{"id":"efee804e-04bc-4193-9a3b-b3f5ea4013cb","arxiv_id":"2606.08257","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Under an injectivity assumption on the gradient, infinity-harmonic functions in the plane have 1/3-Holder continuous gradients via the caloric structure.","lead":"The paper shows that if the gradient of an infinity-harmonic function in the plane is injective, then the gradient is 1/3-Holder continuous by using a known link to the one-dimensional heat equation. A smart generalist might read it to see how an extra structural assumption unlocks a long-standing regularity conjecture in nonlinear PDEs.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption matches the paper's own scoping of the result. With the claim already conditional and no full proof text supplied to inspect for hidden assumptions, no load-bearing technical flaw can be isolated.","tokens_in":1679,"tokens_out":240,"duration_ms":16630,"concrete_test":"Extract the precise injectivity hypothesis and the derivation that converts it into the caloric equation (likely in the first few sections after the introduction); re-derive the 1/3-Hölder estimate from that equation using only standard parabolic regularity and check whether any hidden non-degeneracy or domain restriction was tacitly used.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is explicitly conditional on an injectivity assumption for the gradient map that grants access to the 1D heat equation structure; the abstract states this assumption is sufficient but not necessary and gives counter-examples (planes, cones) where it fails. No internal inconsistency, circularity, or unsupported step is detectable from the scoped statement, and the 1/3-Hölder target matches the known sharpness example.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims that under a sufficient (but not necessary) injectivity condition on the gradient, the known link between the infinity-Laplace equation and the one-dimensional heat equation in the plane becomes accessible, and the resulting caloric structure yields the sharp 1/3-Hölder continuity of the gradient for infinity-harmonic functions. The paper recalls the known C^1 regularity and local α-Hölder continuity (with no uniform positive lower bound on α), notes that Aronsson's example x^{4/3}-y^{4/3} shows 1/3 is optimal, and supplies counter-examples (planes, cones) where the injectivity condition fails.","tokens_in":1732,"tokens_out":376,"duration_ms":27342,"significance":"If the derivation holds, the result supplies a conditional route to the long-standing 1/3-Hölder conjecture for the gradient of plane infinity-harmonic functions by converting the injectivity assumption into access to the heat equation. The approach is proportionate to the known sharpness example and correctly flags that the condition is sufficient rather than necessary. The manuscript gives explicit credit to prior C^1 results and to Aronsson's original observation of the caloric link.","major_comments":[],"minor_comments":[{"comment":"Abstract: the phrase 'a certain injectivity condition' is used without a forward reference to its precise definition or to the section in which it is introduced; adding a parenthetical pointer would improve readability for readers who stop at the abstract.","section":"Abstract"},{"comment":"The manuscript could usefully include a short remark, perhaps in the introduction, on whether the injectivity condition is expected to hold for a dense class of solutions or only for a special subclass.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report.","responses":[],"tokens_in":1239,"tokens_out":46,"duration_ms":7493,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is a conditional result: under an injectivity condition on the gradient, infinity-harmonic functions in the plane have 1/3-Hölder continuous gradients, obtained by accessing the one-dimensional heat equation structure.\n\nThis is new because it provides a concrete way to use Aronsson's observed connection between the infinity-Laplace equation and the heat equation to prove the sharp regularity. The paper shows how the caloric properties yield the Hölder bound once injectivity is assumed.\n\nThe paper does well in being upfront about the limitations. It states clearly that the injectivity condition is sufficient but not necessary, and it gives counterexamples like planes and cones where the gradient is not injective but the functions are still smooth.\n\nThe soft spots are mostly around the conditional nature. This does not solve the general conjecture, only a special case. The derivation from injectivity to the heat equation and then to the 1/3-Hölder is not spelled out in the abstract, so the strength depends on whether those steps are rigorous and free of additional assumptions in the full text. No signs of circularity or fitting appear in the description.\n\nOverall, the argument seems to engage honestly with the literature and the known sharpness example.\n\nThis work is for experts in nonlinear PDEs and viscosity solutions who follow the infinity-harmonic regularity problem. A reader interested in new techniques for conditional regularity would get something from it.\n\nI think it should go to peer review. The idea is worth having referees examine the details of how the injectivity unlocks the caloric structure.","headline":"This paper gives a conditional proof of 1/3-Hölder gradient regularity for plane infinity-harmonic functions by using injectivity to reach the 1D heat equation structure.","tokens_in":2174,"tokens_out":404,"would_cite":false,"duration_ms":15486,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Under an injectivity condition on the gradient, infinity-harmonic functions in the plane have 1/3-Hölder continuous gradients.","keywords":["infinity-harmonic functions","gradient regularity","injectivity condition","Holder continuity","infinity-Laplace equation","one-dimensional heat equation","plane"],"falsifier":"An infinity-harmonic function in the plane whose gradient is injective but fails to be 1/3-Hölder continuous would falsify the claim.","tokens_in":2572,"feed_emoji":"","tokens_out":438,"duration_ms":14925,"temperature":0.7,"pith_summary":"The paper shows that infinity-harmonic functions in the plane whose gradient satisfies an injectivity condition have gradients that are 1/3-Hölder continuous. This follows from accessing the known link to the one-dimensional heat equation and using its caloric structure to obtain the regularity. The result targets a long-standing conjecture on the optimal regularity of these functions, which are already known to be C^1 but lack a uniform positive Hölder exponent on the gradient in general. A sympathetic reader cares because the condition isolates a class where the conjectured exponent is achieved, even though the condition is sufficient but not necessary.","feed_headline":"Injective gradient yields 1/3-Holder continuity for plane infinity-harmonics","feed_subtitle":"The condition unlocks the one-dimensional heat equation link that delivers the conjectured regularity.","key_machinery":"The injectivity condition on the gradient, which grants access to the caloric structure from the connection to the one-dimensional heat equation.","core_discovery":"Under a certain injectivity condition on the gradient, the link between the infinity-Laplace equation and the one-dimensional heat equation becomes accessible, and the resulting caloric structure proves the 1/3-Hölder continuity of the gradient for infinity-harmonic functions in the plane.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Injectivity gives 1/3-Holder to plane infinity-harmonics","Gradient injectivity yields 1/3-Holder plane infinity-harmonics","1/3-Holder continuity via injectivity in plane infinity-harmonics","Plane infinity-harmonics get 1/3-Holder under gradient injectivity"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The gradient of the infinity-harmonic function must satisfy the injectivity condition to access the caloric structure from the heat equation link.","fun_headline_variants_meta":{"raw":{"variants":["Injectivity gives 1/3-Holder to plane infinity-harmonics","Gradient injectivity yields 1/3-Holder plane infinity-harmonics","1/3-Holder continuity via injectivity in plane infinity-harmonics","Plane infinity-harmonics get 1/3-Holder under gradient injectivity"]},"model":"grok-4.3","cost_usd":0.013205,"raw_usage":{"total_tokens":5612,"prompt_tokens":606,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":132053000,"prompt_tokens_details":{"text_tokens":606,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4924,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":606,"tokens_out":82,"duration_ms":34113,"temperature":1.0,"reasoning_tokens":4924,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T19:23:34.892347+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An infinity-harmonic function in the plane whose gradient is injective but fails to be 1/3-Hölder continuous would falsify the claim.","supporting_citations":[],"review_version":1}