{"id":"fe01e0e4-b0c9-44c8-a7e8-61cb2539c379","arxiv_id":"1908.01706","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For known involutive maps on CP1 times CP1, the paper exhaustively lists the 3D space of separated-variable invariants and uses them to derive vertex potentials, recovering the ABS list of integrable difference equations.","lead":"The authors classify all separated-variable invariants for a catalog of known involutive maps, showing they generate a 3-dimensional space of scalar potentials on a lattice. This yields a unified derivation of vertex-based integrable equations, including the ABS list, from edge-based bond systems.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim of exactly three separated invariants per map hinges on an unproved exhaustiveness step in Section 4; a direct functional-equation check would settle it.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing gap: the exhaustiveness of the invariant lists. I read Section 4 as deriving only a necessary condition—equation (18)—and the paper does not supply the final verification that the general solution of (18), after imposing (16), collapses to the three-parameter families (plus an additive constant) displayed in Table 1. This is not an internal contradiction, and the tables are checkable, but it is an unproved completeness assertion on which the central claim depends. A concrete computer-algebra check on one F-map and one cH-map would either validate the dimension count or expose extra invariants. Since the reader already marked the paper CONDITIONAL with moderate confidence, my analysis does not move the verdict; the concern supports keeping it conditional pending such a check. If the check failed, the correct verdict would be REJECT; if it passed, ACCEPT would be justified. I am not raising any objection about authorship or intent, and I credit the paper for its explicit tables and reproducible derivations within the assumptions it states.","tokens_in":13982,"tokens_out":9624,"duration_ms":98965,"concrete_test":"Perform a direct differential-elimination test for the FI map: impose F(U(u,v))+G(V(u,v))=f(u)+g(v) with U,V as in (FI), differentiate with respect to u and v up to third order at a generic point, substitute the inverse map, and eliminate f,g and their derivatives. Solve the resulting ODE system for F and G and verify that its solution space is exactly the four-parameter family a ln(√(q/p) U/V) + b ln(√((q-1)/(p-1)) (U-1)/(V-1)) + c ln(√((q(q-1))/(p(p-1))) (U-p)/(V-q)) + d of Table 1. Repeat for the cHI map to cover the companion/alternating-sign case. If the computed dimension exceeds four, the '3D' claim is refuted; if it equals four, the exhaustiveness step is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that each F/cH map has exactly three linearly independent separated-variable invariants rests on the reduction in Section 4: from F(U)+G(V)=f(u)+g(v), the paper differentiates and asserts that any solution must satisfy (17), hence (18). This is a necessary condition, and Tables 1 and 2 are consistent with it. But the paper does not prove the converse or the exhaustiveness: it never shows that every solution of (18) that also satisfies the original identity (16) is a linear combination of the three listed invariants, nor that no solutions are lost where m(U) or n(V) vanish or at the maps' singularities. The 'complete set' and 'exhaustive list' statements in Sections 1 and 3 are asserted, with the derivation stopping at the ODE; the final step—substituting the general solution of (18) back into (16), verifying which constants survive, and counting the dimension—is omitted. Because the title's '3D space' and the ABS recovery in Table 8 both depend on this count, the gap is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies involutive maps on CP^1 × CP^1, chiefly the degree-two F-list and cH-list of quadrirational Yang-Baxter maps and their companions, and claims to find, for each such map, the complete set of invariants and alternating invariants with separated variables. It argues that these invariants form a three-dimensional affine space and derives them from a first-order ODE obtained by separating variables. The paper then places each map on the edges of a Z^2 lattice, uses the separated invariants to introduce scalar potentials on vertices, and shows that rewriting the edge systems in terms of these potentials yields known discrete integrable equations, including the ABS list. The final sections discuss multidimensional consistency, Bäcklund transformations, and the distinction between bond systems and vertex models.","tokens_in":14193,"tokens_out":10448,"duration_ms":108095,"significance":"If the claimed exhaustiveness is correct, the paper provides a systematic and genuinely useful bridge between Yang-Baxter maps (bond systems) and vertex equations of ABS type via separation of variables, and it gives a unified explanation of the appearance of non-multiaffine and multi-valued relations in this context. The explicit invariant tables are a strength, since the listed invariants can be directly verified by substitution, and the