The Fundamental Core Recursion (FCR) An=αn+βn(An+1+γn)tA n =α n+β n (A n+1+γ n ) t provides a single algebraic template that generates linear recurrences, continued fractions, quadratic maps, and constant sequences as special cases. This paper develops the general theory of the FCR, including its continuum limits to Painlevé equations, its bilinear (Hirota) form, and its Lax integrability. We then analyze two specific parameter choices within this framework: A5 (αn=0,βn=n+1γn=t=−1α =0, β n=n+1, γ n =0, t=−1) and A9 (αn=n,β=n2,γn=0,t=−1α n =n, β =n2 , γ n=0, t=−1). We show that A5 is completely solvable, possesses a closed‑form solution and an explicit conserved quantity, and degenerates to a linear recurrence. The A9 candidate is identified as a new discrete integrable system – distinct from discrete Painlevé I – which admits a Lax pair and a Hirota bilinear form, and exhibits stable, non‑chaotic numerical behavior. Together, these results demonstrate that the FCR serves as a unifying master equation for discrete integrability, capable of generating both known and novel integrable systems.
Kang A. (Wed,) studied this question.