Positive Functions Equimeasurable with Their Derivatives: Construction and Rigidity

Rubel asked for functions whose derivative has the same value distribution as the function (Hayman-Lingham Problem 7.77). We parametrize all strictly positive C1 solutions on a compact interval by continuous Lebesgue-measure-preserving maps satisfying a spectral-radius condition. Twice-differentiable strictly positive solutions are exactly exponentials and an explicitly phased sine family. The tent map gives positive C1 solutions precisely for interval lengths below 2, with exact spectral radius, endpoint value, uniform approximation errors and dense dyadic losses of higher differentiability. At the critical length, a rescaling of the classical Fabius distribution supplies a smooth nonnegative example; that distribution and its smoothness are explicitly credited, not claimed as new. The construction also gives positive functions equimeasurable with any prescribed higher derivative on sufficiently short intervals. This answers the additional-examples request and classifies the strictly positive C1 class. The unrestricted classification of vanishing or sign-changing functions remains open; the whole frozen UnsolvedMath record AMR-022-7077 is not claimed solved. The package contains the five-page proof, exact rational checks and a source/scope audit. AI-assisted self-audited preprint; no independent peer review, formal verification or exhaustive priority certification is claimed.