The moments of the digit transform

• 2026-09-06 [Proved] The even moments of the digit transform are additive energies counted by a carry DP: sum_{a mod base^level} |hat F_level(a/base^level)|^(2r) = base^level E_r(level), E_r the r-fold additive energy modulo base^level of the length-level strings, C-finite in level of order at most r(r+1)/2 with growth constant Lambda(2r) = base rho, rho the certified Perron root of the carry-pair transfer matrix; Lambda(4) = 18 at {0,1} base 3 (x - 6), 2(23 + sqrt 353) at {0,1,2} base 4, (275 + 5 sqrt 2369)/2 at {0,1,2,3} base 5, every value strictly inside [max(fill^4, base fill^2), base fill^3]; brute force at level <= 7, direct grid evaluation at level <= 6, the bounds and the recurrence asserted to level = 60. Witness: lab/rs/rho-decoupling (the riesz module, 19 tests), mobius.md THE METER AND ITS YARDSTICK.
• 2026-09-06 [Proved] The multiplicative energy E_x(level) = #{n_1 n_2 = n_3 n_4} of a digit-restricted column has exponent 2 alpha for every base and digit set: 2K^2 - K <= E_x(level) <= K^2 max_m r(m) with r(m) <= d(m), so E_x(level) = fill^(2L) x^(o(1)); the census reads 1, 15, 111, 655, 3179, 14211, ... to 58760487 at level = 1..12 for {0,1} base 3 with theta_x = 1.475642, 1.410978, 1.356938 at level = 4, 8, 12 falling toward 1.261860; the shift family (base^i u, base^j v, base^(i') u, base^(j') v), i + j = i' + j', i != i', counted in closed form in fill and level when 0 is a digit, is a floor on the excess over the two diagonals, 0.4418 of it at {0,1} base 3, level = 12 and 0.19 to 0.0003 at the other families. Witness: lab/rs/rho-decoupling (the menergy module, 27 tests), mobius.md THE METER AND ITS YARDSTICK.
• 2026-09-06 [Refuted] That a moment of the digit transform alone carries the Type II estimate: Holder with the 2r-th moment and Parseval on the bilinear side gives x^(theta_p/p + 1/2 - 1/p) >= x^(alpha + 1/4) for every even p >= 4 and every digit set, above the trivial x^alpha, so the route needs the bilinear sum on the minor arcs below its own root mean square, which random-sign coefficients defeat on the census (minor arcs carrying 0.79 to 0.86 of the l^2 mass, the supremum 2.8 to 3.2 times x^(1/2)). Witness: lab/rs/rho-decoupling (the arcs lines), mobius.md THE METER AND ITS YARDSTICK.
• 2026-09-06 [Refuted] The sparse large-sieve shape (fill^level + x^beta) x^(o(1)) for the digit set at the points r/base^j: the exact constant is fill^(level-j) base^j = x^(alpha + beta(1 - alpha)), above both x^alpha and x^beta for 0 < beta < 1 (the Gram eigenvalue at base = 3, {0,1}, level = 2, j = 1 is exactly 6). Witness: lab/rs/rho-decoupling.
• 2026-09-07 [Refuted] That a Type II estimate on a digit set is a statement about coefficients whose sums over residue classes mod base^j cancel for base^j up to x^(2 eta/alpha): the Type II coefficients are hypothesised 1-bounded and nothing more, the polytope being a support constraint that supplies a divisor in [X^(9/25), X^(17/40)], and the Cauchy-Schwarz in m spends even that bound, the triangle inequality dropping the coefficient product to 1; residue sums of the coefficient side occur only on the major arcs at base <= (log X)^C. Witness: Maynard 2019 Proposition 7.2, Lemma 13.1 and the reduction (13.2), both read at source and quoted verbatim, with an adversarial pass confirming the wording and the pagination.