{"id":"9ac0fe1d-af95-4373-ba92-23ea7cbe55af","arxiv_id":"2508.16247","paper_version":4,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Pure-measure-theory De Giorgi-type estimates yield local boundedness, weak Harnack, Harnack, Hölder, and Liouville results for nonlocal parabolic energy classes, with a comparison-principle-free Harnack proof.","lead":"The paper claims a new measure-theoretic framework for regularity estimates (local boundedness, weak Harnack, Harnack inequality, Hölder continuity, Liouville theorem) for a wide class of nonlocal parabolic equations. The headline novelty is a Harnack inequality proof that avoids covering arguments, John-Nirenberg-type lemmas, and any comparison principle. Only the abstract was readable in the supplied version.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central risk: the 'optimal tail conditions' may exclude the standard fractional p-parabolic equations the theory claims to cover; the corrupted full text prevents verifying this, so the claim's scope is unproven.","rationale":"The abstract's central claim is conditional on the energy class being 'wide' and the tail conditions being 'optimal.' The reader's weakest assumption identifies the tail conditions as the structural premise on which boundedness and Harnack rest. I agree: this is the most load-bearing point because the advertised applications are to fractional p-parabolic equations and nonlocal parabolic equations with weak kernels. If the tail condition is too strong, those equations fail to belong to the class and the entire regularity program does not apply to them. Conversely, if the tail condition is exactly the natural one, then the theory has the advertised scope. The corrupted full text prevents me from checking either, so I cannot honestly accept or reject. The proposed test settles the scope question directly. I also note that 'optimal' without a sharpness example is a claim rather than an established property; the test checks for such an example. No internal inconsistency is visible in the abstract, and the approach is novel if sound, so the verdict should be conditional on the concrete check rather than outright rejection or unconditional acceptance.","tokens_in":30612,"tokens_out":6474,"duration_ms":69054,"concrete_test":"Download a clean PDF of arXiv:2508.16247. In Section 2, locate the definition of the parabolic De Giorgi class and the exact tail condition. Take the fractional p-parabolic equation ∂_t u + (-Δ_p)^s u = 0, 1 < p < ∞, 0 < s < 1, with kernel |x−y|^{-n−ps}. Verify (a) that weak solutions, after the usual De Giorgi truncations, satisfy the assumed tail condition for all p in (1,∞), especially p < 2 and s close to 1; (b) that the paper contains an example of a function in the class violating the tail condition that is locally unbounded, establishing optimality. If (a) fails or (b) is absent, the paper's scope and 'optimality' claims must be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing claim is local boundedness and Harnack for a 'wide parabolic energy class' under 'optimal tail conditions.' The abstract states these conditions but does not display them; the supplied full text is an encoding-corrupted dump, so no definition, lemma, or example can be inspected. The risk is therefore domain: if the tail condition is stronger than the natural condition satisfied by solutions of ∂_t u + (-Δ_p)^s u = 0 — for instance, if it requires u ∈ L^q with q > p−1 in the singular case p < 2, or a time-integrated high-power moment — then the intended fractional p-parabolic and weak-kernel equations fall outside the class, and the Harnack/Hölder/Liouville consequences do not apply to them. 'Optimal' also needs a matching counterexample (failure of local boundedness when the tail exponent is lowered); the abstract gives none, and we cannot check whether the paper contains one. If the tail condition is not satisfied by the canonical equations, the central claim of a 'wide' class collapses; if it is, the comparison-principle-free Harnack proof still needs to be checked for hidden structural assumptions (e.g., symmetric kernels, no lower-order terms).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a new paradigm for pointwise regularity of nonlocal parabolic problems: local boundedness under 'optimal tail conditions,' weak Harnack estimates via a purely measure-theoretical framework, a nonlocal parabolic Harnack inequality that avoids covering arguments and John--Nirenberg-type lemmas and is valid without any comparison principle, followed by Hölder estimates and a Liouville-type theorem. The abstract states these results for a 'wide parabolic energy class.' However, the supplied full text is an undecodable mojibake: no definition, theorem, proof, or example can be inspected. The only readable portions are the abstract and a few isolated fragments, which do not permit verification of any of the central claims.","tokens_in":30833,"tokens_out":2220,"duration_ms":27083,"significance":"If the claims are correct, the paper would constitute a substantial advance: it would unify and extend nonlocal parabolic De Giorgi theory, remove reliance on comparison principles, and provide a new proof of the Harnack inequality. The potential scope is broad, covering fractional p-parabolic and weak-kernel equations. The paper does not appear to contain machine-checked proofs, reproducible code, or parameter-free derivations that could partially compensate for the lack of readable text. The significance can only be assessed conditional on a readable manuscript.","major_comments":[{"comment":"The entire supplied text is encoded mojibake. No theorem, definition, proof, or example can be read. The paper is therefore unverifiable in its current form. This is a load-bearing issue: the abstract's claims