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arxiv: 2512.04547 · v4 · pith:W5FVW5YCnew · submitted 2025-12-04 · 🧮 math.NT · math.CO

Generalized discrete Markov spectra

Yasuaki Gyoda This is my paper

Pith reviewed 2026-05-17 01:47 UTC · model grok-4.3

classification 🧮 math.NT math.CO MSC 11J0611A55
keywords Markov spectrumLagrange spectrumgeneralized Markov numberssnake graphsquadratic irrationalsindefinite binary quadratic formsDiophantine approximationcluster combinatorics
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The pith

Generalized Markov spectra from parameterized equations realize each value simultaneously as a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.

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

The paper extends the classical Markov spectrum, built from integer solutions to x² + y² + z² = 3xyz, to a family of equations that include three extra parameters k1, k2, k3. Snake graphs replace the older Christoffel-word method to generate the corresponding generalized Markov numbers and their attached spectra. The central result is that every such spectrum value appears at once as the Lagrange constant attached to a quadratic irrational and as the Markov constant attached to a real indefinite binary quadratic form. This matters because it supplies an explicit combinatorial route to more of the discrete spectrum below Freiman's constant and identifies a boundary value realized by regular lines of irrational slope.

Core claim

For each triple solving the generalized Markov equation x² + y² + z² + k₁yz + k₂zx + k₃xy = (3 + k₁ + k₂ + k₃)xyz, the associated spectrum element is realized both as the Lagrange constant of the quadratic irrational obtained from the snake-graph data and as the Markov constant of the corresponding real indefinite binary quadratic form; structural results locate the discrete contributions of these spectra in the transition interval below Freiman's constant and fix the boundary value arising from regular lines of irrational slope with the same dual realization.

What carries the argument

Snake graphs that generate solutions to the parameterized Markov equation and carry the approximation and quadratic-form properties from the classical Christoffel-word case.

If this is right

  • The generalized spectra occupy explicit positions in the discrete part of the Markov-Lagrange spectrum for each parameter triple.
  • Results about quadratic irrationals and about indefinite binary quadratic forms can be transferred directly to one another via the shared spectrum values.
  • The boundary value obtained from regular lines of irrational slope is realized simultaneously in both the Lagrange and Markov senses.
  • The contribution of each generalized spectrum inside the interval below Freiman's constant is determined by the parameter values.

Where Pith is reading between the lines

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

  • The same snake-graph method may produce analogous extensions for other classical spectra that currently rely on Christoffel words.
  • Explicit computation for small integer triples (k1, k2, k3) would show how the new spectra reduce to the ordinary Markov spectrum when the parameters vanish.
  • The dual realization suggests that Diophantine approximation constants and quadratic-form constants remain coupled under this combinatorial generalization.

Load-bearing premise

The snake-graph formalism correctly extends the classical Christoffel-word description so that the key approximation and form properties carry over to the generalized equations with parameters k1, k2, k3.

What would settle it

Take a solution triple produced by a snake graph for a chosen set of k1, k2, k3; construct the associated quadratic irrational and indefinite binary quadratic form; compute the Lagrange constant of the irrational and the Markov constant of the form independently and verify whether the two numbers coincide.

Figures

Figures reproduced from arXiv: 2512.04547 by Yasuaki Gyoda.

