{"id":"93e6c67c-8104-48fa-aeac-9ab0d7c2f8a4","arxiv_id":"1907.09678","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The discontinuity set of any subharmonic function is polar.","lead":"The paper proves that the set of points where a subharmonic function is discontinuous is a polar set. This clarifies regularity properties of subharmonic functions used in potential theory and complex analysis.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flagged that only the abstract was initially visible and therefore withheld verdict. The claim itself matches a textbook theorem with no evident deviation or unsupported step, so the UNVERDICTED status remains appropriate pending full-text inspection; no load-bearing weakness is detectable.","tokens_in":1466,"tokens_out":311,"duration_ms":17672,"concrete_test":"Compare the paper's argument against the standard proof in Ransford, Potential Theory in the Complex Plane, Theorem 3.2.3 (or equivalent in Hörmander or Klimek); confirm that the manuscript's reasoning reproduces the key step that the set {x : liminf_{y→x} u(y) < u(x)} is contained in the polar set where the associated Riesz measure is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts that the discontinuity set of any subharmonic function (in the standard sense: upper semicontinuous function satisfying the submean inequality) is a polar set. This is a classical result in potential theory, following from the fact that subharmonic functions are harmonic (hence continuous) off a polar set where they may take the value −∞ or exhibit jumps, with the exceptional set controlled by the Riesz measure or logarithmic potential. No internal gap, hidden assumption, or definitional circularity is visible from the stated claim; the background notions of subharmonicity and polarity are standard and correctly invoked.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that the set of points where a subharmonic function fails to be continuous is a polar set. The argument relies on the standard definition of subharmonicity (upper semicontinuous functions satisfying the submean property) together with the Riesz representation and properties of polar sets in the plane or in several complex variables.","tokens_in":1548,"tokens_out":258,"duration_ms":13404,"significance":"If the derivation holds, the result is a classical theorem of potential theory: subharmonic functions are harmonic (hence continuous) off a polar set determined by the support of the associated Riesz measure. The paper supplies an explicit proof of this fact; its value lies in the clarity or novelty of the argument rather than in the statement itself.","major_comments":[],"minor_comments":[{"comment":"The manuscript should include a brief comparison with standard references (e.g., Ransford's Potential Theory in the Complex Plane or Hörmander's Notions of Convexity) to situate the argument.","section":null},{"comment":"Notation for the polar set and the Riesz measure should be introduced explicitly in the first section rather than assumed from context.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive report and recommendation to accept the manuscript. The referee correctly notes that the result is classical in potential theory and that the contribution lies in the explicit proof; we have no major comments requiring response or revision.","responses":[],"tokens_in":936,"tokens_out":66,"duration_ms":10692,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The result is already known in potential theory. Subharmonic functions fail to be continuous precisely on a polar set. The paper states this but gives no sign that the proof or the framing is new. The statement is accurate. It follows directly from the definition and the properties of the associated Riesz measure or logarithmic potential. The set where the function is not harmonic is polar, and outside that the function is continuous. This is covered in books on potential theory in the plane and in several variables. What the paper does is restate this fact. If it includes an accessible proof, that could be mildly useful for reference. The weakness is the absence of novelty. Classical potential theory has covered this for a long time. Without a new angle or simplification, the work does not add to the literature. The abstract does not mention any comparison to prior proofs or any extension. No issues with circularity or assumptions show up. The definitions are standard. This paper is for readers who want a short note on the topic, perhaps in a teaching context. Experts will find nothing new. I would skip it for a reading group. I would not cite it. It does not merit sending to referees for a serious journal.","headline":"The result is already known in potential theory and the paper shows no sign of adding anything new.","tokens_in":1975,"tokens_out":305,"would_cite":false,"duration_ms":22301,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Classical potential-theory result on subharmonic discontinuity sets; no overlap with RS forcing chain","alignment":"orthogonal","rationale":"The paper's central machinery (Lemma 3.1 on polar sets defined via subharmonic level sets, Theorem 4.1 proving discontinuity set is polar via harmonic measure and thinness) lives entirely in classical potential theory (Armitage-Gardiner references). RS modules (AbsoluteFloorClosure, AlexanderDuality for D=3, Cost/FunctionalEquation for J-cost uniqueness, ArithmeticFromLogic) derive spacetime, constants, and arithmetic from a single distinction; they contain no subharmonic functions, polar sets, or potential theory. Domain mismatch is total.","tokens_in":41931,"confidence":"high","tokens_out":160,"duration_ms":6963,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Subharmonic functions fail to be continuous only on polar sets.","keywords":["subharmonic functions","continuity","polar sets","complex analysis","potential theory","upper semicontinuous functions"],"falsifier":"An explicit subharmonic function whose set of discontinuities has positive capacity would disprove the claim.","tokens_in":2365,"feed_emoji":"","tokens_out":480,"duration_ms":9237,"temperature":0.7,"pith_summary":"The paper proves that any subharmonic function is continuous at all points outside a polar set. Subharmonic functions are upper semicontinuous and obey the submean property, so they are already known to be somewhat regular; the result identifies precisely how small the exceptional set must be. Polar sets are the standard null sets of capacity zero in potential theory. A reader would care because subharmonic functions appear throughout complex analysis as logarithms of moduli and as potentials, and knowing their discontinuities are negligible lets one work with them pointwise almost everywhere.","feed_headline":"Subharmonic functions continuous except on polar sets","feed_subtitle":"Discontinuities of any subharmonic function form a capacity-zero set.","key_machinery":"The submean inequality together with the definition of polar sets.","core_discovery":"The set of points where a subharmonic function fails to be continuous is polar.","pith_inferences":["The same conclusion may hold for plurisubharmonic functions in several complex variables.","The result supplies a uniform way to pass from semicontinuity to full continuity when working with subharmonic majorants.","It suggests looking for analogous statements about the size of discontinuity sets for other classes of functions obeying mean-value inequalities."],"forward_implications":["Subharmonic functions are continuous almost everywhere with respect to capacity.","Removable-singularity theorems for subharmonic functions can be stated with polar exceptional sets.","Logarithms of holomorphic functions are continuous outside polar sets.","Potential-theoretic arguments that ignore polar sets apply directly to subharmonic functions."],"fun_headline_variants":["Subharmonic continuity fails only on polar sets","Discontinuities of subharmonic functions are polar","Subharmonic functions discontinuous solely on polar sets","Continuity of subharmonic functions breaks on polar sets"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Subharmonic functions are upper semicontinuous and satisfy the submean property, while polar sets are the usual capacity-zero sets.","fun_headline_variants_meta":{"raw":{"variants":["Subharmonic continuity fails only on polar sets","Discontinuities of subharmonic functions are polar","Subharmonic functions discontinuous solely on polar sets","Continuity of subharmonic functions breaks on polar sets"]},"model":"grok-4.3","cost_usd":0.003732,"raw_usage":{"total_tokens":1783,"prompt_tokens":365,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":37324500,"prompt_tokens_details":{"text_tokens":365,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1362,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":365,"tokens_out":56,"duration_ms":7934,"temperature":1.0,"reasoning_tokens":1362,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T17:18:51.152955+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit subharmonic function whose set of discontinuities has positive capacity would disprove the claim.","supporting_citations":[],"review_version":1}