The Chebyshev–Monomial Oscillation Gap: Contact Geometry, Level Sets, and Gaussian Crossovers

We study the uniform norm of a prescribed Chebyshev polynomial against a monomial, \(\|T_m-x^d\|_{\infty,[-1,1]}\), for integers \(d\ge m\ge3\) of matching parity. The continuous contact surface \(\mathcal{C}(p,q)=\max_{0\le\theta\le\pi/2} |\cos^p\theta-\cos(q\theta)|\) separates the power of the envelope from the oscillation frequency. The global superunit maximum is carried by one nondegenerate contact. We use that contact to describe the inverse exponent and frequency at every norm level above one, and to obtain the exact induced Hessian. A naturally defined threshold \(p_{\mathrm c}\in(1,2)\) separates a strictly concave frequency section from one having exactly one simple inflection. The integer ladder has an additional first-wave stationary point precisely when \(d>m^2\); one first-wave monotonicity lemma explains this threshold without separate numerical cutoffs. Gaussian contact asymptotics admit finite-parameter corrections, a sharp worst-case first-order constant, and a quartic relative transition; inverse level curves have an explicit further coefficient. The mechanisms are analytic, with no numerical or interval certificate entering any proof. The diagonal case and the ladder polynomial are credited to earlier CF companion manuscripts; classical polynomial approximation and mod-Gaussian results account for the underlying square-root and quartic scales. Assertions of global mathematical priority are not made here.