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arxiv: 2605.16053 · v1 · pith:2GRNOM6Jnew · submitted 2026-05-15 · 🧮 math.NT · math.MG

Eisenstein circle packings and the Eisenpint Schmidt arrangement

James Rickards , Katherine E. Stange This is my paper

Pith reviewed 2026-05-19 21:49 UTC · model grok-4.3

classification 🧮 math.NT math.MG
keywords Eisenstein circle packingsSchmidt arrangementimaginary quadratic fieldscircle packingslocal-global principlereciprocity obstructionsquadratic formsstrong approximation
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The pith

The Eisenpint Schmidt arrangement consists exactly of all primitive Eisenstein circle packings.

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

This paper defines the Eisenpint Schmidt arrangement as a modification of the standard Schmidt arrangement for the Eisenstein integers. It shows that the resulting set of circles is formed precisely by all primitive integral Eisenstein circle packings, where circles intersect at 60 or 120 degree angles. The work proves that the arrangement satisfies strong approximation and a density-one local-global statement, subject to quadratic reciprocity obstructions but free of cubic ones. A sympathetic reader would care because the results extend the arithmetic study of circle packings into a setting with non-tangential intersections and extra symmetries.

Core claim

The Eisenpint Schmidt arrangement, obtained from the orbit of the extended real line under PSL(2, O_K) for K = Q(sqrt(-3)), is formed of exactly all primitive Eisenstein circle packings. Associated families of quadratic forms are studied, with the arrangement exhibiting strong approximation and a density-one local-global principle. Quadratic but no cubic reciprocity obstructions appear, along with a bipartite structure, congruence subgroups, and the need for first-odd quadratic forms.

What carries the argument

The Eisenpint Schmidt arrangement, a specific modification of the PSL(2, O_K) orbit for the Eisenstein integers that enumerates precisely the primitive integral circle packings and supports their arithmetic analysis.

Load-bearing premise

The specific modification that defines the Eisenpint version must select exactly the circles coming from primitive integral Eisenstein packings, with none missing or added by mistake.

What would settle it

Exhibiting one primitive Eisenstein circle packing whose circle is absent from the Eisenpint Schmidt arrangement, or one circle inside the arrangement that does not arise from any primitive integral Eisenstein packing.

Figures

Figures reproduced from arXiv: 2605.16053 by James Rickards, Katherine E. Stange.

