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Idempotent compatible maps and discrete integrable systems on the triangular lattice

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Pith's one-line read Three new families of idempotent, non-invertible compatible maps yield Yang-Baxter companion maps and integrable difference equations on the triangular lattice.

desk verdict The three maps and the triangular-lattice interpretation are real, but Theorem 2.3 as printed classifies an empty family because equation (6) forces u=v; likely a fixable typo, but the paper needs correction before it is reliable. read the letter →

arxiv 2504.12212 v1 pith:L26VP7HN submitted 2025-04-16 nlin.SI

Pavlos Kassotakis , Maciej Nieszporski This is my paper · ORCID
classification nlin.SI
keywords mapscompatibledifferenceequationssystemsdefinedidempotentintegrable
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies rules that take two numbers and produce two new numbers, but with a special feature: doing the rule twice is the same as doing it once. Such rules are called idempotent, and usually they are not invertible, so they are often set aside. The authors show that, despite being non-invertible, these rules can still be multidimensionally consistent: when applied in different directions on a grid, the order does not matter. Under that condition, each rule has a partial inverse called a companion map, and the companion map satisfies the Yang-Baxter equation, a central algebraic condition in integrability.

The authors classify all rational rules of a certain separable form. They find three equivalence classes, named QI, QII and QIII, and show that any such rule is a Mobius transformation of one of these three. They then reinterpret the rules as difference equations on the edges of a square lattice. Using conservation laws, they introduce potential functions that turn the edge equations into vertex equations on a triangular lattice. For QI, the resulting equation is a discrete analogue of Burgers equation, together with a linear equation that linearizes it. For QII and QIII, similar linearizable systems appear. The equations live on alternating black and white triangles, and the authors describe a well-posed initial value problem.

The work connects non-invertible idempotent maps to the standard machinery of discrete integrable systems, showing that invertibility is not required for multidimensional consistency and integrability.

Extended reading notes

Core claim

The central claim, in Theorem 2.3 plus the surrounding remarks, is that every rational 3D-compatible map of the form (5), (6) with separable multiplicative or additive F is Mobius equivalent to exactly one of QI, QII, QIII, that all three maps are idempotent, and that their companion maps RI, RII, RIII are birational Yang-Baxter maps. If correct, this gives the first systematic bridge from non-invertible idempotent compatible maps to integrable triangular-lattice equations such as a discrete Burgers equation.

Load-bearing premise

The load-bearing modeling assumption is the ansatz in equations (5) and (6): the map has the form Q: (x,p; y,q) maps to (u,p; v,q) with u=v and F(u,y,p,q)=F(v,x,p,q), and F is required to be of separable multiplicative or additive type (Section 2.2, before Theorem 2.3). If F is allowed to be an arbitrary rational function, the claimed exhaustive three-class classification is not established; the theorem, and the unqualified wording of the abstract, hold only inside this ansatz.

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Referee Report

4 major / 5 minor

Summary. The paper introduces three rational, non-invertible maps QI, QII, QIII on CP1×CP1, claims that these are idempotent and 3D-compatible, and asserts in Theorem 2.3 that they represent all equivalence classes of rational 3D-compatible maps of a stated separable form. The authors then construct birational companion maps RI, RII, RIII, which they claim are Yang-Baxter maps, reinterpret the maps as difference systems on edges of Z2, introduce potentials to obtain vertex equations on the triangular lattice Q(A2), and identify one of these as a discrete Burgers equation. The paper also discusses complementary maps, coalescence relations among the three classes, and an initial-value problem on Q(A2). However, the central classification statement is not valid as written: the defining equations (5)-(6) are internally inconsistent with the displayed maps, and the proof of Theorem 2.3 relies on an incorrect algebraic step.

Significance. If the ansatz and classification were corrected, the paper would make a useful contribution by giving explicit idempotent multidimensional compatible maps whose companions are birational Yang-Baxter maps, and by connecting these maps to linearizable triangular-lattice equations, including a discrete Burgers equation. The potential relations in Section 3 and the explicit formulas for the maps and their inverses are checkable and constitute a strength of the paper. That said, the current central theorem does not hold as stated, because equation (6) defines an essentially empty family while the exhibited maps do not satisfy it. The gap is not a matter of presentation: it undermines the claimed classification and, consequently, the interpretation of the lattice systems as derived from that classification.

