# Weighted designs

- 2026-09-06 [Proved] For a weighted design (a probability vector on `|F| >= 2` cells) the Dirichlet root of `sum_f w_f^s = 1` is identically `1`, so it is the arithmetic class of the `log w_f`, never the root, that carries the mass-stopping count `N(t) = #{words of mass >= t}`: `N` is log-periodic exactly when the group generated by the `log w_f` is cyclic (for rational weights, when the prime-exponent matrix has rank 1) and smooth otherwise, by Lalley's renewal dichotomy on `f = -log w`; and every length observable keeps its `log base` ripple at every weight, `mu(B(r)) = w_0 mu(B(qr))` at a corner fixed point for `r < min_{f != 0} |f|/base`. Witness: lab/py/weighted-designs.
- 2026-09-06 [Verified] The multifractal pressure of a weighted design with equal contraction `1/base` under the open set condition is `tau(s) = log(sum_f w_f^s)/log(base)`, so `f(alpha) = inf_s (alpha s + tau(s))` is explicit: the box moments at level `level` carry it exactly as `sum_i mu_i^s = (sum_f w_f^s)^level`, the coarse-grained band sits under the transform at every level (`f_level <= f` from `N_i mu_i^s <= base^(level tau(s))`), exact at both endpoints and deficient by `0.176458, 0.147536, 0.127619` at the band's middle `alpha = 1.077324384` at levels 6, 8, 10 on the three-cell weighted gasket (base 3, cells `(0,0) (2,0) (0,2)`, weights `3/8, 3/8, 1/4`), the deficit matching Stirling's series, and the JSR bracket validates on the hat mask at `alpha = 1` and on D4 at `alpha` in `[0.4929285, 0.5500157]` against the closed form `2 - log_2(1 + sqrt 3) = 0.5500157`. Witness: lab/py/weighted-designs, Cawley and Mauldin 1992.
- 2026-09-06 [Refuted] That `delta`, the Dirichlet root, is a weight observable of a design (it is `1` at every probability vector), and that a norm upper bound may print truncated: D4's norm upper is `0.710581107211`, so the safe print is `0.7105812` and `alpha`'s upper `0.5501` at four digits, `0.5500` sitting strictly below the closed form `0.5500156865`. Witness: lab/py/weighted-designs.
- 2026-09-06 [Refuted] The Type II route through the multiplicative energy of a column: with `|a|, |b| <= 1`, two Cauchy-Schwarz steps give `|Sigma|^2 <= M E_x(M, N) <= 2MN E_x(level)^(1/2) x^(o(1))`, hence `|Sigma| <= x^((1 + alpha)/2 + o(1))`, missing the trivial `x^alpha` by `(1 - alpha)/2` for every digit set with `alpha < 1`; the unbalanced sum has no estimate at all since `a = b = 1` returns the representation count itself, and the balanced form returns the box's own trivial bound on the census (bound over trivial `1.0134` at `level = 12` rising to `1.0730` at `level = 14`). Witness: lab/rs/rho-decoupling (the `menergy` module), mobius.md THE METER AND ITS YARDSTICK.