Sequential Posterior Multiplier Framework in Fourth-Order Linear Recurrence Relations and Its Constant Convergence (Theory-11)

This paper formalizes a newly discovered integer sequence governed by a unique fourth-order linear recurrence relation operating under a deterministic "Sequential Posterior Multiplier" rule. Discovered independently by the author, Nandkishor Prajapat, this sequence utilizes the initial boundary seeds [1, 2, 3, 4] and fields an engineered dependency signature (1, 0, 1, 2). We mathematically demonstrate that despite bypassing immediate historical states via a zero-coefficient drop, the sequence exhibits rigid structural integrity. As n→∞, the limiting ratio of consecutive terms stabilizes identically at a novel cosmic algebraic constant, α ≈ 1.74514. We outline its mathematical proofs, compare it against classical models, provide a Python execution framework, and explore its applications in cryptography and digital signal processing.