) = X^mu_nu d_mu delta (abstract action matrix = X): True
CHECK first order J12: old coordinate matrix = -X^T: True
CHECK first order J12 (rotation): abstract action = old coordinate matrix: True
CHECK first order J12: abstract = eta (old) eta: True
CHECK first order J13: X.(d_nu delta) = X^mu_nu d_mu delta (abstract action matrix = X): True
CHECK first order J13: old coordinate matrix = -X^T: True
CHECK first order J13 (rotation): abstract action = old coordinate matrix: True
CHECK first order J13: abstract = eta (old) eta: True
CHECK first order J23: X.(d_nu delta) = X^mu_nu d_mu delta (abstract action matrix = X): True
CHECK first order J23: old coordinate matrix = -X^T: True
CHECK first order J23 (rotation): abstract action = old coordinate matrix: True
CHECK first order J23: abstract = eta (old) eta: True
CHECK first order K01: X.(d_nu delta) = X^mu_nu d_mu delta (abstract action matrix = X): True
CHECK first order K01: old coordinate matrix = -X^T: True
CHECK first order K01 (boost): abstract action = -(old coordinate matrix): OLD MATRIX WRONG FOR BOOSTS: True
CHECK first order K01: abstract = eta (old) eta: True
CHECK first order K02: X.(d_nu delta) = X^mu_nu d_mu delta (abstract action matrix = X): True
CHECK first order K02: old coordinate matrix = -X^T: True
CHECK first order K02 (boost): abstract action = -(old coordinate matrix): OLD MATRIX WRONG FOR BOOSTS: True
CHECK first order K02: abstract = eta (old) eta: True
CHECK first order K03: X.(d_nu delta) = X^mu_nu d_mu delta (abstract action matrix = X): True
CHECK first order K03: old coordinate matrix = -X^T: True
CHECK first order K03 (boost): abstract action = -(old coordinate matrix): OLD MATRIX WRONG FOR BOOSTS: True
CHECK first order K03: abstract = eta (old) eta: True
# first-order boost K01 on (d0 delta,d1 delta,d2 delta,d3 delta): abstract = [['0', '1', '0', '0'], ['1', '0', '0', '0'], ['0', '0', '0', '0'], ['0', '0', '0', '0']]
#                                                 old coordinate matrix = [['0', '-1', '0', '0'], ['-1', '0', '0', '0'], ['0', '0', '0', '0'], ['0', '0', '0', '0']]
CHECK symbolic test function: K01.(d_1 delta) = + d_0 delta (old matrix gave - d_0 delta): True
CHECK Poly_N = sum H_{N-2k} dims, N=0: True
CHECK Poly_N = sum H_{N-2k} dims, N=1: True
CHECK Poly_N = sum H_{N-2k} dims, N=2: True
CHECK Poly_N = sum H_{N-2k} dims, N=3: True
CHECK Poly_N = sum H_{N-2k} dims, N=4: True
CHECK N=0: number of phi_{k,j} = Schur count 1: True
CHECK N=0: corrected equivariant maps linearly independent: True
# N=0: corrected C = P o phi_{k,j} equivariant on all 6 generators (exact); maps [(0, 0)]
CHECK N=1: number of phi_{k,j} = Schur count 1: True
CHECK N=1: corrected equivariant maps linearly independent: True
# N=1: corrected C = P o phi_{k,j} equivariant on all 6 generators (exact); maps [(0, 0)]
CHECK N=2: number of phi_{k,j} = Schur count 3: True
CHECK N=2: corrected equivariant maps linearly independent: True
# N=2: corrected C = P o phi_{k,j} equivariant on all 6 generators (exact); maps [(0, 0), (1, 0), (1, 1)]
CHECK N=3: number of phi_{k,j} = Schur count 3: True