major comments (4)
  1. [Section 2.2, Eq. (6)] The defining relations (6) are internally inconsistent. The first relation imposes v=u, so the second relation reduces to F(u,y,p,q)=F(u,x,p,q). For the multiplicative separable F in Theorem 2.3, the common factor (pu-P)/(δu-1) cancels and one is left with (qy-Q)/(δy-1)=(qx-Q)/(δx-1), an equation independent of u; the additive case behaves analogously. Consequently equation (6) determines no nonconstant map u(x,y,p,q), and the three maps QI, QII, QIII displayed in Theorem 2.3, which have u≠v in general, are not representatives of the stated type. The classification statement is therefore not meaningful as printed.
  2. [Section 2.2, proof of Theorem 2.3] The derivation of QI from (6) is not valid. The proof asserts that after substituting v=u, the second relation of (6) becomes a quadratic polynomial equation in u whose discriminant must be a square; however, after cancellation of the common u-dependent factor there is no quadratic equation at all. The subsequent relations pu-P=qv-Q and uy=vx, which are used to define QI, do not follow from (6) and are incompatible with v=u. Unless the authors restate the intended defining system, the claimed equivalence of the three maps with the ansatz (5)-(6) is unsupported.
  3. [Section 2.2, Theorem 2.3 and remarks (i)-(v)] The 3D-compatibility of QI is asserted with the phrase 'it can be shown' and the compatibility of QII and QIII is dispatched with 'following the same analysis'. Since 3D-compatibility is both a hypothesis of the classification and a prerequisite for the multidimensional extension in Section 3, explicit verification should be provided, or at least a precise reference for the computations. The formulas are simple enough that the authors can include them without excessive length.
  4. [Section 2.1, Proposition 2.1 and remark (vii)] Proposition 2.1 is stated for quadrirational maps, but the paper applies it to the non-invertible idempotent maps QI-III, whose inverses do not exist as maps. The text notes that 'some points' of Proposition 2.1 remain true in the non-quadrirational case, but no proof or precise statement is given for the specific items used in remarks (v) and (vii). Since the claim that RI-III are Yang-Baxter maps is load-bearing for the paper's main message, this gap should be addressed explicitly.
minor comments (5)
  1. [Abstract] The sentence 'These maps turns out to be idempotent' has a subject-verb agreement error; it should read 'These maps turn out to be idempotent'.
  2. [Section 3.3.1] 'A well possed initial value problem' should be 'A well-posed initial value problem'.
  3. [Theorem 2.3] In the sentence listing singular sets, 'QI, QII and QII' should be 'QI, QII and QIII', and the notation Σ_QII={(∞,∞)^2} is confusing and should be explained or replaced.
  4. [Table 1] The notation T2_id(x)=T1_id(y) in the QI row is not defined; the meaning of 'id' in this notation should be clarified, and 'separably' should be 'separately' in the sentence preceding Table 1.
  5. [Appendix A] The proof of Proposition 2.1 outsources the equivalence of items (1) and (3) to [7] and then says the rest 'follows in a similar manner'. For a proposition used as a black box, a more precise citation or a brief indication of how the non-quadrirational case is handled would improve the paper.
Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on an explicit ansatz and on standard Yang-Baxter and Mobius equivalence theorems. There are no fitted data parameters and no new physical entities. The main limitations are the separable-F ansatz and proof steps deferred to prior literature or to 'similar manner'.

assumptions (4)
  • domain assumption Map Q has the form Q: (x,p; y,q) -> (u,p; v,q) with u=v and F(u,y,p,q)=F(v,x,p,q).
    This ansatz is what makes the maps non-invertible; the classification in Theorem 2.3 applies only within it (Section 2.2, equations (5) and (6)).
  • domain assumption F is restricted to separable multiplicative or additive rational forms.
    The theorem excludes generic rational F, so the exhaustive 'three classes' claim is conditional on this restriction (Section 2.2, before Theorem 2.3).
  • standard math Mobius equivalence (7) preserves 3D-compatibility and idempotency.
    Used to normalize the maps; preservation of 3D-compatibility is cited to [31], and idempotency preservation is shown in remark (ii).
  • standard math For a quadrirational map, 3D-compatibility is equivalent to the companion map being a Yang-Baxter map (items (1) and (3) of Proposition 2.1).
    This known result from [7] underpins the claim that the companion maps are Yang-Baxter maps; the proof is not reproduced.

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Pith. "Pith review of Idempotent compatible maps and discrete integrable systems on the triangular lattice." pith.science (2026). https://pith.science/paper/L26VP7HN

@misc{pith2026250412212,
  author       = {Pith},
  title        = {Pith review of: Idempotent compatible maps and discrete integrable systems on the triangular lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L26VP7HN}},
  note         = {Machine review of arXiv:2504.12212}
}
abstract

We present three equivalence classes of rational non-invertible multidimensional compatible maps. These maps turns out to be idempotent and by construction they admit birational partial inverses (companion maps) which are Yang-Baxter maps. The maps in question can be reinterpreted as systems of difference equations defined on the edges of the $\mathbb{Z}^2$ graph. Finally, we associate these compatible systems of difference equations with integrable difference equations defined on the triangular lattice $Q(A2)$.

Figures

Figures reproduced from arXiv: 2504.12212 by the authors.

Figure 1
Figure 1. (a):Mapping Q : (x, y) 7→ (x2, y1), assigned on an elementary quad of the Z 2 graph. (b): The inverse map Q−1 of Q. (c): The companion map R, associated with mapping Q. (d): The inverse map R−1 of the companion map R. (3) The companion map R := Qc is a Yang-Baxter map; (4) The inverse R−1 of the companion map R is a Yang-Baxter map; (5) It holds R12 ◦ Q13 ◦ Q23 = Q23 ◦ Q13 ◦ R12; (6) It holds R −1 12 ◦ Q −1 13 ◦ Q −… view at source ↗
Figure 2
Figure 2. The difference system (15) assigned at the edges of the Z 2 graph [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Consider the difference system QI that in conservation form (see [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: l = 0 n = 0 m = 0 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Representation on the cube of the 3D−compatible map Q In the following Figures we prove that provided that Q is a 3D−compatible map, the map R−1 := (Qc ) −1 , that is the inverse of the companion map Qc is a Yang-Baxter map (see item (4)). x y z R −1 12 7−−→ x y z y−1 …
Figure 6
Figure 6. Figure 6: The chain of maps R −1 23 R −1 13 R −1 12 applied on the initial data (x, y, z) [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: The chain of maps R −1 12 R −1 13 R −1 23 applied on the initial data (x, y, z). The assumption that Q is 3D−compatible assures that two ways (see [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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