A C-Point Force-Center Reformulation with Candidate-Reference State Evaluation for the Newtonian Three-Body Problem

We give an exact force-center reformulation of Newtonian three-body dynamics. For general N, each body's Newtonian acceleration can be written as a force toward a body-dependent weighted centroid C_i, r̈_i = G A_i (C_i − r_i). For N = 3, the bodywise force lines have the additional concurrence property that all three nonzero net-acceleration lines pass through a shared instantaneous C-point: C = (Σ_i m_i s_i³ r_i) / (Σ_i m_i s_i³), and the equations become r̈_i = G W / (s_j³ s_k³) · (C − r_i), where s_i = ‖r_j − r_k‖ is the side opposite body i and W = Σ_ℓ m_ℓ s_ℓ³. These identities are exact algebraic reformulations of Newtonian gravity on the no-contact domain, not new force laws. We also define a normalized C-flow state-evaluation operator for the three-body initial-value problem. Physical initial data are reduced by center-of-mass removal, Galilean drift, total-mass normalization, and scale normalization to a dimensionless canonical flow Φ_3, and the dimensional state is reconstructed algebraically. This notation packages normalization, normalized flow, and reconstruction into a single physical state-evaluation operator S_3(t; G, m, r⁰, v⁰). It is not claimed to be an elementary closed-form solution or a proof of classical integrability. Finally, we describe a finite Chebyshev-slab evaluator for the normalized C-flow and report candidate-reference error estimates and numerical consistency tests. These estimates are not rigorous interval enclosures. A rigorous error-enclosure implementation would require interval arithmetic, radii-polynomial bounds, rigorous defect bounds, or an equivalent a posteriori method.