interval-digits

• The numbers of the half interval on mobius: the digit set F = {0..fill-1} at the odd base base = 2 fill - 1, its transfer certificate, the walls it sets at the bars 1/5 and 1/4, a falsification of every inequality behind it, readings of the true rate, and a sanity meter.
• wall: the constant c_inf of the tail bound lambda/fill <= (2/pi) log base + c_inf + 3.4/base, proved from base 101 and checked directly at every odd base 101..3001 and at 94939, 200001, 10^6 + 1, 10^8 + 1, the least base where the tail bound clears each bar, the closed form lambda/fill certified at 120 bits at every odd base from the wall to that point and failing at the base below, bar - alpha_1 and the gap at the wall, the bars read as the exact intervals 1/5 and 1/4, the lower bound min G/fill >= (2/pi) log base - 1.31 at every odd base 9..9999 with its tail from 101, the density floor 1 - alpha_base < 1/5, alpha_1 at three larger bases, and the base where the constant of the Section 9 sketch of Maynard 2022 clears 1/5 on the same set, calibrated against its one-missing-digit crossing 1520573.
• check: the three sums of the certificate's proof, the bP, bQ and signed aP sums, against their bounds at every odd base 3..401 and at 1001, 10001, 100001 on 201 shifts; then at every odd base 3..401 on 801 shifts and at five larger bases on 401 shifts, the closed form of S(t) against the direct sum, the bound on G(t), the inequality T phi <= lambda phi and the lower bound on G(t), each asserted; the readings G(0)/fill - (2/pi) log base against 1 + gamma' and max G/fill - (2 sqrt2/pi) log base; the level sums at 40 digits at nine small bases, four shifts and every level up to base^level <= 9261 against (3/2) lambda^level; and the transfer identity at level 2.
• rate: power iteration of the transfer operator on a grid of cells with linear interpolation, readings of the true rate near the wall, and the one-step constant G(1/2)/fill against both bars.
• meter: the Mertens meter and the prime count on the half interval at the prime bases 101, 1009 and 10007 below 10^7, a sanity print far below every wall and never evidence for the theorem.

THE ROUNDING

• wall evaluates every closed form in mpmath interval arithmetic at 120 bits; lower bounds print their interval's lower end truncated down, upper bounds the upper end rounded up.
• check prints every upper bound rounded up and every lower bound truncated down at its last printed digit; it runs in float64 and asserts strict inequalities whose margins sit far above float error except at t = 0, where the bound on G is tight by design to 1.5 * 10^-4 of its size; the level sums run at 40 digits.
• rate prints readings: the interpolated power iteration and the one-step scan bound nothing.

RUN

• uv run python research/lab/py/interval-digits/interval.py wall in under a second.
• uv run python research/lab/py/interval-digits/interval.py check in 13 seconds.
• uv run python research/lab/py/interval-digits/interval.py rate in about 20 seconds.
• uv run python research/lab/py/interval-digits/interval.py meter in 2 seconds.
• Prints only, writes nothing; every check raises if it fails.

WITNESSES

• The tail constant c_inf <= 2.60043004, the direct check's smallest gap >= 1.3332 * 10^-7; the wall at the bar 1/5 is 94939, the closed form certified at every odd base 94939..94947 and the tail bound from 94946, failing at 94937 with gap <= -2.6733 * 10^-5, 1/5 - alpha_1 >= 1.3678 * 10^-8 at the wall.
• The lower bound holds at every odd base from 9, its tail margin >= 0.016518 from 101, and the factor is negative at 7; the density floor 1 - alpha_base < 1/5 holds at every odd base from 27 and fails at 25.
• The wall at the bar 1/4 is 3789, certified at 3789..3793, failing at 3787, 1/4 - alpha_1 >= 7.9625 * 10^-6 at the wall.
• alpha_1 <= 0.1993872 at 100003, 0.1761232 at 1000003, 0.1331636 at 10^9 + 7.
• The Section 9 constant of Maynard 2022 at the consecutive half interval clears 1/5 from 7777884825.
• The check: the bP sum under its bound by at least 3.99 * 10^-5, largest G/bound 0.999857, largest T phi/(lambda phi) 0.999866, smallest G/lower 1.534638, largest level sum over (3/2) lambda^level 0.557408, transfer identity to 10^-15.
• Readings: G(0)/fill - (2/pi) log base is 1.962430 at 100001 against 1 + gamma' = 1.962523; the true rate reads (2/pi) log base + 2.26 near 7 * 10^4 and crosses base^(1/5) between 70001 and 80001; the one-step constant reads (2 sqrt2/pi) log base + 1.19 and clears 1/4 from 7075 and 1/5 from 317063.