Finite Algebraic Non-Compressibility of the Planar Equal-Mass Newtonian Three-Body Flow

Abstract

We give a reproducible finite-degree algebraic obstruction to polynomial compression of an orbit germ of the planar equal-mass Newtonian three-body flow. The question is deliberately narrower than global integrability or the existence of a closed-form solution: after quotienting the exact geometric distance constraints and the fixed energy and angular-momentum levels, can a nonzero polynomial of bounded total degree vanish identically on the orbit germ? We answer this question by an exact polynomial realization, a filtered invariant quotient, and a finite matrix of Lie derivatives evaluated at an exact rational witness. For the asymmetric witness u = (3, 0), v = (0, 4), U = (1, 2), V = (−1, 1), (cu, cv, cw) =  1 3 , 1 4 , 1 5  , the degree-four quotient has dimension 1207. An executable modular implementation regenerates the 1207×1207 Taylor-jet matrix directly from the polynomial vector field and obtains rank 1207 modulo both 1 000 003 and 1 000 033. Since a nonzero maximal minor modulo one good prime implies the corresponding characteristic-zero minor is nonzero, this is an exact characteristic-zero certificate. Independently, the same Lie-jet construction on the three squared mutual-distance variables has full rank through total degree 10, with dimensions 4, 10, 20, 35, 56, 84, 120, 165, 220, 286. A nonzero-minor argument yields a componentwise genericity statement for each fixed degree. The constituent mathematics is standard differential algebra, Lie differentiation, and modular linear algebra; the contribution is their explicit, invariant-aware, machine-reproducible assembly into a finite orbit-germ certificate.

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