THE ABÏON METRIC DERIVATION OF THE SECOND-ORDER COEFFICIENT C₃

We present a symbolic derivation of the second-order coefficient C₃ that governs the non-linear correction to translational frame-dragging in the ABÏON metric. Using the Hartle-Thorne slow-rotation formalism, we show that the second-order correction to the frame-dragging function ω(r) has a universal functional structure shared with the Kerr metric, with a single geometric factor K that encodes the specific topology of the source. We demonstrate that: (1) the exponent n = 11 for the effective source S_ω(r) is required by dimensional analysis and asymptotic flatness; (2) the coefficient C₃ takes the form C₃ = K·c⁴/(992·M²·J), where K is a dimensionless geometric factor; (3) in the Kerr limit, K reduces to K_Kerr = 1488·M²·J/c⁴, recovering C₃ = 3/2 as required; (4) the roadmap to computing K numerically is explicit and tractable. This analysis converts an assumed value (C₃ = 3/2, inherited from Kerr) into a derived structural result with one geometric parameter K identified for future numerical computation. The sign of C₃ is positive for any physical source, confirming that non-linear effects amplify frame-dragging. A companion SymPy script reproduces all symbolic and numerical results. Keywords: ABÏON metric, second-order perturbation theory, Hartle-Thorne formalism, frame-dragging, C₃ coefficient, Kerr limit, geometric factor, slow-rotation expansion, gravitomagnetism, General Relativity