{"id":"5684787d-68e5-4c29-aeda-34056abe199f","arxiv_id":"2504.12212","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Three new families of idempotent, non-invertible compatible maps yield Yang-Baxter companion maps and integrable difference equations on the triangular lattice.","lead":"This paper classifies three families of rational, non-invertible maps that are multidimensional consistent, idempotent, and linked to Yang-Baxter equations. The authors turn these maps into difference equations on a triangular lattice, giving new discrete versions of integrable systems such as Burgers equation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (6) defines u=v, making the defining equation F(u,y)=F(u,x) independent of u for the stated F, so the theorem's maps are not of the printed form.","rationale":"The reader's weakest assumption is the separable ansatz itself, and I agree that the classification is not established outside that ansatz. My concern is more basic: even inside the printed ansatz, equation (6) does not define the maps that Theorem 2.3 claims to classify. Setting u = v collapses the defining relation to F(u,y) = F(u,x), and for the stated multiplicative F the u-dependent factor cancels, leaving only x = y or a constraint on parameters. Hence there is no quadratic equation in u, and the proof's discriminant argument does not apply. The maps QI-QIII are displayed with u ≠ v, so either the displayed maps are not of the stated type or equation (6) is misprinted. This is an internal-consistency problem rather than a scope limitation; the central theorem cannot be evaluated as written. I do not recommend outright rejection because the explicit maps, the multidimensional compatibility formulas in Section 3.2.1, and the associated triangular-lattice equations can be checked directly and may be correct; the issue may be fixable by replacing (6) with the intended defining relations. For that reason I would mark the current version as unverdictable pending a corrected statement of the ansatz, rather than accepting the classification claim as it stands.","tokens_in":12507,"tokens_out":9445,"duration_ms":95555,"concrete_test":"Use a computer algebra system to impose v = u and F(u,y,p,q) = F(u,x,p,q) with F = (px-P)/(δx-1) * (qy-Q)/(δy-1), then solve for u. The expected result is that u cancels and the only solutions are x = y or q = δQ. If that outcome is confirmed, the proof step in Theorem 2.3 is invalid under the printed definition. To distinguish a typo from a substantive gap, repeat the classification proof with (6) replaced by the two equations defining QI, namely pu-P = qv-Q and uy = vx; if that corrected ansatz yields QI-QIII and the paper's remaining computations, the issue is typographical and the central construction can be salvaged.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"Section 2.2 defines the objects of Theorem 2.3 by (5)-(6), where (6) reads 'u = v, F(u,y,p,q) = F(v,x,p,q)'. With v = u, the second relation becomes F(u,y,p,q) = F(u,x,p,q). For the multiplicative separable F in the theorem, F(u,y) and F(u,x) share the same u-dependent factor, so after cancellation the equation reduces to (qy-Q)/(δy-1) = (qx-Q)/(δx-1), i.e. (q-δQ)(x-y) = 0. No nontrivial function u(x,y,p,q) is determined, in contrast to the proof's claim that substitution of v = u gives a quadratic in u whose discriminant must be a square. The three maps QI-QIII exhibited in the theorem have u ≠ v, so they are not of the form (5)-(6) as printed. Thus the central classification statement is internally inconsistent: either (6) is a typo for the actual defining relations (e.g. pu-P = qv-Q and uy = vx for QI), or the theorem classifies a different family than the one presented. This is more fundamental than the separable-ansatz limitation: the ansatz as written defines no nontrivial maps of the claimed kind.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12776,"tokens_out":9189,"duration_ms":96204,"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":[{"comment":"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.","section":"Section 2.2, Eq. (6)"},{"comment":"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.","section":"Section 2.2, proof of Theorem 2.3"},{"comment":"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.","section":"Section 2.2, Theorem 2.3 and remarks (i)-(v)"},{"comment":"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.","section":"Section 2.1, Proposition 2.1 and remark (vii)"}],"minor_comments":[{"comment":"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'.","section":"Abstract"},{"comment":"'A well possed initial value problem' should be 'A well-posed initial value problem'.","section":"Section 3.3.1"},{"comment":"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.","section":"Theorem 2.3"},{"comment":"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.","section":"Table 1"},{"comment":"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.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about equation (6) lands: the ansatz as written defines no nontrivial maps of the claimed kind, and the displayed maps QI-QIII contradict the printed defining relations. This is a corrigible issue only if the authors can supply the correct defining system and re-prove the classification from it; as it stands, the main theorem is not valid. I would be willing to review a revised version, but the revision must restate the ansatz, provide the missing derivation, and give the explicit 3D-compatibility checks."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: Theorem 2.3, as printed, classifies an empty family. Equation (6) states u=v, so the second relation becomes F(u,y,p,q)=F(u,x,p,q). For the multiplicative separable F used in the theorem, the u-dependent factor cancels and you're left with (q-δQ)(x-y)=0. No nontrivial u is determined. Yet the three maps QI-QIII displayed in the theorem all have u≠v. The proof actually derives different defining relations—for QI it ends with pu-P=qv-Q and uy=vx. So (6) is almost certainly a typo in the written ansatz. But as it stands, the central classification statement is self-contradictory, and the abstract overstates what is proved.\n\nThat is the main problem. The rest of the paper is more solid. The three explicit maps are concrete and checkable. Their idempotency is directly verified, the companion maps are explicit birational Yang-Baxter maps, and the translation to edge variables on Z^2 and then to vertex equations on the triangular lattice is worked out in detail. The discrete Burgers connection for QI is spelled out with the potential transformation, and that part holds up on inspection. No fitted parameters, no post-hoc selection. The multidimensional compatibility formulas for QI-QIII are written down explicitly and are invariant under j↔k, so those are independently checkable.\n\nThe soft spots besides the typo: the proof of Proposition 2.1 mostly outsources to Adler–Bobenko–Suris and says the rest 'follows in a similar manner'; the proof of Theorem 2.3 says 'it can be shown' for the 3D-compatibility; the additive case is dispatched with 'following the same analysis.' Those are gaps a referee would want filled, but they are not necessarily wrong. The classification also only covers separable F, so the claim of three equivalence classes is not a full classification of all rational maps of the general form. The abstract should be qualified accordingly.\n\nBottom line: this is a promising contribution to the theory of non-invertible compatible maps, with a real lattice interpretation. The typo is load-bearing but looks fixable in a revision. It deserves a serious referee, provided the referee is asked to verify the corrected ansatz and the omitted computations. I would not cite it in its current form, and I'd hold off on the reading group until a corrected version appears.","headline":"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.","tokens_in":13294,"tokens_out":6814,"would_cite":false,"duration_ms":57168,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-16T12:35:45.020973+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}