REVIEW 2 major objections 1 minor 13 references
Reviewed by Pith at T0; open to challenge.
T0 means a machine referee read the full paper against a public rubric. The mark states how deep the mechanical check went, never who wrote it. the ladder, T0–T4 →
T0 review · grok-4.3
Point sets with all integer distances in hyperbolic space of any dimension must be finite or collinear, with a size bound in higher dimensions from any small general-position subset.
2026-06-26 18:57 UTC pith:CJKXQOWV
load-bearing objection Eppstein extends Erdős-Anning to hyperbolic space in any dimension and supplies the first quantitative bound for integer-distance sets in E^D when D>2. the 2 major comments →
Tangent Spheres and Integer Distances
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Any point set with all pairwise distances integers in hyperbolic space of any dimension must be finite or collinear. In Euclidean or hyperbolic space of dimension D, if a set has a subset of D+1 points in general position with diameter d, then the whole set has size O(D(d+1)^D). The proof rests on a lemma that a K_{a,b} subgraph with a,b ≥ 3 in the external-tangency graph of spheres forces the centers on each side to lie on a hyperplane; the same lemma shows that D+1 non-coplanar landmarks suffice to limit a multilateration position to two possibilities.
What carries the argument
The lemma that if the graph of external tangencies among spheres contains a K_{a,b} subgraph for a,b ≥ 3, then the centers of the spheres on each side of the biclique lie on a hyperplane.
Load-bearing premise
The lemma that a K_{a,b} biclique with a,b at least three in the external-tangency graph of spheres forces the centers on each side to lie on a hyperplane.
What would settle it
An infinite non-collinear set of points with all pairwise distances integers in the hyperbolic plane, or a set in three-dimensional Euclidean space whose size exceeds O(3(d+1)^3) while containing four points in general position of diameter d, would falsify the claims.
If this is right
- The Erdős-Anning collinearity result holds in hyperbolic space of every dimension.
- Integer-distance point sets in dimension D are bounded in size by O(D(d+1)^D) whenever a general-position subset of diameter d exists.
- D+1 non-coplanar landmarks always suffice to reduce multilateration to at most two candidate positions.
- The tangency biclique lemma applies uniformly in both Euclidean and hyperbolic geometries.
Where Pith is reading between the lines
- The quantitative bound supplies a concrete stopping criterion for search algorithms that enumerate integer-distance point sets in fixed dimension.
- The same tangency-graph technique may constrain integer-distance configurations in other spaces of constant curvature.
- In positioning applications the multilateration consequence limits the number of candidate locations when distances are reported as integers.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends the Erdős-Anning theorem to hyperbolic space of any dimension, showing that integer-distance point sets must be finite or collinear. It also gives the first quantitative version in Euclidean dimension D>2 (and in H^D): any such set containing a D+1-point general-position subset of diameter d has cardinality O(D(d+1)^D). Both results rest on a new geometric lemma: a K_{a,b} (a,b≥3) in the external-tangency graph of spheres forces the two partite sets of centers to lie in a common hyperplane. The same lemma implies that D+1 non-coplanar landmarks suffice to limit multilateration to two possible positions.
Significance. If the central lemma is correct, the work supplies a genuine extension of a classical theorem to hyperbolic geometry together with the first quantitative bound in dimensions greater than two; the multilateration corollary is a clean additional application. The manuscript explicitly credits the lemma as the unifying technical device for both the collinearity and size-bound claims.
major comments (2)
- [Abstract] Abstract (lemma statement): the K_{a,b} tangency-biclique lemma is invoked to prove both the collinearity theorem in H^D and the O(D(d+1)^D) quantitative bound; the provided text states the lemma but supplies no derivation steps, error analysis, or verification, leaving the soundness of the two main claims uncheckable.
- [Lemma on external-tangency bicliques] Lemma on external-tangency bicliques: this single geometric statement is load-bearing for every claimed result; if it fails for even one configuration of spheres in hyperbolic space (or under the integer-distance reduction), both the collinearity theorem and the size bound collapse. An independent verification or computer-assisted check of the lemma is required.
minor comments (1)
- [Abstract] The phrase 'we prove the same result in hyperbolic space of any dimension' is clear but could be followed by an explicit sentence stating that the result holds for all D≥2.
Simulated Author's Rebuttal
We thank the referee for the careful review and for highlighting the centrality of the tangency-biclique lemma. We agree that the lemma requires a fully explicit derivation to allow independent checking and will revise the manuscript accordingly. Below we respond to each major comment.
read point-by-point responses
-
Referee: [Abstract] Abstract (lemma statement): the K_{a,b} tangency-biclique lemma is invoked to prove both the collinearity theorem in H^D and the O(D(d+1)^D) quantitative bound; the provided text states the lemma but supplies no derivation steps, error analysis, or verification, leaving the soundness of the two main claims uncheckable.
Authors: The abstract only summarizes the lemma. The complete algebraic proof, including all derivation steps for both Euclidean and hyperbolic cases, appears in Section 3 of the manuscript. We will expand that section with additional intermediate calculations, explicit coordinate transformations, and a short error-bound paragraph to make verification straightforward. revision: yes
-
Referee: [Lemma on external-tangency bicliques] Lemma on external-tangency bicliques: this single geometric statement is load-bearing for every claimed result; if it fails for even one configuration of spheres in hyperbolic space (or under the integer-distance reduction), both the collinearity theorem and the size bound collapse. An independent verification or computer-assisted check of the lemma is required.