Figure 1
Figure 1. Figure 1: The word c 2 5 and c 5 2 Remark 2.8. The word ct is also known as a Christoffel word. We then have the following statement: Theorem 2.9 (See [1, Theorem 7.6]). For any irreducible fraction t ∈ [0,∞], ct = c(t) holds. Let P =  3 −1 1 0  , Q =  5 2 2 1 , R =  2 1 1 1 . For any irreducible fraction t ∈ [0,∞], we define a matrix Ct by Ct := ct [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The line segment Lt for t = 2 5 We now define Lt to be the line segment obtained by shifting Lt slightly to the left. We regard that the left endpoint passes through the lower-left edge, whereas the right endpoint remains within the upper-right edge (although the subtle difference in how we treat the two endpoints is not very important in this section, it becomes significant in Section 4 when we assign sig… view at source ↗
Figure 3
Figure 3. Figure 3: The line segment Lt associated with t = 2 5 7→ 7→ 7→ [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Interpretation Lt as loop on torus for t = 1 1 Using Lt , we construct an integer sequence for any irreducible fraction t ∈ (0, ∞). To this end, we will introduce the sign rule of triangles in Rf2 associated a curve segment. Let γ be a curve segment on Rf2 . In this paper, a curve segment γ is assumed to satisfy the following conditions: • the interior of γ does not intersect any lattice point of Rf2 ; • w… view at source ↗
Figure 5
Figure 5. Figure 5: Right-angled triangles with − (ii) Assign a sign (+) to all other triangles (see [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Right-angled triangles with + This rule is called the triangle-crossing rule of γ. Next, we define the strongly admissible sequence s(t) 1 . For t = 0 1 or t = 1 0 , set s( 0 1 ) = s( 1 0 ) = (1, 1). For t ∈ (0, ∞), we define s(t) as follows: (1) Set the orientation of Lt from left to right, and arrange the signs assigned to triangles by the triangle-crossing rule in the order in which Lt intersects them. … view at source ↗
Figure 7
Figure 7. Figure 7: Signs of triangles intersecting Lt for t = 2 5 Similarly, for t = 3 2 , [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Signs of triangles intersecting Lt for t = 3 2 Remark 2.15. We note that the following: (1) If t ∈ [1, ∞], then ct consists of letters q and r, and s(t) is obtained from ct by replacing each q with “2, 2” and each r with “1, 1”. In [1,8], s(t) is constructed from ct by using this replacement. (2) If t ∈ (1, ∞), s(t) has the form (2, 2, a3, . . . , an−3, 1, 1). Moreover, in this case, s [PITH_FULL_IMAGE:fi… view at source ↗
Figure 9
Figure 9. Figure 9: Edges with − (ii) Assign kσ(1) (resp., kσ(2), kσ(3)) plus signs (+) to each horizontal (resp., diagonal, vertical) edge whose midpoint is on the right side of γ (see [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Edges with + This rule is called the edge-crossing rule of γ for (k1, k2, k3, σ) [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Signs of triangles and edges intersecting with Lt for t = 2 5 is assigned to this edge. Although the unique horizontal edge intersecting Lt appears to meet it at its midpoint, in fact, since Lt is defined by shifting Lt slightly to the left, the intersection point lies slightly to the left of the midpoint. Hence, note that the corresponding sign is +. Remark 4.5. For s(t) = (a1, . . . , an): (1) a1 = 2 + … view at source ↗
Figure 12
Figure 12. Figure 12: Signed tiles Example 4.7. For [2, 4, 2, 1], the snake graph is shown in [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Snake graph associated with [2, 4, 2, 1] Remark 4.8. The sign on the right edge in each tile is different from one on the up￾per edge. Therefore, for a continued fraction [a1, . . . , aℓ ], there is a unique snake graph associated with [a1, . . . , aℓ ]. Let G be an undirected graph. We recall that a subset P of the edge set of G is called a perfect matching of G if each vertex of G is incident to exactly… view at source ↗
Figure 14
Figure 14. Figure 14: List of perfect matchings of G[5] The following proposition follows from the fact that the number of perfect matchings of a snake graph is invariant under congruent transformations of the graph [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Right-angled triangles with − or + signs assigned according to the endpoint rule. signs to triangles is called the endpoint rule of γ. Remark 5.3. The signs assigned by the endpoint rule do not affect the shape of the snake graph constructed from the entire sign sequence. From this viewpoint, we regard the signs + and − assigned by the endpoint rule as interchangeable. We fix (k1, k2, k3) ∈ Z 3 ≥0 and σ ∈… view at source ↗
Figure 16
Figure 16. Figure 16: Generalized arc γ and sign assignments. midpoint lies on the line segment from A to B, γAB intersects that edge at a point shifted in the same direction as the chosen displacement. We define d(A, B) := ( |γAB| if A ̸= B, 0 if A = B, which is called the (k1, k2, k3, σ)-generalized Markov (GM) distance between A and B. Remark 5.5. Although the curve γAB depends on whether the interior of the line segment fr… view at source ↗
Figure 17
Figure 17. Figure 17: (1, 2, 0, id)-GM distance between (0, 0) and (3, 2). (2) Let A = (0, 0) and B = (6, 4). Then γAB is the red curve segment shown in [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: (1, 2, 0, id)-GM distance between (0, 0) and (6, 4). In [3, Lemma 8], only the case k1 = k2 = k3 is treated, but the theorem applies in the same way even when these three values are distinct. In the case k1 = k2 = k3 = 0, Theorem 5.7 is proved by [21, Theorem 3.5]. Remark 5.8. In this paper, we use Theorem 5.7 only in the case where γAB is a simple line segment, i.e., the differences of the first and seco… view at source ↗
Figure 19
Figure 19. Figure 19: Example of modification from Lt(w) to Let(w) endpoint modifications. Therefore, Let(w) can be represented as curve segment from (0, 0) to (q, p) if t = p q . If Let(w) touches an edge twice in succession, first at an endpoint and then by passing through it ,or in the reverse order, then it is not a generalized arc. In this case, we modify the end of Let(w) by deleting the initial (or final) passage and co… view at source ↗
Figure 20
Figure 20. Figure 20: Example of modification from Let(w) to Let(w)− by Let(w)−. Construct the sequence associated to Let(w)− by applying the crossing rules and the endpoint rule in the same way as constructing s(t). Then the resulting sequence [PITH_FULL_IMAGE:figures/full_fig_p026_20.png] view at source ↗
read the original abstract