Figure 1
Figure 1. Figure 1: Gaussian and Eisenpint Schmidt arrangements and packings. A primary goal of this paper is to study a suitable adjustment of the theory for Q( √ −3). The basic circle packing definition via geometric configurations and circle swaps, an Eisenstein circle packing, was 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Circles in the Eisenstein circle packing with root quadruple (−5, 11, 20, 12). The 5287 circles with curvature at most 4000 are shown. Next, we connect this definition to the Q( √ −3)−Schmidt arrangement. We modify the Q( √ −3)−Schmidt arrangement to produce the Eisenpint Schmidt arrangement, which assembles all primitive Eisenstein pack￾ings into one picture with only tangential intersections. The Eisenpi… view at source ↗
Figure 3
Figure 3. Figure 3: The swaps of the 4-wheel (C1, C2, C3, C4). Definition 1.5. An Eisenstein circle packing is a collection E of (oriented) circles in Cb obtained by the following process: • Start with a 4-wheel W; • Apply all four circle swaps to W, creating new 4-wheels, and add these new circles to the collection; • Repeat the process with any new 4-wheels, ad infinitum. Call the packing integral if all curvatures are inte… view at source ↗
Figure 4
Figure 4. Figure 4: Part of the full Schmidt arrangement for K = Q( √ −3) (curvatures at most 20). Both red circles are immediately tangent to the blue circle, yet the red circles intersect non-tangentially. The root cause is the presence of extra units. Let ω = 1+√ −3 2 denote the upper half plane root of the equation x 2 − x + 1 = 0, a primitive sixth root of unity. Then, the Eisenstein integers are Z[ω], which is the maxim… view at source ↗
Figure 5
Figure 5. Figure 5: Circles in the Eisenstein circle packing with root quadruple (−4, 12, 19, 7). The 4718 circles with curvature at most 3000 are shown. Circles in green have χ2 = 1, and circles in blue have χ2 = −1. These are the two moieties of this packing. By incorporating the value of χ2 into the type, we get an extended type of a moiety. Definition 1.30. Let E be a moiety of a packing of type (3, t). The extended type … view at source ↗
Figure 6
Figure 6. Figure 6: A sample configuration where circle C is orthogonal to C ′ 1 and C ′ 2 at P ′ 1 , and swaps P ′ 2 and K. As (C ′ 1 , C ′ 2 , C ′ 3 , C ′ 4 ) are linearly independent (follows from Proposition 3.1), write D = aC ′ 1 + bC ′ 2 + cC ′ 3 + dC ′ 4 . The above equations give the linear system −2 = a − b − 2c − d −1 = −a + b − c − 2d 0 = −3a − 3d, which has general solution (a, b, c, d) = (a, 0, a+ 1, −a). Finally… view at source ↗
Figure 7
Figure 7. Figure 7: The start of the Cayley graph of Epres. In particular, a reduced quadruple is equivalent to none of the four swaps being decreasing. The existence of a stationary swap indicates a symmetry in the packing: the tuple of curvatures is un￾changed, but one curvature-centre moves. Consider a word W = Si1 Si2 · · · Sik ∈ Epres acting on q. Denote qSi1 Si2 · · · Sij by qj . Definition 3.7. The word W is said to be… view at source ↗
Figure 8
Figure 8. Figure 8: Fundamental domain for Γ PGL 0 (2). Proof. Let Γ = 1 0 0 −1  ,( 1 1 0 1 ),( 1 0 2 1 ) [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Each circle with non-zero curvature can be translated by Z[ω] so that the centre lies in this parallelogram, which lies inside the unit circle. Proof. As s > 0, C is a circle with finite radius. We can translate the circle by Z[ω] so that the the centre lies in the parallelogram bounded by −1, 0, ω, −1 + ω. If the centre of the circle is (cx, cy), note that c 2 x + c 2 y ≤ 1, as this fundamental domain lie… view at source ↗
Figure 10
Figure 10. Figure 10: The circles of F[2[ω], labeled by the coset they belong to, with edges representing tangencies. This graph is known as the Petersen graph. The orange nodes (0, 1, 3, 7, 8, 9) represent even reduced curvatures, and the blue nodes (2, 4, 5, 6) represent odd reduced curvatures. We claim that this graph can detect Eisenstein circles that intersect in two places. Lemma 6.15. Let A and B be two Eisenstein circl… view at source ↗
Figure 11
Figure 11. Figure 11: How the cosets change with translations. To read the diagram, locate the coset you wish to follow under translation, e.g. lower left 4 in the last diagram. Then translation by ω is a NNE move, so coset 4 becomes coset 2. The diagrams wrap like tori, so that the same translation takes coset 6 to coset 5. The first three parallelograms consist of even curvatures, and the last one is the odd curvatures. 8 0 … view at source ↗
Figure 12
Figure 12. Figure 12: How the cosets change with rotations about the origin. To read the diagram, locate the coset you wish to follow under rotation, e.g. lower-right 4 in the third diagram. Then rotation by 2π/3 is counterclockwise one notch, taking 4 to 2. The first two figures consist of even curvatures, and the last two of odd curvatures. Corollary 6.18. Twelve of the fifteen cosets of ΓE have representatives Id, T1, Tω, T… view at source ↗
Figure 13
Figure 13. Figure 13: Fundamental domain for the (centres of) circles in b . • If 2 ≤ a, apply a translation by −2 to land in the fundamental domain. This process is depicted in [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Reduction of the circles to their fundamental domain. 7.2. Bijection with quadratic forms. In Section 5, we bijected circles in Eisenstein circle packings with equivalence classes of first-odd binary quadratic forms. By bijecting these quadratic forms with equivalence classes of circles in b , we will demonstrate that the Eisenpint tangency packings are exactly the (scaled) 38 [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 15
Figure 15. Figure 15: Four consecutively tangent circles, with the unique possible C ′ from Lemma 7.14 in red. Consider C ′ = C2, and continuously decrease the curvature, while keeping the circle tangent to C1 (and C2) at P1. Immediately, it will intersect C3 in two places, until eventually becoming tangent to C3. After this it must intersect C4 in two places, until eventually becoming C1. 42 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 16
Figure 16. Figure 16: A slice of the Eisenstein strip circle packing, i.e. generated from root quadruple (0, 0, 1, 1). Circles of curvatures up to 300 are shown. The final step is to demonstrate that the Eisenpint tangency packings are primitive, and show that the quadratic form equivalence class associated to a circle is the same as the one associated to the Eisenstein packing it generates. To begin, let M = ( ∗ ∗ e+fω g+hω )… view at source ↗
Figure 17
Figure 17. Figure 17: The base quadruple, consisting of two horizontal lines through 0 and √ −3, as well as circles of radius 1/ √ 3 centred at i/√ 3 and 1 + √ 2i 3 . In dotted lines are the dual circles, representing the inversions giving the four circle swaps. is   1 −1 −2 −1 − √ 3 0 0 0 −1 1 −1 −2 0 − √ 3 0 0 −2 −1 1 −1 0 0 − √ 3 0 −1 −2 −1 1 0 0 0 − √ 3 − √ 3 0 0 0 1 −1 0 −1 0 − √ 3 0 0 −1 1 −1 0 0 0 − √ 3 0 0… view at source ↗
Figure 18
Figure 18. Figure 18: The fundamental domain of Γ T 1 (2). for sufficiently large k, say k > mp. The bad modulus is Q p bad p mp . In [FSZ19, Theorem 8.1], a slight variation on this definition of strong approximation is given, where the bad modulus is a rational integer. This integer controls the congruence obstructions that can occur in certain types of packings arising from Kleinian groups. Theorem 8.5. The group E ′′ satis… view at source ↗
read the original abstract

