{"id":"97f5bfde-3272-42b4-88ce-f488a3df9555","arxiv_id":"2605.29258","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves existence, uniqueness and flow convergence for weak solutions of generalized Monge-Ampère and supercritical dHYM equations on boundary cohomology classes via viscosity and pluripotential techniques.","lead":"The paper proves existence and uniqueness of weak solutions to the generalized Monge-Ampère equation and the supercritical deformed Hermitian-Yang-Mills equation when cohomology classes sit on the boundary of the solvable region, plus convergence of associated geometric flows to these solutions in the sense of currents. A smart generalist might read it to track progress on nonlinear PDE solvability questions that arise when constructing special metrics on complex manifolds.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flagged the boundary applicability as the key assumption but could not verify it due to abstract-only access. Since the present query likewise supplies no manuscript text for scrutiny, no new or stronger concern emerges and the UNVERDICTED verdict is left unchanged.","tokens_in":1581,"tokens_out":263,"duration_ms":22122,"concrete_test":"Retrieve the full manuscript from the referenced source and scan the main existence proof (likely in the section combining viscosity and pluripotential theory) for any implicit non-degeneracy or strict positivity assumption that would fail exactly on the boundary of the solvable region; if none appears, the claim holds as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The query provides only the abstract and states that the full manuscript is available in a separate cacheable source, but supplies no actual proof details, equations, or section references. Without those, no internal inconsistency, hidden assumption failure, or technical gap in the argument (existence/uniqueness of weak solutions on the boundary via viscosity + pluripotential methods, plus flow convergence in currents) can be located. The central claim is a natural boundary extension of known techniques; no load-bearing concern is identifiable from the given material.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves existence and uniqueness of weak solutions to the generalized Monge-Ampère equation and the supercritical deformed Hermitian-Yang-Mills equation for cohomology classes on the boundary of the solvable region. It further establishes convergence of the associated geometric flows to these weak solutions in the sense of currents, via a combination of viscosity-theoretic and pluripotential-theoretic techniques.","tokens_in":1665,"tokens_out":214,"duration_ms":26272,"significance":"If the arguments hold, the result would complete the picture of solvability by handling the boundary locus, which is typically the most singular case. The dual use of viscosity and pluripotential methods is a methodological strength for treating weak solutions without interior non-degeneracy.","major_comments":[],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":"The query supplies only the abstract and states that the full text resides in a separate cacheable source that is not reproduced here; without access to the actual sections, equations, or estimates, verification of the central claims is not possible from the given material."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary of the manuscript. We are pleased that the significance of completing the solvability picture for boundary classes and the dual use of viscosity and pluripotential methods are recognized as strengths. The recommendation is listed as uncertain, but no specific major comments are provided in the report. We therefore have no point-by-point responses to offer at this stage and would welcome any concrete concerns the referee may have so that we can address them directly.","responses":[],"tokens_in":1022,"tokens_out":109,"duration_ms":17956,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core advance is proving existence and uniqueness of weak solutions in cohomology classes on the boundary of the solvable region, plus convergence of the associated flows in the sense of currents. This finishes the solvability picture for both equations where previous work had covered the interior.\n\nThe combination of viscosity-theoretic and pluripotential-theoretic techniques is a reasonable way to handle the weak setting, and the claim that the boundary can be treated without extra non-degeneracy assumptions looks plausible on the surface. The result is incremental rather than foundational, but it is the natural next step once the interior is settled.\n\nThe main soft spot is that the abstract gives almost no detail on how boundary singularities are controlled or whether any new estimates appear. If the argument reduces to routine adaptation of interior techniques, the paper is short and clean; if hidden regularity issues arise exactly at the boundary, that would need careful verification. No circularity or invented entities are visible.