The direction spectrum of no-three-in-line solutions: a line model derives its shape, not its scale

A solution of the no-three-in-line problem is a set of 2n points of the n×n grid with no three collinear. For a primitive direction v, c_v(n) is the mean number of point pairs of a solution whose difference is a positive multiple of v, divided by n. Measured on Flammenkamp's database of all known solutions, the short directions are nearly constant in n (c_(1,1) = 0.731, c_(1,2) = 0.564) and depleted against the null model with fixed row and column sums, while long directions are slightly enriched and grow with n.

The note shows that the shape of this spectrum — the ratios between directions and their order — follows from a single rule with no fitted parameter: distribute 2n points over the lines of one direction with at most two points per line, a line of L cells carrying x points having weight C(L, x). The expected number of doubly occupied lines is given exactly by a generating function (Theorem 1, with proof); one global renormalisation to the exact number of non-axial pairs of a solution, 2n² − 3n, fixes the scale. The model reproduces fifteen measured constants within 12% (model/data 0.89–1.05 at n = 20) and their order at matched n up to one adjacent pair; seven of them were predicted before being measured by a second agent who had not seen the model.

What the model does not do is stated with equal care: the scale is imposed, not derived (the model's own total falls 17% short of 2n² − 3n, and the shortfall it redistributes is exactly the deficit seen in the measured directions); the growth of the long directions with n is only partly tracked; a first-order pair model fails; and no prior-art search beyond the classical sources has yet been run. Data by size strata (15 direction classes; n = 19–20 complete enumeration, n = 29–31, n = 32–57) and the model values are attached. Research pipeline: https://github.com/iwasborninbali/lemma-atelier (lemma 005, need 008).

v1.1 (3 Sep 2026): the sentence "no prior-art search has yet been run" is replaced by a prior-art section: a multi-channel search (record attached as deep_research_14_answer.md) found no published measurement of pairs per direction on actual solutions, no κ-like constant, no per-direction decomposition of the Guy–Kelly count and no model of this kind; nearest objects delimited (Guy–Kelly 1968; Flammenkamp's frequency maps; Prellberg's capacity-two LP bound, arXiv:2605.09215; Simkin's diagonal occupation for n-queens; an unrefereed web-posted analysis of July 2026 with the same qualitative conclusion — hence the qualitative observation is not claimed as new). The identity of Theorem 1 has no established name we could find (nearest: restricted occupancy, Freund 1956). Limits of the search stated. Data and model unchanged.