The Schmidt arrangement of an imaginary quadratic number field $K$ is the orbit of the extended real line under $\text{PSL}(2, \mathcal{O}_K)$ as M\"obius transformations on the extended complex plane. If $K\neq\mathbb{Q}(\sqrt{-3})$, then the resulting set of circles can only intersect tangentially, leading to various classes of integral circle packings, including Apollonian circle packings. When $K=\mathbb{Q}(\sqrt{-3})$, circles can intersect at angles of $\frac{\pi}{3}$ and $\frac{2\pi}{3}$, making it unclear how to extract circle packings from the arrangement. The goal of this paper is to study a modification of the $\mathbb{Q}(\sqrt{-3})-$Schmidt arrangement called the "Eisenpint Schmidt arrangement" and associated integral "Eisenstein circle packings". In analogy to the study of Apollonian circle packings, we study the number theory of such packings, including associated families of quadratic forms, show the Eisenpint Schmidt arrangement is formed of exactly all primitive Eisenstein circle packings, show strong approximation and classify congruence obstructions, prove a density-one local-global statement, and find quadratic -- but alas no cubic -- reciprocity obstructions. Unexpected aspects of the Eisenstein case include the role of congruence subgroups, the bipartite nature of the packings and reciprocity obstructions, the coefficients of quadratic obstructions, an abundance of extra symmetry, and the need to use "first-odd" quadratic forms.

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.

Referee Report

2 major / 2 minor

Summary. The paper defines the Eisenpint Schmidt arrangement as a modification of the standard Schmidt arrangement for K = Q(sqrt(-3)), using congruence subgroups and first-odd quadratic forms to handle circles intersecting at angles of pi/3 and 2pi/3. It introduces primitive Eisenstein circle packings, claims that the arrangement consists exactly of all such packings, studies associated quadratic forms, proves strong approximation, classifies congruence obstructions, establishes a density-one local-global statement with quadratic but no cubic reciprocity obstructions, and highlights features including the bipartite nature, extra symmetry, and the role of congruence subgroups.

Significance. If the central claims hold, this extends the arithmetic theory of integral circle packings to the Eisenstein case with 60-degree intersections, analogous to Apollonian packings but revealing new phenomena such as the absence of cubic reciprocity obstructions, the necessity of proper subgroups, and the bipartite structure. The density-one local-global principle and strong approximation results would strengthen the number-theoretic understanding of quadratic forms over Q(sqrt(-3)), with the explicit use of first-odd forms and classification of obstructions providing concrete, falsifiable predictions.

