The Stratoverso Hierarchy: A Discrete Multi-Layer Cosmological Structure from Non-Perturbative Scalar Field Dynamics

This paper presents the complete mathematical architecture of the Stratoverso Hierarchy — the discrete, 140-layer cosmological structure that underlies the Stratoverso Framework introduced in Berti (2026a). Key results: (i) derivation of N=140±3 discrete cosmological layers from three independent routes using only parameters fixed by observational data; (ii) formalization of the primordial genesis as a spontaneous Z₂ symmetry breaking in the null manifold V×S×T=0, with explicit derivation of the initial phase φ₀=+π/4; (iii) identification of the toroidal topology as the result of the attraction between the imprinting and outprinting poles of the primordial sphere, with explicit interpolating metric S³→T²; (iv) the Holographic Spherical Membranes (MSO) grounded in the Bousso (1999) covariant entropy bound; (v) BH→WH information transfer anchored to Haggard-Rovelli (2014) ; (vi) formal Stirling thermodynamic engine analogy; (vii) perturbative stability analysis — all 140 layers linearly stable with Re (μ) <0; (viii) Universal Train formalized as a Hamiltonian flow on T². Global parameters derived from first principles: Tcycle ≃ 2000 Gyr and Dₑq ≃ 800 Gly, in excellent agreement with Berti (2026c). New in this version: a dedicated section discusses the independent convergence with Koushiappas (2026, arXiv: 2604. 27771), who derives toroidal topology T²=S¹×S¹ from a deformed cosmological commutation relation â, â̇. The two frameworks reach the same topological structure from completely different approaches — non-perturbative classical scalar field dynamics vs. quantum kinematic deformation — confirming that toroidal topology is a robust and physically fundamental feature of cosmological dynamics. Key distinctions: the Stratoverso derives N=140±3 (no free parameter), predicts log-periodic multi-channel signatures, and partially resolves the Hubble tension; the Koushiappas model has n as a free parameter, predicts power-law deviations in H (z) only, and worsens the Hubble tension. Falsifiable by DESI Year-5, Euclid, and CMB-S4 within 2027–2033. Third paper in the Stratoverso series: Paper A (framework): https: //doi. org/10. 5281/zenodo. 20266625 Paper B (predictions): https: //doi. org/10. 5281/zenodo. 20267494 Paper C (this work): https: //doi. org/10. 5281/zenodo. 20288763