{"id":"0e416f56-eeae-4e6e-a60c-2536788e7a69","arxiv_id":"2412.12080","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Mordell-Schinzel conjecture is proven for cubic equations of the form xyz = A(x) + B(y) - c when A and B have degree 3.","lead":"This paper proves that a broad family of cubic equations in three variables has infinitely many integer solutions. It settles a special case of a classic conjecture by Mordell and Schinzel, using the infinite symmetry groups of associated cubic surfaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The finite exception list is the load-bearing step: it rests on an uncertified computer enumeration and on unverified [KVP24] structural claims used to interpret it.","rationale":"The reader's weakest assumption was the dependence on [KVP24, Thm.29] and [KVP24, 51]. I agree that those results are unverified and are used in the written proof of Proposition 12. However, a careful reading shows that the theorem could be repaired without them: an infinite Sigma_A,B-orbit on any of the four companion surfaces already gives infinitely many integral points on SA,B, because the companion surfaces are isomorphic over Z via the explicit maps sigma_x, sigma_y. Thus the truly indispensable component is the finite classification of all coefficient patterns with sums at most 6 whose trivial solutions have finite Sigma-orbits. That classification is currently backed only by an uncertified computer program. The internal typos in Lemma 9 and Corollary 11 (reversed inequalities and swapped labels of sigma_x/sigma_y) are real but easily corrected and do not threaten the argument. The four displayed points check out arithmetically, and the growth lower bound in Corollary 10 is elementary. The honest status is therefore CONDITIONAL: the central idea is plausible and mostly self-contained, but the finite enumeration must be independently reproduced and the role of [KVP24] clarified before the proof is complete.","tokens_in":6922,"tokens_out":18430,"duration_ms":160012,"concrete_test":"Independently rerun the enumeration behind Proposition 12: for every (a1,a2,b1,b2) in Z^4 with |a1|+|a2| <= 6 and |b1|+|b2| <= 6, compute the Sigma_A,B-orbit of each trivial solution (x0,y0,z0) with x0,y0 in {1,-1} using the explicit formulas (7.3) and (7.4), and confirm that only the companion classes (12.1)-(12.4) have all such orbits finite. Also verify that the four displayed points satisfy their equations and have max{|x|,|y|} >= 6, so that Corollary 10 applies. If the enumeration reproduces the list exactly, the central claim is supported; if any additional exception appears, Theorem 3 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3 is reduced to two finite tasks: Corollary 11 handles every surface with |a1|+|a2| >= 7 or |b1|+|b2| >= 7, while Proposition 12 and Section 14 handle the remaining finite set. Proposition 12 asserts that only the companion classes (12.1)-(12.4) have the property that every trivial solution has finite orbit, and it justifies this assertion by citing [KVP24, Thm.29] and [KVP24, 51] (an arXiv preprint by the first author) to pass from finiteness of the Sigma_A,B-orbit to finiteness of the Aut(SA,B)-orbit. This passage is not proved in the present paper, and it is not self-evident: it requires that the stabilizer of a surface in the groupoid be exactly the cyclic group generated by sigma_A,B, and that the finite-index statement in [KVP24] be correct. If that structural input fails, the stated classification of exceptions has no proof. More fundamentally, the completeness of the finite enumeration itself is load-bearing: the paper supplies only a URL, with no certificate or detailed enumeration protocol. If a surface outside the list had finite Sigma-orbits for all trivial solutions and no large point was found, Theorem 3 would fail. The four explicit points in Section 14 are easily checked and the growth criterion in Corollary 10 is elementary, so the vulnerable part is exactly the finite, computer-assisted classification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that every cubic diophantine equation of the form xyz = ax^3 + by^3 + c + a2 x^2 + a1 x + b2 y^2 + b1 y with a,b nonzero has infinitely many integer solutions. This is the m=n=3 case of the Mordell-Schinzel conjecture. The proof builds on Schinzel's theorem for |abc|>1, reduces the remaining |abc|=1 cases by sign changes to a=b=c=1, and uses an infinite automorphism groupoid of companion surfaces. An elementary growth estimate (Corollary 10) shows that an orbit is infinite once a point exceeds a coefficient-dependent size; Corollary 11 handles all surfaces with coefficient sums at least 7, leaving a finite list that is dispatched by a computer enumeration (Proposition 12) and four explicit large points (Section 14).","tokens_in":7182,"tokens_out":12657,"duration_ms":112223,"significance":"If correct, Theorem 3 settles the cubic case of a conjecture that has been open in this form since Mordell and that Schinzel only resolved for |abc|>1. The overall strategy is attractive: the automorphism groupoid converts an infinite question