# 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.