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 θ_p = 2π·φ⁻ᵖ for odd primes (proven via the Lucas identityφ^p = L_p + φ⁻ᵖ), generating a multi-scale hierarchy without free parameters.Agreement table: p=2 (222.5°) → strong chaos; p=11 (1.81°) → quasi-crystalstability (φ¹¹ ≈ 199.005 ≈ L_11). (2) π expressed as a Viète-φ product: π = limK→∞ (φ+φ⁻²)K+1 /∏ₖ₌₁^K (φ+φ⁻²)_k. Exact (Viète, 1593); phinaire reformulation is original. (3) e expressed as a golden growth limit: e = limk→∞(1+φ⁻ᵏ)φ^k. 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)−χ_k,with open problem PO-χ: construct χ_k = f({φ^k}) 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