Compute-Optimal Allocation Between Model Size and Data Under a Parametric Scaling Law: Closed-Form Solution, Convexity Proof, Misallocation Penalty, and Sensitivity to the Fitted Constants

Under the parametric loss L(N,D) = E + A/N^α + B/D^β with training compute C = 6ND, I derive the compute-optimal model size and token count in closed form: N* = G(C/6)^{β/(α+β)} with G = (αA/(βB))^{1/(α+β)}. I prove that the constrained problem is strictly convex in log N, so the solution is unique, and derive a quadratic approximation to the loss from misallocation. Two published fits are evaluated: the rounded constants of Hoffmann et al. (2022) and the re-estimate of Besiroglu et al. (2024), both read directly from the papers' text. They imply different allocation rules. At the Gopher budget of 5.76 × 10²³ FLOPs, the first gives N* = 32 B parameters and 93 tokens per parameter; the second gives N* = 72 B and 18 tokens per parameter. The ratio drifts with compute in opposite directions under the two fits (50 → 153 against 22 → 16 tokens per parameter over 10²¹–10²⁶ FLOPs). Misallocation is cheap near the optimum: a model 2× too large raises loss by about 0.005 nats, and offsetting a 4× oversize needs 1.8–1.9× more compute. The prescriptive content of "compute-optimal" depends on second-decimal differences in fitted exponents.