Algorithmic exploration of the unit distance problem in the rational plane

Panteleimon Rodis

arxiv: 2606.29415 · v1 · pith:LI6MEJWGnew · submitted 2026-06-28 · 💻 cs.CG · math.CO

Algorithmic exploration of the unit distance problem in the rational plane

Panteleimon Rodis This is my paper

Pith reviewed 2026-06-30 01:48 UTC · model grok-4.3

classification 💻 cs.CG math.CO

keywords unit-distance graphsrational planescaling exponentgraph densitylocal-breadth searchalgorithmic generationcomputational geometry

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The pith

A local-breadth search on finite rational points generates unit-distance graphs whose scaling exponent exceeds recent theoretical lower bounds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an algorithmic method to explore the rational plane by performing local-breadth search on a bounded finite set of rational elements and thereby constructs unit-distance graphs. It shows that this method overcomes the restrictions of earlier grid-based generators and yields graphs whose density scaling surpasses known lower bounds. A sympathetic reader would care because the experiments supply reproducible numerical evidence about how dense unit-distance configurations can become when coordinates are restricted to rationals, a concrete setting inside the larger unit-distance problem.

Core claim

The paper claims that an efficient algorithm performing local-breadth search on a bounded finite set of rational elements generates unit-distance graphs that encompass the general properties of such graphs without being affected by generation restrictions, and that the scaling exponent of the generated graph surpasses recent theoretical lower bounds while overcoming limitations of grid-based structures.

What carries the argument

local-breadth search on a bounded finite set of rational elements, which produces representative unit-distance graphs free of grid restrictions

If this is right

The generated graphs supply reproducible experimental evidence on unit-distance graph density in the rational plane.
The scaling exponent of these graphs exceeds the values given by recent theoretical lower bounds.
The local search approach overcomes the structural limitations of grid-based generators used in prior work.
The resulting graphs are claimed to reflect general unit-distance properties despite the finite bounded generation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the exponent remains stable under larger bounds, the method could serve as a practical probe for constructing candidate dense configurations that theorists might then try to prove or disprove.
The same bounded-search technique might be applied to other discrete point sets, such as algebraic integers, to compare achievable densities across different fields.
Experimental density records obtained this way could guide the search for matching upper-bound constructions or revised lower-bound proofs in the rational case.

Load-bearing premise

Performing a local-breadth search on a bounded finite set of rational elements produces graphs whose properties are not affected by the generation restrictions and therefore reflect general unit-distance graph behavior.

What would settle it

Demonstrating that the scaling exponent of graphs generated by the same search on successively larger bounded rational sets falls to or below the cited theoretical lower bounds would falsify the claim that the method captures unrestricted behavior.

read the original abstract

This paper presents reproducible experimental evidence on unit-distance graph density that surpasses recent theoretical lower bounds. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures used in the literature for generating unit-distance graphs. Furthermore, the scaling exponent of the generated graph surpasses recent results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a local-breadth search on finite rational points yields unit-distance graphs whose scaling exponent beats prior lower bounds, but offers no check that the exponent survives larger bounds or less local search.

read the letter

The main thing here is an algorithmic method that does local-breadth search over a bounded finite subset of Q² to build unit-distance graphs, with the claim that the resulting scaling exponent exceeds recent theoretical lower bounds and that the graphs capture general properties without being limited by the search constraints.

What stands out is the move away from grid-based constructions toward rationals. If the code really produces reproducible graphs and the exponent is measured cleanly, that supplies concrete experimental data points for people studying densities in the rational plane.

The soft spot is exactly the one the stress-test flags. The construction is explicitly finite and local, yet the abstract asserts the output is unaffected by those restrictions. No scaling test with larger bounds, no invariance argument, and no comparison of exponents across different search radii appear in the description. Without those, any density gain could be an artifact of the chosen cutoff rather than evidence about the infinite case.

This is for people running computational searches on unit-distance or chromatic-number problems who want new heuristics. A reader already working on experimental generation methods might extract a usable search routine, but the central claim about surpassing bounds is not yet on solid ground.

I would not send this to peer review until the authors add explicit checks that the exponent remains stable when the finite bound is increased.

Referee Report

1 major / 1 minor

Summary. The paper claims to deliver reproducible experimental evidence on unit-distance graph density in the rational plane via a novel local-breadth search algorithm over a bounded finite subset of Q². It asserts that the generated graphs overcome the limitations of prior grid-based constructions, are not affected by the imposed generation restrictions, and exhibit a scaling exponent that surpasses recent theoretical lower bounds.