CHECK N=3: corrected equivariant maps linearly independent: True
# N=3: corrected C = P o phi_{k,j} equivariant on all 6 generators (exact); maps [(0, 0), (1, 0), (1, 1)]
CHECK N=4: number of phi_{k,j} = Schur count 6: True
CHECK N=4: corrected equivariant maps linearly independent: True
# N=4: corrected C = P o phi_{k,j} equivariant on all 6 generators (exact); maps [(0, 0), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2)]
CHECK N=0 old vector trivially equivariant: True
CHECK OLD (uncorrected) vectors: those changed by P fail exactly the three boost equations; unchanged ones (image only at even spatial order) pass: True
# old vectors failing boost equivariance: ['(1, 0, 0)', '(2, 0, 0)', '(3, 0, 0)', '(3, 1, 0)', '(3, 1, 1)', '(4, 0, 0)', '(4, 1, 0)', '(4, 1, 1)'] ; unchanged by P: ['(0, 0, 0)', '(2, 1, 0)', '(2, 1, 1)', '(4, 2, 0)', '(4, 2, 1)', '(4, 2, 2)']
CHECK equivariant dims 1,1,3,3,6: True
CHECK N=0: exact kernel of the equivariance constraint map (pull-back action) has dim 1, rank 0: True
CHECK N=1: exact kernel of the equivariance constraint map (pull-back action) has dim 1, rank 19: True
CHECK N=2: exact kernel of the equivariance constraint map (pull-back action) has dim 3, rank 147: True
# cokernel = dim codomain - rank, codomain = (+)_6 Hom(H_{4+N},W_N): {0: 6, 1: 101, 2: 753, 3: 3503, 4: 12256}
CHECK equivariant freedom 14 = 5 (K, N<=2) + 9 (C_reg, N>=3): True
CHECK P1, P2, P1-(89/140)P2 are so(1,3)-invariant: True
# --- exact V = Equiv cap SA (14 equivariant coordinates; adjoint defect map; exact QQ kernel) ---
CHECK formal: all 1 basis vectors of V re-verified from scratch: zero adjoint defect AND equivariance (6 generators, all N): True
CHECK formal: dim(V cap K)=dimV-dim pi_reg(V), dim(V cap C_reg)=dimV-dim pi_K(V): True
CHECK formal: dim(V cap K)+dim(V cap C_reg)=1 <= dim V=1: True
CHECK formal: V cap K <= pi_K(V), V cap C_reg <= pi_reg(V): True
CHECK formal: pi_K(V) cap SA = V cap K and pi_reg(V) cap SA = V cap C_reg (projections NOT relabelled as SA): True
# formal          dim V=1 | pi_K(V)=0  V cap K=0 | pi_reg(V)=1  V cap C_reg=1 | dim Coll(V)=1 (2.6s)
CHECK chi_plateau: all 1 basis vectors of V re-verified from scratch: zero adjoint defect AND equivariance (6 generators, all N): True
CHECK chi_plateau: dim(V cap K)=dimV-dim pi_reg(V), dim(V cap C_reg)=dimV-dim pi_K(V): True
CHECK chi_plateau: dim(V cap K)+dim(V cap C_reg)=1 <= dim V=1: True
CHECK chi_plateau: V cap K <= pi_K(V), V cap C_reg <= pi_reg(V): True
CHECK chi_plateau: pi_K(V) cap SA = V cap K and pi_reg(V) cap SA = V cap C_reg (projections NOT relabelled as SA): True
# chi_plateau     dim V=1 | pi_K(V)=0  V cap K=0 | pi_reg(V)=1  V cap C_reg=1 | dim Coll(V)=1 (0.7s)
CHECK double_plateau: all 12 basis vectors of V re-verified from scratch: zero adjoint defect AND equivariance (6 generators, all N): True
CHECK double_plateau: dim(V cap K)=dimV-dim pi_reg(V), dim(V cap C_reg)=dimV-dim pi_K(V): True
CHECK double_plateau: dim(V cap K)+dim(V cap C_reg)=10 <= dim V=12: True
CHECK double_plateau: V cap K <= pi_K(V), V cap C_reg <= pi_reg(V): True
