Geometric Emergence of Spacetime from Prime Spectral Scattering: Complex Phase Rotations, Sakharov Induced Gravity, and Critical Line Vacuum Stability (Complete Research Suite: Core Monograph + 3 Companion Guides)

Abstract & Overview of the Complete Research Suite (Part IV) This repository hosts the fourth and culminating release of the Prime Spectral Scattering Research Suite, delivering 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 and vacuum stability. This publication is organized as an integrated 4-Document Research Suite (45 pages in total) consisting of a central synthesising monograph and three granular companion guides: [00] Core Monograph (12 pp): Geometric Emergence of Spacetime from Prime Spectral Scattering: Complex Phase Rotations, Sakharov Induced Gravity, and Critical Line Vacuum Stability. Provides the unified axiomatic synthesis 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 and lattice weight w(k), pinning the cosmological constant to Apéry's vacuum condensation 𝒵diag = 4ζ(3) − 19/4 ≈ 0.058228 with 99.3865% doublet concentration. [03] Companion Guide III (12 pp): S-Matrix Unitarity, Bianchi Conservation, and Cosmological Selection. Connects the imaginary effective action Im(W) to the continuous scattering phase shift 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. Demonstrates that off-critical excursions (σ ≠ 1/2) induce catastrophic complex pitchfork bifurcations and vacuum pair-production runaway. 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.