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Abstract

We prove a corrected fourth-root Jensen inequality for unimodular integer fibers whose central affine dimension is at most three. For an integrable integer-valued demand A with feasible fibers almost surely and integral mean b, a bounded nonempty central fiber satisfies E[(N(A)−1/12)^(1/4)]≤(N(b)-1/12)^(1/4). This implies a 2/11-power inequality and logarithmic Jensen, with finite expectations and countable support. The proof constructs a monotone continuous extension of type-A₃ lattice counts with all twelve directed bounds varying. A quartic Lorentzian certificate gives the corrected fourth root; shortest-path closure and a common integral rank-three reduction transfer it to arbitrary unimodular systems. For this specified continuous count extension, 2/11 is the largest concavity power and 1/12 is the least fourth-root correction. Endpoint-count thresholds give stronger powers tending to 1/4. We also prove sharp corrected square-root Jensen in dimension two and its optimal leading large-count exponent deficit. For the complete acyclic four-vertex digraph, the classical continuous count extension has optimal power 4/13, while exact integral examples obstruct cube-root Jensen. Directed rounding and a full-dimensional local obstruction explain the role of a global continuous extension.