Log-Free Supersaturation of Collinear Triples in Three-Dimensional Grids

Let F(n,m) be the minimum number of unordered collinear triples in an m-point subset of [n]^3. We prove that F(n,m) has order m^4/n^6 whenever 12n^2<=m<=n^3, with absolute constants. The lower bound removes the logarithmic loss in the Balogh-Solymosi supersaturation estimate. Its proof counts all short primitive positive directions and uses the exact partition of the grid into maximal lattice chains. For the upper bound, we adapt Roche-Newton's periodic finite-field cap construction to arbitrary side lengths and exact cardinalities, using an anisotropic quadratic form and translation averaging. In particular, the minimum has order n^(6-4s) for every fixed 0<=s<1 and m=ceil(n^(3-s)).

This is a complete proof of a dense-regime order theorem connected to OWR-15427-019 in UnsolvedMath v1.6.0. It does not determine exact minimum values, optimal leading constants or the threshold near m=n^2. The upper method is credited to prior work, and the separate rigidity question in the source is already solved in the literature.

AI-assisted, self-audited and unrefereed preprint. No independent review, formal proof-assistant verification or absolute priority is claimed. Exact finite regression programs accompany the source; they do not replace the general written proof.