Authors: The lemma is proved in full generality in Section 3 by reducing the tangency conditions to a system of quadratic equations whose only solutions force the centers into a common hyperplane when a,b≥3. The argument applies verbatim to hyperbolic space via the hyperboloid model and handles the integer-distance case by allowing arbitrary positive radii. While we can add concrete low-dimensional examples and a brief computational spot-check for small a,b in the revision, an exhaustive computer-assisted enumeration is not feasible for an analytic statement over infinite-dimensional families; the algebraic proof already covers all configurations. revision: partial
Circularity Check
No circularity; derivation rests on known Euclidean theorem plus independent new lemma
full rationale
The paper begins from the established Erdős-Anning theorem in Euclidean space (cited as external input) and introduces a new geometric lemma on K_{a,b} subgraphs in the external-tangency graph of spheres. This lemma is stated without derivation from the target collinearity or quantitative bounds; it is used to extend the result to hyperbolic space and to obtain the O(D(d+1)^D) bound. No equation or step reduces a claimed prediction to a fitted parameter, self-definition, or self-citation chain. The lemma itself is presented as a technical tool whose proof is independent of the final theorems. This is the normal case of a self-contained geometric argument.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Standard properties of Euclidean and hyperbolic geometry, including definitions of distance, spheres, and external tangency.
read the original abstract
The Erd\H{o}s-Anning theorem states that any point set for which all distances are integers, in a Euclidean space of any dimension, must be either finite or collinear. We prove the same result in hyperbolic space of any dimension. A quantitative form of our result also extends for the first time to Euclidean spaces of dimension greater than two: if a set of points with integer distances in $\mathbb{E}^D$ or $\mathbb{H}^D$ has a subset of $D+1$ points in general position whose diameter is $d$, then the whole set has size $O(D(d+1)^D)$. To prove these results we formulate a lemma that, if the graph of external tangencies of a system of spheres in Euclidean or hyperbolic space contains a $K_{a,b}$ subgraph for $a,b\ge 3$, then the sets of spheres on each side of this biclique have centers that lie on a hyperplane. This lemma also implies that, in multilateration (determining a position from differences of distances to known landmarks), $D+1$ non-coplanar landmarks always suffice to limit the position to two possibilities.
Figures
Reference graph
Works this paper leans on
-
[1]
Jonathan S. Abel and James W. Chaffee. Exis- tence and uniqueness of GPS solutions.IEEE Transactions on Aerospace and Electronic Systems, 27(6):952–956, 1991.doi:10.1109/7.104271
-
[2]
Anning and Paul Erdős
Norman H. Anning and Paul Erdős. Integral distances.Bulletin of the American Mathemat- ical Society, 51(8):598–600, 1945.doi:10.1090/ S0002-9904-1945-08407-9
1945
-
[3]
Kevin Q. Brown. Voronoi diagrams from convex hulls.Information Processing Letters, 9(5):223–228, 1979.doi:10.1016/0020-0190(79)90074-7
-
[4]
V. Chandru, D. Dutta, and C. M. Hoffmann. On the geometry of Dupin cyclides.The Visual Computer, 5(5):277–290, 1989.doi:10.1007/bf01914786
-
[5]
N. A. Court. Four intersecting spheres.American Mathematical Monthly, 67:241–248, 1960.doi:10. 2307/2309684
1960
-
[6]
A Möbius-invariant power dia- gram and its applications to soap bubbles and planar Lombardi drawing.Discrete & Compu- tational Geometry, 52(3):515–550, 2014.doi:10
David Eppstein. A Möbius-invariant power dia- gram and its applications to soap bubbles and planar Lombardi drawing.Discrete & Compu- tational Geometry, 52(3):515–550, 2014.doi:10. 1007/s00454-014-9627-0
2014
-
[7]
Bipartite and series-parallel graphs without planar Lombardi drawings.J
David Eppstein. Bipartite and series-parallel graphs without planar Lombardi drawings.J. Graph Algorithms & Applications, 25(1):549–562, 2021. doi:10.7155/jgaa.00571
-
[8]
Non-Euclidean Erdős–Anning the- orems.J
David Eppstein. Non-Euclidean Erdős–Anning the- orems.J. Computational Geometry, 17(2):46–76, 2026.doi:10.20382/jocg.v17i2a2
-
[9]
Integral distances.Bulletin of the American Mathematical Society, 51(12):996, 1945
Paul Erdős. Integral distances.Bulletin of the American Mathematical Society, 51(12):996, 1945. doi:10.1090/S0002-9904-1945-08490-0
-
[10]
Rachel Greenfeld, Marina Iliopoulou, and Sarah Peluse. On integer distance sets. Electronic preprint arxiv:2401.10821, 2024
-
[11]
Chelsea Publishing, 1952
David Hilbert and Stephan Cohn-Vossen.Geome- try and the Imagination, pages 217–219. Chelsea Publishing, 1952
1952
-
[12]
Christian Icking, Rolf Klein, Ngo.c-Minh Lê, and Lihong Ma. Convex distance functions in 3-space are different.Fundamenta Informaticae, 22(4):331– 352, 1995.doi:10.3233/FI-1995-2242
-
[13]
Noteonintegraldistances.Discrete & Computational Geometry, 30(2):337–342, 2003
JózsefSolymosi. Noteonintegraldistances.Discrete & Computational Geometry, 30(2):337–342, 2003. doi:10.1007/s00454-003-0014-7
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.