Weighted Designs

• One exact generator for a weighted design, the refinement equation phi(x) = sum_f c_f phi(base x - f) on the base grid with c_f = base^dim w_f, taken as the measure mu = sum_f w_f mu . S_f^(-1), S_f(x) = (x + f)/base. Input is (base, dim, F, w) with F the filled digits and w a probability vector of exact rationals; output is safe-rounded floats.
• The object it runs by default is base 3, dim 2, cells (0,0) (2,0) (0,2), weights 3/8, 3/8, 1/4, level 10. Every mass at level is an integer over 8^level.
• THE ARITHMETIC CLASS is decided exactly, never fitted: each w_f becomes its vector of prime exponents, and the group generated by the log w_f is cyclic exactly when that integer matrix has rank 1. Rank 1 prints the span, rank 2 or more prints nonlattice. This is the input to the renewal dichotomy and is computed before any sweep runs.
• The rank test holds for positive rational weights and for nothing else: only there does a weight carry a prime-exponent vector, and the criterion rests on the Q-linear independence of the logs of the primes, so the exponent lattice and the group generated by the log w_f have the same rank. An irrational weight has no such vector and the test does not run on it; the printed header carries the hypothesis and the run asserts every weight is a rational.
• THE MASS SIDE is the renewal variable -log w. delta is the root of sum_f w_f^s = 1 by bisection, and N(a) counts words of mass at least e^(-a) by exact multinomial enumeration over the distinct weight values with their multiplicities. The study prints the oscillation of ln N(a) - delta a over a window of ln base at a = 20, 40, 60, 80, 100, and asserts flat against a lattice class and decaying against a nonlattice one, the class having been fixed by the exponent-matrix rank.
• THE LENGTH SIDE is the mass-weighted spin ripple: M(r) is the mass of cells within radius r of the corner fixed point, read as an exact integer shell histogram, detrended by alpha = -log_base w_0 and folded into 24 bins of log_base r over whole periods of R in base^4..base^8. The drift bar is that fold on the first half of the window against the second, over the bins both halves populate, and a second bar is taken on a whole-period split. The log base periodicity residual is printed by decade, and the cross-level identity M_level(r) = n_0 M_(level-1)(r) for r < 2 base^(level-1) is asserted as exact integer equality.
• THE LOCAL DIMENSIONS are alpha_min = -log_base max_f w_f, alpha_max = -log_base min_f w_f and the information exponent alpha(1) = -sum_f w_f log_base w_f.
• THE JSR LADDER runs one solver on two rungs. Rung 0 is the designs: F inside {0,...,base-1}^dim puts supp phi in the unit cell, exactly one integer translate meets it, the Daubechies-Lagarias matrices are 1 x 1 with T_f = [base^dim w_f], and JSR = base^dim max_f w_f in closed form with alpha_Holder(phi) = -dim - log_base(max_f w_f). Rung 1 is the first overlap at base 2, T_0 = (c_(2i-j-1)), T_1 = (c_(2i-j)), restricted to sum_i v_i = 0 by an exact change of basis whose defining identity B C = T B is asserted entry by entry.
• The bracket is a certificate, not a float. All matrix arithmetic runs in Q(sqrt m) over exact rationals; spectral radii and spectral norms of the 1 x 1 and 2 x 2 blocks are enclosed by rational bounds through their trace and determinant, the Gripenberg lower end scans words to length 8 and truncates its root down, and the norm upper end scans to length 14 and rounds its root up. The same solver is run on rung 0 and asserted to collapse onto the closed form.
• THE CONTROLS are pre-registered and printed with their verdicts: equal weights must reproduce the 0/1 design exactly, log-commensurable weights must keep every ripple, and only an irrational log ratio may move a mass observable. The study asserts that the length ripple survives all three and that exactly one of the three moves the mass side. A mass observable moving under all three would fail here.
• THE MULTIFRACTAL SPECTRUM closes the mass side. For equal contraction 1/base under the open set condition the pressure equation sum_f w_f^s (1/base)^tau(s) = 1 gives tau(s) = log_base sum_f w_f^s in closed form, and f(alpha) = inf_s (alpha s + tau(s)). Three things are checked: tau(0) = log_base |F| and tau(1) = 0; the box partition at level carries the moments exactly, sum_i mu_i^s = (sum_f w_f^s)^level in exact rational arithmetic at s = -2..3 and levels 6, 8, 10; and the coarse-grained band f_level(alpha) = log_base N(alpha)/level sits under the transform at every level and every achievable alpha, which follows in one line from N_i mu_i^s <= base^(level tau(s)).
• The gap between the band and the transform is the Stirling volume term of the multinomial count. The expectation is Stirling's series, derived independently of the sweep, and the study asserts the band matches it to better than 1e-3 at all three levels while the deficit itself falls with the level. Both endpoints of the band are exact at every level, deficit 0.000000000.
• The whole band at level 10 is printed row by row by the generator, never assembled by hand.
• Every printed number is asserted before it prints, and the study exits nonzero on any failure.

RUN

• uv run python research/lab/py/weighted-designs/weighted.py
• About four seconds; prints only, writes nothing, and holds one level-10 cell list at a time.

WITNESSES

• weights.md What weights move, the class table: the seven rank rows, 1/2,1/2,1/2, 1/2,1/2,1/4, 1/3,1/3,1/3 and 1/2,1/4,1/4 at rank 1 with spans ln 2, ln 2, ln 3, ln 2, and 1/2,1/2,1/3, 2/5,2/5,1/5, 3/8,3/8,1/4 at rank 2.
• weights.md What weights move, the class table's last column: every delta and every five-point oscillation of ln N(a) - delta a, the four flat and the three decaying.
• weights.md What weights move, the Dirichlet root: 1.0000000000 at all four probability vectors.
• weights.md What weights move, the controls: the three verdicts, length kept under all three and mass moved under exactly one.
• weights.md What weights never move, the ripple table: the four swings, drift bars, whole-period bars and fold gaps, the periodicity residual by decade, and the cross-level identity exact on all four.
• weights.md The ladder, rung 0: T_f, JSR, the solver bracket, alpha_min, alpha_max and alpha_Holder(phi) on all four sets.
• weights.md The ladder, rung 1: the hat mask at [0.5000000, 0.5000000] and D4 at [0.6830127, 0.7105812], hence alpha in [0.4929285, 0.5500157].
• weights.md The pressure: the tau(s) table at s = -2..3, the exact moments at levels 6, 8, 10, the deficits 0.176458, 0.147536, 0.127619 at alpha = 1.077324384, the Stirling match and the exact endpoints.
• The whole coarse-grained band at level 10 is printed row by row by the run and is not carried on the page.