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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.