The digit-restricted Mobius exponent

• 2026-09-01 [Conjecture] theta(F) = 1/2 for every digit set with 2 <= |F| <= base - 1 and squarefree digit gcd - square-root cancellation against the set's own counting function: the 47 running-maximum exponents across base = 3, 4, 5, 10 read 0.4465..0.5358 with last-five-level drifts 0.0157..0.1056, while the full-set controls, whose limiting exponent is 1/2 under RH and at least 1/2 unconditionally, read 0.4413..0.4517 at the same depths; the finite tables are consistent and decide nothing, single-cut exponents scattering 0.22..0.53 on the same data. Witness: lab/rs/mobius-designs, mobius.md.
• 2026-09-19 [Conjecture] An unconditional Mertens-shape bound on the dense column: for F omitting exactly one digit, base >= 92317 and x = base^level with level past a point depending on base alone, abs(M_F(x)) <= C(base) A_F(x) exp(-c(base) sqrt(log x)) with C(base) and c(base) > 0 effective, through a Dirichlet-approximation dissection whose region A is the ladder rung b = 4/5 and whose one load-bearing minor-arc input is unread. Witness: lab/rs/mertens-numerology for the wall 92317, Maynard 2022 for the imported lemmas.
• 2026-09-19 [Proved] That dissection is dead at fixed digit count: {0,1} at base 3 has l^1 exponent log 2 / log 3 = 0.630929, above every bar the dissection sets, region B's 1/4 and region A's own ask included, the l^1 floor again in arc-local form. Witness: exact arithmetic in the sentence that prints it.
• 2026-09-19 [Proved] Vaughan's identity is circular here at power strength: the mu_{<=U} * mu_{<=U} * 1 piece carries the main term fill^level M_1(U)^2, so bounding the pieces one by one at power strength forces M_1(U) << U^(-delta), which continues 1/zeta into sigma > 1 - delta. Witness: the identity, carried out in the sentence that prints it.