This version refines the previous empirical framework by establishing the exact arithmetic backbone and structural identities governing the distribution of symmetric prime pairs. Key Advancements in this Revision Exact Parity Decomposition: This version formally proves that the full nesting depth, N(m; R), decomposes exactly into even-center and odd-center sectors. This parity-resolved approach provides a rigorous bridge to companion studies on spacing statistics. Exact First-Moment Identities: The research replaces heuristic approximations with exact identities that relate the averaged nesting depth directly to prime-pair counting functions, π₂ᵣ(M - r). This establishes the primary arithmetic link between local nesting density and the global distribution of prime gaps. Finite-Primorial Admissibility Theorem: To explain the "scaffold" patterns observed in sparse regimes, the paper proves a theorem quantifying the radius classes modulo primorials (Wₚ) available to a fixed center. This provides an exact count—∏ₚ (p - δₚ(m))—of the local supply of admissible radii, clarifying why centers with specific divisibility properties (e.g., divisible by 3 and 5) exhibit higher nesting. Expansion of the Empirical Program: The computational scope is extended to M = 10⁹ across a radius ladder ranging from R = 1024 to R = 8192. This allows for a more granular analysis of how the distribution shifts and reconcentrates as the radius increases. Refined Reporting Metrics: The revision introduces disciplined diagnostics, including threshold-density discrepancy (Θₖ) and support-fraction statistics (Σₖ), to precisely quantify the transition from sparse tails to reconcentrated distributions. Explicit Layering of Assertions: The methodology now strictly separates exact arithmetic theorems from heuristic models and empirical observations. This ensures a clear distinction between proven structural facts and conjectural scaling laws. Mathematical Notation (Unicode) N(m; R): Full nesting depth at center m with radius R. π₂ᵣ(x): Prime-pair counting function for gap 2r. Wₚ = 2 × ∏ pᵢ: Primorial modulus for a set of primes P. 𝒜ₚ(m): Set of admissible radius classes available to center m. δₚ(m): Local exclusion factor (1 if p | m, 2 otherwise). Θₖ(M, R): Threshold density for centers with depth at least k. Σₖ(M, R): Support fraction of the threshold set relative to the interval M Python Included in ZIP
David Betzer (Thu,) studied this question.