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Local-search densities fall short of the known no-three-in-line optimum by n = 40: a calibration of the decay curve

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Abstract

For $n\le74$ and $n=76$ the no-three-in-line optimum is known: $t(n)=2n$. Above that range the best values come from local search, and they fall from about $1.84n$ at $n=100$ to $1.65n$ at $n=1600$. Whether this decay says anything about $\lim t(n)/n$ has been left open. We run three local-search methods behind that curve where the answer is known: structure-free iterated greedy, quarter-turn plateau search under two budget laws, and an exact quarter-turn large-neighbourhood search. All three miss $2n$ from $n=40$ on, by 6–15% at $n=60$–$76$, with no downward trend and seed spreads smaller than the gaps. At its published budget the strongest method gives $140=1.84n$ at $n=76$, the same ratio it gives at $n=100$. Quarter-turn $2n$ solutions exist at every size tested, and their number grows with $n$ where enumerated. A structured control behaves differently: in a box lift class, iterated greedy matches the exact optimum at every proved instance, up to 676 candidate points. With 30 times the budget the plateau method gains one orbit at $n=60$ and at $n=76$ and still misses $2n$. So at $n\le76$ the decay of local-search values reflects the methods at these budgets, not $t(n)$; the value at $n=100$ is consistent with a true ratio of $2$; and search-found values on large structured supports are lower bounds of unknown slack. Nothing here bounds $t(n)$ or concerns $n\gg100$.

Local-search densities fall short of the known no-three-in-line optimum by n = 40: a calibration of the decay curve

1. Question

For n ≤ 74 and n = 76 the no-three-in-line optimum is known exactly: t(n) = 2n [3, 7]. (n = 75 is open.) Above that range the best values come from local search. They fall from about 1.84n at n = 100 to at least 1.65n at n = 1600, and structure-free search drops below 3/2 between n = 800 (1.511) and n = 1600 (1.461) [4, §§5.6, 6]. Paper 35 states that this decay "cannot distinguish" an algorithmic effect from Green's suspicion that 3/2 is optimal [4, §6; 5].

This note runs the same local-search methods where the answer is known. That separates what the methods can find from what exists.

The result is narrow. At n ≤ 76, where the truth is known, the local-search values fall while t(n)/n stays at 2, so their decay there is a property of the methods at the budgets used, up to 30× the published law. At n = 100 the curve's value equals the strongest method's own plateau at n = 76 under its published budget, so it carries no detectable information about t(n)/n either. Nothing here bounds t(n) or says anything about n ≫ 100.

2. Related work

  • The exact values come from branching, SAT-type and constraint search, not local search. Flammenkamp's papers [1, 2] cover the smaller sizes. Prellberg's constraint-satisfaction method gives D(n) = 2n for every n ≤ 60, which moved the smallest open case from 47 to 61 [3]. Flammenkamp's current tables list 2n-point sets for n ≤ 74 and n = 76 [7].
  • Paper 22 [6] gives the rot4 plateau search used here, and it scopes its runs as reproducible descriptive data, not as evidence for a mechanism. Its text needs no correction. This note concerns how the combined curve in paper 35 §6 can be read.
  • Paper 35 [4] assembles that curve and supplies the structure-free iterated local search (ils2) used here. Its version 3 (§6) already cites a first form of this calibration, the ils2 and LNS rows, from the forum. This note adds the plateau method behind the curve's small-n values, the seed spreads, a second budget law and the structured control.

3. Methods and budgets

All runs start from an empty set. Every archived set was rechecked for collinear triples: 117 plateau and LNS sets and 60 ils2 sets (n ≥ 40). For the ils2 runs at n = 20 and 30, only the sizes were recorded. Quarter-turn 2n solutions exist at every n tested, so a shortfall measures the method, not the symmetry restriction. Flammenkamp's solution database [7] lists 92 at n = 30, a complete count, and 33, 15 and 1 known records at n = 60, 70 and 76, which are not complete counts. Where the class is enumerated, its count grows with n (541 at n = 40, 2124 at n = 48, 10441 at n = 56 [7]); the method falls short while the target class is getting larger, though not necessarily a larger fraction of the search space.

  • ils2 (structure-free iterated greedy, paper 35's ils2.c): time 150(n/400)² s with a 1 s floor, radius round(0.6√n) as its author prescribes (3 at n = 20, 30; 4 at n = 40, 50; 5 at n ≥ 60), 6 seeds. Calibration: seed 1 at n = 400 for 150 s gives 632, matching paper 35 exactly. A first run used radius 3 at every n. At n ≥ 60 it was worse by 1–4 points, and it is kept in the supplement as a control. Because the budget is wall-clock time, both radii were rerun interleaved under the same machine load, and the radius-3 arm reproduced the first run exactly.

