Relative equilibria on R³ × T^k under Gauss-law gravity: a two-body stability threshold, certified stable rings containing sub-threshold pairs, and an instability theorem for centred rings

Abstract. We consider the classical (non-relativistic) N-body problem on R³ × T^k — three large dimensions and k compact ones, a flat square torus of period 1 — with the attraction that follows Gauss's law in 3+k dimensions: the potential of a point mass is the sum over images of the Green's function of the (3+k)-dimensional Laplacian. We study relative equilibria in which all bodies sit at the same point of the torus; perturbations are taken in the full phase space, including the compact directions. For two bodies on a circular orbit and k = 1, the orbit is linearly stable for every radius (and Lyapunov stable as an equilibrium of the system reduced at fixed angular momentum). For k ∈ {2, 3} there is exactly one threshold radius r_c(k): circular orbits of smaller radius are spectrally unstable and those of larger radius are linearly stable (and Lyapunov stable in the same sense); the case of radius equal to r_c(k) is not decided. Interval arithmetic certifies r_c(2) ∈ [0.704632163781, 0.704632163783] and r_c(3) ∈ [0.858944741464, 0.858944741468]. For three bodies at the vertices of an equilateral triangle (a single mutual distance) and k ∈ {2, 3}, the relative equilibrium is spectrally unstable for all masses whenever the side is at most r_c(k); the criterion is an application of an inequality of known form. With mixed distances, stable relative equilibria containing pairs inside the threshold exist: for three configurations — a heavy centre (mass ratio 10^4:1) and a regular ring of 7 or 8 light bodies revolving at a radius outside the threshold; one example for k = 2 and two for k = 3 — the distance between neighbouring ring bodies is smaller than r_c(k), yet the relative equilibrium is spectrally and linearly stable (certified by interval arithmetic). The pair formed by the centre and a ring body, in contrast, is not rescued: for k ∈ {2, 3}, a relative equilibrium consisting of a central body and a regular ring of n ≥ 2 bodies of equal mass is spectrally unstable whenever the centre–vertex distance is at most r_c(k), for every n and every mass ratio (a theorem, proved analytically). Hence, within this ring family, a centre–vertex distance larger than r_c(k) is necessary for spectral stability (its sufficiency is not claimed: stability outside the threshold is asserted only for the three certified examples), and the boundary of stable sizes is not of the form "all pairs outside the threshold". We do not claim novelty: the forms of the criteria and the type of the configurations are known, and prior work was compared mostly at the level of titles and abstracts (the reservations are stated in the text). Formal stability and nonlinear confinement, k ≥ 4, the stability of rings other than the three certified examples, and pairs of two heavy bodies orbiting each other are left open. Nothing is said about the reality of extra dimensions or about observational bounds. All statements were drafted by an AI system and were confirmed only after review by fresh, context-free instances of another AI model of the same family, which re-derived the proofs and, for statements that depend on computation, checked them with independent re-implementations — 3 rounds of proof review and 3 narrow difference reviews of the revisions — with no mandatory fix remaining; no human refereeing or external peer review has taken place (Section 6, Appendix E). The proofs are reproduced in Japanese, byte for byte from the project's normative documents (Appendix B); Appendix A translates the eleven confirmed statements. English outlines of the proofs of Theorems WR-T2 and WR-T3 precede Appendix B; they are reading guides and are not stronger than the transcribed proofs. Version. v0.3.1 (2026-09-28): the note v0.3.1, after the manuscript review and its narrow difference review; Project WHITROW. This record holds the note (note.pdf), the English version of the statement (onepage-en.pdf), a README, and an archive of the project repository at the deposit commit (whitrow-v0.3.1-repository.tar.gz; files without the git history, with the excluded files and masked copies listed in the README), with a MANIFEST.sha256. The normative texts are the Japanese project documents; the note is a derived product. Contact: [email protected]