What question did this study set out to answer?

The main aim is to establish a unified analytical framework for the stability and dynamics of polygonal configurations under various potentials.

May 14, 2026Open Access

The N-Gon Harmonization Programme: A Unified Framework for Cyclic-Symmetric Relative Equilibria (Papers I–IV)

Key Points

The main aim is to establish a unified analytical framework for the stability and dynamics of polygonal configurations under various potentials.
Developed closed-form vortex spectrum and asymptotic structure at A = 2.
Utilized linear isomorphism to connect perturbation spaces across configurations.
Proved the Planarity Theorem for equilibrium structures using non-perturbative arguments.
Established an explicit vortex master spectrum and refuted an earlier conjecture on odd N setups.
Demonstrated structural identity between bifurcation and Nyquist frequencies for polygons.
Classified all results by proof status in the Claim-Status Register for further validation.

Abstract

This deposit contains the complete four-paper submission package for the N-Gon Harmonization Programme: a unified analytical framework for the stability, bifurcations, and dynamics of regular polygon central configurations across the smooth Riesz family of homogeneous potentials V (r) ∝ r^ (2−A), A ∈ (0, 2), with A = 2 the Kirchhoff–Routh vortex limit. The package consists of four papers (91 pages total) plus 10 Python validation scripts running 305+ numerical checks: Paper I (25 pp) — Closed-form vortex spectrum and asymptotic structure. Establishes the all-mode vortex master spectrum at A = 2, the sharp two-term Hampton asymptotic, and the universal A* (N) ·N → 4 log 2 threshold across both parities (refuting a prior 4π²/ (N log² N) conjecture for odd N). Paper II (33 pp) — Linear stability and bifurcation structure. The programme's headline result: an explicit linear isomorphism Φₙ intertwining the Maxwell two-ring perturbation space at μ = ρ = 1 with the regular 2n-gon perturbation space at every A ∈ (0, 2), yielding the structural identity Abif (n) = A*Nyq (2n) for every n ≥ 4. Paper III (16 pp) — Spatial structure and domain of validity. The Planarity Theorem: every smooth-Riesz relative equilibrium with positive masses under single-axis rotation is strictly planar, proved non-perturbatively via a maximum-z-body argument under V' (r) > 0. Perturbative consistency framework with J₂ oblateness, post-Newtonian corrections, and Kozai–Lidov. Paper IV (17 pp) — Programme synthesis. Hosts a Claim-Status Register classifying every headline result by proof status: analytically proved, reduced to standard identity, numerically validated conjecture, or heuristic/application-level. A single methodological backbone is used uniformly throughout: Fejér-kernel decomposition with Fourier-of-log-sin expansions and Gauss's digamma special-value formula, yielding closed forms in elementary special functions (harmonic numbers, digamma values, ζ (3) ) suitable for symbolic and numerical verification. The 10 validation scripts (Python 3. 10+, numpy/scipy/mpmath) reproduce every numerical claim in the papers. To run: cd validation && python3. py. Comments, collaboration inquiries, and citation requests are welcome. Contact: rommagael@gmail. com (ORCID: 0009-0006-5109-1549).

The N-Gon Harmonization Programme: A Unified Framework for Cyclic-Symmetric Relative Equilibria (Papers I–IV)

Key Points

Abstract

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