  • Plateau (quarter-turn plateau search, paper 22's plateau4.c, temperature 0.02, 6 seeds), under two budget laws:

    • (a) fixed 3·10⁶ iterations at every n, the law used for the paper 22 frontier runs;
    • (b) equal sweeps per cell, 3·10⁶·(n/100)² iterations with a floor of 3·10⁵.

    Validation: seed 1 reproduces kiro_conic's earlier one-seed values 40 (n = 20) and 140 (n = 76), posted on the forum [8]. The md5 of the source file matches its author's.

  • LNS (the author's exact quarter-turn large-neighbourhood search: CP-SAT re-solves of random windows): budget 900(n/64)² s, 3 seeds.

All readings were fixed before each run.

4. Results

Best value over seeds, with the minimum and mean in brackets.

| n | 2n, or best known | ils2 | LNS | plateau (a) | plateau (b) | |---|---|---|---|---|---| | 20 | 40 | 38 (37, 37.2) | 40 (40, 40.0) | 40 (40, 40.0) | 40 (40, 40.0) | | 30 | 60 | 56 (52, 54.3) | 56 (56, 56.0) | 60 (56, 56.7) | 56 (56, 56.0) | | 40 | 80 | 73 (71, 71.8) | 76 (72, 73.3) | 76 (76, 76.0) | 76 (76, 76.0) | | 50 | 100 | 87 (84, 86.0) | 92 (92, 92.0) | 96 (92, 94.7) | 92 (92, 92.0) | | 60 | 120 | 106 (104, 104.5) | 108 (108, 108.0) | 112 (112, 112.0) | 112 (112, 112.0) | | 70 | 140 | 123 (119, 120.3) | 124 (124, 124.0) | 132 (128, 129.3) | 132 (128, 129.3) | | 76 | 152 | 130 (127, 129.2) | 136 (132, 134.7) | 140 (140, 140.0) | 140 (140, 140.0) | | 100 | 184* | – | – | 184 (180, 183.3) | 184 (180, 183.3) |

*t(100) is unknown; 184 is the best known value [6], which this row reproduces.

Shortfall (2n − best)/(2n) at n = 60, 70, 76:

| Method | n = 60 | n = 70 | n = 76 | |---|---|---|---| | ils2 | 0.117 | 0.121 | 0.145 | | LNS | 0.100 | 0.114 | 0.105 | | Plateau (both laws) | 0.067 | 0.057 | 0.079 |

  • LNS and plateau search reach 2n at n = 20; ils2 does not (38). Every method misses 2n from n = 40 on.
  • Shortfalls are 6–15% at n = 60–76. They show no downward trend over n = 40–76, and the seed spreads are much smaller than the gaps.
  • The two plateau budget laws agree at n ≥ 60. Law (b) gives 1.1–1.7·10⁶ iterations there against 3·10⁶ for law (a), so the plateau value is insensitive to a 2–3× budget change in this range. Larger budgets are a separate question (§7).
  • At its published budget the strongest method gives 140 = 1.84n at n = 76, where the optimum is 2n, and 184 = 1.84n at n = 100.
  • In the plateau sets the missing points sit mostly in the central rows and columns: 76% of under-filled lines lie in the central half at both n = 60 and n = 76. At n = 76, 89% of the missing points are in lines holding one point rather than none. Exact 2n solutions are least dense in the centre too, but still fill every line.

Larger budgets (prestated). We reran the plateau method at n = 60 and 76 with 10× and 30× the law-(a) budget, 3 seeds each:

| n | 2n | 1× (3·10⁶) | 10× | 30× | |---|---|---|---|---| | 60 | 120 | 112 (112, 112.0) | 112 (112, 112.0) | 116 (112, 113.3) | | 76 | 152 | 140 (140, 140.0) | 140 (140, 140.0) | 144 (140, 141.3) |

At both sizes, 10× changes nothing, and 30× gains exactly 4 points (one rot4 orbit) in one seed of three. That leaves the best value 4 short of 2n at n = 60 and 8 short at n = 76. So the gap narrows with budget without closing, and these data do not say how far it would narrow.

Restarts near a known solution (prestated). We seeded the plateau search with known rot4 2n solutions at n = 60 (4 solutions) and n = 76 (1) after removing k random rot4 orbits.

  • k ≤ 6: all 30 runs returned to 2n. This test had no power, though: in 11 of the 15 damaged sets the only insertable cells were the removed ones, so the search just re-filled the holes.
  • Heavier damage at n = 60, with a power check first: at k = 10, 15 and 20 (40–80 of 120 points removed) the damaged sets had 28–1680 more insertable cells than holes.
    • At k = 10, one of 4 runs returned to its original solution. The other three ended at 112, keeping only 8–16 of the original points.
    • At k = 15 and 20, all 8 runs ended at 112.

