This paper provides a function field model for the motivic rigidity conjecture formulated in 1. Over Fq (t), we study families of elliptic curves Ef: y² = x (x+f) (x+f+g) parametrized by monic irreducible polynomials f of degree d, and establish unconditional equidistribution of their Frobenius traces via the Deligne equidistribution theorem when SL (2) monodromy holds. The key result is a density threshold: equidistribution requires the density of irreducible polynomials (approximately 1/d by the Prime Polynomial Theorem) to exceed the spectral gap of the monodromy group, which is satisfied for all d ≥ 1 in the function field setting. Over Z, the analogous density 1/log x for rational primes falls below effective equidistribution thresholds, identifying the density deficit as a fundamental obstruction to transferring function field results to number fields.
Ruqing Chen (Wed,) studied this question.