{"id":"7d90fd8f-a540-4c60-b53e-5fb2558a7c94","arxiv_id":"2508.13353","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature, with additional critical point and monotonicity results for other constant curvature triangles and polygons.","lead":"This paper extends the hot spots conjecture to triangles and polygons in spaces of constant curvature. It proves that on non-acute geodesic triangles of constant negative curvature, the hot spots conjecture holds, and that mixed eigenfunctions have no non-vertex critical points.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Mixed eigenfunction lemma's 'under certain circumstances' may exclude the right-angle splits on which the non-acute triangle theorem depends","rationale":"The reader identified the unspecified 'under certain circumstances' and the subtle definition of 'non-acute' as the weakest assumptions. I agree, and sharpen the issue: in hyperbolic geometry, non-acute triangles are those with at least one angle ≥ π/2, and the proof almost certainly uses a splitting argument whose validity at the right-angle boundary case is the fragile point. The abstract's qualification leaves this unresolved. However, because the full text is corrupted, this is a concern about unverifiability rather than a demonstrated error. The reader's UNVERDICTED verdict is exactly right, and my critique does not move it. The concrete test—reading the clean source and checking the lemma's corner-angle hypotheses—would settle whether the concern lands.","tokens_in":12510,"tokens_out":4684,"duration_ms":55482,"concrete_test":"Obtain the uncorrupted LaTeX source (arXiv:2508.13353) and locate the mixed Dirichlet–Neumann lemma (likely in §4 or §5). Check its hypotheses: does it allow a Dirichlet and Neumann side meeting at an angle of exactly π/2, or does it require strict inequality? If strict, verify whether the main theorem's proof handles right hyperbolic triangles by a separate argument; if not, the theorem's statement overreaches. As an independent numerical check, solve the Neumann Laplace eigenvalue problem on a right isosceles hyperbolic triangle in the Poincaré disk using a high-order FEM; examine the second eigenfunction for interior extrema. If any interior maximum/minimum appears, the theorem is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem claims the hot spots conjecture for all non-acute hyperbolic triangles. In the hyperbolic plane a non-acute triangle necessarily has exactly one angle ≥ π/2 (right or obtuse), and the standard proof strategy for hot-spot results on triangles is to split the domain along an altitude/median, obtaining two triangles with mixed Dirichlet–Neumann boundary conditions. The abstract states this mixed-eigenfunction result holds only 'under certain circumstances,' a hypothesis left unspecified. If those circumstances require, for example, that the two boundary segments meeting at a Dirichlet–Neumann interface do so at a strictly acute angle, then the borderline right-triangle case (angle exactly π/2) would be excluded from the main theorem even though 'non-acute' includes it. Since the full text is corrupted and unreadable, this possible gap cannot be ruled out from the arXiv version. This is the single most load-bearing concern: the central theorem's advertised class may exceed the class for which the key lemma is actually proven.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims several results on the hot spots conjecture for Laplace eigenfunctions on two-dimensional domains of constant curvature. The main theorem asserts that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature. Additionally, the abstract announces: (i) under unspecified 'certain circumstances', first mixed Dirichlet–Neumann Laplace eigenfunctions on constant-curvature triangles have no non-vertex critical points; (ii) each such eigenfunction is monotonic with respect to some Killing field; and (iii) for general simply connected polygons of non-zero constant curvature, with exactly one family of exceptions, second Neumann eigenfunctions have at most finitely many critical points. The supplied full text is corrupted and unreadable, so the assessment below is based almost entirely on the abstract.","tokens_in":12764,"tokens_out":1592,"duration_ms":19187,"significance":"If the main theorem is correct, it would be a substantial advance: the hot spots conjecture for all non-acute geodesic triangles in the hyperbolic plane is a natural and nontrivial extension of known Euclidean results. The mixed-boundary critical-point theorem and the finiteness result for polygons would also be useful contributions. However, because the manuscript text is unreadable in the version provided, I cannot verify the proofs, definitions, or hypotheses. The significance of the claims is high, but the evidentiary basis for accepting them is presently absent.","major_comments":[{"comment":"The supplied manuscript is corrupted: the body consists of unreadable replacement characters and does not permit verification of any theorem, lemma, or derivation. Since the central claim is a proof-carrying mathematical assertion, the unreadable text is a load-bearing obstacle. I cannot assess whether the proofs are correct, whether the hypotheses are consistent, or whether the stated theorems follow. This prevents acceptance and even substantive review.","section":"Full text (all sections)"},{"comment":"The abstract states that first mixed Dirichlet–Neumann eigenfunctions have no non-vertex critical points 'under certain circumstances,' but the circumstances are not specified. This is load-bearing because the main theorem on non-acute hyperbolic triangles is plausibly proved by splitting a triangle along an altitude