Significance. If the central experimental claim holds after verification that the observed exponent is invariant under relaxation of the bound and locality constraints, the work would supply concrete, reproducible data that could guide further theoretical investigation of the unit-distance problem over the rationals. The emphasis on reproducibility and the explicit algorithmic construction are positive features.

major comments (1)

[Abstract] Abstract, paragraph 3: the claim that the algorithm 'generate[s] a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation' is load-bearing for the headline result. No scaling check, invariance argument, or comparison of exponents across increasing bounds is supplied to show that the reported exponent is independent of the finite-set and locality constraints; without such evidence the surpassing of theoretical bounds cannot be interpreted as applying to the unrestricted rational plane.

minor comments (1)

The description of the local-breadth search procedure would benefit from an explicit pseudocode listing or complexity statement to allow independent reproduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful review and for highlighting the need for stronger evidence supporting the independence of our reported scaling exponent from the finite-set and locality constraints. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 3: the claim that the algorithm 'generate[s] a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation' is load-bearing for the headline result. No scaling check, invariance argument, or comparison of exponents across increasing bounds is supplied to show that the reported exponent is independent of the finite-set and locality constraints; without such evidence the surpassing of theoretical bounds cannot be interpreted as applying to the unrestricted rational plane.

Authors: We agree that the manuscript does not supply explicit scaling checks, invariance arguments, or exponent comparisons across increasing bounds, and that this evidence is required to substantiate the abstract claim. The current text shows only that the local-breadth search overcomes grid-based limitations for the chosen bound; it does not demonstrate invariance. In the revised version we will add a dedicated subsection that reports the scaling exponent for a sequence of successively larger bounds and locality radii, together with a brief argument that the local search rule produces statistically similar local configurations once the bound exceeds a modest threshold. This addition will directly address the referee's concern and allow the headline result to be interpreted for the unrestricted rational plane. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental output from bounded search compared to external bounds

full rationale

The paper reports an algorithmic procedure that runs local-breadth search over a finite bounded subset of Q² to produce a unit-distance graph, then computes its scaling exponent and states that the exponent exceeds known external theoretical lower bounds. No equations, fitted parameters, self-citations, or derivations appear in the supplied text; the reported exponent is a direct computational result of the algorithm rather than a quantity defined in terms of itself or obtained by renaming a fitted input. The claim that the finite construction “encompasses the general properties” is an unproven modeling assumption, but it does not reduce any derived quantity to the input by construction, nor does any load-bearing step rely on a self-citation chain. The work is therefore self-contained as experimental evidence against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that the search procedure faithfully samples unit-distance properties.

pith-pipeline@v0.9.1-grok · 5627 in / 1054 out tokens · 24782 ms · 2026-06-30T01:48:32.057521+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 1 canonical work pages · 1 internal anchor

[1]

The American Mathematical Monthly 53(5), 248–250 (1946)

Erd¨ os, P.: On sets of distances of n points. The American Mathematical Monthly 53(5), 248–250 (1946)

1946
[2]

https://cdn.openai.com/ pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf

OpenAI: Planar Point Sets with Many Unit Distances. https://cdn.openai.com/ pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf. Accessed: 2026-06-02 (2026)

2026
[3]

An explicit lower bound for the unit distance problem

Sawin, W.: An explicit lower bound for the unit distance problem. arXiv preprint arXiv:2605.20579 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[4]

Graph theory and combinatorics, 293–303 (1984)

Spencer, J., Szemer´ edi, E., Trotter, W.T.: Unit distances in the euclidean plane. Graph theory and combinatorics, 293–303 (1984)

1984
[5]

Experimental Mathematics, 1–13 (2025)

Engel, P., Hammond-Lee, O., Su, Y., Varga, D., Zs´ amboki, P.: Diverse beam search to find densest-known planar unit distance graphs. Experimental Mathematics, 1–13 (2025)

2025
[6]

https://rodispantelis.github.io/UnitDistanceGraphs/ 10

Unit-Distance Graphs. https://rodispantelis.github.io/UnitDistanceGraphs/ 10

[1] [1]

The American Mathematical Monthly 53(5), 248–250 (1946)

Erd¨ os, P.: On sets of distances of n points. The American Mathematical Monthly 53(5), 248–250 (1946)

1946

[2] [2]

https://cdn.openai.com/ pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf

OpenAI: Planar Point Sets with Many Unit Distances. https://cdn.openai.com/ pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf. Accessed: 2026-06-02 (2026)

2026

[3] [3]

An explicit lower bound for the unit distance problem

Sawin, W.: An explicit lower bound for the unit distance problem. arXiv preprint arXiv:2605.20579 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[4] [4]

Graph theory and combinatorics, 293–303 (1984)

Spencer, J., Szemer´ edi, E., Trotter, W.T.: Unit distances in the euclidean plane. Graph theory and combinatorics, 293–303 (1984)

1984

[5] [5]

Experimental Mathematics, 1–13 (2025)

Engel, P., Hammond-Lee, O., Su, Y., Varga, D., Zs´ amboki, P.: Diverse beam search to find densest-known planar unit distance graphs. Experimental Mathematics, 1–13 (2025)

2025

[6] [6]

https://rodispantelis.github.io/UnitDistanceGraphs/ 10

Unit-Distance Graphs. https://rodispantelis.github.io/UnitDistanceGraphs/ 10