• • Bidirectional Reversible Coefficient Logic, Unitary Seed-Value Convergence, and Fixed Genetic Attenuation in Fourth-Order Linear Recurrence Relations (Theory-1)

This research paper formalizes a newly discovered fourth-order linear recurrence relation operating under a unique deterministic multiplier paradigm, designated as Theory-1. Discovered independently by the author, Nandkishor Prajapat, on October 30, 2026, this sequence utilizes a minimal unitary boundary initialization matrix (). We mathematically demonstrate that despite the simplicity of its single-digit seed configuration, the sequence exhibits immense structural stability across high-indexed dimensions. As \(n \to \infty\), the limiting ratio of consecutive terms stabilizes identically at a novel cosmic algebraic constant, \(\alpha \approx 1.917173\). This paper outlines the manual rule derivation, proves the constant's convergence using a characteristic quartic polynomial, contrasts it against historical models, verifies its emergence across global physical sciences, and provides a production-ready computational Python engine. Keywords: Number Theory, Quartic Polynomials, Prajapat's 12th Constant, Digital Pulse Generation, Cryptographic Key Networks.