Positivity in a Spectral Inverse Problem for Free Transport

We study a spectral inverse problem for the one-dimensional free transport equation on non-negative initial states with a prescribed spatially averaged velocity profile. The central point is that positivity couples the blind frequency regions of the observation symbol to this known profile, so identifiability is not determined by the symbol alone. We establish an exact static injectivity criterion and a modewise counterpart for centred finite-time histories. Beurling–Malliavin majorants lying below the profile produce distinct non-negative states with identical data, whereas a Denjoy–Carleman moment condition yields exact identifiability. In the family fα,c(v)∝ exp[-c(1+v²)α/2], these mechanisms give a sharp transition: hard-cutoff observations are non-injective for 0<α<1 and injective for α≥ 1. If m denotes the observation symbol, the modewise Gramian over the centred interval $[-T,T]$ is gm,T(k,η)=∫-TT|m(η-kt)|²\,dt. This formula makes the invisible frequency geometry explicit. Under a global algebraic lower envelope, we derive sharp static and finite-time ambiguity scales in a prescribed finite spatial sector, together with optimal-recovery bounds for centred linear noisy fibres in that same sector. For quasi-analytic profiles, we also prove qualitative compact conditional stability on a fixed mixed space–velocity Sobolev class without a finite-mode restriction: the worst-case diameter of non-negative noisy fibres tends to zero as the noise level vanishes. No explicit stability modulus is claimed. The model therefore separates exact injectivity, discontinuity of unrestricted inversion, quantitative ambiguity, and compact conditional stability.