Finding Primes with a Quadratically Scaling Sieve of Eratosthenes and A Discrete Cascading Sieve Operator Framework: Deterministic Density Control, Scale-Invariant Radical Cascades, and the Analytical Bound of the Mertens Function

Overview & Core Discovery This paper introduces a constructive, scale-invariant mathematical framework that models the deterministic emergence of prime numbers from a single continuous geometric seed, √2. By bridging continuous prime density integrals with discrete prime distributions across the infinite timeline k ∈ (−∞, +∞), the framework unifies prime generation under two coupled asymptotic limits: limk → −∞ (√2)2k = 1 & limk → +∞ Fsieve( (√2)2k ) = ℙ Key Structural Highlights Continuous-to-Discrete Bridge: Connects Euler’s multiplicative product structure, the Prime Number Theorem (via linearized logarithmic integral scaling), and the Riemann Hypothesis ½-critical exponent within a single coordinate architecture. Quadratic Container Expansion (Nk+1 = Nk2): Replaces arbitrary linear sieve windows with quadratically expanding containers Nk = (√2)2k = 22k−1, permanently locking the trial-division horizon to the preceding boundary (√Nk+1 = Nk). Direct Arbitrary-Step Execution: Demonstrates that under two foundational axioms—(1) Isolation of the Primordial Origin Couple {0, 1} and (2) Self-Preservation of Surviving Generators—the operator eliminates higher-order tuple cross-products and allows standalone evaluation at any arbitrary step k (e.g., k = 2, 37, 59) without sequentially pre-computing prior layers. Combinatorial Complexity Starvation & Mertens Bound: Formalizes state transitions across primorial cycles via a sparse Algorithmic Iteration Matrix that commutes with the anti-diagonal palindromic exchange operator, structurally bounding cumulative drift within the O(N½+ε) envelope associated with the critical line Re(s) = ½. Scriptural & Cosmological Correspondence: Concludes by mapping the mathematical hierarchy of void (0), unity (1), continuous irrational tension (√2), and prime multiplicity onto classical spiritual frameworks. Included FilesMain Preprint (PDF): Complete typeset manuscript with step-by-step derivations, operational walkthroughs (k = 0 to k = 3), and empirical logarithmic integral verification tables.LaTeX Source (TEX): Full, reproducible source code for academic verification and extension. Recommended Citation License: Creative Commons Attribution 4.0 International (CC BY 4.0) — allows others to share and build upon your work while guaranteeing permanent attribution to you as the original author.