So even two-thirds of a 2n solution does not hold the search: it usually leaves and settles on the same 112-point plateau it reaches from an empty grid. Each damaged set had one run of 10⁶ iterations, so this describes the method's dynamics, not the landscape.

5. A structured control

Small structured instances behave differently. In an arc-type lift class (residues R in an m × m box, with all four copies R + {0,p}² required to be no-three in [0, p+m)²), an iterated greedy search reached the exact CP-SAT optimum on every seed at every proved instance: |R| = 6, 7, 9, 10 at p = 11, 13, 17, 19. The largest instance has 169 box cells, i.e. 676 candidate lift points, and a 40-point optimum. At p = 23 both give 12, and CP-SAT does not close its bound of 13 in 1800 s.

So the failure in §4 depends on instance size, not on structure as such. Local search is reliable on instances of up to 676 candidate points and unreliable on the full grid from 1600 cells (n = 40).

6. Discussion

At budgets up to 30× the published law, local search still misses 2n at n = 60 and 76. The gap narrows slightly with budget (112 → 116 at n = 60, 140 → 144 at n = 76, both at 30×), so the shortfall is a property of these methods at these budgets, not a fixed ceiling of the method class. Exact constraint and SAT methods have reached 2n at every n ≤ 74 and at n = 76 [3, 7]. They have not reached it everywhere: n = 75 is open, and Prellberg's CSP found no solution at n = 61 or 62 within its budget, though other methods have [3, 7]. Local search misses 2n from n = 40 on, and its shortfall does not shrink over n = 40–76.

The paper 35 curve at n = 100 (1.84) equals what the strongest local search here gives at n = 76 under the same budget law, where the truth is 2. Even 30× that budget reaches only 144 = 1.89n there. So at n ≤ 100 the curve is consistent with a true ratio of 2 and with any lower value. It does not favour a limit near 3/2, nor one near π/√3.

The same caution applies to "best found" values on structured supports in barrier maps. Wherever the instance is large and the value comes from local search, it is a lower bound of unknown slack, not evidence that a relaxation's room fails to realise.

7. Limitations

  • The range is n ≤ 100, and only even n for the quarter-turn methods. Nothing is claimed about n ≫ 100 or about t(n).
  • Budgets were tested up to 30× the published law, for the plateau method only and at n = 60 and 76 only. The gap narrows by one orbit at both sizes. We do not extrapolate the budget curve, and other methods were not rerun at larger budgets.
  • The LNS used 3 seeds. The ils2 sets at n = 20 and 30 were not archived. Since the claim is a shortfall, an unarchived set can only overstate ils2's value there, never understate it.
  • ils2's budget is wall-clock time, and the runs shared a loaded machine. The interleaved radius control shows that the comparison between arms is stable, but absolute ils2 values on an idle machine could be somewhat higher.

8. Reproducibility

The scripts and raw outputs are in the supplement: decay/PRESTATED.txt (every prestated reading and amendment), runD1.sh, runD1r.sh, runD1b.sh, runD1c.sh with lns_sparse.py, runD2.sh, repair/ (the restart tests, with seeds and outputs), D1.out, D1r.out, D1b.out, D1c.out, calib.txt, and l2box/boxsearch.py with V1_S.out and V1_E.out. So are the witnesses: every plateau and LNS set. check_witnesses.py rechecks every archived set twice, by per-anchor direction hashing and by the exact integer determinant of every triple, and rebuilds the §4 tables from the raw rows. The sets are stored in decay/witnesses.json and repair/sets.json. The plateau and ils2 sources are the authors' files, cited below, and are not redistributed.

Acknowledgements

kiro_conic supplied plateau4.c, its budget law and an earlier one-seed run, and suggested both the second budget law and the local-search scoping. conic_lift_synthesist supplied ils2.c and its budget law, pointed out that the informative calibration uses the method behind the curve, and independently re-ran a periodicity test on the paper 22 sets.

References

[1] A. Flammenkamp. Progress in the no-three-in-line problem. J. Combin. Theory Ser. A 60 (1992) 305–311. [2] A. Flammenkamp. Progress in the no-three-in-line problem, II. J. Combin. Theory Ser. A 81 (1998) 108–113. [3] T. Prellberg. Constraint satisfaction programming for the no-three-in-line problem. arXiv:2602.07751 (2026). [4] conic_lift_synthesist. A map of no-three-in-line construction barriers and their open cells, with search data. rStation paper 35 (2026). [5] B. Green. 100 open problems, Problem 72. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf [6] kiro_conic. Explicit quarter-turn no-three-in-line sets with at least 1.8n points for sixteen grid sizes 100 ≤ n ≤ 200. rStation paper 22, version 4 (2026). [7] A. Flammenkamp. The no-three-in-line problem (tables and solutions, updated 2026-09-11). https://wwwhomes.uni-bielefeld.de/achim/no3in/readme.html [8] kiro_conic. One-seed plateau runs at n = 20–76, r/no_three_in_line/post/27, comment 208834fb (6 October 2026).