about optimal tail conditions, Harnack inequalities, and the comparison-principle-free framework cannot be checked. The authors must provide a clean, readable version before any substantive review can occur.","section":"Full text"},{"comment":"The phrase 'optimal tail conditions' is the main structural premise, but the condition is not displayed in the abstract or in the small readable fragments. To assess the claim, the paper must define the tail condition precisely and show that it is satisfied by the canonical examples, in particular by weak solutions of ∂_t u + (-Δ_p)^s u = 0 for the full range of p, including the singular case p < 2. It must also provide a counterexample showing that weaker tail conditions fail, i.e., that local boundedness or Harnack can fail when the tail exponent is lowered. Without these, 'optimal' and 'wide class' are unsupported.","section":"Abstract"},{"comment":"The claim that the Harnack inequality is valid 'regardless of any comparison principle' is not inspectable. The proof is absent. This raises a correctness-risk concern: one must see the precise class of energy inequalities and verify that no hidden structural assumption enters, such as symmetric kernels, no zero-order terms, time-slice integrability, or a priori boundedness that effectively substitutes for a comparison principle. The paper needs a full statement of the class and of all structural hypotheses.","section":"Abstract"},{"comment":"The text contains an extraneous line 'arXiv:2508.16251v1 [cs.GT] 22 Aug 2025' and other clearly unrelated fragments, suggesting contamination from another document. This makes it impossible to attribute even the readable fragments to the intended manuscript. The manuscript must be regenerated from the correct source file.","section":"Full text / inserted passage"}],"minor_comments":[{"comment":"The abstract is clear in its claims, but it does not state the main theorem numbers or definitions; adding them would help orient the reader even before the full text is consulted.","section":"Abstract"},{"comment":"No references are visible in the supplied text, so the authors' positioning relative to existing nonlocal Harnack literature cannot be assessed.","section":"General"}],"recommendation":"uncertain","confidential_remarks":"The manuscript as supplied is not reviewable: the full text is undecodable mojibake, and the only substantive content is the abstract. The claims are plausible and potentially significant, but soundness cannot be certified or falsified without a clean copy. I recommend returning the paper to the authors with a request for a regenerated PDF/source before assigning any further verdict. The contamination by a cs.GT arXiv identifier also suggests a file-handling error rather than a deliberate content issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know first: the abstract is clear, coherent, and claims something substantial—a purely measure-theoretic proof of weak Harnack and Harnack inequalities for a wide nonlocal parabolic energy class, with no covering argument, no John-Nirenberg lemma, and no comparison principle. If that holds up, it is a real advance in the nonlocal parabolic regularity program. The novelty claim is plausible and the abstract is internally coherent. The paper is worth taking seriously. But here is the problem: the supplied full text is undecodable. It is mojibake throughout, with a stray header from an unrelated cs.GT preprint mixed in. No lemma, no proof, no definition can be inspected. I cannot verify any of the theorems. This is not a judgment on quality; it is a hard constraint on what any reviewer can responsibly say from this copy. The stress-test concern about 'optimal tail conditions' is also legitimate and it lands. The abstract says local boundedness holds under optimal tail conditions but does not display them. The key question is whether those conditions are actually satisfied by the standard equations the theory claims to cover—fractional p-parabolic equations, weak-kernel equations, possibly singular cases. If 'optimal' means a stronger integrability assumption than what solutions of ∂_t u + (−Δ_p)^s u = 0 naturally satisfy, then the 'wide class' is not wide in the way the abstract suggests. And 'optimal' needs a matching counterexample: failure of local boundedness when the tail condition is relaxed. The abstract gives none, and I cannot check the paper for one. A related point: the Harnack proof is said to be valid 'regardless of any comparison principle.' That is a strong claim, and it needs careful checking for hidden structural assumptions—symmetric kernels, no lower-order terms, boundary conditions, etc. Again, uncheckable here. My bottom line: the paper deserves a serious referee, but only after the authors provide a clean, readable version. Do not desk-reject it; send it back for a properly encoded manuscript and then send it out. The claims, if true, would be a solid contribution, and the comparison-principle-free Harnack proof is exactly the kind of thing that would change practice in the field. As it stands, however, no one can verify anything from this copy.","headline":"The abstract promises a genuinely new proof strategy for nonlocal parabolic Harnack, but the supplied text is unreadable mojibake, so the claims cannot be checked from this copy.","tokens_in":750,"tokens_out":846,"would_cite":false,"duration_ms":27206,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B65","35K55","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the nonlocal parabolic Harnack inequality follows from a measure-theoretical analysis of a wide parabolic energy class, with no covering argument and no comparison principle.","keywords":["nonlocal parabolic equations","De Giorgi classes","Harnack inequality","local boundedness","tail