== run2_core: NMAX_CLOSED=40 NMAX_INV=8 NMAX_SOLVE=10
ok   Lemma 2.1(a): (p,q) bijection onto [0,2n]^2 with p+q odd; colour rule; n(n+1) squares of each colour
ok   Lemma 2.1(b): E,W,N,S = (p+1,q+1),(p-1,q-1),(p+1,q-1),(p-1,q+1); exist iff in [0,2n]^2
ok   Lemma 2.1(c): edge squares have exactly the stated two neighbours, n per edge, no square on two edges
ok   Lemma 2.1(d): colours of the four edges
ok   (2.2): K(b,w) = -1 iff w = N(b); K^T(w,b) = -1 iff b = S(w)
ok   Lemma 4.1(a): (s,t) from s = n+eps x+y, t = n+eps x-y equals (p,q) / (2n-q,2n-p); parities; bijection
ok   Lemma 4.1(b): N = (s+1,t-1), S = (s-1,t+1), {E,W} = {(s+1,t+1),(s-1,t-1)}, existence
ok   Lemma 4.1(c) and Section 3: Sigma_S = {s=0} = black squares on p=0 or q=2n; Sigma_N = {t=0}
ok   Lemma 3.1: K(1_W) = 2(1_B - 1_{Sigma_S}), K^T(1_B) = 2(1_W - 1_{Sigma_N})
ok   Lemma 4.2: kappa(m,k)+kappa(m,k+1) = 2 lambda(m,k), kappa(m+1,k+1)-kappa(m+1,k) = -2 lambda(m,k), lambda(0,k)=1, lambda(n,k)=0
ok   kappa_n(m,k) by polynomial multiplication = binomial convolution sum_i (-1)^i C(k,i) C(n-k,m-i)
ok   Proposition 4.3: K(2 phi) = 2 1_{Sigma_S} with 2phi(s,t) = -kappa_n((s+1)/2, t/2)
ok   Corollary 1.3 (proof): K^T(2 psi) = 2 1_{Sigma_N} and psi(s,t) = phi(t,s)
ok   Theorem 1.2 via Proposition 3.2: values C(n,m) on Sigma_N, sum -(2^n-1)/2, S_n = formula
ok   Remark 5.1: sum of phi over W = -(2^n-n-1)/2; n(n+1)/2 + it = formula; sum_k sum_{m>=1} kappa = 2^n-n-1
     n=1: N=2 det=-2 entry sum=1 block=-1/2
     n=2: N=6 det=-8 entry sum=5/2 block=-3/2
     n=3: N=12 det=64 entry sum=4 block=-7/2
     n=4: N=20 det=1024 entry sum=9/2 block=-15/2
     n=5: N=30 det=-32768 entry sum=2 block=-31/2
     n=6: N=42 det=-2097152 entry sum=-15/2 block=-63/2
     n=7: N=56 det=268435456 entry sum=-32 block=-127/2
     n=8: N=72 det=68719476736 entry sum=-175/2 block=-255/2
ok   Proposition 2.2 (n <= 8): |det K_n| = 2^(n(n+1)/2); K K^-1 = I exactly
ok   Theorem 1.2 (n <= 8): sum of all entries of the exact inverse = formula
ok   Proposition 3.2 (n <= 8): Sigma_N x Sigma_S block sum = 1/2 - 2^(n-1); S_n = n(n+1)/2 + n/2 + it
ok   Corollary 1.3 (n <= 8): row sums (1-kappa((s+1)/2,t/2))/2, column sums (1-kappa((t+1)/2,s/2))/2, all in Z/2
ok   Proposition 4.3 by the inverse (n <= 8): K^-1 1_{Sigma_S} = phi
ok   Theorem 1.2 (n <= 10): S_n by exact solution of K y = 1_B equals the formula   values: 1, 5/2, 4, 9/2, 2, -15/2, -32, -175/2, -206, -903/2
ok   Lemma 2.3 (n <= 10): both colourings, white on top / at the bottom, and the transposes: same sum
     Table 1 (formula, n = 1..12): 1, 5/2, 4, 9/2, 2, -15/2, -32, -175/2, -206, -903/2, -952, -3927/2
     wrong formula (n+1)^2/2 - 2^n, n = 1..12: 0, 1/2, 0, -7/2, -14, -79/2, -96, -431/2, -462, -1927/2, -1976, -8023/2
TOTAL: 22 checks, 0 failed -- ALL OK  (1.8 s)
