Claims · 14 dated claims on Half interval, each with its tag and its witness; newest 2026-10-03.
Proved At an odd base base = 2 fill - 1 the half interval F = {0..fill-1} has 2 S_F equal to the design on the even digits, and at a prime base p its set S_F is {k >= 1 : p does not divide C(2k,k)}, since Legendre's formula makes v_p(C(2k,k)) the number of carries in k + k. Witness: mobius The half interval, bullet The object.
Proved For every digit set the shifted grid sums satisfy Sigma_i(s) = (T^i 1)(base^i s) with (T phi)(t) = sum_(r < base) abs(hat F((t+r)/base)) phi((t+r)/base), so any phi with 1 <= phi <= Phi and T phi <= lambda phi is a shifted-grid certificate with C_F = Phi and base^(alpha_1) = lambda/fill. Witness: mobius The half interval, bullet The transfer operator.
Proved At every odd base >= 3 the half interval has T phi <= lambda phi for phi = 1 + abs(sin(pi t))/2 and lambda = fill + X_0 + csc(pi/(2 base))/2, X_0 = (base/pi)(log(base + 3) + gamma + log tan(3 pi/8 + pi/(4 base))) + (sqrt 2 - 4/pi)(base + 1)^2/(8 base), so Sigma_i(s) <= (3/2) lambda^i at every level and shift. Witness: mobius The half interval, bullet The certificate.
Proved The per-digit l^1 cost of the half interval is (2/pi) log base + O(1) from both sides: lambda/fill <= (2/pi) log base + 2.60043004 + 3.4/base at every odd base >= 101, checked directly at odd 101..3001 and four larger bases, and at every odd base >= 9 every grid sum at level i is at least (((2/pi) log base - 1.31) fill)^i. Witness: mobius The half interval, bullet The constant is 2/pi; lab/py/interval-digits verb wall.
Proved The half interval carries a shifted-grid certificate with alpha_1 < 1/5 at every odd base >= 94939, where 1/5 - alpha_1 >= 1.3678 * 10^-8, and with alpha_1 < 1/4 at every odd base >= 3789, where 1/4 - alpha_1 >= 7.9625 * 10^-6, each certified at 120 bits up to a monotone tail bound. Witness: lab/py/interval-digits verb wall; mobius The half interval, bullet The wall.
Proved At every odd base >= 94939 the half interval has abs(M_F(x)) <= C A_F(x) exp(-c sqrt(log x)) and abs(sum_(n <= x, n in S_F) Lambda(n) - kappa_F A_F(x)) <= C A_F(x) exp(-c sqrt(log x)) at every x >= 2, kappa_F = (base/phi(base)) #{1 <= f < fill : gcd(f, base) = 1}/fill, with C, c > 0 computable from base alone. Witness: mobius The half interval, bullet The theorem on the half interval; coprime The half interval.
Proved At every prime p >= 94939 the sum of mu(k) over k <= x with p not dividing C(2k,k) is at most C A(x) exp(-c sqrt(log x)) in absolute value, and the sum of Lambda(k) over the same k is p A(x)/(p+1) within C A(x) exp(-c sqrt(log x)), A(x) the count of such k. Witness: mobius The half interval, bullet The theorem on the half interval.
Proved Under the generalized Riemann hypothesis for every Dirichlet character, at every odd base >= 3789 the half interval has abs(M_F(x)) <<_(base, eps) A_F(x)^(1 - delta + eps) with delta = (1/4 - alpha_1)/alpha_base > 0, and 1 - delta tends to 3/4 as the base grows. Witness: mobius The half interval, bullet The theorem on the half interval; coprime bullet What the bar 1/4 buys.
Proved If for every eps > 0 some C_eps gives abs(M_F(x)) <= C_eps A_F(x)^(1/2 + eps) on the half interval at every odd base and every x >= 1, the Riemann hypothesis holds, trivially, since S_F contains every integer below fill; no converse is claimed. Witness: mobius The half interval, bullet Uniform square-root cancellation over the bases implies RH.
Verified Maynard 2022 Theorem 1.3 at consecutive excluded digits with q - s >= q^(4/5 + eps), q large in terms of eps, contains the prime asymptotic on the half interval with a log-power error; it prints the main term with q - 1 where its 2019 constant has q - s, giving p/(2(p-1)) against p/(p+1) at a prime p, and its Section 9 constant clears 1/5 there only from q = 7777884825. Witness: arXiv:1510.07711v1 read at source; lab/py/interval-digits verb wall.
Verified Maynard 2019 Theorem 1.2 gives the order of magnitude of the primes avoiding the top block B = {q - s..q-1} at s <= q - q^(57/80) for q large, a range holding the half interval, and its remark gives the asymptotic only for the primes avoiding the bottom block B = {0..s-1} at s <= q - q^(3/4 + delta); neither Maynard paper carries a Mobius sum. Witness: arXiv:1604.01041v2 read at source.
Verified The three sums of the certificate's proof hold on grids of shifts at every odd base 3..401 and at 1001, 10001, 100001, and the bounds on G and T phi and the lower bound on G at every odd base 3..401 and at five larger bases 1001..100001, G reaching at most 0.999857 of its bound, and the 40-digit grid sums at nine bases 3..21 stay at most 0.557408 of (3/2) lambda^i. Witness: lab/py/interval-digits verb check.
Conjecture The true growth rate of the half interval's transfer operator reads (2/pi) log base + 2.26 near base 7 * 10^4 and meets base^(1/5) between 70001 and 80001, and its one-step constant reads (2 sqrt2/pi) log base + 1.19, clearing 1/4 from 7075 and 1/5 from 317063. Witness: lab/py/interval-digits verb rate.
Proved At every odd base >= 3 the half interval F = {0..fill-1}, base = 2 fill - 1, has kappa_F = (base/phi(base)) #{f in F : gcd(f, base) = 1}/fill = base/(base + 1), since the coprime residues pair as f and base - f, never equal at odd base, with exactly one of each pair at most fill - 1, so phi(base)/2 of them lie in F. Witness: mobius.md, The half interval.