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.