Cosmological Dynamics of the PEW Mother Wave: The Phi-Prime Fractal, Big Bounce, and the Constants e, π, φ

We present the cosmological fractal equation of the PEW (Primordial Energy Waves) framework, unifying Mother Wave dynamics with the Big Bang–Big Bounce cycle. Starting from the PEW Lagrangian augmented by a φ-asymmetric potential, wederive emergent Friedmann-like equations and establish a natural Big Bounce (nosingularity) via the topological barrier λA⁻². Three analytically established results A form the core of this article: (1) Theorem θₚ = 2π·φ^−p for odd primes (proven via the Lucas identityφᵖ = Lₚ + φ^−p), generating a multi-scale hierarchy without free parameters. Agreement table: p=2 (222. 5°) → strong chaos; p=11 (1. 81°) → quasi-crystalstability (φ^11 ≈ 199. 005 ≈ L₁1). (2) π expressed as a Viète-φ product: π = lim₊→∞ (φ+φ^−2) ^K+1 /∏₊=₁K (φ+φ^−2) ₖ. Exact (Viète, 1593) ; phinaire reformulation is original. (3) e expressed as a golden growth limit: e = lim₊→∞ (1+φ^−k) ^φᵏ. Exact. The Viète parameter b encodes CPT symmetry breaking: b=−1 places the criticalline at Re (s) = +1/2 (matter universe), connecting the phinaire structure tothe Riemann Hypothesis as an algebraic consequence. The prime–structure correspondence (p=2→quarks, p=3→hadrons, …) is explicitlyidentified as a structural analogy B, not a derivation. The case p=23 (chromosomes) is noted as a probable coincidence. A new conjecture is proposed: the φ-Euler product ζ (s) ≈ ∏ₖ (1−φ^−sk) ^−χₖ, with open problem PO-χ: construct χₖ = f (φᵏ) without circular reference to primes. Epistemic statuses explicitly distinguished: A proven; B structural analogy;D open conjecture. MSC: 83F05 · 11B39 · 11A41 · 37F50 · 83C55PACS: 98. 80. -k · 02. 10. De · 02. 10. Ox