Primes as a Developmental–Dynamical System

Primes as a Developmental–Dynamical System — Series Overview This upload contains the first three 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 —Primes as a Developmental–Dynamical System: 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: strong‑instability basin density (~0.19–0.20) inflation scaling law gstrong≈1.6ln⁡p dual‑attractor geometry 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 all scans up to N = 100,000,000 (≈5.76 million primes), every constellation exhibits the same phase architecture: overwhelming Corridor dominance exactly one bifurcation at p=5 zero strong instability bounded curvature linear drift rare non‑accelerating turbulence pockets complete developmental stability under scale extension Associated Files: ACCR_on_primes.ipynb Operator_Geometry_for_Prime_Constellations.ipynb Paper 3 —Prime Gaps as a Developmental Dynamical System: Global Stability, Attractor Energy, and Multi‑Scale Geometric Constraints Paper 3 analyzes prime gaps up to N = 2,040,000,000 (≈100M primes) and demonstrates that prime gaps form a multi‑scale developmental dynamical system with stable attractor windows, structured turbulence, and collapse‑resistant behavior. Key results include: attractor width ≈ 490,998 gaps attractor completeness ≈ 0.999992 curvature–jerk coupling near 1 across scales medium‑scale basin density ≈ 0.814 high‑scale basin density ≈ 0.163 structured turbulence: 20 curvature pulses micro‑corridors (18, 42, 70) and macro‑corridors (54, 102, 184) micro‑cluster and macro‑cluster geometry inflation ratio: 2.87 inside attractor vs. 1.01 outside no ceiling violations under the logarithmic envelope A 9‑point threshold sweep confirms topological stability and parameter‑invariant attractor geometry. Associated File: PrimeGap_AttractorGeometry_Test_v3.ipynb Included Notebooks Summary These notebooks reproduce all figures, diagnostics, and operator analyses referenced in Papers 1–3, including: curvature, jerk, and basin‑density diagnostics attractor‑window analysis maximal‑gap geometry multi‑scale developmental structure constellation‑phase architecture ACCR‑based stability measurements