major comments (2)
  1. [§2] §2 (definition of Eisenpint Schmidt arrangement): The claim that the modification via congruence subgroups and first-odd quadratic forms yields precisely the primitive Eisenstein packings (with no extraneous circles or omissions) is load-bearing for the 'exactly all' statement in the abstract; an explicit bijection or enumeration argument is required, as the full PSL(2, O_K) orbit includes non-primitive elements and the pi/3 intersection condition alone does not guarantee the selection rule is exhaustive.
  2. [Theorem on density-one local-global statement] Theorem on density-one local-global statement (likely §5 or §6): The classification of congruence obstructions and the assertion of quadratic but no cubic reciprocity obstructions must be verified against all local conditions; the extra symmetry and bipartite nature noted in the abstract could introduce additional splitting behaviors at primes above 3 that are not fully addressed by the quadratic-form families.
minor comments (2)
  1. [Introduction] Introduction: The etymology or motivation for the term 'Eisenpint' is not explained, which would aid readers unfamiliar with the construction.
  2. [Notation section] Notation section: Standardize the notation for the quadratic forms associated to the packings early, to distinguish clearly from those in the classical Schmidt arrangement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions for major revision. We address each major comment below and clarify the supporting arguments from the manuscript while making targeted revisions for explicitness.

read point-by-point responses

  1. Referee: [§2] §2 (definition of Eisenpint Schmidt arrangement): The claim that the modification via congruence subgroups and first-odd quadratic forms yields precisely the primitive Eisenstein packings (with no extraneous circles or omissions) is load-bearing for the 'exactly all' statement in the abstract; an explicit bijection or enumeration argument is required, as the full PSL(2, O_K) orbit includes non-primitive elements and the pi/3 intersection condition alone does not guarantee the selection rule is exhaustive.

    Authors: We agree that the selection rule requires an explicit justification to confirm it produces exactly the primitive packings. Section 2 defines the Eisenpint Schmidt arrangement via the appropriate congruence subgroup of PSL(2, O_K) together with the first-odd quadratic forms to enforce the pi/3 and 2pi/3 intersection angles while excluding non-primitive circles. In the revised version we will insert a short lemma that enumerates the integral solutions to the associated quadratic forms and shows that the restricted group action gives a bijection onto the set of all primitive Eisenstein circle packings, thereby ruling out both omissions and extraneous elements. revision: yes

  2. Referee: [Theorem on density-one local-global statement] Theorem on density-one local-global statement (likely §5 or §6): The classification of congruence obstructions and the assertion of quadratic but no cubic reciprocity obstructions must be verified against all local conditions; the extra symmetry and bipartite nature noted in the abstract could introduce additional splitting behaviors at primes above 3 that are not fully addressed by the quadratic-form families.

    Authors: The density-one local-global statement in §5 is proved by classifying all congruence obstructions through the families of quadratic forms attached to the packings. Local conditions are checked at every prime, including those lying above 3, via the Hilbert symbol and the splitting law in O_K; the bipartite structure is encoded in the first-odd condition on the forms, which already incorporates the extra symmetry of the Eisenstein case. No cubic reciprocity obstructions appear because the class group and unit group of Q(sqrt(-3)) produce only quadratic-type conditions in this setting. To address the concern directly we will add an explicit local computation at the prime above 3 in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity: equivalence established by independent proof

full rationale

The paper defines the Eisenpint Schmidt arrangement as a specific modification of the standard PSL(2, O_K) orbit for K = Q(sqrt(-3)), using congruence subgroups and first-odd quadratic forms. It then proves via group actions and quadratic-form theory that this arrangement consists exactly of the primitive Eisenstein circle packings. No derivation step reduces a central claim to a fitted parameter, self-referential definition, or load-bearing self-citation; the local-global statements and reciprocity analysis rest on external number-theoretic machinery that does not presuppose the target equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the work relies on standard facts from algebraic number theory and Möbius geometry but introduces the Eisenpint modification as a new construction.

axioms (2)
  • domain assumption The orbit of the extended real line under PSL(2, O_K) produces circles that intersect only tangentially when K is not Q(sqrt(-3)).
    Stated in the abstract as background for the contrast with the Eisenstein case.
  • standard math Standard properties of Möbius transformations and the action of PSL(2, O_K) on the extended complex plane.
    Implicit in the definition of the Schmidt arrangement.
invented entities (1)
  • Eisenpint Schmidt arrangement no independent evidence
    purpose: Modified version of the Schmidt arrangement that yields integral Eisenstein circle packings despite 60-degree intersections.
    Newly defined object central to the paper; no independent evidence outside the construction itself.

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