Explicit Transcendental Numbers with Arbitrarily Large Finite Irrationality Measure

Fix an integer M ≥ 2, a bounded sequence of positive integer coefficients 1 ≤ aₙ ≤ K, and a strictly increasing sequence of positive integers g₁ < g₂ < ⋯. This paper studies the lacunary base-M series α = ∑ₙ≥1 aₙM⁻ᵍⁿ through its natural truncations and the continued fraction expansion of the resulting real number. The main result is a uniform exact formula for the irrationality exponent. If gₙ₊₁/gₙ → s ≥ φ², where φ = (1 + √5)/2, then every bounded positive integer coefficient source produces a transcendental number satisfying μ(α) = s. The theorem is uniform over all sequences 1 ≤ aₙ ≤ K, including automatic, morphic, recurrence-generated, and eventually constant coefficient sources. In particular, for every s ≥ φ² and every bounded positive integer sequence (aₙ), ∑ₙ≥1 aₙM⁻⌊ˢⁿ⌋ has irrationality exponent exactly s. The proof combines standard continued-fraction estimates with a uniform block decomposition. The Nth truncation satisfies α − ∑ₙ₌₁ᴺ aₙM⁻ᵍⁿ ≍₍M,K₎ M⁻ᵍᴺ⁺¹, giving the lower bound determined by lim supₙ→∞ gₙ₊₁/gₙ. If λ = lim infₙ→∞ gₙ₊₁/gₙ > 2, then the reduced truncations are eventually continued-fraction convergents. For the corresponding continued-fraction index mₙ, the truncation convergents and their successors satisfy the asymptotic denominator and partial-quotient relations qₘₙ = Mᵍⁿ⁺ᴼ⁽¹⁾, cₘₙ₊₁ = Mᵍⁿ⁺¹⁻²ᵍⁿ⁺ᴼ⁽¹⁾, qₘₙ₊₁ = Mᵍⁿ⁺¹⁻ᵍⁿ⁺ᴼ⁽¹⁾. The remaining convergents between consecutive truncations form intervening blocks. Their endpoint denominators yield the uniform bound Θ(α) ≤ B(λ), where B(λ) = 1 + λ/(λ − 1) for 2 < λ < ∞, and B(∞) = 2. The decisive inequality B(s) ≤ s holds exactly when s ≥ φ². Thus, above the golden-ratio-squared threshold, the truncation branch determines the irrationality exponent. The paper also establishes a metric smallness result for the associated exponential-gap families: they have Hausdorff dimension zero and zero h-Hausdorff measure for every dimension function h dominated by a superlinear log-log gauge.