Rational-Height Staircases for Sturmian Collatz Seeds

We study a quantitative rational-approximation invariant attached to accelerated Collatz valuation words. For an irrational mechanical slope α ∈ (1, 2) and intercept ρ, let ξα,ρ ∈ Z2 be the unique 2-adic seed whose accelerated valuations are the mechanical word, and let Hα,ρ(N) be the minimum projective height of a rational 2-adic seed realizing the first N valuation letters. Writing β = log2 3, we prove for the characteristic word and every α ≥ β a continued-fraction staircase law: on the convergent interval qk + qk+1 − 1 < N ≤ qk+1 + qk+2 − 1 one has |log2 Hα,0(N) − αqk+1| ≤ log2 (qk+1 + 1) + 2α + 3. Consequently, if L = lim sup qk+1/qk, then λ− = α 1 + L , λ+ = αL 1 + L , λ− + λ+ = α. For arbitrary intercepts in the same sector, the height profile is governed by the minimum eventual-periodic approximation cost and hence by the Sturmian exponent of repetition. In particular, positivity of the lower height exponent is equivalent to bounded continued-fraction partial quotients. We further prove an exact lattice structure for consecutive standard words. If Vk = corr(sk), 2 pk − 3 qk is the cycle vector of the standard block associated with the convergent pk/qk, then det(Vk, Vk+1) = ±2 pk+pk+1−2 , and every intermediate 2-adic cylinder lattice is generated explicitly by Vk+1 and a dyadic multiple of Vk. This exact basis yields a lattice-conditioning phase transition at α = log2 3. The Sturmian repetition input is classical; the Collatz-height and explicit cylinder-lattice statements are the arithmetic content studied here.