Reversible Negative Coefficient Logic, Unitary Seed-Value Damping, and Stable Binary Attractor Convergence in Fourth-Order Linear Recurrence Relations (Theory-2)

This research paper formalizes an exceptionally unique and groundbreaking fourth-order linear recurrence relation, designated as Theory-2, independently discovered by the author, Nandkishor Prajapat, on October 30, 2026. This system operates under a specific bidirectional negative configuration, where the development of the sequence initiates from an array of unitary seed values (). Mathematically and structurally, this theory demonstrates an exceedingly rare mathematical phenomenon: the sequence exhibits sharp dynamic chaos during its initial phases, but its internal negative friction forces (damping force) completely regulate its trajectory across high-indexed dimensions. As the sequence approaches infinity (\(n \to \infty\)), the limiting ratio of consecutive terms stabilizes identically at a fixed attractor point, converging precisely to the absolute constant \(\alpha \approx 1.000000\) instead of escaping to infinity. This paper provides the formal mathematical proofs, derives the constant via its characteristic quartic polynomial, contrasts the model against established classical sequences, highlights its relevance within global physical sciences, and presents a production-ready computational Python engine. Keywords: Number Theory, Stable Attractor, Self-Stabilizing System, Digital Signal Damping, Cryptography.