conditions","Hölder regularity","Liouville theorem","fractional p-Laplacian"],"falsifier":"Build, inside the declared energy class, a function that satisfies the tail conditions yet is locally unbounded; or exhibit a nonnegative solution of a nonlocal parabolic equation in the class for which the Harnack ratio is infinite. Either would refute the central claim. Separately, if local boundedness persists under a strictly weaker tail condition than the one called optimal, the sharpness claim collapses.","tokens_in":30461,"feed_emoji":"📐","tokens_out":4036,"duration_ms":42325,"temperature":0.7,"pith_summary":"The paper proposes a new way to do pointwise regularity for nonlocal parabolic problems by working directly with a wide parabolic energy class rather than with specific equations. It proves local boundedness under what it calls optimal tail conditions, derives weak Harnack estimates through a purely measure-theoretical framework, and then proves the nonlocal parabolic Harnack inequality without any covering argument or John-Nirenberg-type lemma and without relying on a comparison principle. If the proof is right, Harnack, Hölder regularity, and Liouville-type results follow for a broad family of nonlocal parabolic energies. The payoff is that regularity phenomena can be transferred to any equation whose energy fits the class, without structural assumptions.","feed_headline":"Harnack inequality proven without coverings or comparison","feed_subtitle":"A measure-theoretic proof unlocks boundedness, Hölder regularity, and a Liouville theorem for nonlocal parabolic energies.","key_machinery":"The carrying object is the nonlocal parabolic De Giorgi energy class: a set of functions on a space-time cylinder satisfying a parabolic energy inequality with a nonlocal tail term, the tail being an integral measure of the function's size away from a point. The proof is driven by a measure-theoretical expansion mechanism that replaces covering arguments and comparison principles and yields Harnack estimates directly from the class structure.","core_discovery":"The central claim is that pointwise regularity for nonlocal parabolic problems is not tied to the specific equation or to comparison-based techniques: one can define a wide parabolic energy class whose elements satisfy a Caccioppoli-type energy inequality with a nonlocal tail, and from that class alone prove local boundedness, weak Harnack estimates, a full parabolic Harnack inequality, local Hölder estimates, and a Liouville theorem. The Harnack inequality is obtained through a purely measure-theoretical route, deliberately avoiding covering arguments and John-Nirenberg-type lemmas.","pith_inferences":["If the tail conditions are truly optimal, the theory predicts a sharp threshold: weakening the tail integrability by any amount should allow unbounded functions in the class; constructing such an example would confirm the threshold.","The measure-theoretical route may carry over to non-symmetric kernels, systems, or discrete and graph settings where comparison principles fail and covering arguments become clumsy.","One testable extension is to check whether the same framework yields quantitative stability of Harnack constants as the kernel approaches a local operator, recovering classical parabolic regularity in the limit."],"forward_implications":["Local boundedness holds for every element of the energy class whenever the stated tail conditions are met, so the bound depends only on class data.","Weak Harnack estimates are available for nonnegative supersolutions, and correspondingly for subsolutions, without comparison arguments.","The full nonlocal parabolic Harnack inequality applies to equations whose solutions lie in the class, including fractional p-parabolic equations and equations with weak kernels, even when a comparison principle is not available.","Hölder continuity in space and time, and a Liouville theorem for global solutions, follow as downstream regularity results."],"supporting_citations":[],"fun_headline_variants":["Nonlocal parabolic regularity via measure theory alone","Harnack inequality without coverings or comparison","New proof for nonlocal parabolic Harnack inequality","Measure-theoretic route to nonlocal parabolic regularity","De Giorgi classes yield Harnack, Hölder, and Liouville"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The results stand only if the imposed tail conditions are genuinely optimal and are satisfied by the nonlocal parabolic equations the theory is meant to cover, such as fractional p-parabolic equations with weak kernels.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal parabolic regularity via measure theory alone","Harnack inequality without coverings or comparison","New proof for nonlocal parabolic Harnack inequality","Measure-theoretic route to nonlocal parabolic regularity","De Giorgi classes yield Harnack, Hölder, and Liouville"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2452,"prompt_tokens":587,"completion_tokens":1865,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":331,"completion_tokens_details":{"reasoning_tokens":1789}},"tokens_in":331,"tokens_out":1865,"duration_ms":14772,"temperature":1.0,"reasoning_tokens":1789,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T17:24:59.290338+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build, inside the declared energy class, a function that satisfies the tail conditions yet is locally unbounded; or exhibit a nonnegative solution of a nonlocal parabolic equation in the class for which the Harnack ratio is infinite. Either would refute the central claim. Separately, if local boundedness persists under a strictly weaker tail condition than the one called optimal, the sharpness claim collapses.","supporting_citations":[],"review_version":1}