mobius-dissection

• The numbers of the unconditional dissection on mobius: the wall its region A sets, the block split that carries it off the powers of the base, and a falsification of every region's bookkeeping on the whole grid of three small designs.
• wall: the least base with PB < (base - 1) base^(-4/5) at one excluded digit, for the step 3 constant PB_base(1) and for the chord form PB'_base(1, e_0), the latter at the worst digit and at e_0 in {0, base - 1}; the up-set flag and the held count over the scan, the least float gap above each wall, the gap at the wall and one below at 40 digits, and 1/5 - alpha_1 at each wall; then readings of the unshifted mass ratio c_k/c_(k-1) at base 33 and base 17 against 2(base - 1).
• blocks: the split of S_F below x into blocks P base^k + D_k, checked against the exact M_F(x) at 400 random x below 2 * 10^6 and at every base^e - 1, in five one-missing-digit families, with the count per scale against fill + 1 and the mass floor A_F(x) >= fill^(L-1) - 1.
• regions: every grid point a mod y of base 10 missing 5 and missing 0 at level 6 and base 5 missing 2 at level 9, assigned its Dirichlet fraction at Q = y^(3/5) by the last convergent, and cut into regions A, B, C1, C2 at the printed Z; then the region sums against the exact sum of mu over the strings, the two counts, the C2 algebra, the second approximation at every point of B with h >= 1, and the hybrid l^1 bound at every class meeting the two conditions its proof uses, V_1 V_2 <= y and 16 D H <= y, a wider set than its hypothesis 16 base^2 D (D + H) <= y, with alpha_1 from the step 3 constant, printed past the class of a = 0, whose ratio is a closed form; and the Gallagher step of the proof at every such class with V_1 <= 10^5, the fractions' largest |hat F_(i_1)| at their residues summed against 4 D^2 ||f||_1 + ||f'||_1, both norms read on 4 V_1 points, with every fraction's residues checked distinct mod V_2. The perturbed Lemma A' is then checked at seeded grid points of level 30 in base 10 and level 45 in base 5, where its perturbation hypothesis holds, with the phases base^i a mod y in exact integers.

THE ROUNDING

• The walls are float scans with the least gap above each wall printed; that gap sits at the wall, above 10^-6, and the wall and the base below it are re-read in mpmath at 40 digits, so the sign at each is certified far past float error.
• 1/5 - alpha_1 is printed from the 40-digit gap through log(1 + gap/PB)/log base, never by differencing two numbers of size 1, its mantissa floored at four decimals in mpmath.
• The hybrid ratio and the alpha_1 of the small designs add 5 * 10^-7 before a six-digit print, so each printed value is an upper bound.
• The minor-arc bound carries an unstated constant, so regions prints its readings on the grid as readings; they bound nothing.

RUN

• uv run python research/lab/py/mobius-dissection/dissection.py wall in under a second.
• uv run python research/lab/py/mobius-dissection/dissection.py blocks in 18 seconds.
• uv run python research/lab/py/mobius-dissection/dissection.py regions in five seconds, peak resident memory 0.59 GB.
• Prints only, writes nothing; every check raises if it fails.

WITNESSES

• The wall: 39363 over every excluded digit and 28352 at e_0 in {0, base - 1} with the chord form, 92317 with the step 3 constant, each an up-set of its scan to 2 * 10^5; gaps <= -2.3195 * 10^-6 at 39362 and >= 2.3677 * 10^-5 at 39363, <= -1.3539 * 10^-5 at 28351 and >= 1.8837 * 10^-5 at 28352; 1/5 - alpha_1 >= 2.6965 * 10^-7 at 39363, 2.3643 * 10^-7 at 28352 and 1.8712 * 10^-8 at 92317.
• The unshifted mass: c_4/c_3 = 74.1654 at base 33 missing 16 and 70.5661 missing 0, c_5/c_4 = 35.5234 at base 17 missing 8, readings above 2(base - 1).
• The block split: exact at every tested x in all five families, at most fill blocks at one scale.
• The regions: the four sums meet the exact total at all three designs; 74778 and 128942 second approximations in range; 73 hybrid classes at each base-10 design and 86 at base 5 under the proof's two conditions, largest ratio past the class of a = 0 at most 0.001356, the Gallagher step at most 0.110814; the perturbed Lemma A' bound exceeds |hat F_k(a/y)| by at least 27.9 in the logarithm at all 6000 draws.