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.