Machine-verified Lean 4 / Mathlib formalization of the Casoratian closed form (Theorem 6.3) for a 4/pi polynomial continued fraction

A Lean 4 / Mathlib formalization of the Casoratian closed form (Theorem 6.3) underlying the rapidly convergent series identity for π/4 in the polynomial continued fraction PCF(−n(2n−3), 3n+1).

At the level of the reduced three-term recurrence (2n−1)·x_n = (3n+1)·x_n−1 − n·x_n−2, the development proves that for any two solutions h, g with Casoratian normalization W₁ = 1 one has W_n = n!/(2n−1)!! for all n ≥ 1 (casoratian_step, casoratian_closed_form). A concrete instantiation takes h_n = n²+3n+1 and g_n = q_n/(2n−1)!!, where q_n are the convergent denominators of the continued fraction (gate-checked against the q₀..q₅ = 1, 4, 26, 224, 2392, 30432 values), and formalizes the rational telescoping bridge to the π/4 series.

Soundness. Verified by axiom-cone inspection: every theorem (casoratian_step, casoratian_closed_form, casoratian_concrete, g_isSolution, g_baseValue, and the telescoping lemma) has the axiom set {propext, Classical.choice, Quot.sound} with no occurrence of sorryAx. The build reports zero sorry and zero errors. Toolchain: Lean v4.29.0, Mathlib pinned via the committed lake-manifest.json.

Scope. This verification covers the unconditionally proved core only. It does not address the ratio conjecture (Conjecture 3.3), the conditional Gamma-function identity (Section 5), or the numerical catalogue of 482 constants (Section 12) of the source paper.