into a finite verification, and the four large points in Section 14 are immediately checkable by hand. The proof is not fully self-contained, however, because the structural facts about the automorphism group are imported from an arXiv preprint by the first author ([KVP24]) and the finite classification rests on an uncertified computer enumeration. If those inputs are valid, the argument is sound; the gaps identified below are what currently prevent the paper from being a complete proof.","major_comments":[{"comment":"The reduction in Proposition 12 relies on [KVP24, Thm.29] and [KVP24, 51], which are not proved in this paper and are stated as results of an arXiv preprint coauthored by the first author. In particular, the finite-index statement for <sigma_A,B> in Aut_C(S_A,B) is needed to pass from finiteness of all Sigma_A,B-orbits of trivial solutions to finiteness of all Aut(S_A,B)-orbits. If those structural results are flawed or do not apply to the surfaces (4.1), the classification of exceptions in Proposition 12 has no proof. The paper should either include proofs of these facts for m=n=3, or explicitly state and verify them for the finite set of surfaces actually used.","section":"§8, Proposition 12"},{"comment":"The completeness of the finite enumeration underlying Proposition 12 is load-bearing but not supported by the manuscript. The proof only gives a URL and states that a computer program is available; there is no description of the enumeration algorithm, no code listing, no output, and no certificate that all surfaces with |a1|+|a2| <= 6 and |b1|+|b2| <= 6 were checked. The theorem depends on the assertion that (12.1)-(12.4) are the only surfaces with finite Sigma_A,B-orbits for all trivial solutions, so this gap must be closed. Please provide a verifiable certificate, including the program and its output, or an independent computational protocol.","section":"Proposition 12"},{"comment":"Lemma 9.2 is stated in a form that is false as written: the inequality |f(x)| > |x^m| cannot hold for |x| <= -1 + sum |a_i| (for example, f(x)=x^3+x+1 and x=0,m=0 gives 1 > 1, which is false). The condition should presumably be |x| >= 1 + sum |a_i| (or similar), which is exactly what Corollary 10 needs when it applies (9.2). Please correct the statement of Lemma 9.2 and adjust the proof of Corollary 10 accordingly.","section":"Lemma 9.2, Corollary 10"}],"minor_comments":[{"comment":"In the proof of Corollary 11, the names sigma_x and sigma_y appear to be interchanged relative to the definitions in (7.3). The displayed computation makes sense if the first step is sigma_y (giving x1 = B(y0)/x0) and the second is sigma_x (giving y2 = A(x1)/y1); please correct the labels or the formulas.","section":"Corollary 11"},{"comment":"The manuscript contains numerous typos: 'stricty' in Corollary 10, missing carets in equations such as 'ax3' in the abstract and Section 1, and 'A(x−1)' in (7.1) should be 'A(x^{-1})'. A careful proofreading pass is needed.","section":"Notation and typos"},{"comment":"In Remark 7.5, the sentence 'Thus applying sigma_y as in (7.4) we get points with x1 arbitrarily large' is correct because z0 can be chosen arbitrarily, but this should be stated explicitly to make the argument transparent.","section":"Remark 7.5"},{"comment":"The statement of Proposition 12 lists four equations and says 'There are only 3 such, up to isomorphism.' Since (12.3) and (12.4) are isomorphic, this is consistent, but a sentence making the count explicit would avoid confusion.","section":"Proposition 12"},{"comment":"The four points in Section 14 are easily checked to lie on the respective surfaces, but the text should state explicitly that each point satisfies the size condition of Corollary 10 (max{|x0|,|y0|} >= 6), so that the conclusion follows directly.","section":"Section 14"},{"comment":"The reference to [KVP24, 51] uses an unusual numbering; please identify clearly whether '51' refers to an equation, a theorem, or a numbered paragraph in that preprint.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's proof is heavily dependent on [KVP24], a preprint by the first author that is not independently verified in this manuscript, and on an uncertified computer enumeration. If the editor can arrange independent verification of the relevant structural statements in [KVP24], or if the authors include a self-contained proof for the cubic case, the result would be much more solid. The four explicit points and the elementary growth argument are, by contrast, verifiable by hand and are strengths of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Kollár and Li prove the first fully proven instance of the Mordell–Schinzel conjecture for degrees at least 3: the case m=n=3 for equations xyz = x^3+y^3+1+a2x^2+a1x+b2y^2+b1y (and their general a,b,c reductions). That is a real result; no (m,n) with m,n>=3 had been settled before. The strategy is attractive: instead of using the huge automorphism generator, they work with a 4-surface groupoid and elementary growth estimates to show that any sufficiently large point has infinite orbit. The estimates in Corollaries 10 and 11 are clean and checkable, and the four exceptional surfaces are each supplied with an explicit large integral point.