== date (UTC): 2026-10-10T06:15:30Z
== machine: Darwin arm64, Apple M3 Max, 16 cores
== python3: Python 3.13.5;  venv python: Python 3.13.5, numpy 2.5.2, scipy 1.18.1
== sha256 of certificates and checkers
47f44020b60bc90595ca244254cc1559c3f5b35c3a6b9e608a1bcf57c5eac854  cert_ABBB_q20.txt
406968b0f4c733a3f76f66bb83f3a2e07cc05d5b528b6ff78de90c834b518bc7  cert_ABBB_q280.txt
9ab036c692744db447329db6a191ac7b0d4fc0c47d56dedf1b01eaba9951fe06  cert_ABBB_density_q280.txt
f4e10419c97335ec00b117d2318301baefdf7c54b645f4562262d1a560f6f2a2  cert_ABB_3D.json
80ae2ad2d778490494dbff1f572acde732d54357466abbb2eb42390991fe6a18  check_gcert.py
9dcf6e86be56490988f34903a3dcc48a70363692d9348cb254c3a23dd495a02e  check_gcert_numpy.py
efa8664a7eb589f69a51df907d9c000a8168390691e619a7a5f4c01b74f5bb26  check_extremal.py
0304d079b0cdba825b9ca13088e05a5a1263caa39bfcfd2e99baf00a0fc34fe6  check_density.py
dbac281aeca869c580dac8c2e56055931a50bd18be453e36f9ed266c4eb380d7  identity_test.py
e389933cac1c66469d79c57732ddf4e388752d94d1f5bb18c75194413ccf107c  rejection_tests.py
9fbc578af368678ef78eef4dc06fc87397c0d114f3670717ab9475caf2f0aa9c  check_abb3d.py
3c7bd49a74b703702f7396112d402e3a5532e680a84d6925a55cf9c6d17bf9ad  rejection_tests_3d.py
9df336750c12f4d2c9f6eac7f1e13ffa1edfe7ac9ae3b974717c8fe40badf0fd  identity_test_3d.py
6de429508e2b3a0aa34b59ca13ea5bf947ffde299a2afb514e8b3f062661fd89  check_abb2d.py

== $ python3 check_gcert.py cert_ABBB_q20.txt
(L) appearances of ABBB in the 5x5 grid: 40; size 25; concentration = 40/25
(U) q = 20; nonzero table entries: 1222; max |G| = 29
(U) max over all 65536 patterns of 20*q*(Psi0 + D(X) + D(X^T)) = 640  <=  32*q = 640
(U) number of patterns with equality: 568
CERTIFICATE VALID: every 4x4 pattern satisfies Psi0 + D(X) + D(X^T) <= 8/5; hence C_2(ABBB) <= 8/5, and (L) gives equality.
   [exit status 0, 1 s]

== $ python3 check_gcert.py cert_ABBB_q280.txt
(L) appearances of ABBB in the 5x5 grid: 40; size 25; concentration = 40/25
(U) q = 280; nonzero table entries: 1300; max |G| = 404
(U) max over all 65536 patterns of 20*q*(Psi0 + D(X) + D(X^T)) = 8960  <=  32*q = 8960
(U) number of patterns with equality: 10
CERTIFICATE VALID: every 4x4 pattern satisfies Psi0 + D(X) + D(X^T) <= 8/5; hence C_2(ABBB) <= 8/5, and (L) gives equality.
   [exit status 0, 0 s]

== $ PYTHON check_gcert_numpy.py cert_ABBB_q20.txt
q = 20; max of 20*q*(Psi0 + D(X) + D(X^T)) over 65536 patterns = 640; 32*q = 640; valid = True; tight patterns = 568
   largest value below the bound: 8/5 - 1/20
   [exit status 0, 0 s]

== $ PYTHON check_gcert_numpy.py cert_ABBB_q280.txt
q = 280; max of 20*q*(Psi0 + D(X) + D(X^T)) over 65536 patterns = 8960; 32*q = 8960; valid = True; tight patterns = 10
   largest value below the bound: 8/5 - 3/70
   [exit status 0, 0 s]

== $ python3 check_extremal.py cert_ABBB_q280.txt
(1) q = 280: patterns with Psi = 8/5: 10; all other patterns have Psi <= 8/5 - 3/70
(2) tight set equals the ten windows of the two 5-queens lattices: True
(3) gluing rule (unique continuation to the right and downwards, inside the same lattice): True
(4) every non-A cell of the lattice configurations lies on an appearance: True
ALL CHECKS PASSED
   [exit status 0, 1 s]

== $ python3 check_density.py cert_ABBB_density_q280.txt
(T) q = 280; max over all 65536 patterns of 28*q*(Psi0 + nA/28 + D(X) + D(X^T)) = 13440 <= 48*q = 13440; tight patterns: 206
(S) 5x5 lattice grid: concentration 8/5, density of A 1/5, c + (4/7) alpha = 12/7
(S) 4x4 grid: concentration 3/2, density of A 3/8, c + (4/7) alpha = 12/7
(Q) 4x4 grid: delta = 1 - c/(8/5) = 1/16; total variation distance to (1/5,4/5) = 7/40; ratio = 14/5
ALL CHECKS PASSED
   [exit status 0, 0 s]