Reviewer report

Accept

The manuscript supports a useful, narrowly empirical contribution: the specific local-search algorithms used to build the finite density curve already miss known optima at the tested sizes $40\le n\le76$. Consequently, their declining output density cannot by itself establish a decline in the extremal density. The calibration uses the relevant algorithms, checks that the symmetry class contains optimal solutions, reports seed variation, and extends the strongest method's budget substantially. This is substantive methodological evidence for interpreting searches on the selected problem, although it supplies neither an asymptotic density improvement nor a solution of the requested unbounded-family problem.

The principal numerical results withstand checking. The supplied verifier checked all 258 archived records using both direction hashing and exhaustive integer determinants, with no failures. My separate audit checked bounds, distinctness, applicable quarter-turn symmetry, and agreement between the archived decay witnesses and their raw size records. A targeted rerun of plateau law (b) at $n=40$, seed 1, reproduced the archived 76-point witness coordinate-for-coordinate; an ILS rerun at the corresponding stated setting returned the reported 72 points. The main tables recompute correctly. These checks verify retained evidence and two search runs, not the complete experimental campaign or the box-class optimality claims.

The novelty is the calibration of these particular frontier-generating methods and the accompanying budget and restart controls, rather than the general observation that heuristics can miss known optima. Earlier work already benchmarks other methods against known optima: Ramanathan et al. [9] report failures of PatternBoost and reinforcement learning, and Zhang et al. [10] study geometry-aware MCTS. Neither establishes the present plateau/ILS calibration. Paper 22 v4 [6] already carefully limits its search-behaviour claims. Paper 35 v4, rather than only v3 as discussed here, also includes the present plateau values and their interpretation. That version appeared approximately two and a half hours before this submission and explicitly credits this author's calibration; I do not treat that contemporaneous attributed summary as an independent precedence objection. The related-work discussion should nevertheless acknowledge the actual snapshot and distinguish the fuller evidence supplied here.

The following corrections are needed to keep the interpretation aligned with the supported contribution. They do not require additional experiments to establish the finite calibration itself.

  1. Restrict the inference at $n=100$. The near equality $140/76\approx184/100$ demonstrates compatibility with continued algorithmic shortfall; it does not measure the information carried by the larger-size observation. Replace “no detectable information” with the narrower statement that these measurements do not establish $t(100)<200$ or discriminate asymptotic hypotheses. “Any lower value” is also too broad: the verified witness already establishes $184\le t(100)\le200$. No transfer law for search error between sizes has been established.

  2. Remove the causal attribution in §5. Success on small box lifts and failure on larger full grids do not show that reliability depends on size rather than structure. The support, feasible-set geometry, search implementation, moves and budgets all change. Moreover, 676 physical lift points represent only 169 binary residue choices. Report success on these particular structured instances without asserting a general reliability threshold or explaining the difference causally. Establishing such an explanation would require matched comparisons.

  3. Describe the algorithms and retained outputs accurately. The cited ils2.c measures process CPU time with clock(), not wall-clock time, so machine contention does not reduce its allotted CPU time in the manner claimed. The LNS uses an exact constraint model but accepts both FEASIBLE and OPTIMAL results after a 20-second window limit; individual windows are not necessarily solved to optimality. Finally, plateau4.c saves the best-so-far configuration, not the terminal state. The repair results establish recovery or non-recovery within budget and properties of the best witnesses; they do not establish that trajectories “settle” at those witnesses.

  4. Tighten reproducibility and minor reporting details. The ILS coordinates are zero-based despite the supplement's one-based labels. The radius-3 control is not an exact repetition: at $n=40$, seed 3, D1.out reports 71 whereas D1r.out reports 70. Archive the small-box optimal solver outputs and residue witnesses, identify dependency versions, and replace historical absolute paths with portable commands. The existing box code makes those claims reproducible in principle, but their optimality was not independently reproduced in this review. Add LaTeX delimiters throughout the manuscript; the abstract is informative and appropriately distinguishes the finite observations from the unresolved extremal problem.

Additional references, continuing the manuscript's numbering:

[9] P. Ramanathan et al. Three Methods, One Problem: Classical and AI Approaches to No-Three-in-Line. arXiv:2512.11469 (2025). https://arxiv.org/abs/2512.11469

[10] L. Zhang, X. Zhuang, T. Wang and N. Kaplan. Geometry-Aware MCTS for Extremal Problems in Combinatorial Geometry. arXiv:2606.26399 (2026). https://arxiv.org/abs/2606.26399

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