\n\nThis is for people already working on Kähler geometry and these specific equations. A reader who follows the interior results will get direct value from seeing the boundary completed. It is narrow in scope but technically grounded enough to warrant referee time rather than a desk reject.","headline":"This paper closes the boundary cases for weak solutions of the generalized Monge-Ampère and supercritical dHYM equations using viscosity plus pluripotential methods.","tokens_in":2125,"tokens_out":315,"would_cite":false,"duration_ms":18906,"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":"Weak solutions exist and are unique for the generalized Monge-Ampère and supercritical deformed Hermitian-Yang-Mills equations on the boundary of the solvable region.","keywords":["weak solutions","Monge-Ampère equation","deformed Hermitian-Yang-Mills equation","viscosity methods","pluripotential theory","geometric flows","cohomology classes","boundary cases"],"falsifier":"A specific cohomology class on the boundary of the solvable region for which no weak solution exists, or for which the associated geometric flow fails to converge in the sense of currents, would falsify the claim.","tokens_in":2476,"feed_emoji":"📐","tokens_out":590,"duration_ms":23032,"temperature":0.7,"pith_summary":"The paper proves existence and uniqueness of weak solutions to the generalized Monge-Ampère equation and the supercritical deformed Hermitian-Yang-Mills equation for cohomology classes on the boundary of the solvable region. It also establishes that the associated geometric flows converge to these weak solutions in the sense of currents. The argument relies on combining viscosity-theoretic and pluripotential-theoretic techniques. A sympathetic reader would care because this completes the solvability picture by handling the previously excluded boundary cases in these nonlinear equations from complex geometry.","feed_headline":"Weak solutions exist on boundary for Monge-Ampère and dHYM equations","feed_subtitle":"Existence, uniqueness and flow convergence hold for cohomology classes at the edge of the solvable region.","key_machinery":"The combination of viscosity-theoretic and pluripotential-theoretic techniques applied directly to boundary cohomology classes.","core_discovery":"We prove the existence and uniqueness of weak solutions for the generalized Monge-Ampère equation and the supercritical deformed Hermitian-Yang-Mills equation in cohomology classes lying on the boundary of the solvable region. Moreover, we prove that the associated geometric flows converge to the weak solutions in the sense of currents. The proof combines viscosity-theoretic and pluripotential-theoretic techniques.","pith_inferences":["The solvable region in cohomology space may be closed in the appropriate topology.","The boundary analysis could extend to related fully nonlinear equations on Kähler manifolds.","Convergence of flows at the boundary may allow study of limiting behavior exactly at the solvability threshold."],"forward_implications":["Weak solutions exist uniquely in boundary cohomology classes for both equations.","The geometric flows converge to these weak solutions in the sense of currents.","The combined techniques extend solvability results from the interior to the boundary without additional assumptions."],"fun_headline_variants":["Boundary weak solutions for Monge-Ampère and dHYM","Weak solutions at boundary for generalized Monge-Ampère","Supercritical dHYM gets weak solutions on boundary","Weak solutions for boundary cases in Monge-Ampère and dHYM"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The boundary of the solvable region in cohomology space remains a locus where viscosity and pluripotential techniques continue to apply without extra regularity or non-degeneracy assumptions.","fun_headline_variants_meta":{"raw":{"variants":["Boundary weak solutions for Monge-Ampère and dHYM","Weak solutions at boundary for generalized Monge-Ampère","Supercritical dHYM gets weak solutions on boundary","Weak solutions for boundary cases in Monge-Ampère and dHYM"]},"model":"grok-4.3","cost_usd":0.006054,"raw_usage":{"total_tokens":2701,"prompt_tokens":505,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":60540500,"prompt_tokens_details":{"text_tokens":505,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2125,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":505,"tokens_out":71,"duration_ms":23110,"temperature":1.0,"reasoning_tokens":2125,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T06:00:42.988970+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A specific cohomology class on the boundary of the solvable region for which no weak solution exists, or for which the associated geometric flow fails to converge in the sense of currents, would falsify the claim.","supporting_citations":[],"review_version":1}