The Amari-Schwarzian Cubic: Nodal Realisation and Empirical G_m Structure of Symplectic Integrators

We study the algebraic geometry of the Amari-Schwarzian identity (A + S')^2 = 2(D + S)(D + 2S)^2 previously established in (Maino, 2026, DOI: 10.5281/zenodo.19425788). Under the affine substitution X = D + S, Y = A + S' the identity becomes the equation of a rational nodal cubic C_S : Y^2 = 2X(X + S)^2 with a unique singular point at (-S, 0) and a smooth locus naturally isomorphic to the multiplicative group G_m. We give the explicit rational parametrisation X = s^2/2, Y = s(s^2/2 + S) and verify the entire algebraic content symbolically. Numerical integration of the Riccati flow associated with any non-trivial Q(t) confirms that trajectories remain on C_S to floating-point cancellation accuracy (median relative residual 7.6 x 10^-17, max 3.3 x 10^-9).We then establish empirically that any second-order symmetric integrator (Stormer-Verlet/leapfrog) applied to omega'' + Q(t) omega = 0 produces a discrete spectrum {u_i = (omega'/omega)_i / sqrt(Q)} whose distribution matches the half-Cauchy spectrum |tan Theta| with Theta uniform on (0, pi/2). Across constant, smoothly varying, and large-complex-structure-flavoured (Q ~ 1/t^2) potentials, we measure median ~ 1.000, IQR ~ [sqrt(2) - 1, sqrt(2) + 1], inverse-pair product Q1*Q3 ~ 1.000, and Var[log|u/sqrt(Q)|] ~ 2.43 vs. the predicted pi^2/4 ~ 2.467, the small systematic shortfall attributable to finite-sample tail truncation. The IQR endpoints sqrt(2) +/- 1 are exact multiplicative inverses in R^x, reflecting the G_m structure of the smooth locus, with multiplicative IQR (sqrt(2) + 1)^2 = 3 + 2 sqrt(2) coinciding with the fundamental unit of Z[sqrt(2)]. This bridges the continuous integrable structure of the Amari-Schwarzian identity with the discrete statistical behaviour of the integrators that compute it. The paper distinguishes rigorously proven results (the algebraic theorems, the trajectory residual at machine precision) from empirically supported observations (the half-Cauchy emergence), and lists open problems for the next steps.