Sharp fourth-root concavity for lattice counts on smooth toric Fano threefolds

Abstract

The fourth root of the ordinary Riemann–Roch count polynomial is concave on the entire closed nef cone of every smooth toric Fano threefold. The exponent 1/4 is uniformly optimal for these real polynomial extensions. Nine of the eighteen threefolds have concave cube root; the other nine have genuine cube-root failure. We prove a sufficient criterion for cubic full-count polynomials using classical Lorentzian volume theory and verify its hypotheses by exact fan and polynomial certificates. The result includes integral nonbig nef classes. For the split bundles P_{P²}(O ⊕ O(a)), the ordinary polynomial has optimal power 1/3 at twists a=0,1,2 and no positive concave power at any integer twist a≥3. For geometrically weighted lattice counts, biases r≥1 admit a nonpolynomial concave logarithmic extension for every twist, whereas every fixed 0<r<1 has integral midpoint counterexamples at large twists. The proof uses sharp coefficient conditions for general quadratic prefix weights and uniform rectangular-moment tail identities. It also gives exponential-quadratic extensions. No positive power works uniformly in the unweighted twist family, or at any fixed bias r>1 over prefixes. Ordinary-polynomial optimality and lattice-only sharpness are kept separate.