Certified moment bounds for transported threshold quantities

Moment observations may leave a transported threshold quantityuncertain even when numerical optimization is accurate. We compute its admissible range over nonnegative measures, including atoms, from boundederror phase-space moments for one-dimensional periodic free transport. Anexact affine partition preserves boundary membership and reduces the problemto coupled univariate measures. Physical atomic witnesses and rational polynomial dual certificates bound both endpoints and separate information diameter from endpoint optimization error. Exact moment implications can removeboundary relaxations without new observations. A complementary transporttube construction bounds the target through rectangular zeroth and secondmoments. Its width depends on weighted boundary mass and measurementnoise, yielding conditional convergence and explicit atomic obstructions. In atwelve-case synthetic study, early reactive measurements beat a fixed initialreference in eight cases and lost in four; the normalized mean improvement’s95% case-bootstrap interval contained zero. At a prespecified 48-purchase setting, a 192-state development census and an independently frozen 192-stateholdout each certify smaller tube information diameter than two equal-costgrids in all 32 cases. A separate 36-state adversarial study certifies a smallpolicy-order reversal despite convergence of every endpoint solve. Finer resolution can worsen noisy bounds, while exact implications can resolve structural boundary gaps. The results support this computational framework andrestricted enclosure construction, without establishing universally superior acquisition, uniform atomic convergence, or physical discharge validation.