Published June 15, 2026
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A Discrete-Wronskian Pearl: One Order-2 Casoratian Certificate, Reused at Catalan's Constant and ζ(4)
Authors/Creators
Description
A homogeneous order-2 linear recurrence lead(n)·s(n+2) + mid(n)·s(n+1) + trail(n)·s(n) = 0 has, for any two solutions u, v, a discrete Wronskian (Casoratian) W_n = u_n v_{n+1} − u_{n+1} v_n obeying the first-order step law lead(n)·W_{n+1} = trail(n)·W_n — in which the middle coefficient cancels identically. The pearl is the consequence: trail(n) ≠ 0 for all n together with W_0 ≠ 0 already force W_n ≠ 0 for all n (so (u, v) is a fundamental system at every index), with no closed form, no determinant, and no hypothesis on the leading coefficient. We give this as one field-general Lean 4 / Mathlib theorem (PROVEN, clean cone) and reuse it unchanged at two deliberately mismatched Apéry-like recurrences: W. Zudilin's equation for Catalan's constant G (irrationality open) and the Schneider–Zudilin recurrence for ζ(4) = π⁴/90 (known irrational, degree-13 coefficient polynomials). In the latter the heavy degree-13 middle polynomial is carried as an opaque atom and provably never enters the proof term. Both instances are re-derived as a single exact application of the abstraction and add zero axioms; both are fully sorry-free. We make no claim on the irrationality of G (it stays CONJECTURED); ζ(4) = π⁴/90 is cited, not reproved.
Notes
Formal development public at github.com/papanokechi/project-fingerprint commit 3158337 (origin/main), path lean/pcf-apery3/. Toolchain pins: Lean v4.30.0, Mathlib c5ea00351c28e24afc9f0f84379aa41082b1188f. Module SHA-256 at commit 3158337: CasoratianFramework.lean 79BD43DD31452F6101D1A2DDB5DE8BAED3201E1D292B7AB9A4047C60E310BE21; Apery3Catalan.lean 663C2225F7B2D332E767F026674F3DF62C25EC556BD39488B78689ED80CF1408; Zeta4Caso.lean D986A271681C42CBE7FAE7F563628DFB2A29F721984C498E1FBBCF39811FED04; FrameworkInstances.lean F5B8F37B3110A320B62D3E9FA1E35DA1CBBA87B62C6AFCC814584157BE2240B9. Axiom cones (in-package #print axioms): step_law, caso_ne_zero, catalan_caso_ne_zero, zeta4_caso_ne_zero (and the discharged auxiliaries caso_closed_form, W_eq_det) each depend only on [propext, Classical.choice, Quot.sound]; no sorryAx; all four modules fully sorry-free. The bundled Lean source has NO separate LICENSE file at 3158337; absent one it is covered by this record's CC-BY-4.0 (operator may add an MIT/Apache LICENSE and re-archive before mint if a code/paper split is desired). Prepared with AI assistance under human-directed falsification-first discipline (disclosed in the manuscript). OPERATOR: recompute the four module SHA-256 from the staged snapshot zip to confirm before mint.