# 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](https://doi.org/10.1007/s00222-019-00865-6) 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.