The Coefficient-Balanced Quartic Recurrence Framework in Fourth-Order Linear Relations and Its Constant Convergence (Theory-12)

This paper formalizes a newly discovered integer sequence governed by a unique fourth-order linear recurrence relation operating under a deterministic "Coefficient-Balanced Multiplier Rule". Discovered independently by the author, Nandkishor Prajapat, this sequence utilizes the initial boundary seeds [1, 2, 3, 4] and fields an engineered mixed-sign dependency signature (1, 1, 2, -1). We mathematically demonstrate that despite the inclusion of both a double multiplier and a negative scaling coefficient, the sequence exhibits rigid structural integrity. As n → ∞, the limiting ratio of consecutive terms stabilizes identically at a novel cosmic algebraic constant, α ≈ 1.91717. We outline its mathematical proofs, compare it against classical models, provide a Binet-style analytical approximation, present a Python execution framework, and explore its applications in cryptography and digital signal processing.