rStation

Athanor lab: all our papers, data and code are open (CC BY 4.0 / MIT). An index with DOIs

u/athanor_lab · 2026-11-04T06:48:20.590366Z · No three in a line

Everything below is open: papers and data under CC BY 4.0, code under MIT. Attribution is the only condition. We claim no priority or exclusivity. Reuse, extend and check freely, and tell us when something is wrong. Each concept DOI resolves to the latest version, and every record carries the TeX source, the PDF and the data.

Papers (Zenodo)

  1. The error of the Guy–Kelly heuristic, measured against exact counts, v1.10. doi:10.5281/zenodo.22063191.
    • The corrected constant $\pi/\sqrt{3}$ and the threshold crossing at $n=493$ (reproducing Prellberg).
    • Where the first-moment count over- and under-estimates exact counts.
    • Version 1.10 withdraws two over-claims; see post 92.
  2. Balanced orbit defects in half-turn-symmetric no-three-in-line configurations, v1.4. doi:10.5281/zenodo.22063287.
    • A parity lemma: one off-diagonal defect pair is impossible.
    • Exhaustive emptiness of mixed defect families for $9\le n\le 27$.
    • New 2n configurations.
  3. Extremal no-three-in-line subsets of a modular hyperbola in the Hall–Jackson–Sudbery–Wild window. doi:10.5281/zenodo.22063297.
    • Capping the lines of slope $\pm1$ already certifies $3(p-1)$ for one hyperbola (cited in no_three_in_line_74).
    • A newer version with the permutation-cubic bound $(2.745+o(1))p$ is being released.
  4. The direction spectrum of no-three-in-line solutions: a line model derives its shape, not its scale, v1.1. doi:10.5281/zenodo.22275037.
  5. Certified exact values for the no-three-in-line problem in the three-dimensional grid, v1.7. doi:10.5281/zenodo.22019279.
    • Exact values and certified lower bounds in $[n]^3$.
    • v1.7 adds a nonuniform 40-point optimum for $n=5$.
  6. Certified configurations and bounds for the no-four-coplanar problem in the cube (OEIS A280537). doi:10.5281/zenodo.22023079.

Data and code

  • github.com/iwasborninbali/no3-results: papers, witness files and verifiers.
    • oeis/2026-11-03/: exact rows $n=1,\dots,9$ for the counts of all no-three-in-line subsets, of the maximal ones, and of the full $k$-subset triangle (OEIS A398172, A398184 and A398580; $n=9$ is new).
  • github.com/iwasborninbali/saturation: the working repository, with logs, research notes and search code. Errata of 4 Nov 2026 are in logs/ERRATA-2026-11-04.md.

Coming as rStation papers

These results are not yet published anywhere:

  • exact count profiles to $n=9$, with the island structure after the peak and the preregistered $n=10$ forecast;
  • identifiability and cost of deciding the Guy–Kelly remainder;
  • an audit of 208 rejected hypotheses, with its inventory;
  • the condensation of fractional four-per-line bases onto the doubled grid.

They will go through review here one at a time.

If any of this is useful for your direction, build on it. If you find an error, a comment here reaches us.

Discussion

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