Geometric Emergence of Spacetime from Prime Spectral Scattering: Complex Phase Rotations, Sakharov Induced Gravity, and Critical Line Vacuum Stability (Complete Research Suite: Twin Core Monographs + 4 Companion Guides)

### Abstract & Overview of the Expanded Research Suite (Part IV, Version 2)This repository hosts the expanded, comprehensive second release (v2) of the Prime Spectral Scattering Research Suite. The suite establishes a mathematically rigorous, non-perturbative unification bridging microscopic prime-phase spectral scattering on a 3-point harmonic discrete lattice with the macroscopic emergence of four-dimensional Einsteinian spacetime, vacuum stability, and non-perturbative gauge dynamics. This publication is organized as an integrated 6-Document Research Suite (68 pages in total) consisting of Twin Core Monographs and four modular Companion Guides: • [00-A] Core Monograph I (15 pp): Microscopic Many-Body Kinetics, Automorphic Theta Synthesis, and Invariant Spectral Phasing.Establishes the microscopic quantum foundations: analyzes prime phase logarithmic oscillators ωp = ln p, the emergence of GUE level repulsion (β ≈ 2.0), the super-exponential 5-body EFT closure (R_≥6 < 2.3 × 10^(-7)), and the boundary theta synthesis operator. • [00-B] Core Monograph II (11 pp): Geometric Emergence of Spacetime from Prime Spectral Scattering: Complex Phase Rotations, Sakharov Induced Gravity, and Critical Line Vacuum Stability.Provides the macroscopic spacetime emergence architecture connecting 3D complex spatial rotations to warped hyperbolic throat geometries (R = −2/ℓ₀²), Seeley-DeWitt heat kernel renormalization, and the critical line exclusivity theorem (Re(s) = 1/2). • [01] Companion Guide I (10 pp): Differential Geometry and Metric Reductions.Delivers unabridged, step-by-step tensor derivations of Christoffel symbols, Riemann curvature components, finite throat volume integration (Vol(ℳ₂) = 2πℓ₀²), and the WKB proof of double-exponential transverse centrifugal barrier shielding isolating the isotropic S-wave channel (l = 0). • [02] Companion Guide II (11 pp): Quantum Vacuum Energy and Induced Action.Details Sakharov's induced gravity mechanism via the Seeley-DeWitt heat kernel expansion (a₀, a₁). Provides the complete algebraic partial fraction decomposition of the diagonal self-energy operator [Ŵw]kk, proving that the Basel constant ζ(2) = π²/6 is identically purged via exact telescoping differences, pinning the cosmological constant to Apéry's vacuum condensation 𝒵diag = 4ζ(3) − 19/4 ≈ 0.058228 with 99.3865% base doublet concentration. • [03] Companion Guide III (13 pp): S-Matrix Unitarity, Bianchi Conservation, and Cosmological Selection.Connects the imaginary effective action Im(W) to continuous scattering phase shifts via Krein's spectral shift formula and the Birman-Krein relation. Proves that the Schwinger vacuum stability condition (Γ = 0) and the contracted Bianchi identity (∇μGμν = 0) demand Im(W) ≡ 0, which holds if and only if Re(s) = 1/2, precluding off-critical complex pitchfork bifurcations. • [04] Companion Guide IV (8 pp): 105-DPS Numerical Certification, Scale Dilation RG Flow, and Non-Perturbative Yang-Mills Mass Gap.Provides the algorithmic and gauge-theoretic verification suite:(i) Audits the 105-DPS Tanh-Sinh quadrature certifying the 72-digit permanently frozen invariant core with linear freezing slope d(Locked)/d(DPS) = 1.0;(ii) Formulates the continuous Abel-Plana contour integration proving continuum Apéry condensation;(iii) Resolves the gauge-gravity hierarchy via scale dilation RG flow μ∂_μ l(μ) = β(l), deriving the non-perturbative Yang-Mills 0++ glueball mass gap Δ ≈ 1.6500 GeV and color confinement from the DIS core radius l_core ≈ 0.02886 fm;(iv) Includes the complete, self-contained Python numerical benchmark script. --- ### Companion Preprints in the Research Suite:• Part I: DOI: 10.5281/zenodo.22763956 — Dynamical Origin of GUE Level Repulsion, Fock-Space Entanglement Transitions, and Critical Boundary Decoupling in Prime Phase Networks.• Part II: DOI: 10.5281/zenodo.23075918 — A 3-Point Harmonic Discrete Laplacian Lattice Architecture: Part II: Spectral Phasing, Boundary Theta Synthesis, and a 105-DPS Numerical Audit.• Part III: DOI: 10.5281/zenodo.23100306 — Autonomous Spectral Scattering on a 3-Point Harmonic Discrete Lattice: Geometric Rigidity, Boundary Funnel Dynamics, and Critical Line Exclusivity.