CHECK double_plateau: pi_K(V) cap SA = V cap K and pi_reg(V) cap SA = V cap C_reg (projections NOT relabelled as SA): True
# double_plateau  dim V=12 | pi_K(V)=5  V cap K=3 | pi_reg(V)=9  V cap C_reg=7 | dim Coll(V)=9 (0.9s)
CHECK formal and chi-plateau: dim(Equiv cap SA) = 1: True
CHECK formal: the SA equivariant line lies in C_reg (N=4): True
CHECK formal: Coll(V) = span of E[P1-(89/140)P2] (ratio 17920); V = phi_{2,0} at N=4 (Box^2, gamma=0): True
CHECK formal: supplementary mod-p kernel dims [1, 1, 1] agree with the exact dim V (correct rational reduction): True
CHECK chi_plateau: supplementary mod-p kernel dims [1, 1, 1] agree with the exact dim V (correct rational reduction): True
CHECK double_plateau: supplementary mod-p kernel dims [12, 12, 12] agree with the exact dim V (correct rational reduction): True
# --- linear (gauge) divergence identities: d_mu E^{mu nu}, E with UPPER indices, no metric factor ---
CHECK Identity A: eta^{mu rho} d_rho G^lin_{mu nu} = 0 (G LOWER indices, metric factor correct here): True
CHECK double_plateau: d_mu E^{mu nu}[R2] == 0 identically (weight w=1): True
CHECK double_plateau: d_mu E^{mu nu}[RC2] == 0 identically (weight w=1): True
CHECK double_plateau: d_mu E^{mu nu}[R2-(188813/88456)RC2] == 0 identically (weight w=1): True
CHECK chi_plateau: d_mu E^{mu nu}[R2] == 0 identically (weight w=s_): True
CHECK chi_plateau: d_mu E^{mu nu}[RC2] == 0 identically (weight w=s_): True
CHECK chi_plateau: d_mu E^{mu nu}[R2-(188813/88456)RC2] == 0 identically (weight w=s_): True
CHECK formal: d_mu E^{mu nu}[R2] == 0 identically (weight w=s_*x_): True
CHECK formal: d_mu E^{mu nu}[RC2] == 0 identically (weight w=s_*x_): True
CHECK formal: d_mu E^{mu nu}[R2-(188813/88456)RC2] == 0 identically (weight w=s_*x_): True
CHECK diagnosis: the OLD formula sum_mu eta^{mu mu} d_mu E^{mu nu} is nonzero (this produced the false 'does not vanish'): True
CHECK independent metric linearization: R_lin(metric) = +1 x R_lin(manuscript formula): True
CHECK independent metric linearization: Ric_lin(metric) = +1 x Ric_lin(manuscript formula): True
CHECK independent route: d_mu E^{mu nu}[R2] == 0 (Functions, euler_equations): True
CHECK independent route: old eta^{mu mu}-weighted formula nonzero for R2: True
CHECK independent Euler tensor of int R2 equals the matching_probe/jet_euler target exactly: True
CHECK independent route: d_mu E^{mu nu}[RC2] == 0 (Functions, euler_equations): True
CHECK independent route: old eta^{mu mu}-weighted formula nonzero for RC2: True
CHECK independent Euler tensor of int RC2 equals the matching_probe/jet_euler target exactly: True
# independent curvature route done (13.8s); sign(R)=+1 sign(Ric)=+1 (squares are sign-insensitive)
CHECK preimage [double_plateau C_coh R2]: Coll(C) equals the INDEPENDENTLY built target exactly: True
CHECK preimage [double_plateau C_coh R2]: d_mu E_C^{mu nu} == 0: True
CHECK preimage [double_plateau C_coh R2]: support meets K-block (N<=2): True
CHECK preimage [double_plateau C_coh RC2]: Coll(C) equals the INDEPENDENTLY built target exactly: True
CHECK preimage [double_plateau C_coh RC2]: d_mu E_C^{mu nu} == 0: True
