Listing: Listing

# kernel_relations.py output   (python 3.13.5, sympy 1.14.0, exact QQ arithmetic)
# M.2 numerical-kernel relations: presentation C^{Lambda_adm} -> V_adm, degree by degree
# --- finite symbolic check behind the MOTIVATION for (S-bv) (Euclidean calculation; does not certify (S-bv)) ---
CHECK int_0^oo e^{-p tau} sin(p r) dp = r/(tau^2+r^2): True
CHECK XVI-K eq:W-fam integrand at z0=-i*tau gives 1/(4 pi^2 (tau^2+r^2)) (radial integration): True
CHECK d^alpha (1/Q) = p_alpha / Q^{1+|alpha|} with p_alpha homogeneous of degree |alpha| (all 70 |alpha|<=4): True
CHECK p_0 = 1: True
CHECK p_{e_0} = -2 z0  (coordinate derivative d_0 Q = 2 z0): True
CHECK p_{e_1} = +2 z1  (coordinate derivative d_1 Q = -2 z1): True
CHECK sign example: p_{e0} p_{e1} = -4 z0 z1: True
CHECK sign example: p_{e0+e1} = -8 z0 z1: True
CHECK sign example: (d0 S)(d1 S) = (1/2) S d0 d1 S  (numerators -4 z0 z1 = (1/2)(-8 z0 z1)): True
# labelled generators per N: [1, 4, 20, 60, 170]  total 255
CHECK label counts 1,4,20,60,170 (=Lambda_adm, XVI-L): True
CHECK qo_lambda = c^2 n_lambda (Q-i0)^{-2-N}, n_lambda=p_alpha p_beta homogeneous of degree N, for all 255 labels: True
CHECK N=0: greedy basis has rank-many elements (1): True
CHECK N=0: exact reconstruction n_lambda = sum_b R_{lambda b} n_b for all 1 labels: True
CHECK N=0: every exported relation vector annihilates the numerator map exactly (0 relations): True
CHECK N=0: dim ker pi_N = #labels - rank = 0: True
CHECK N=1: greedy basis has rank-many elements (4): True
CHECK N=1: exact reconstruction n_lambda = sum_b R_{lambda b} n_b for all 4 labels: True
CHECK N=1: every exported relation vector annihilates the numerator map exactly (0 relations): True
CHECK N=1: dim ker pi_N = #labels - rank = 0: True
CHECK N=2: greedy basis has rank-many elements (10): True
CHECK N=2: exact reconstruction n_lambda = sum_b R_{lambda b} n_b for all 20 labels: True
CHECK N=2: every exported relation vector annihilates the numerator map exactly (10 relations): True
CHECK N=2: dim ker pi_N = #labels - rank = 10: True
CHECK N=3: greedy basis has rank-many elements (20): True
CHECK N=3: exact reconstruction n_lambda = sum_b R_{lambda b} n_b for all 60 labels: True
CHECK N=3: every exported relation vector annihilates the numerator map exactly (40 relations): True
CHECK N=3: dim ker pi_N = #labels - rank = 40: True
CHECK N=4: greedy basis has rank-many elements (35): True
CHECK N=4: exact reconstruction n_lambda = sum_b R_{lambda b} n_b for all 170 labels: True
CHECK N=4: every exported relation vector annihilates the numerator map exactly (135 relations): True
CHECK N=4: dim ker pi_N = #labels - rank = 135: True
# === DEGREE-BY-DEGREE PRESENTATION (conditional on (S-bv)) ===
#  N | labelled generators | dim Poly_N | d_N = dim H_{4+N} | dim ker pi_N
#  0 |   1 |  1 |  1 |   0
#  1 |   4 |  4 |  4 |   0
#  2 |  20 | 10 | 10 |  10
#  3 |  60 | 20 | 20 |  40
#  4 | 170 | 35 | 35 | 135
CHECK d_N = dim Poly_N = C(N+3,3) = 1,4,10,20,35 : numerator map is ONTO homogeneous polynomials of degree N: True
# dim V_adm = 70 (vs 255 labels); dim ker pi = 185
CHECK dim V_adm = 70 (conditional on (S-bv)): True
CHECK every label with N<=2 is an XVI-K label (V_K block = C_2 block): True
# V_K = H_4 + H_5 + H_6 ; dim V_K = 15 = {0: 1, 1: 4, 2: 10} ; V_K cap H_7 = V_K cap H_8 = 0
CHECK dim V_K = 1+4+10 = 15 (25 XVI-K generators, 10 relations, all at N=2): True
CHECK selected basis of H_4+H_5+H_6 consists of XVI-K labels (basis extends a basis of V_K): True
# === UNCONDITIONAL STRUCTURAL RELATIONS (square S = 0 on R^4\{0}; no (S-bv)) ===
#  N=0: wave-operator relations generated 0, span dim 0 ; dim ker pi_N = 0 ; unconditional upper bound d_N <= 1
#  N=1: wave-operator relations generated 0, span dim 0 ; dim ker pi_N = 0 ; unconditional upper bound d_N <= 4
CHECK N=2: every wave-operator relation lies in ker pi_N (consistency with (S-bv)): True
#  N=2: wave-operator relations generated 1, span dim 1 ; dim ker pi_N = 10 ; unconditional upper bound d_N <= 19
CHECK N=3: every wave-operator relation lies in ker pi_N (consistency with (S-bv)): True
#  N=3: wave-operator relations generated 8, span dim 8 ; dim ker pi_N = 40 ; unconditional upper bound d_N <= 52
CHECK N=4: every wave-operator relation lies in ker pi_N (consistency with (S-bv)): True
#  N=4: wave-operator relations generated 36, span dim 36 ; dim ker pi_N = 135 ; unconditional upper bound d_N <= 134
# unconditional upper bound: dim V_adm <= 210 ; (S-bv) value: 70
CHECK wave-operator span is contained in the full kernel (dimension check): True
# relations NOT generated by the wave-operator family (need (S-bv)): {0: 0, 1: 0, 2: 9, 3: 32, 4: 99}
# sample N=2 relations (reconstruction form  e_lambda - sum_b R e_b = 0):
#   (-1)*q{(), 33} + (-1)*q{(), 22} + (-1)*q{(), 11} + (1)*q{(), 00}
#   (-1)*q{(), 33} + (-1/2)*q{(), 22} + (-1/2)*q{(), 11} + (1)*q{3, 3} + (1)*q{0, 0}
#   (-1/2)*q{(), 01} + (1)*q{1, 0}
#   (-1/2)*q{(), 02} + (1)*q{2, 0}
# dim D_{<=N} = {0: 1, 1: 5, 2: 15, 3: 35, 4: 70}
# dim C_coh = sum_N d_N dim D_{<=N} = 3321   (raw labelled slots 14321, XVI-L count 14321)
# dim C_reg (Delta R|_{V_K}=0) = d_3*35 + d_4*70 = 3150
CHECK raw slot count reproduces XVI-L 14321: True
CHECK dim C_coh = 3321, dim C_reg = 3150 (conditional on (S-bv)): True
# wrote kernel_basis_table.txt  sha256=267344b69a488ec7f99fa3a86180fe1cf25431c60a31beeec3d777da579d88e8
# wrote kernel_relation_matrix.txt  sha256=f25684da36b2f19709e344ab3cb599c093bf8d4e9b236ba3ef069ac2fc8dcbec
# wrote kernel_basis.json  sha256=fcbe2876abd2a6b410813f732356f829c970c22c3a333f7e4ec34909c0e0793b
# total runtime 2.9s
ALL CHECKS PASSED