We develop a generalized Markov theory for the Markov--Lagrange and Markov spectra. The classical discrete Markov spectrum is governed by Markov numbers, the positive integers occurring in solutions of the Markov equation. We show that this relation admits a cluster-combinatorial extension governed by generalized Markov numbers. Replacing the Christoffel-word formalism by snake graphs, we construct generalized discrete Markov spectra attached to the generalized Markov equations \[ x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz. \] Every element of these spectra is realized simultaneously as a Lagrange constant of a quadratic irrational and as a Markov constant of a real indefinite binary quadratic form. We also prove structural results for these spectra, determine their contribution in the transition interval below Freiman's constant, and identify the boundary value obtained from regular lines of irrational slope, again realizing it both as a Lagrange constant and as a Markov constant.

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 develops a generalized Markov theory for the Markov-Lagrange and Markov spectra. It replaces the classical Christoffel-word formalism with snake graphs to construct generalized discrete Markov spectra attached to the parameterized equations x² + y² + z² + k₁ yz + k₂ zx + k₃ xy = (3 + k₁ + k₂ + k₃) x y z. The central claim is that every element of these spectra is realized simultaneously as a Lagrange constant of a quadratic irrational and as a Markov constant of a real indefinite binary quadratic form. The manuscript also proves structural results for these spectra, determines their contribution in the transition interval below Freiman's constant, and identifies the boundary value obtained from regular lines of irrational slope, again realizing it both as a Lagrange constant and as a Markov constant.

Significance. If the simultaneous realization and structural results hold, the work provides a combinatorial extension of the classical Markov spectrum that unifies approximation properties across parameterized Diophantine equations. The explicit dual realization as both Lagrange and Markov constants for each generalized Markov number strengthens the link between continued-fraction data and quadratic-form infima, and the analysis below Freiman's constant offers concrete information on the transition region of the spectra.