== $ PYTHON check_gcert_numpy.py --density cert_ABBB_density_q280.txt
q = 280; max of 28*q*(Psi0 + nA/28 + D(X) + D(X^T)) over 65536 patterns = 13440; 48*q = 13440; valid = True; tight patterns = 206
   [exit status 0, 0 s]

== $ python3 identity_test.py cert_ABBB_q280.txt 400
400 grids tested: identity (ii) exact in every case; ct <= N_bin <= (8/5)|G| in every case; largest N_bin/|G| seen = 1.6
   [exit status 0, 0 s]

== $ PYTHON rejection_tests.py cert_ABBB_q280.txt
original certificate: pure-python checker accepts, numpy checker accepts  [expected: accept]  ok
G[65] increased by 1 (top block of the tight window W(2,0)): pure-python checker REJECTS, numpy checker REJECTS  [expected: reject]  ok
denominator q replaced by q+20: pure-python checker REJECTS, numpy checker REJECTS  [expected: reject]  ok
table negated: pure-python checker REJECTS, numpy checker REJECTS  [expected: reject]  ok
zero table (Psi0 alone): pure-python checker REJECTS, numpy checker REJECTS  [expected: reject]  ok
entries permuted (index y -> y xor 1): pure-python checker REJECTS, numpy checker REJECTS  [expected: reject]  ok
40 random entries changed by +-q/2: pure-python checker REJECTS, numpy checker REJECTS  [expected: reject]  ok
REJECTION TESTS PASSED
   [exit status 0, 1 s]

== $ nice -n 10 PYTHON check_abb3d.py cert_ABB_3D.json
terms: 160; monomials after expansion: 4941; Qden = 1451520; max |Z| = 5362624
(a) all congruence-class sums vanish: True
(b) max over all 2^27 patterns of 9*Qden*(Psi0 + Q) = 78382080;  54*Qden = 78382080;  valid = True
    patterns with equality: 39;  largest value below the bound = 54*Qden - 155376
(L) appearances of ABB in the 3x3x3 grid with A on a coordinate plane: 162; size 27; concentration 162/27
(E) the patterns with equality are exactly the 39 hyperplane windows: True
(E) unique continuation of hyperplane windows along the three axes: True
CERTIFICATE VALID: Psi0 + Q <= 6 on all patterns; hence C_3(ABB) <= 6, and (L) gives equality.
   [exit status 0, 18 s]

== $ python3 check_abb2d.py
max over the 512 patterns of 3*(Psi0 + Q) = 6 (bound 6); patterns with equality: 72
VALID: Psi0 + Q <= 2 on all 3x3 patterns, hence C_2(ABB) <= 2.
   [exit status 0, 0 s]

== $ nice -n 10 PYTHON identity_test_3d.py cert_ABB_3D.json 10
shape (1, 1, 3): ct = 0, N_bin = 0, cells = 3, N_bin/cells = 0.000; identity exact
shape (1, 2, 3): ct = 24, N_bin = 24, cells = 6, N_bin/cells = 4.000; identity exact
shape (2, 2, 2): ct = 0, N_bin = 0, cells = 8, N_bin/cells = 0.000; identity exact
shape (3, 3, 3): ct = 162, N_bin = 162, cells = 27, N_bin/cells = 6.000; identity exact
shape (2, 3, 4): ct = 56, N_bin = 56, cells = 24, N_bin/cells = 2.333; identity exact
shape (3, 4, 5): ct = 163, N_bin = 216, cells = 60, N_bin/cells = 3.600; identity exact
shape (1, 5, 6): ct = 102, N_bin = 102, cells = 30, N_bin/cells = 3.400; identity exact
shape (4, 4, 4): ct = 228, N_bin = 228, cells = 64, N_bin/cells = 3.562; identity exact
shape (3, 3, 6): ct = 200, N_bin = 200, cells = 54, N_bin/cells = 3.704; identity exact
shape (2, 5, 3): ct = 108, N_bin = 108, cells = 30, N_bin/cells = 3.600; identity exact
ALL IDENTITY TESTS PASSED
   [exit status 0, 1 s]

== $ nice -n 10 PYTHON rejection_tests_3d.py cert_ABB_3D.json SCRATCH/rej3d
original: checker accepts [expected: accept] ok   | CERTIFICATE VALID: Psi0 + Q <= 6 on all patterns; hence C_3(ABB) <= 6, and (L) gives equality.
one_coefficient_plus_1: checker REJECTS [expected: reject] ok   | AssertionError
all_coefficients_doubled: checker REJECTS [expected: reject] ok   | AssertionError
all_coefficients_negated: checker REJECTS [expected: reject] ok   | AssertionError
denominator_doubled: checker REJECTS [expected: reject] ok   | AssertionError
REJECTION TESTS PASSED
   [exit status 0, 69 s]