\n\nThe soft spots are exactly where the stress-test puts them. The reduction to the finite list in Proposition 12 is load-bearing, and its proof is thin: it cites [KVP24, Thm.29] and [KVP24, 51] to argue that finiteness of the sigma_A,B-orbit implies finiteness of the full Aut-orbit. That step is not proved here and is not self-evident; it requires the groupoid stabilizer to be the cyclic group generated by sigma. If that structural result from the first author's preprint is wrong, the exception list collapses. The computer enumeration behind Proposition 12 is also not certified: a URL is not a certificate, and the protocol is not described. It may be perfectly correct, but for a theorem that reduces to a finite check, the check should be reproducible without trusting the program.\n\nThere are also small presentational issues: some apparent sign/label errors and typos, and the paper is candid that it has no direct proof of Remark 13. None of this undermines the central claim, which I think is probably true. But the paper is not self-contained, and the missing pieces are structurally important rather than cosmetic.\n\nWho is this for? Anyone working on Diophantine equations or arithmetic geometry will want to know this result. It deserves a serious referee: send it out, with a request that the authors either prove or independently verify the needed [KVP24] facts and make the finite enumeration reproducible. I would not desk-reject it; I would engage with it, but acceptance should wait until those two pillars are solid.","headline":"First fully proven case for degrees at least 3, but the load-bearing reduction rests on an unverified preprint and an uncertified computation; send to referees, expect revision.","tokens_in":7725,"tokens_out":3044,"would_cite":true,"duration_ms":26422,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D25","14G05","14J50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every cubic Diophantine equation of the form $xyz=ax^3+by^3+c+a_2x^2+a_1x+b_2y^2+b_1y$ has infinitely many integer solutions.","keywords":["Mordell-Schinzel conjecture","cubic Diophantine equations","integral points","cubic surfaces","infinite automorphism groups","companion surfaces","orbit growth"],"falsifier":"The claim would be falsified by exhibiting any equation of the stated form with nonzero $a,b$ and integer coefficients that has only finitely many integer solutions; a more targeted check is to rerun the enumeration behind Proposition 12 and find a surface with $|a_1|+|a_2|\\le6$ and $|b_1|+|b_2|\\le6$ whose trivial solutions all have finite $\\Sigma_{A,B}$-orbits but which is not isomorphic to one of the four listed surfaces.","tokens_in":6709,"feed_emoji":"♾️","tokens_out":12172,"duration_ms":96137,"temperature":0.7,"pith_summary":"The paper proves the cubic case of the Mordell-Schinzel conjecture: every equation of the form $xyz=ax^3+by^3+c+a_2x^2+a_1x+b_2y^2+b_1y$ with nonzero integer $a,b$ and arbitrary integer coefficients has infinitely many integer solutions. This settles a claim about equations $xyz=G(x,y)$ that was first made for all polynomials in 1952 but had known counterexamples in degree 2, leaving the cubic case as the natural next step. The proof shows that once one integer point is sufficiently large, symmetries of the underlying cubic surface generate infinitely many other integer points; the few remaining small cases are handled by explicitly listed large solutions. If the proof is right, a whole infinite family of cubic Diophantine equations is completely understood, with an explicit mechanism producing the solutions.","feed_headline":"Cubic equations of this type all have infinitely many solutions","feed_subtitle":"Hidden symmetries of cubic surfaces turn one integer solution into infinitely many, closing the cubic case.","key_machinery":"The key machinery is the isomorphism groupoid $\\Sigma_{A,B}$ attached to the surface and its three companion surfaces. It is generated by two involutions $\\sigma_x$ and $\\sigma_y$, given in rational form by $(x,y)\\mapsto (B(y)/x,y)$ and $(x,y)\\mapsto (x,A(x)/y)$, and the composition $\\sigma_{A,B}=\\sigma_y\\circ\\sigma_x\\circ\\sigma_y\\circ\\sigma_x$ is an infinite-order automorphism. The workhorse statement is Corollary 10: if a point satisfies $\\max\\{|x|,|y|\\}>1+\\max\\{1,\\sum|a_i|,\\sum|b_j|\\}$, then one of $\\sigma_x,\\sigma_y$ strictly increases the norm, so the $\\Sigma_{A,B}$-orbit of the point is infinite. This reduces the whole theorem to finding one sufficiently large integral point: for most surfaces a trivial point $(\\pm1,\\pm1,z)$ works, and the few exceptions are settled by an explicit large solution.","core_discovery":"The central claim of the paper is Theorem 3: for every integer choice of $a,b\\ne 0$ and $a_1,a_2,b_1,b_2,c$, the cubic surface defined by $xyz=ax^3+by^3+c+a_2x^2+a_1x+b_2y^2+b_1y$ contains infinitely many integral points. The paper reduces the problem to the normalized case $a=b=c=1$ by sign