A Deterministic Cascade Framework: Generator Partitioning, Upper-Zone Dynamics, and Prime Void Persistence in Quadratic Intervals

This record presents Version 2.0 of the deterministic cascade framework, providing a formalized combinatorial model for composite number generation and prime void preservation across short quadratic intervals I_n = (n^2, (n+1)^2).  

Key Updates & Formal Results in Version 2.0:

Zero-Redundancy Multiplier Partitioning: Implements an exact canonical factor decomposition C = p \cdot m subject to \min(\mathbb{P}(m)) \ge p, eliminating composite coordinate overlap and bypassing traditional inclusion-exclusion parity barriers.  

The X/3 Late-Born Cutoff Invariance: Demonstrates that for X = (n+1)^2, primes strictly exceeding X/3 cannot instantiate odd composite branches below X (as 3p > X), proving that target quadratic intervals lie entirely within an inactive generator zone and restricting active generators strictly to p \le n.  

Multiplier Horizon Sparsity & Collision Dynamics: Details branch step dilation (2p > \sqrt{8n}) for upper generators (\sqrt{2n} < p \le n), forcing active multiplier sets \mathcal{M}(p, n) into singleton or empty states under canonical filtering.  

Empirical Validation (n \in [9, 200]): Provides exact computational data confirming that composite saturation remains structurally capped (\le 83.5\%) while prime voids scale monotonically from N_{\text{void}} = 3 up to 33.  

Analytical Research Target: Outlines the remaining objective to establish an unconditional proof of Legendre's Conjecture by bounding prime gap variance via exponential phase sums.