The Hidden Symmetry Unique to n=2: Why Inverse-Square Orbits Close — The Laplace–Runge–Lenz Vector and the Dynamical Twin of Paper 01.
Paper 01 showed that the density exponent for which the spherical-shell mass is radius-independent in 3D space is uniquely n=2 (inverse-square) — the GEOMETRIC uniqueness of n=2. This paper presents its DYNAMICAL twin: the same 1/r² force possesses an extra conserved quantity absent from other central forces — the Laplace–Runge–Lenz (LRL) vector A = p×L − mk·r̂ — which makes the orbit close (a non-precessing ellipse). A is conserved pointing along the major axis; this is the answer to “why do planetary orbits close?” This extra conservation generates, for bound states, a hidden symmetry SO(4) (Fock 1935; in momentum space the bound Kepler problem is free motion on the 3-sphere S³). The key discipline is consistency with Bertrand’s theorem: exactly two central forces give closed orbits — 1/r² and the harmonic oscillator r² — yet their hidden symmetries DIFFER (SO(4) vs SU(3)). “The orbit closes” is a shared value, but the origins are distinct and must not be merged (distinct roots, Papers 58/48). Thus n=2’s specialness propagates from 3D geometry (Paper 01) into dynamics — not as the sole winner, but as the one bearing its own symmetry. (Mercury’s residual perihelion precession is a real instance of A’s non-conservation, i.e. gravity deviating from exact 1/r² — general relativity.) IMPORTANT (scope): the LRL vector, Fock’s SO(4), and Bertrand’s theorem are all standard; no new mathematics, no new predictions; n=2 is not claimed to be the sole closed orbit; Fock’s S³ and the corpus’s Hopf S³ (67/68) are NOT identified (a rhyme, not a fusion).
A note on the aim of this work. This is a structural interpretation — the dynamical twin of Paper 01’s geometric uniqueness — not a solution and not a new physical law. The word “Universe OS” is an interpretive lens, not a testable claim. Nothing here is intended to contradict existing results.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Hidden Symmetry Unique to n=2 [Paper 79]. Zenodo.
DOI: 10.5281/zenodo.21732820
ORCID: 0009-0006-8587-3450