and applying the mixed-eigenfunction result to the two pieces. If the unspecified circumstances exclude the case where the Dirichlet–Neumann interface meets the boundary at a right angle, then the advertised class of all non-acute triangles (which includes right triangles) may exceed what the proof covers. The manuscript must state the exact hypotheses and verify that they include every split used in the main theorem.","section":"Abstract, mixed eigenfunction result"},{"comment":"The main theorem is stated without the precise definition of 'non-acute' for geodesic triangles in constant negative curvature. In the hyperbolic plane, a triangle can have multiple angles greater than or equal to π/2 only under angle-sum restrictions, so the term is presumably unambiguous, but the manuscript should explicitly define it and clarify whether right-angled triangles are included. More importantly, the proof structure—especially the role of the mixed-eigenfunction lemma—must be visible to confirm that the theorem covers the full stated class. At present this is unverifiable.","section":"Abstract, main theorem"}],"minor_comments":[{"comment":"The phrase 'constant (positive or negative) curvature triangles' should specify whether Euclidean (zero-curvature) triangles are included or excluded in each result, since the abstract later distinguishes 'non-zero constant curvature' for polygons.","section":"Abstract"},{"comment":"The header line 'arXiv:2508.13350v2 [math.OC] 14 Mar 2026' appears inconsistent with the stated arXiv identifier 2508.13353 and the subject classification math.SP. This may be a corruption artifact, but it should be corrected in a resubmission.","section":"Full text header"},{"comment":"The 'exactly one family of exceptions' for polygons is not described. Even a brief characterization of the exceptional family would help readers assess the scope of the result.","section":"Abstract, finiteness result"}],"recommendation":"uncertain","confidential_remarks":"The major issue is not mathematical content but the unreadable full text. If this is a submission artifact, the authors should be asked to provide a readable version. Until then, no referee can verify the proofs. The stress-test concern about the mixed-eigenfunction lemma's 'under certain circumstances' possibly excluding right-angle splits is plausible and should be checked once the text is available. I recommend treating this as 'uncertain' rather than 'reject' because the abstract-level claims are coherent and potentially true, but they are entirely unverified in the supplied version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this paper is unreadable as submitted—the full text is a corrupted byte stream—so no referee can verify a single line. All I have is the abstract, and on that basis the paper is plausible and potentially valuable, but it cannot be evaluated in this form.\n\nThe abstract's main theorem—hot spots for all non-acute geodesic triangles in constant negative curvature—is a meaningful target. The supporting results about mixed eigenfunctions, Killing-field monotonicity, and finite critical sets for second Neumann eigenfunctions on polygons would be useful if proved. None of that is checkable here.\n\nThe soft spot, beyond the corruption, is the phrase \"under certain circumstances\" attached to the mixed Dirichlet-Neumann lemma. The stress-test note raises the obvious question: if those circumstances exclude the right-angle split, then the main theorem, which advertises all non-acute triangles, is narrower than stated. I can't tell from the abstract, but the burden is on the authors to state those hypotheses precisely. The \"exactly one family of exceptions\" for polygons is similarly vague. Novelty is also unassessable without references, though the abstract does not read like a review.\n\nFor a reader: this is for specialists in the hot spots conjecture and constant-curvature spectral theory. In its current state, I would not bring it to a reading group—there is nothing to read—and I would not cite it. If a clean version appears, it deserves a serious referee; the claims are important enough. But the present submission should be desk rejected and the authors asked to upload a readable PDF and to spell out the mixed-lemma hypotheses.","headline":"An abstract-only paper: the full text is corrupted and unreadable, so the claims cannot be verified; the real open question is whether the mixed-eigenfunction lemma covers the right-angle case.","tokens_in":13160,"tokens_out":4250,"would_cite":false,"duration_ms":45688,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","58J50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the hot spots conjecture for every non-acute geodesic triangle of constant negative curvature.","keywords":["hot spots conjecture","Laplace eigenfunctions","Neumann boundary conditions","constant curvature","hyperbolic geometry","geodesic triangles","Killing fields","critical points"],"falsifier":"Compute the second Neumann eigenfunction on a specific non-acute hyperbolic geodesic triangle—for example, one with angles $(\\pi/2, \\pi/4, \\pi/6)$—using high-accuracy numerical methods, and check whether its gradient vanishes at any interior point. Any interior critical point, especially an interior local maximum or minimum, would directly contradict the main theorem.","tokens_in":12451,"feed_emoji":"📐","tokens_out":8867,"duration_ms":89907,"temperature":0.7,"pith_summary":"This paper proves that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature: the second Neumann eigenfunction of the Laplace–Beltrami operator on any such triangle attains its maximum and minimum on the boundary, not in the interior. It also shows that, under