{"id":"da278889-6ae0-4bf1-91d4-d23cc341202c","arxiv_id":"2607.06698","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Infinity-harmonic functions on planar domains are C^{1,1/3} with isolated critical points and unique quasiradial blow-ups, via a p-to-infinity duality that produces inverse mean curvature flow clusters.","lead":"Planar infinity-harmonic functions are proved C^{1,1/3}, with isolated critical points and unique quasiradial blow-ups; polynomial-growth entire solutions have unique quasiradial blow-downs fixed by Fourier modes at infinity. The proofs rest on a new duality limit that turns infinity-harmonic functions into piecewise inverse mean curvature flows with obstacle interfaces (IMCF clusters).","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s central claim (Thm 1.1) rests on converting infinity-harmonic functions into simple/mixed IMCF clusters via the simultaneous p\to∞, q\to1 limit of conjugates (process (1.15), Thm 6.1), then obtaining C^{1,1/2} edges of the level sets of |\nabla u| from the heat equation for support functions (Thm 7.1) and converting that into C^{0,1/3} normals via Lem. 2.10/2.14. The reader correctly identifies this chain as the weakest link. Re-reading §§6–8 and the reductions in §§10–12 shows that the limiting cluster is constructed carefully (ridges from zero sets of v_p, outer-obstacle condition from Dirichlet data, continuous calibration \nu = -e^w \nabla^\top u), the viscosity comparison for h is proved by geometric maximum principle against sub/supersolutions generated by test functions, and the ε-regularity (1.5) follows from Sturmian control of roots of κ^{-1}. No missing hypothesis or circular step was located. The ordinary residual risk of a long comparison-principle argument remains, but it does not rise to a concrete load-bearing objection that would move the verdict. Hence the reader’s ACCEPT (MODERATE confidence) stands.","tokens_in":84721,"tokens_out":647,"duration_ms":7921,"concrete_test":"Independently re-derive the viscosity inequality of Thm 7.1(ii) for a non-compact segment \theta^{-1}(t,\theta) that is a half-line (using only the local models of Lem. 3.7 and the maximum principles of Rem. 2.13, without the compact-box reductions of Case 1). If the inequality fails for any admissible test function \theta, the support-function regularity used in §8 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly flags the p–q\to IMCF-cluster bridge (Thm 6.1 + viscosity heat equation for support functions across cusps in Thm 7.1) as the architectural load-bearing step for C^{1,1/3} and isolation. After re-examining the constructions, the comparison arguments in §7 (Cases 1–2, Claims 1–8, maximum principles of Rem. 2.13), the C^{1} equicontinuity of p-harmonics (Thm 5.1), and the reduction of sublevel sets of |\nabla u| to IMCF-and-splitting (Thm 9.1, §10), no concrete internal gap or missing hypothesis was found that would break the bridge. Residual risk is the ordinary one for a long geometric-PDE manuscript: a subtle comparison failure at non-compact segments or an overlooked jump configuration, but nothing that can be isolated as a load-bearing flaw from the given text.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper develops a duality between planar ∞-harmonic functions and inverse mean curvature flow (IMCF) clusters—piecewise weak IMCFs with outer-obstacle conditions on ridges (and optional valleys/poles). Via the simultaneous p\to∞, q\to1 limit of the classical p–q conjugate pair (process (1.15), Theorem 6.1), the author extracts a simple/mixed cluster whose support functions satisfy the heat equation ∂_t h=∂_{θθ}h+h across cusps in the viscosity sense (Theorem 7.1). From the resulting C^{1,1/2} regularity of the cuspidal level curves of |\nabla u| and the normal-field Hölder conversion (Lemma 2.10), the paper obtains the sharp local regularity u∈C^{1,1/3}_loc(Ω) (Theorem 1.1 / ε-regularity Theorems 8.1–8.2). Structural consequences include isolation of critical points with unique quasiradial blow-ups of degree d≥2 (Theorem 1.2) and unique quasiradial blow-downs for entire polynomial-growth solutions, determined by finitely many Fourier modes of the support function (Theorem 1.3). Degree-3 entire solutions are constructed explicitly by polygon splitting of a perturbed support function (Subsection 4.1).","tokens_in":85011,"tokens_out":1055,"duration_ms":11440,"significance":"If correct, the work settles the long-standing question of the sharp C^{1,1/3} regularity of planar ∞-harmonic functions (previously only C^{1,α} for some α>0 was known) and supplies