Height-Stable Lacunary Rigidity, Mahler Fibers, and Exact Irrationality-Exponent Correspondences

Let x = A/B ∈ (0,1) be reduced and set κ(x) = log(B/A)/log B. This paper develops a height-stable lacunary rigidity theorem for rational coefficients and applies it to exact irrationality-exponent problems for Mahler, Hone, Sturmian, Hecke–Mahler, Engel–Lüroth, and related families. For α = Σⱼ≥1 cⱼxhⱼ with cⱼ ∈ ℚ× and log H(cⱼ) = o(hⱼ), write λ = liminf hⱼ₊₁/hⱼ and ρ = limsup hⱼ₊₁/hⱼ. If λ > 2/κ, the reduced truncations are eventually continued-fraction convergents and the exponent splits into a truncation branch and an intervening-block branch. The resulting threshold s*(κ) = (2κ + 1 + √(4κ + 1))/(2κ²) gives λ ≥ s*(κ) ⇒ μ(α) = κρ. At reciprocal integer points κ = 1, this recovers the golden threshold s*(1) = φ² and extends the bounded-positive-coefficient theorem to signed rational coefficients, zeros, and subexponential coefficient height on the effective nonzero support. Applications include an exact Skolem–Mahler–Lech law for recurrence-weighted Mahler series, μ(F(1/M)) = dᴸ where L is the eventual maximal gap between consecutive nonzero recurrence terms; the rational-point formula μ(K_d(A/B)) = d·log(B/A)/log B above the explicit threshold; explicit Mahler-function values with irrational exact irrationality exponent at every nonreciprocal rational point for all sufficiently large d; infinite-dimensional ℚ-vector spaces whose nonzero elements all lie in a single exact exponent fiber; and an exact logarithmic Hausdorff gauge for fixed-support-ratio lacunary Cantor sets. The paper also develops complete exponent correspondences with generalized Hone, Sturmian, Hecke–Mahler, Engel–Lüroth, exponent-2, and Liouville families, together with a same-base Sturmian–lacunary value-separation theorem.