stated conditions, first mixed Dirichlet–Neumann eigenfunctions on constant-curvature triangles have no non-vertex critical points and are monotone along a suitable Killing field. A further theorem says that for general simply connected polygons of non-zero constant curvature, with exactly one family of exceptions, second Neumann eigenfunctions have at most finitely many critical points. If correct, this gives a geometric, symmetry-based explanation for where hot spots can and cannot appear on curved surfaces.","feed_headline":"Hot spots conjecture proven for all non-acute hyperbolic triangles","feed_subtitle":"Second Neumann eigenfunctions on such geodesic triangles can reach extremes only on the boundary.","key_machinery":"The central object is a Killing field—a vector field whose flow is a local isometry of the surface, here of the hyperbolic plane or the sphere. The paper chooses a Killing field that sweeps the triangle in a single direction, and proves that the first mixed Dirichlet–Neumann eigenfunction is strictly monotone along it. Monotonicity along this field controls the sign of directional derivatives, forces level sets to be graphs over a boundary arc, and prevents the existence of interior extrema. The same mechanism, supplemented by a curve-counting argument on level sets, yields the finiteness statement for second Neumann eigenfunctions on polygons.","core_discovery":"The central claim is that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature. In such a triangle, the second Neumann eigenfunction of the Laplace–Beltrami operator has no critical points in the interior; its maximum and minimum are attained on the boundary. The proof rests on constructing, for each such triangle, a Killing field (an infinitesimal isometry of the hyperbolic plane) along which the first mixed Dirichlet–Neumann eigenfunction is strictly monotone; this monotonicity rules out interior extrema and, together with a boundary maximum principle, places the extrema on the boundary. The same monotonicity mechanism is used to show that, und","pith_inferences":["A natural next question—not asked in the paper—is whether the same Killing-field monotonicity proof can be pushed to hyperbolic polygons with more than three sides; if the real condition is the existence of a global sweeping isometry, 'non-acute' may be replaceable by a broader geometric notion.","The 'one family of exceptions' to the finiteness theorem is likely the family of polygons that admit a continuous symmetry (a Killing field). If so, the exception is precisely where the paper's monotonicity mechanism breaks, and checking whether hot spots still hold there by a different argument would be a natural follow-up.","One could test the robustness of the method by taking a curvature-degeneration limit (hyperbolic curvature going to zero) to see whether the Euclidean triangle hot-spots result emerges as a limiting case; if it does, the paper unifies Euclidean and curved hot-spots proofs through one geometric mechanism.","The monotonicity of mixed eigenfunctions along Killing fields may also constrain nodal lines—for instance, forcing the nodal line to connect the Dirichlet and Neumann boundary arcs—which could be checked numerically on the same triangles."],"forward_implications":["Every non-acute hyperbolic geodesic triangle now has the hot spots property: second Neumann eigenfunctions attain extrema only on the boundary.","First mixed Dirichlet–Neumann eigenfunctions on constant-curvature triangles have no interior critical points, so their extrema are confined to vertices or boundary arcs.","For all but one family of simply connected polygons of non-zero constant curvature, second Neumann eigenfunctions have only finitely many critical points.","Any future counterexample to the hot spots conjecture in constant negative curvature must be an acute geodesic triangle; the non-acute case is closed."],"supporting_citations":[],"fun_headline_variants":["Hyperbolic triangles: no interior hot spots","Hot spots conjecture true for non-acute hyperbolic triangles","Second eigenfunction extrema on boundary in hyperbolic triangles","Killing fields move hot spots to the boundary","Non-acute hyperbolic triangles: extremes only at edges"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The proof assumes that every non-acute geodesic triangle in constant negative curvature admits a Killing field whose flow is monotone across the entire triangle; if any such triangle lacks a suitable sweeping isometry, the proof's central mechanism breaks.","fun_headline_variants_meta":{"raw":{"variants":["Hyperbolic triangles: no interior hot spots","Hot spots conjecture true for non-acute hyperbolic triangles","Second eigenfunction extrema on boundary in hyperbolic triangles","Killing fields move hot spots to the boundary","Non-acute hyperbolic triangles: extremes only at edges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1300,"prompt_tokens":626,"completion_tokens":674,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":370,"completion_tokens_details":{"reasoning_tokens":601}},"tokens_in":370,"tokens_out":674,"duration_ms":7476,"temperature":1.0,"reasoning_tokens":601,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T19:04:21.263923+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the second Neumann eigenfunction on a specific non-acute hyperbolic geodesic triangle—for example, one with angles $(\\pi/2, \\pi/4, \\pi/6)$—using high-accuracy numerical methods, and check whether its gradient vanishes at any interior point. Any interior critical point, especially an interior local maximum or minimum, would directly contradict the main theorem.","supporting_citations":[],"review_version":1}