C-Point Force-Center Reformulation w/ Candidate-Ref State Eval for the Newtonian 3-Body Problem

We give an exact force-center reformulation of Newtonian three-body dynamics. For an arbitrary number of gravitating bodies, each body's Newtonian acceleration can be rewritten as a single attraction directed toward a body-dependent, mass-weighted centroid. For three bodies, the three nonzero net-acceleration lines have an additional property of concurrence: they all pass through one common instantaneous point, an explicit inverse-cube, side-weighted barycentric combination of the three positions. This concurrence holds for every non-collinear, collision-free configuration, not only the equilateral (Lagrange) case. The shared point is generally distinct from the center of mass, coinciding with it exactly when the triangle is equilateral, and in the equal-mass case it is the catalogued triangle center X(31). We are not aware of a prior statement of this inverse-square, all-configuration, mass-weighted concurrence. These identities are exact algebraic reformulations of Newtonian gravity on the collision-free domain, not new force laws.

We also define a normalized state-evaluation operator for the three-body initial-value problem: physical data are reduced by center-of-mass removal, Galilean drift, and total-mass and scale normalization to a dimensionless canonical flow, from which the dimensional state is reconstructed algebraically. This is a flow-map notation, not an elementary closed-form solution and not a proof of classical integrability.

Finally, we describe a finite Chebyshev-slab evaluator for the normalized flow and report candidate-reference error estimates and numerical consistency tests; these are empirical estimates, not rigorous interval enclosures.