Saturation Constants of the Exact-Approximation Method

Closed-form analysis of Marsaglia's exact-approximation method (1984) as presented by Fishman (1996). This paper establishes three closed-form identities in the Ahrens–Dieter generators for the exponential and Cauchy distributions, and identifies — at modern arbitrary-precision resolution — three structurally distinct constants that share the leading digits of one historically printed value. For Algorithm EA, the saturation constant is shown to equal K_AUD = √2·ln(2) exactly — derivable through an endpoint-balancing equation (b−1)² = b²·e^(−L) which collapses at the binary cell L = ln 2 to the quadratic b² − 4b + 2 = 0. For Algorithm CA, two distinct closed forms are derived: a clean rational expression in π² for the induced density at the structural midpoint y = 1/4 (f_X(1/4) = 16(9π² + 88)/(9π⁴ + 192π² + 64) ≈ 0.99774), and an explicit Cardano–Ferrari radical closure for the locked formula's true saturation minimum (p_CA(true) ≈ 0.99731). The framework's high-precision analysis resolves the published p = 0.997738 in Fishman's illustration as the implementation's design probability 2√W (Algorithm CA setup, p. 187) rather than the locked formula's true minimum (p_CA(true) ≈ 0.99731). The two values agree to four decimal places — the resolution typically displayed in 1996 — and separate cleanly only at the sixth decimal place, a distinction made visible by dps ≥ 100 arbitrary-precision evaluation. The three closed forms sit at distinct radical degrees of closure — K_AUD at degree 2 (single sqrt over ℚ), f_X(1/4) at degree 0 (rational in π²), p_CA(true) at degree 6+ (Cardano cube root nested with two Ferrari square roots) — defining a ladder of tractability. The work was produced through a multi-AI-session collaborative review using a Stack/Exchanges architecture and five procedural verification standards (M1–M5) documented in §7. Verified across seven independent AI architectures (Claude, Grok, GPT, Gemini, Perplexity, DeepSeek, Mistral) at decimal precision up to dps = 50000. The novel mathematical content (the K_AUD identity for Algorithm EA, the rational-in-π² closed form for f_X(1/4), the Cardano–Ferrari closed form for p_CA(true)) is independent of Fishman's presentation choices and stands as an algebraic property of the locked formulas as mathematical objects. The framework Main OSF project (overview of all framework papers, reading order, version history): https://osf.io/zx4g7 (DOI: 10.17605/OSF.IO/ZX4G7) Framework overview (reading order, methodology, all papers): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/about.html Direct downloads (current PDF and text files for every paper): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/direct-documents.html AI Readers hub (verification-first entry point with executable mpmath): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/ai-readers.html GitHub: https://github.com/Gap-geometry