Certified Gyroscopic Stabilization in the Four-Dimensional 4-Body Problem: A Three-Layer Atlas of Balanced Configurations and the First Examples of L4/L5-Type Stable Saddles

To the best of our knowledge, this paper presents the first examples of gyroscopic stabilization in the four-dimensional 4-body problem: relative equilibria that are saddle points of the restricted potential U|_S yet spectrally stable, established with interval-arithmetic proofs. For the balanced-configuration equation M⁻¹∇U + U(q)Ŝq = 0 with S = diag(s, s, 1, 1) (s being the square of the angular-velocity ratio ω₁/ω₂ = √s), we build a three-layer atlas over 24 mass points × a deterministic two-tier s-grid: an existence layer (483,222 entries = 478,502 from the main scan + 4,720 audit recoveries, after excluding 9 entries at the G3 gate; every entry carrying a Krawczyk existence-and-uniqueness certificate), an energy layer (Morse-type classification via the B/D block decomposition: 462,452 saddles / 20,770 minima), and a spectral layer (numerical eigenvalues, with interval certification for all candidates). 53 entries (8 mass points, 9 branches, two mass-pattern families) are energy-saddle and spectrally certified: all roots of the associated degree-10 μ = λ² polynomial are proven real and negative (worst root-box upper bound −1.096×10⁻⁵ < 0). The Krein signature is indefinite for all 53, as it must be for a spectrally stable saddle, confirming the defining signature of genuine gyroscopic stabilization. Each stable branch connects linearly to a spatial central configuration as s → 1 (a Routh-type continuous deformation in R⁴, reported at the level of numerical evidence). The certified catalog, the full implementation, and a browser simulator are released. This record contains the complete release bundle: the paper (English version of record; a Japanese version is published on Jxiv), the certified catalog (all shards with per-shard SHA-256), the full Julia implementation with tests, the deterministic grid rules and exclusion logs, and a browser-based simulator. Code repository: https://github.com/yukie-lab/tessera-atlas (paper-pinned tag: v1.0-paper).