reduction to the ordinary differential equation (18) is a clean and testable mechanism. The potential formulation in Tables 3 and 4 and the recovery of ABS equations in Table 8 are concrete, falsifiable results. The main weakness is that the exhaustive character of the invariant lists is asserted rather than proved, and that assertion is load-bearing for the title's '3D space' claim.","major_comments":[{"comment":"Sections 3 and 4 assert that the three invariants per map in Table 1 exhaust all invariants (or alternating invariants) with separated variables, and this exhaustiveness is load-bearing for the title's '3D space' and for Table 8. The derivation, however, only establishes a necessary condition: differentiating (16) and separating variables leads to (17) and hence to (18) for any sufficiently smooth solution. It is not shown that every solution of (18) with arbitrary separation constants actually solves the original identity (16), nor that the space of solutions of (18) is exactly spanned by the three functions of Table 1. In addition, the derivation divides by quantities such as m(U)n(V) and by denominators of the inverse map, so the zeros and singularities of the maps in Section 2.2, where logarithmic invariants are singular, require a separate argument. Please supply a proof of the converse and exhaustiveness, for instance by substituting the general solution of (18) back into (16) and counting admissible constants, or explicitly state the completeness claim as conditional.","section":"Section 4, Eqs. (16)-(18)"},{"comment":"Equation (2) states that for arbitrary a,b,c,d the function I(u,v)=a(u-v)+b(u^2+p-v^2-q)+c(u^3+3pu-v^3-3qv)+d is an alternating invariant of map (1). This is not correct for d≠0: for the involution (1), I(U,V)=-I(u,v)+2d, so the alternating property holds only for d=0. The same slip underlies the phrase '3-dimensional affine space' immediately afterwards: the space is 3-dimensional only after one subtracts the trivial constant, that is, it is the affine space of potentials rather than the linear space of alternating invariants. Please correct the statement and the associated dimension count.","section":"Section 1, Eq. (2)"}],"minor_comments":[{"comment":"The connection between the ODE (15), with coefficients b0,...,c4, and the polynomials m(u), n(v) in Table 2 is not explicitly written out. A short derivation showing how (18) follows from (15) under a homographic transformation would make this section much easier to check.","section":"Section 4, Eq. (15) and Table 2"},{"comment":"Proposition 2.1 is stated as an 'iff' classification of degree-one involutive maps of the form (6), but no proof or reference for this classification is provided. If it is a new result, a proof should be included; if it is taken from earlier work, that reference should be given.","section":"Section 2.1, Proposition 2.1"},{"comment":"The sentence 'within ABS list only the above equations admit existence of the potentials ψ' is a classification claim that is not proved or referenced in the manuscript. Either provide a derivation or qualify the statement.","section":"Section 6.2, after Eq. (26)"},{"comment":"There are typographical errors, including 'sytems' in the abstract and 'integrable sytems' in the introduction; the paper would benefit from a careful proofreading pass.","section":"Abstract and Section 1"},{"comment":"The notation p_i is used both for the lattice parameter and for the function on edges, and Tables 3 and 4 list four constants a,b,c,d without explaining which of them can be absorbed into the definition of ψ. A brief remark on the affine freedom in the potentials would avoid confusion.","section":"Tables 3 and 4"}],"recommendation":"major_revision","confidential_remarks":"The paper has substantial verifiable content and the main idea is attractive. The decisive issue is the unproved exhaustiveness in Section 4; if the authors supply a rigorous proof that the listed invariants are complete, I would be willing to reconsider. The d≠0 error in Eq. (2) is a genuine but local slip that should be corrected in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper does something useful and mostly honest. It takes a known class of involutive maps, mostly from Adler–Bobenko–Suris and Papageorgiou et al., and gives an orderly derivation of all separated-variable invariants, then shows how those invariants become potentials that rewrite edge-based difference systems as vertex equations, recovering the ABS list. Tables 1, 3, and 4 are directly checkable, and the structural observation that the invariants come from an ODE of the form (15) is a clean way to organize the classification. The point about non-single-valued evolution being resolved by going back to the bond system is genuinely nice, and the Bäcklund transformation section, though sketchy, points in an interesting direction.