Primes as a Developmental–Dynamical System

Abstract

Primes as a Developmental–Dynamical System — Series Overview This upload contains the first two papers in the Primes as a Developmental–Dynamical System series. The series develops a geometric–operator and developmental interpretation of prime gaps and prime constellations, supported by executable notebooks. Paper 1 — Operator Geometry, Invariants, and Phase Structure Paper 1 introduces the curvature–instability operator framework for prime gaps. It defines the normalized operators Jn and Hn, establishes dimensionless phase boundaries, and identifies six structural invariants including the strong‑instability basin density (~0.19–0.20) and the inflation scaling law gstrong≈1.6ln⁡p. The paper presents dual‑attractor geometry and a developmental transition from explosive early behavior to mature ln⁡p-governed packing. Associated File: Primes.ipynb Paper 2 — Prime Constellations as a Developmental Dynamical System Paper 2 extends the developmental framework to prime constellations (twin primes, triplets, quadruplets, sexy primes). Across 5M–100M primes, all constellations exhibit the same phase architecture: overwhelming Corridor dominance, exactly one bifurcation at p=5, and zero strong instability. ACCR diagnostics confirm bounded curvature, linear drift, rare non‑accelerating turbulence pockets, and complete developmental stability under scale extension. Associated Files: ACCR_on_primes.ipynb Operator_Geometry_for_Prime_Constellations.ipynb Included Notebooks Summary These notebooks reproduce all figures, diagnostics, and operator analyses referenced in Papers 1–2.