the first systematic structural theory near critical points and at infinity. The IMCF-cluster formalism, the C^{1} equicontinuity of p-harmonics (Theorem 5.1), the viscosity heat equation across cusps, and the explicit degree-3 construction are substantial new tools. The arguments are pure analysis (viscosity, weak IMCF, Sturmian theory) with no free parameters or numerical fitting; the reduction is self-contained once the classical p–q duality and Huisken–Ilmanen weak IMCF are granted. These features make the paper a high-impact contribution to the regularity theory of the ∞-Laplacian and to the interface between free-boundary IMCF and absolute minimizers.","major_comments":[],"minor_comments":[{"comment":"The manuscript is very long (∼140 pages). A short roadmap paragraph at the end of the introduction that lists which sections are logically independent would help readers who only need the ε-regularity or the isolation theorem.","section":"§1.1"},{"comment":"Notation for the two families of sublevel sets (Yt,Zt for |\nabla u| versus Y't,Z't for the cluster w) is introduced late (Theorem 6.1(xii)). A brief reminder table or a consistent superscript convention earlier would reduce cross-referencing load.","section":"§6"},{"comment":"Figures 15–17 illustrate the degree-3 family well, but the caption of Figure 5 (polygon splitting) could explicitly mark the valley σ and the two poles so that the mixed-cluster definition is immediately visible.","section":"§4.1"},{"comment":"A few typographical slips remain (e.g., occasional missing spaces after punctuation, and the arXiv identifier appears as 2607.06698). A final copy-edit pass would clean these.","section":null}],"recommendation":"accept","confidential_remarks":"The architectural load-bearing step is the p–q\to IMCF-cluster bridge (Thm 6.1 + viscosity heat equation Thm 7.1). After checking the comparison arguments in §7 (Cases 1–2, Claims 1–8, maximum principles of Rem. 2.13), the C^{1} equicontinuity of p-harmonics, and the sublevel-set reduction in §§9–10, I found no concrete internal gap. Residual risk is the ordinary one for a long geometric-PDE paper (subtle comparison failure at non-compact segments or an overlooked jump configuration), not a load-bearing flaw that can be isolated from the text. The paper is suitable for a top analysis journal; length is the only practical concern for production."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles the long-open sharp C^{1,1/3} regularity for planar infinity-harmonic functions (Thm 1.1), plus isolation of critical points with unique quasiradial blow-ups (1.2) and a Fourier-mode description of polynomial-growth entire solutions (1.3). That is the real news.\n\nWhat is new is the IMCF-cluster formalism: simple and mixed clusters (piecewise weak IMCF with outer-obstacle ridges and optional valleys/poles) obtained as the simultaneous p\to∞/q\to1 limit of the classical p–q conjugate pair. The support-function heat equation is shown to hold across cusps in the viscosity sense (Thm 7.1), and the C^{1,1/2} regularity of the edges then yields the 1/3 Hölder for the joint normal field, hence for \nabla u. The architecture is clean: C^{1} equicontinuity of p-harmonics (5.1), cluster extraction (6.1), viscosity comparison for support functions (§7), ε-regularity (8.1–8.2), then the isolation and Liouville arguments (10–12). Quasiradial models and the degree-3 construction with polygon splitting are worked carefully; the alternating-length identity and Sturmian root-counting are used correctly.\n\nThe load-bearing step is exactly the one the reader flagged: that every infinity-harmonic function produces such a cluster and that the viscosity comparison for h works at non-compact segments. After looking at the comparison cases in §7, the maximum principles of Rem. 2.13, and the reduction of sublevel sets of |\nabla u| (Thm 9.1, §10), I do not see a concrete gap that breaks the bridge. Residual risk is the ordinary one for a 140-page geometric-PDE manuscript—an overlooked jump configuration or a subtle comparison failure—not a structural flaw visible from the text. Self-citations supply background tools and do not circularly force the main claims.