\n\nThe soft spot is exactly where your stress-test note lands. The exhaustiveness of the invariant lists is asserted rather than proved. Section 4 derives a necessary condition: any invariant of the form F(U)+G(V)=f(u)+g(v) must satisfy the separated ODE (17)-(18). But the paper never completes the loop by substituting the general solution of (18) back into the original functional equation, checking which constants survive, and counting the dimension. Without that, the claim that the space is exactly three-dimensional is not fully established. The same goes for possible exceptional cases where m(U) or n(V) vanish or at singularities of the map. I suspect the claim is true, and the tables are consistent with it, but it is load-bearing for the title and for Table 8, so it should be a lemma, not an assertion. Proposition 2.1, the classification of degree-one involutive maps, is also stated without proof; it looks plausible, but again the reader is asked to take it on faith.\n\nThe reliance on the authors' own earlier work is not a flaw here—the earlier papers are background, not the input of the derivation. The novelty is moderate but real: the systematic method, the exhaustive tables for the full F and cH lists, and the explicit connection to the ABS list via potentials go beyond what was in [19] or in the authors' previous papers.\n\nWho is this for? Anyone working on discrete integrable systems, Yang–Baxter maps, or the ABS classification. It deserves a serious referee. My recommendation: send it to review, and make the referee's main job the completeness proof. If that gap gets filled, this becomes a solid reference point.","headline":"A systematic, checkable account of separated-variable invariants for Yang–Baxter maps that connects bond systems to the ABS list; the 'exhaustive' claim needs a completeness proof before the central count is fully earned.","tokens_in":14729,"tokens_out":1390,"would_cite":true,"duration_ms":16975,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["39A14","37K10","37K60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Separated invariants of Yang-Baxter maps form a 3D potential space.","keywords":["Yang-Baxter maps","separated-variable invariants","discrete integrable systems","quadrirational maps","ABS equations","scalar potentials","bond systems","vertex equations"],"falsifier":"Find, for any map in the F-list or cH-list, a separated-variable invariant that is not a linear combination of the three invariants listed in Table 1. If such an invariant exists, the assertion that the space is three-dimensional is false; equivalently, solving the separated ODE (17)-(18) and finding more than a three-parameter family of solutions would disprove the classification.","tokens_in":13769,"feed_emoji":"🧮","tokens_out":8984,"duration_ms":88884,"temperature":0.7,"pith_summary":"For every map in the F-list and the cH-list of involutive quadrirational maps, the paper determines all invariants and alternating invariants that separate variables, and finds that they form a three-dimensional affine space. These invariants are the potentials of the associated edge-based difference systems: an invariant $\\psi_i+\\psi=f(u_i,p_i)$ (or $\\psi_i-\\psi$ in the companion case) turns the edge variables into discrete differences or products of one scalar vertex potential. Rewriting the systems on vertices recovers the ABS equations H1, H2, H3, Q1, Q3, A1, A2, and also produces difference relations that are not single-valued; the paper argues these are made well-posed by remembering the underlying bond system. A sympathetic reader would care because this gives a systematic origin for a large part of the discrete integrable landscape: maps, potentials, and vertex equations are linked by one dimensionality statement about separated invariants.","feed_headline":"Three invariants turn edge difference systems into vertex equations","feed_subtitle":"For the F- and cH-lists of maps, separated invariants recover the ABS equations and resolve multi-valued relations.","key_machinery":"The load-bearing object is the separation-of-variables equation for the invariant, written as $F(U)+G(V)=f(u)+g(v)$. Differentiating it and using the inverse map converts it into a relation that separates into $m(U)F'''(U)+2m'(U)F''(U)+m''(U)F'(U)=c$, which integrates twice to $m(U)F'(U)=\\tfrac{c}{2}U^2+dU+e$; the polynomial $m(U)$ for each map is given explicitly. The three constants $c,d,e$ (plus the additive constant) are exactly the three-dimensional space of potentials, and the discrete quadrature $\\psi_i+\\psi=f(u_i,p_i)$ (or $\\psi_i-\\psi$) is the mechanism that carries these invariants from maps to lattice potentials.","core_discovery":"The central claim is that the separated-variable invariants of these maps are not accidental: each map carries exactly a three-dimensional space of them, and that space is the full set of scalar potentials for the corresponding lattice system. The derivation reduces the search to a master differential equation $m(u)F'(u)=\\tfrac{c}{2}u^2+du+e$, whose integration yields the three independent invariants listed in the tables. From these potentials, every edge system is rewritten as a vertex equation, with the ABS list appearing when the potential is fractional linear, and higher-degree or multi-valued vertex relations appearing otherwise. The paper treats the edge (bond) systems as primary