A Certified Enumeration of the Planar Four-Body Central Configurations with Three Equal Masses

We give the first interval-arithmetic–certified enumeration of the planar central configurations of the Newtonian four-body problem for the one-parameter mass family (1,1,1,μ). At seven values of μ spanning a thirty-fold range, together with two further values bracketing the symmetry-breaking window about μ=1, we certify the count exactly: 38 outside that window and 50 inside it, counted as labelled configurations modulo translation, rotation and scaling. The numbers themselves are not new — they appear in Simó's 1978 numerical survey and are tabulated by Albouy; the full count-versus-mass curve is given numerically by Rusu and Santoprete, who state explicitly that they are "unable to prove" it. What is new is the proof of exhaustiveness: each count is obtained by a Krawczyk interval-Newton branch-and-bound over a search box that we prove contains every central configuration of the given masses, so no configuration can have been missed.

Three ingredients make this possible. First, a formulation in Cartesian positions in which the multiplier λ is eliminated as U/I; the residual system is then four equations in four unknowns, and we show that its zero set coincides exactly with the set of central configurations, the obstruction being a linear system whose determinant is independent of the configuration. Second, an explicit a priori bound D ≤ κ(m)·d on the ratio of largest to smallest mutual distance, derived from a sub-cluster virial identity; the literature supplies compactness but no usable constant. Third, the observation that the residual map is not proper, so that a bounded domain is a necessity rather than a convenience.

We also measure the cost of certification near a degeneracy, giving a second and independent measurement of the certification-cost law χ introduced in a companion paper (Lovecký 2026, doi:10.5281/zenodo.21505198). We derive χ ≈ log₂(1/gap) from the Krawczyk contraction condition and confirm it at a corank-one fold (1.18 measured); at the corank-two degeneracy of Palmore we measure 1.56, which the derivation does not explain, and we report the discrepancy rather than fit it.

Reproducibility. Every certified count is emitted as a disposition-tree certificate whose verifier re-derives the count leaf by leaf and shares no code with the search; the one unverified premise (the a priori domain bound) is printed on every run. The attached artifact snapshot (code license Apache-2.0) reproduces the counts: cargo test --release runs the suite, and the per-mass certificate summaries carry the SHA-256 of each disposition tree.

Methodology disclosure. The results were produced using an agentic AI system directed and adjudicated by the author, under a methodology of the author's design: adversarial review of every derivation and a numeral-tracing gate on the manuscript. Section 6 documents it, including the five earlier claims the reviews refuted and this paper corrects.