changes. It then studies the surface together with its three companion surfaces, obtained by replacing $A(x)=x^3+a_2x^2+a_1x+1$ with $x^3A(1/x)$ and similarly for $B$. The isomorphisms between these surfaces generate the groupoid $\\Sigma_{A,B}$, and an elementary estimate shows that any point with $\\max\\{|x|,|y|\\}$ larger than a bound depending only on the coefficients has an infinite $\\Sigma_{A,B}$-orbit. For the normalized cubics this covers all cases with $|a_1|+|a_2|\\ge 7$ or $|b_1|+|b_2|\\ge 7$; the finitely many remaining surfaces form three isomorphism classes, and for each the authors exhibit an explicit integral point with $\\max\\{|x|,|y|\\}\\ge 6$, whose orbit is then infinite and supplies infinitely many integral points.","pith_inferences":["A natural testable extension is to treat 'find one sufficiently large integral point, then let the automorphism group produce the rest' as a general strategy for Diophantine equations attached to surfaces with infinite automorphism groups; the exceptional list shows the only bottleneck is producing that first large point.","If the structural automorphism-group results used from the separately circulated preprint are independently verified, the same reduction should apply to all degrees $m,n\\ge3$, but the quartic examples suggest the exceptional families can be infinite in higher degree, so a different source of starting points would then be needed.","Because the proof is effective—the finite exceptions are listed and the required large points are explicit—a reader could convert the argument into a terminating procedure that, for any concrete cubic equation of the stated shape, either generates infinitely many solutions or identifies it as one of the exceptional surfaces."],"forward_implications":["Every cubic equation of the form $xyz=ax^3+by^3+c+a_2x^2+a_1x+b_2y^2+b_1y$ with $a,b\\ne 0$ has infinitely many integer solutions, so the cubic case of the Mordell-Schinzel conjecture is settled.","Apart from three isomorphism classes listed in Proposition 12, the trivial points $(\\pm1,\\pm1,z)$ already generate infinitely many solutions through the automorphism group.","For the auxiliary equations $mxyz=x^3+y^3+1+a_2x^2+a_1x+b_2y^2+b_1y$ with fixed $m\\ne\\pm1$, the same orbit method proves infinitude whenever a single solution with $\\max\\{|x|,|y|\\}>2+\\max\\{|a_1|+|a_2|,|b_1|+|b_2|\\}$ exists.","The method does not directly extend to quartic equations, since the paper exhibits infinitely many quartic examples where all trivial-solution orbits are finite, so higher-degree cases will need another source of starting points."],"supporting_citations":[{"why":"The original statement that equations $xyz=G(x,y)$ have infinitely many solutions; its monomial-solution plan motivates the normalization used here.","marker":"[Mor52]"},{"why":"Proves all cases with $|abc|>1$ using Fibonacci-type recursions, leaving exactly the normalized cases treated here.","marker":"[Sch15]"},{"why":"Supplies the structural automorphism-group results (infinite automorphism group, finite-index cyclic subgroup, Zariski-dense orbits) used to reduce the proof to a finite exceptional list.","marker":"[KVP24]"},{"why":"Records the unresolved quadratic case, marking the cubic case as the next problem to attack.","marker":"[Sch18]"}],"fun_headline_variants":["Mordell-Schinzel conjecture proven for cubic equations","Cubic equations of form xyz=G(x,y) have infinite solutions","Hidden symmetries show infinite integer solutions in cubic family","Every cubic surface in this family has infinitely many points"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on an unproved structural result about the automorphism groups of these cubic surfaces, namely that a single explicit automorphism generates a finite-index subgroup and that every infinite orbit is Zariski dense; if that result is wrong, the reduction to the finite list of exceptional surfaces collapses.","fun_headline_variants_meta":{"raw":{"variants":["Mordell-Schinzel conjecture proven for cubic equations","Cubic equations of form xyz=G(x,y) have infinite solutions","Hidden symmetries show infinite integer solutions in cubic family","Every cubic surface in this family has infinitely many points"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000272,"raw_usage":{"total_tokens":1586,"prompt_tokens":851,"completion_tokens":735,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":666}},"tokens_in":467,"tokens_out":735,"duration_ms":7046,"temperature":1.0,"reasoning_tokens":666,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:17:08.772565+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The claim would be falsified by exhibiting any equation of the stated form with nonzero $a,b$ and integer coefficients that has only finitely many integer solutions; a more targeted check is to rerun the enumeration behind Proposition 12 and find a surface with $|a_1|+|a_2|\\le6$ and $|b_1|+|b_2|\\le6$ whose trivial solutions all have finite $\\Sigma_{A,B}$-orbits but which is not isomorphic to one of the four listed surfaces.","supporting_citations":[],"review_version":1}