# Moment ladder and Lemma B

- 2026-08-28 [Conjecture] The moment ladder's rows above the tenth approach the wall and stop: `beta_0^(2K)` reads `0.447838092, 0.447904613, 0.447923402, 0.447928788, 0.447930346` at `2K = 12, 14, 16, 18, 20` from Perron roots `59307.487289, 532101.317617, 4784678.13057, 43051182.4466, 387432198.159`, the twentieth row lying only `6.42e-7` below the universal wall `2/(3 + log_3 5) = 0.447930987882`; these are floating eigenvalues of exact integer matrices, not interval-certified, and the wall above them is separately settled and unaffected. Witness: lab/rs/dimension-one-ladder.
- 2026-08-31 [Proved] The energy cap `E_2K(G_a) <= lambda_2K^a` holds at every order with constant exactly 1, which is what turns a ladder rung from a growth rate into a master inequality: the carry box `{-r,...,r}^2` with `r = floor((K-1)/2)` is closed because a digit difference lies in `[-K, K]` and `floor((r+K)/3) <= r` for every `K >= 1`, every walk from the zero state back to itself stays inside `S`, the strongly connected component of that state, `M_S` is irreducible by the definition of a component and carries a self-loop at the zero state, hence is primitive with Perron root `lambda_2K` and positive right eigenvector `u`, and `e_0 <= u/u_0` componentwise with `M_S >= 0` gives `(M_S^a)_(0,0) <= lambda_2K^a`; the attempt to break it looked for the constant `C > 1` a reducible matrix would force and found none, since the reduction to the component is free and `lambda_2K = lim E_2K(G_a)^(1/a)` is the component's own root, checked equal to the full matrix's Perron root at `2K = 4, 6, 8, 10` with the component sizes `1, 7, 7, 19` inside `1, 9, 9, 25` states and the ratios `E_2K(G_a)/lambda_2K^a` falling monotonically to `1, 0.942327, 0.790590, 0.643725` at `a = 6`. Witness: lab/rs/dimension-one-ladder.
- 2026-08-31 [Proved] The master bound at order `2K` is one formula for every rung: with `a = floor(log_3(p/2))`, `d_K = ceil(log_3(K/2))`, a digit window of `n` digits with `n >= 2a` and `b = min(a - d_K, n - 2a)`, Hoelder over three blocks of the digit window of lengths `a, a, b` at exponents `4K/(2K-1), 4K/(2K-1), 2K` gives `L_n(p) <= p^2 3^(-((2K-2+kappa)a + kappa_2K b)/(2K))`, the outer blocks interpolated between the exact `L^2` and `L^4` identities and the inner block supplied by the energy cap, both admissible since `K 3^(a-d_K) <= 2*3^a <= p`; the exponent gain exceeds `a` exactly when `beta < kappa_2K/Lambda_2K` with `Lambda_2K = 2 - kappa + 2 kappa_2K`, which is the rung formula, and the seam `3a` against `n` is the same at every order with the two branches agreeing at `n = 3a - d_K`, so the feared order-10 crossing does not exist; the adversarial pass tried to break it by hunting a violation over every prime `5 <= p <= 199` and `2 <= n <= 24`, running the order-10 block alone in its 833 applicable cases as well as the min over all orders, and found none, worst ratio `0.7839` at `(p, n) = (11, 2)` for the order-10 block alone and `0.8755` at `(13, 4)` for the min, and by hunting an uncovered or negative-gain case in the main range `3a > n` over `eta` in `0.001..0.1` and `n = 6..400`, finding none, worst geometric sum over cap `0.8630`. Witness: lab/rs/dimension-one-ladder.
- 2026-08-31 [Proved] The order-10 rung is unconditional and the exponent needs no root-finding to be trusted: with `Lambda_10 = 4.436585106`, main-range decay `Lambda_10/10 = 0.443658511` and geometric constant `2/(1 - 3^(-Lambda_10/10)) = 5.1842` rounded to `6`, the ladder reads `Sum_(z < p <= 3^((beta_0^(10) - eta) n)) T*_p(n)/3^n <= 2/z + 35 z^(1-kappa) + 40 * 3^(-(kappa-1)n/8) + 6 * 3^(-0.443658511 eta n)` for `eta in (0, beta_0^(10))`, `z >= 5`, `n >= 1`, the first three terms being the unchanged order-4 bookkeeping; a Sturm count on the exact quartic `x^4 - 7833x^3 + 7916949x^2 - 850684437x + 13054946580` places no root above `66641136626/10^7` and exactly one root in the bracket of width `10^-7` below it, so `lambda_10 < 6664.1136626`, `kappa_10 > 1.985805792698` and `beta_0^(10) > 0.447597813453`, every digit truncated down, never rounded, so the short form printed everywhere is `0.4475978`; rows 12 through 20 stay Conjecture for a different reason, their `lambda_2K` being floating eigenvalues and not certified algebraic numbers, so the energy cap alone does not promote them. Witness: lab/rs/dimension-one-ladder.
- 2026-08-31 [Proved] The dimension-one moment ladder has a tenth rung and the lower wall is `0.4475978`, not `0.446717`: the exact 25-by-25 integer carry matrix `M_10` on the box `{-2,-1,0,1,2}^2` satisfies `E_10(G_a) = (M_10^a)_((0,0),(0,0))` with first energies `1, 4653, 28967859, 190911254427, 1270015973323281, 8461182216374750493` (matched by direct convolution of the digit set at `a = 1, 2, 3`), its characteristic polynomial factors symbolically as `x^6 (x-120)(x^2-450x+12231)(x^3-2190x^2+282096x-5186835)^2 (x^3-990x^2+116154x-2569725)^2 (x^4-7833x^3+7916949x^2-850684437x+13054946580)`, the Perron root is the largest root of the quartic `lambda_10 = 6664.113662506`, so `kappa_10 = 1.985805792712` and `beta_0^(10) = kappa_10/(2 kappa_10 + 2 - (3 - log_3 5)) = 0.447597813454`, above the eighth rung by `0.000880502992`, with Holder block exponents `20/9, 20/9, 10`; "dimension one" means the similarity condition `log(fill)/log(base) = 1` at base 3 and not the base-2 gasket of density `16/(3 Pi^2)`, and the order-10 three-block master bound with explicit constants and checked regime seams is now written, so the rung is a theorem and a re-proof of target-uniform Lemma B at that edge. Witness: lab/rs/dimension-one-ladder; lemma-b-pincer.