check_note.py (OPG-59927) -- exact checks; Python 3.13.5
A  t=0 k=2 N=3  file enum_N3_t0_prove3_certified.json  sha256 63b4905a8e520fbf8dc7a134ed561415853dff7c5c88c787937157f8d43cac29
   certificates: 2, invalid: 0, smallest total sum(y) = 3 (> 2), distinct orbits: 2, duplicates: 0
A  t=1 k=3 N=7  file enum_N3_t1_prove7_certified.json  sha256 0d866535c09e026df0abe8bdc68f72526686843cae26ae8b206619d6f4be0319
   certificates: 18, invalid: 0, smallest total sum(y) = 15/2 (> 6), distinct orbits: 18, duplicates: 0
A  t=2 k=4 N=9  file enum_N3_t2_prove9_certified.json  sha256 253209c9553f7f666b4f1e9e6bc80dc93cb6c2a3b0fd524ec5002f0574593ea6
   certificates: 435, invalid: 0, smallest total sum(y) = 16 (> 8), distinct orbits: 435, duplicates: 0
B  t=0 k=2 N=3  vectors generated 11 (DP count 11, expected 11); >= 3 size-k configurations: 7;
   remaining 4 vectors fall into 2 orbits, all covered by certificates; certificate orbits not needed: 0;
   orbits of all vectors under Sym(S) x Sym(X): 5 (expected 5); fast count = full enumeration on 11 vectors; 0.0 s
B  t=1 k=3 N=7  vectors generated 551 (DP count 551, expected 551); >= 7 size-k configurations: 486;
   remaining 65 vectors fall into 18 orbits, all covered by certificates; certificate orbits not needed: 0;
   orbits of all vectors under Sym(S) x Sym(X): 125 (expected 125); fast count = full enumeration on 551 vectors; 0.0 s
B  t=2 k=4 N=9  vectors generated 297955 (DP count 297955, expected 297955); >= 9 size-k configurations: 294180;
   remaining 3775 vectors fall into 435 orbits, all covered by certificates; certificate orbits not needed: 0;
   orbits of all vectors under Sym(S) x Sym(X): 27871 (expected 27871); fast count = full enumeration on 11918 vectors; 19.1 s
C  K_{3,3}+K_0 (k=2)        lists of size 2: proper colouring exists: False; colourable after dropping any one Y-vertex: True (0.00 s)
C  K_{3,7}+K_1 (k=3)        lists of size 3: proper colouring exists: False; colourable after dropping any one Y-vertex: True (0.00 s)
C  K_{3,9}+K_2 (k=4)        lists of size 4: proper colouring exists: False; colourable after dropping any one Y-vertex: True (0.00 s)
C  K_{3,16}+K_3 (k=5)       lists of size 5: proper colouring exists: False; colourable after dropping any one Y-vertex: True (0.00 s)
C  K_{3,19}+K_4 (k=6)       lists of size 6: proper colouring exists: False; colourable after dropping any one Y-vertex: True (0.01 s)
C  K_{4,4}+K_1 (k=3)        lists of size 3: proper colouring exists: False; colourable after dropping any one Y-vertex: True (0.00 s)
D  phi(3,b) = phi(b,3) = min{t : N_3(t) > b}, using N_3(0)=3, N_3(1)=7, N_3(2)=9 and N_3(t) >= 9 for t >= 2:
   b=2: 0; b=3: 1; b=4: 1; b=5: 1; b=6: 1; b=7: 2; b=8: 2; b=9: >=3; b=10: >=3; b=11: >=3
TOTAL failures: 0   (20.3 s)
ALL CHECKS PASSED: N_3(0) >= 3, N_3(1) >= 7, N_3(2) >= 9 (Parts A, B); N_3(t) <= 1 + k*ceil(k/2) for k = 2..6 and N_4(1) <= 4 (Part C).