\n\nThis is for people who work on infinity-Laplace, absolute minimizers, or weak IMCF. It deserves a serious referee. I would accept it for peer review and would cite the regularity and isolation theorems once they clear.","headline":"Sharp planar C^{1,1/3} for infinity-harmonic functions via a new IMCF-cluster duality; the bridge looks solid enough to send to referees.","tokens_in":85613,"tokens_out":563,"would_cite":true,"duration_ms":10055,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35J70","53E10","35B65"],"pacs":[],"model":"grok-4.5","headline":"Infinity-harmonic functions in the plane are always C^{1,1/3}, with isolated critical points of unique quasiradial shape.","keywords":["infinity-harmonic functions","inverse mean curvature flow","IMCF clusters","C^{1,1/3} regularity","quasiradial solutions","p-q duality","critical points","support functions"],"falsifier":"Either exhibit a planar infinity-harmonic function whose gradient fails to be C^{0,1/3}, or produce an infinity-harmonic function that cannot be reconstructed from any simple or mixed IMCF cluster obtained by the p–q limiting procedure.","tokens_in":85598,"feed_emoji":"📐","tokens_out":665,"duration_ms":7343,"temperature":0.7,"pith_summary":"The paper proves that every infinity-harmonic function on a planar domain is C^{1,1/3} locally. Critical points form a discrete set, and near each of them the function admits a unique quasiradial blow-up of integer degree. Entire solutions of polynomial growth likewise admit unique quasiradial blow-downs fixed by finitely many Fourier modes of their support functions at infinity. These statements follow from a duality that converts the infinity-Laplacian into piecewise inverse mean-curvature flows glued along ridges and valleys (IMCF clusters). The same clusters recover classical examples such as Aronsson’s |x|^{4/3}–|y|^{4/3} and produce new non-quasiradial entire solutions of degree 3. A sympathetic reader cares because the result settles the long-standing question of optimal Hölder regularity for the gradient in two dimensions and supplies a geometric mechanism that organises the otherwise opaque structure of absolute Lipschitz minimisers.","feed_headline":"Planar infinity-harmonic functions are always C^{1,1/3}","feed_subtitle":"Critical points are isolated and entire solutions of polynomial growth are fixed by Fourier modes at infinity","key_machinery":"IMCF clusters: continuous functions w together with a continuous unit vector field that, on each piece of a C^1 partition of the domain, solve the weak inverse-mean-curvature-flow equation with outer-obstacle (ridge/valley) boundary conditions. These clusters arise as the simultaneous p→∞, q→1 limit of the classical p–q conjugate pair and convert the infinity-Laplace equation into a heat equation for support functions of cuspidal curves.","core_discovery":"Every infinity-harmonic function u on a domain in the plane is locally C^{1,1/3}; its critical set is discrete; each critical point admits a unique degree-d quasiradial C^1 blow-up; and every nonlinear entire solution of polynomial growth admits a unique quasiradial blow-down determined by finitely many Fourier modes of the support function of its sublevel sets of |∇u|.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Planar ∞-harmonics are always C^{1,1/3}","Critical points of 2D ∞-harmonics are isolated","∞-harmonics dual to IMCF clusters in the plane","Entire poly-growth ∞-harmonics fixed by Fourier modes","Unique quasiradial blow-ups for ∞-harmonic critical points"],"cache_read_input_tokens":82048,"weakest_assumption_plain":"The whole theory rests on the claim that every infinity-harmonic function can be recovered from the p–q limiting IMCF cluster, including that the support functions satisfy the heat equation across cusps and that ridges arise exactly from zero sets of the conjugates.","fun_headline_variants_meta":{"raw":{"variants":["Planar ∞-harmonics are always C^{1,1/3}","Critical points of 2D ∞-harmonics are isolated","∞-harmonics dual to IMCF clusters in the plane","Entire poly-growth ∞-harmonics fixed by Fourier modes","Unique quasiradial blow-ups for ∞-harmonic critical points"]},"model":"grok-4.5","effort":"low","cost_usd":0.008674,"raw_usage":{"total_tokens":2033,"prompt_tokens":785,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":86740000,"prompt_tokens_details":{"text_tokens":785,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1155,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":785,"tokens_out":93,"duration_ms":10462,"temperature":1.0,"reasoning_tokens":1155,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T23:10:50.258266+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Either exhibit a planar infinity-harmonic function whose gradient fails to be C^{0,1/3}, or produce an infinity-harmonic function that cannot be reconstructed from any simple or mixed IMCF cluster obtained by the p–q limiting procedure.","supporting_citations":[],"review_version":1}