prime-dissection

• The numbers of the dissection for the von Mangoldt function on coprime, section PRIMES ON A DESIGN: the wall where a shifted-grid l^1 certificate below 1/5 holds at one excluded digit, the singular series as the principal characters of region C2, the four regions for Lambda on the whole grid of three small designs, and a prime count against the main term.
• It imports pb_step3, strings, convergent and smooth from mobius-dissection and uniform_alpha from digit-uniform-bound, read only.
• wall: the digit-uniform chain (z-1)^3 = (2/pi)(log base) z + gamma'(z-1) + (2/pi)(z-1)^2/(base z - 1) against z < base^(1/5)(1 - 1/base), certified in mpmath.iv at 120 bits from the least base where it clears to the base where the closed-form cap 1 + sqrt(2 (2/pi) log base + 0.97) takes over, with the slope test that keeps the cap below from base 100 on; the least base where Maynard's written constant log((q/(q-1)) log q + 3q/(q-1))/log q drops below 1/5, and its value at 2000001; the missing-digit budget at base 10^7 and 10^8 for Maynard's C_(q,s) = 1 + (2+s)/log q, for the wall condition (W) through pb_step3, and for the chain; the chain's float wall at two and three excluded digits.
• walls: one wall per number m = 1..13 of excluded digits. The chain at m digits, (z - m)(z - 1)^2 = m((2/pi)(log base) z + gamma'(z - 1) + (2/pi)(z - 1)^2/(base z - 1)) against z < base^(1/5)(1 - m/base), is located in floats, certified in mpmath.iv at 120 bits from the wall to the base where the cap m + sqrt(m((2/pi) log base + 0.97)) takes over (point by point near the wall, on interval blocks split until they clear above it), with the slope test base^(1/5) sqrt((2/pi) log base + 0.97) > (5/2) sqrt(m)(2/pi) there and a failing base below the wall; the (W) wall sqrt(m) + Phi_base/base < base^(1/5)(1 - m/base) is certified at its points and above by a smooth upper bound on Phi_base/base whose gap grows from base 327; the margins of both certificates at and one below every wall, the thinnest over all 26; then the better certificate per m, and the m >= 14 argument through the smooth (W) gap at base m^5.
• window: the digit-uniform window at two window digits against 1/5, one outward-rounded Collatz-Wielandt certificate per base covering every excluded digit, scanned down from 583 to the first base that fails.
• series: sum_(d | base) mu(d)/phi(d) sum_f c_d(f) over F, divided by fill, against kappa_F = (base/phi(base)) #{f in F : gcd(f, base) = 1}/fill in exact rationals at every set missing one or two digits of every base 3..30, and the constant printed in Maynard 2022 Theorem 1.3 of arXiv v1 beside it at six sets; the identity is checked against its own closed form, so the prime count of count is the test of the main term.
• regions: every grid point a mod y of base 10 missing 5 and missing 1 at level 6 and base 5 missing 2 at level 9, cut into regions A, B, C1, C2 as on mobius, with Lambda in place of mu; the region sums against the exact sum of Lambda over the strings, the principal characters y mu(d)/phi(d) at the C2 points with h = 0 against kappa_F fill^k, and each region's share.
• count: sum_(n <= x, n in S_F) Lambda(n) over a sieve to 10^8 against kappa_F A_F(x), A_F by digit counting checked against enumeration to 10^5, at the powers of the base and twelve seeded x, on eight sets, two of them without two consecutive digits.

THE ROUNDING

• 1/5 - alpha_1 at the wall is read from the float root's upper endpoint and floored at three digits.
• The chain margin, the cap gap and the slope test are mpmath.iv intervals at 120 bits, and a row passes only on the sign of the unsafe endpoint; gamma' = 0.9625229 is asserted above the true (2/pi)(gamma + log(8/pi)) first.
• The printed alpha_1 of the chain is log_base(z base/(base - 1)) rounded up at six decimals from the float root; the window rows are the rounded-up exponents of digit-uniform-bound.
• series is exact; regions and count are floats and their ratios are readings of the main term's shape at bases far below any wall, never a bound.

RUN

• uv run python research/lab/py/prime-dissection/primes.py wall in 2 seconds.
• uv run python research/lab/py/prime-dissection/primes.py walls in one second.
• uv run python research/lab/py/prime-dissection/primes.py window in 21 seconds.
• uv run python research/lab/py/prime-dissection/primes.py series in under a second.
• uv run python research/lab/py/prime-dissection/primes.py regions in one second.
• uv run python research/lab/py/prime-dissection/primes.py count in 3 seconds the first time, writing count-100000000.json under data/, then read from that cache.
• Every check raises if it fails.

WITNESSES

• The chain: certified on [584, 1272], tightest margin 6.0170 * 10^-3 at 584 in the units of the root equation cleared of denominators, 1/5 - alpha_1 >= 1.79 * 10^-5 there, margin -8.3138 * 10^-3 at 583; cap gap 4.6127 * 10^-4 at 1272, slope test 3.3175 at 100; alpha_1 < 0.199983 at 584.
• The walls: chain 584, 4692, 17596, 46798, 102133, 195891 at m = 1..6, certified up to 1272, 5477, 27761, 87303, 212356, 439640; (W) 92317, 124332, 153738, 182147, 210153, 238044 at m = 1..6 and 265981, 294064, 322357, 350906, 379742, 408886, 438358 at m = 7..13; the chain better at m <= 6, (W) from m = 7; smooth (W) gap at least 0.2947 at base 14^5; over all 26 walls the thinnest pass 1/5 - alpha_1 >= 8.42 * 10^-10, (W) at m = 12, and the closest failure one below -9.835 * 10^-8, the chain at m = 11; w - z = -1.083 * 10^-7 at 4691, m = 2, and w - PB = -9.992 * 10^-8 at 265980, m = 7.
• The window: alpha_1 < 1/5 at every base 301..583, largest 0.199923 at 301, 0.200021 at 300.
• Maynard's constant below 1/5 from 1520573, 0.197309 at 2000001; budgets s <= 7, (W) m <= 176, chain m <= 16 at 10^7 and 19, 703, 29 at 10^8; the chain's float walls 4692 and 17596 at two and three excluded digits.
• The series: equal at all 4956 sets; 5/4 at base 10 missing {0, 5} against the printed 10/9.
• The regions: the four sums meet the exact total at all three designs; the principal characters return 1.000000 kappa_F fill^k; C2 on the grid 0.999546, 0.999021, 1.000427.
• The count: 0.9994..1.0001 at the largest power of the base and 0.9984..1.0009 at the seeded x on the six sets with two consecutive digits; below 0.0006 at base 3 missing 1 and base 5 missing {1, 3}.