Complete Classification and Analysis of the Core Recursion Formula for the Family t = − 1 t=−1

Within the framework of the Fundamental Core Recursion (FCR) and fixing t=−1, γn=0t=−1, γn=0, we systematically analyze six typical parameter choices (αn,βn)(αn,βn): A5 (n,1n,1), A9 (n,n2n,n2), A8 (0,n2+10,n2+1), A11 (0,n(n+1)0,n(n+1)), A12 (0,2n0,2n) and A13 (0,n!0,n!). For each case we derive conserved quantities, bilinear forms, closed‑form solutions, continuum limits and integrability properties. A clear hierarchy emerges: when αn=0αn=0 the system decouples into even and odd subsequences and becomes linearizable (trivial integrability); when αn≠0αn=0 the systems correspond to discrete Painlevé equations and possess non‑trivial integrability. This work provides a complete set of benchmark examples for the FCR theory, covering growth types from polynomial to super‑factorial.