and the vertex equations as 'idolons' derived from them, so that non-single-valued evolution of a vertex relation is resolved by the underlying bond system.","pith_inferences":["The paper's completeness claim could be made fully rigorous by classifying all solutions of equation (15) on the projective line; the argument given integrates the equation up to homographic equivalence but does not prove that equivalence exhausts the solution set.","The same invariant-as-potential mechanism should extend to quadrirational maps outside the F and cH lists, where the dimension of the separated-invariant space might still be three but with higher-degree potentials, producing new vertex relations.","The 'bond system as primary, vertex equation as idolon' hierarchy offers a test for integrability of an arbitrary vertex equation: ask whether it admits a completion by a compatible bond system."],"forward_implications":["Each difference system in Tables 3 and 4 is multidimensionally consistent, extending to $n$ dimensions with the compatibility $u^i_{jk}=u^i_{kj}$.","The ABS equations arise as derived objects, not as input: their potentials coincide with specific choices of the three-dimensional invariant spaces.","Vertex relations that are not single-valued become a well-posed evolution once the corresponding bond system is imposed.","In each family there is a multilinear idolon, and non-auto Bäcklund transformations send solutions of that multilinear equation to solutions of all other equations in the family."],"supporting_citations":[{"why":"supplies the F-list of quadrirational Yang-Baxter maps whose separated-variable invariants are classified in this paper.","marker":"[2]"},{"why":"provides the companion H-list of maps (the cH-list) and the symmetry reduction to the homographic-squared action.","marker":"[18]"},{"why":"established the link between Yang-Baxter maps and symmetric lattice equations that the potential rewriting systematizes.","marker":"[19]"},{"why":"introduces the method of finding separated-variable invariants and the associated non-multiaffine lattice equations.","marker":"[12]"},{"why":"gives the ABS classification and the consistency-around-the-cube framework used to characterize the vertex equations.","marker":"[1]"},{"why":"derives the multi-quadratic relation H2* from the nonlinear superposition principle, one of the idolons obtained here.","marker":"[3]"}],"fun_headline_variants":["Three potentials link edge and vertex integrable systems","Three separated invariants give all scalar potentials","Edge maps with three potentials yield vertex equations","Three potentials resolve multi-valued vertex evolution","Three invariants convert bond systems to vertex models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification of invariants is complete only if every separated-variable invariant satisfies the derived differential equation (15) and every solution of that equation is captured up to homographic equivalence; the paper assumes this exhaustiveness rather than proving it.","fun_headline_variants_meta":{"raw":{"variants":["Three potentials link edge and vertex integrable systems","Three separated invariants give all scalar potentials","Edge maps with three potentials yield vertex equations","Three potentials resolve multi-valued vertex evolution","Three invariants convert bond systems to vertex models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3108,"prompt_tokens":849,"completion_tokens":2259,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":2192}},"tokens_in":465,"tokens_out":2259,"duration_ms":18908,"temperature":1.0,"reasoning_tokens":2192,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:05:01.185605+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find, for any map in the F-list or cH-list, a separated-variable invariant that is not a linear combination of the three invariants listed in Table 1. If such an invariant exists, the assertion that the space is three-dimensional is false; equivalently, solving the separated ODE (17)-(18) and finding more than a three-parameter family of solutions would disprove the classification.","supporting_citations":[{"cited_title":"Adler, A.I","cited_arxiv_id":null,"evidence_quote":"supplies the F-list of quadrirational Yang-Baxter maps whose separated-variable invariants are classified in this paper."},{"cited_title":"Papageorgiou, Yu.B","cited_arxiv_id":null,"evidence_quote":"provides the companion H-list of maps (the cH-list) and the symmetry reduction to the homographic-squared action."},{"cited_title":"Papageorgiou, A.G","cited_arxiv_id":null,"evidence_quote":"established the link between Yang-Baxter maps and symmetric lattice equations that the potential rewriting systematizes."},{"cited_title":"Adler, A.I","cited_arxiv_id":null,"evidence_quote":"gives the ABS classification and the consistency-around-the-cube framework used to characterize the vertex equations."},{"cited_title":"Adler and A.P","cited_arxiv_id":null,"evidence_quote":"derives the multi-quadratic relation H2* from the nonlinear superposition principle, one of the idolons obtained here."}],"review_version":1}