A Transition-Lifted Geometric Embedding and Structural Analysis of History-Dependent Integer Sequences: The Recaman Case

Abstract

Recam´an’s sequence is a canonical deterministic recurrence governed by nonlocal historical memory. This paper introduces an exact transition-lifted coordinate system, Sn = an+an+1, and embeds the scalar sequence into a three-dimensional space via Pn = (an,n,Sn). We prove the exact identity |Sn−2an| = n+1, showing that the full 3D trajectory lies entirely on two affine planes, Π± : S = 2a±(n+1). This lifting proves that the transition state En = (an,Sn) is injective across time, rigorously distinguishing coordinate recurrence (a20 = a24 = 42) from state recurrence (E20 = (42,105) vs. E24 = (42,59)).Finite computational observations across N = 2,000,000 iterations reveal long-lived alternating regimes boundedby historical exclusion.