Nyman-Beurling

• The Nyman-Beurling distance restricted to a digit design: d_N^2(S) = inf over real c of norm(chi - sum_(n in S, n <= N) c_n rho_n)^2 in L^2(0, infinity), rho_n(t) = {1/(nt)}, chi = 1_(0,1], for the full set S = N and for S = S_F at base 3 {0,1}, N = 3^level.
• The Gram entries <rho_h, rho_k> for coprime h, k are Vasyunin's formula as printed by Bettin and Conrey, (log 2 pi - gamma)/2 (1/h + 1/k) + ((k - h)/(2hk)) log(h/k) - (pi/(2hk)) (V(h/k) + V(k/h)) with V(h/k) = sum_(m=1)^(k-1) {mh/k} cot(pi m/k), and <rho_(gh), rho_(gk)> = <rho_h, rho_k>/g; the target is <chi, rho_n> = (log n + 1 - gamma)/n and d_N^2 = 1 - b^T G^(-1) b.
• The (0,1) variant drops int_1^infinity, so its Gram matrix is G - v v^T with v_n = 1/n.
• Every Gram system is solved in PARI at 60 digits and again at 90, and the 1-norm condition number norm(G)_1 norm(G^(-1))_1 is printed beside each distance.
• The local floor delta_K(S): the least value of sigma^2 + sum_(k=1)^(K-1) int_k^(k+1) (sigma u - 1 - sum_(n in S, n <= k) c_n floor(k/n))^2 du/u^2 over real sigma and c, a finite least squares; d_N^2(S) >= delta_K(S) for every N and K, and delta_K(S) > 0 exactly when a squarefree integer below K is missing from S. q_0(S) is the least squarefree integer outside S.

RUN

• uv run python research/lab/py/nyman-beurling/nyman_beurling.py gram: Vasyunin's formula against exact quadrature, (1/(hk)) int_0^1 {hw}{kw} zetahurwitz(2, w) dw on the pieces between the breakpoints, at eleven pairs, and d_N^2 at four small sets against the quadrature of the residual itself, with a Hurwitz tail. About 2 seconds.
• uv run python research/lab/py/nyman-beurling/nyman_beurling.py table: the table, full set at N = 3^2..3^6 and the design at N = 3^1..3^7, both precisions. About 165 seconds.
• uv run python research/lab/py/nyman-beurling/nyman_beurling.py floor: q_0 for eight designs, delta_K for base 3 {0,1} at K = 2..40, the full set at K = 2..12 and seven other designs at K = 40. About 6 seconds; floor 100 runs the same to K = 100 in about 125 seconds and reads 0.100877 at K = 60 and 0.101926 at K = 100.
• uv run python research/lab/py/nyman-beurling/nyman_beurling.py zeros: 2 sum_(0 < gamma < T) 1/(1/4 + gamma^2) plus the smooth tail 2 (log(T/2 pi) + 1)/(2 pi T) against 2 + gamma - log(4 pi), T = 1000. About 5 seconds; zeros 3000 takes about 215 seconds and reads 0.04619150 against 0.04619142.
• Prints only; the PARI script it runs is written to the study's own scratch folder under a gitignored data/ at the repo root.

WITNESSES

• zeta.md, THE CLOSURE PROBLEM, the formula row: the eleven pairs agree to 1e-57 at 40 digits, norm(rho_1)^2 = log 2 pi - gamma, and the four residual quadratures agree with 1 - b^T G^(-1) b to every printed digit.
• zeta.md, THE CLOSURE PROBLEM, the table row: 0.024015, 0.015314, 0.010949, 0.008254, 0.006738 on the full set, 0.116043, 0.106008, 0.104951, 0.104689, 0.104610 on the design, rounded up, the condition numbers, d_N^2 log N falling through 0.046191.
• zeta.md, THE CLOSURE PROBLEM, the floor rows: delta_3 = 0.070873, delta_40 = 0.098689, delta_100 = 0.101926 against d_2187^2 = 0.104579, the full set at 1e-75, and the seven designs at K = 40.
• zeta.md, THE CLOSURE PROBLEM, the constant: 0.0461922 from 649 zeros against 0.0461914.