CHECK preimage [double_plateau C_coh RC2]: support meets K-block (N<=2): True
CHECK preimage [double_plateau C_reg (1)R2+(-188813/88456)RC2]: Coll(C) equals the INDEPENDENTLY built target exactly: True
CHECK preimage [double_plateau C_reg (1)R2+(-188813/88456)RC2]: d_mu E_C^{mu nu} == 0: True
CHECK preimage [double_plateau C_reg (1)R2+(-188813/88456)RC2]: support inside C_reg (N>=3): True
CHECK double_plateau: EH identity d_mu E^{mu nu} = (d_mu w) G_lin^{mu nu}[h]  (zero iff w constant): True
CHECK chi_plateau: EH identity d_mu E^{mu nu} = (d_mu w) G_lin^{mu nu}[h]  (zero iff w constant): True
CHECK formal: EH identity d_mu E^{mu nu} = (d_mu w) G_lin^{mu nu}[h]  (zero iff w constant): True
CHECK double_plateau: ultralocal P1-(89/140)P2: d_mu E^{mu nu} = 2 d_mu[w(h^{mu nu}-(89/140)eta^{mu nu} h)] != 0 (not gauge invariant): True
CHECK chi_plateau: ultralocal P1-(89/140)P2: d_mu E^{mu nu} = 2 d_mu[w(h^{mu nu}-(89/140)eta^{mu nu} h)] != 0 (not gauge invariant): True
CHECK formal: ultralocal P1-(89/140)P2: d_mu E^{mu nu} = 2 d_mu[w(h^{mu nu}-(89/140)eta^{mu nu} h)] != 0 (not gauge invariant): True
# wrote covariance_matrix.txt covariance_certificates.txt covariance_summary.json covariance_exact_certificates.json completion_ledger_M89.json
CHECK M-C/M-D not claimed: True
# total runtime 241.8s
ALL CHECKS PASSED
# ---------------- covariance_matrix.txt ----------------
# covariance_matrix.txt -- XVI-M M.7 compatibility matrix (CONDITIONAL on (S-bv)), AUDITED REVISION
# Contact maps C: H_{4+N} -> W_N = span{d_gamma delta : |gamma|<=N}, pull-back action (eta intertwiner)
# rank = rank of the constraint map C -> (C rho_V(X) - rho_delta(X) C)_X = codim of the solution space
# codomain = (+)_{6 generators} Hom(H_{4+N},W_N), dim 6*slots ; coker = dim codomain - rank
#
N  dimH  dimW  slots  rank(=codim)  dim_codomain  dim_coker  dim_sol  block
0     1     1      1             0             6          6        1  K
1     4     5     20            19           120        101        1  K
2    10    15    150           147           900        753        3  K
3    20    35    700           697          4200       3503        3  C_reg
4    35    70   2450          2444         14700      12256        6  C_reg
#
K_block_dim_sol = 5
C_reg_dim_sol = 9
total_equivariant = 14
#
# Exact V = Equiv cap SA and the four objects:
V formal dim_V=1 pi_K(V)=0 V_cap_K=0 pi_reg(V)=1 V_cap_Creg=1 dim_Coll(V)=1
V chi_plateau dim_V=1 pi_K(V)=0 V_cap_K=0 pi_reg(V)=1 V_cap_Creg=1 dim_Coll(V)=1
V double_plateau dim_V=12 pi_K(V)=5 V_cap_K=3 pi_reg(V)=9 V_cap_Creg=7 dim_Coll(V)=9
#
# Linear divergence identities (d_mu E^{mu nu}, E upper indices):
ID_A eta^{mu rho} d_rho G^lin_{mu nu} = 0 : HOLDS (not Ward)
ID_B ultralocal P1-(89/140)P2 : d_mu E^{mu nu} = 2 d_mu[w(h^{mu nu}-(89/140)eta^{mu nu}h)] != 0 (not gauge invariant)
ID_Bprime R2, RC2, R2-(188813/88456)RC2 : d_mu E^{mu nu} = 0 in formal, chi-plateau, double plateau (linear gauge identity; not Ward)
ID_EH : d_mu E^{mu nu} = (d_mu w) G_lin^{mu nu} ; zero in double plateau
#
# Gate S-bv CONDITIONAL. M-C not claimed. M-D not claimed.