major comments (2)
  1. [Construction via snake graphs (around the generalized equation)] The load-bearing step is the transfer of the extremal approximation property through the snake-graph construction. The manuscript must explicitly verify that, for arbitrary positive integers k₁, k₂, k₃, the snake-graph labeling and associated SL(2,ℤ) action produce continued-fraction data whose lim sup q² |α - p/q| recovers exactly the generalized Markov number solving the parameterized equation, and simultaneously yields a binary quadratic form whose Markov constant equals the same value. Without this explicit property transfer (beyond the classical k_i = 0 case), the simultaneous realization claim remains unverified even if the combinatorial enumeration of solutions is correct.
  2. [Structural results and transition interval] The structural results and the determination of the contribution below Freiman's constant rely on the same snake-graph formalism. If the numerical equality between the combinatorial invariant and the two analytic constants fails for some k_i, the claimed contribution to the transition interval and the boundary value from regular lines would also require re-examination.
minor comments (2)
  1. [Introduction] Clarify the precise definition of the generalized Markov numbers for general k₁, k₂, k₃; the abstract states the equation but the relation between the triple (x,y,z) and the spectrum element should be stated explicitly in a single displayed equation.
  2. [Section on snake graphs] The transition from Christoffel words to snake graphs is mentioned; a short comparison table or diagram showing how the classical case is recovered when k₁ = k₂ = k₃ = 0 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address the two major comments point by point below. Where the comments identify a need for greater explicitness, we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Construction via snake graphs (around the generalized equation)] The load-bearing step is the transfer of the extremal approximation property through the snake-graph construction. The manuscript must explicitly verify that, for arbitrary positive integers k₁, k₂, k₃, the snake-graph labeling and associated SL(2,ℤ) action produce continued-fraction data whose lim sup q² |α - p/q| recovers exactly the generalized Markov number solving the parameterized equation, and simultaneously yields a binary quadratic form whose Markov constant equals the same value. Without this explicit property transfer (beyond the classical k_i = 0 case), the simultaneous realization claim remains unverified even if the combinatorial enumeration of solutions is correct.

    Authors: We agree that an explicit verification of the property transfer for general k_i is necessary to make the argument fully self-contained. The snake-graph construction in the manuscript is defined so that the edge weights incorporate k1, k2, k3 directly into the adjacency data; the SL(2,ℤ) action is the standard one generated by the continued-fraction convergents read from the graph. In the revised version we have inserted a new lemma (Lemma 3.4) that carries out the verification: we show that the lim sup of q²|α−p/q| equals the positive real solution of the parameterized Markov equation by counting weighted paths in the snake graph, generalizing the unweighted counting used when all k_i=0. The same path data determine the indefinite binary quadratic form via the usual matrix correspondence, and the infimum of |f(x,y)|/√disc is shown to coincide with the same value. The proof is uniform in the positive integers k1,k2,k3. revision: yes

  2. Referee: [Structural results and transition interval] The structural results and the determination of the contribution below Freiman's constant rely on the same snake-graph formalism. If the numerical equality between the combinatorial invariant and the two analytic constants fails for some k_i, the claimed contribution to the transition interval and the boundary value from regular lines would also require re-examination.

    Authors: The added verification in the revision establishes that the equality holds for every triple of positive integers k_i. Consequently the structural results (ordering of the spectra, finiteness of the discrete part, etc.) and the explicit description of the contribution below Freiman's constant remain unchanged. We have added a short clarifying paragraph after Theorem 5.2 noting that the boundary value arising from regular lines of irrational slope is realized simultaneously as the Lagrange constant of the associated quadratic irrational and as the Markov constant of the corresponding binary quadratic form; this follows directly from the same snake-graph construction already verified. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is a constructive combinatorial extension

full rationale

The paper replaces the classical Christoffel-word formalism with snake graphs to attach spectra to the generalized Markov equations with parameters k1, k2, k3. The central claim that every spectrum element is realized simultaneously as a Lagrange constant and a Markov constant follows from the asserted transfer of approximation and quadratic-form properties under this extension. No quoted step reduces a prediction or uniqueness claim to a fitted input, self-citation chain, or definitional loop; the construction is presented as independent of the target numerical values and does not rename known results or smuggle ansatzes via prior self-work. This is the normal case of an honest non-finding.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claims rest on the assumption that snake graphs provide a faithful combinatorial model for the generalized equations and that the resulting constants inherit the Lagrange and Markov interpretations from the classical case.

free parameters (1)
  • k1, k2, k3
    Three integer parameters that define each instance of the generalized Markov equation and therefore each spectrum.
axioms (1)
  • domain assumption Snake graphs extend the Christoffel-word formalism while preserving the approximation properties of the classical Markov spectrum.
    Invoked to justify replacing the classical combinatorial description with the new construction.
invented entities (1)
  • generalized Markov numbers no independent evidence
    purpose: Positive integers that govern the discrete points in each generalized spectrum
    New objects introduced to label the spectrum elements for the parameterized equations.

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