{"id":"f1223bcd-8484-44f0-bb5d-e52c2539d7a5","arxiv_id":"2605.02610","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Under a minimum-degree condition, for large enough ground sets there is an extremal hypergraph containing an isolated K^k_{t+1}, with a smaller size threshold than prior work for triples.","lead":"The paper improves a threshold guaranteeing that minimum-degree extremal hypergraphs contain an isolated complete k-uniform clique. A smart generalist might care because it tightens classical Kruskal–Katona-type shadow bounds under degree constraints.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Provided full text is the wrong paper (ML, not hypergraphs), so the central extremal claim cannot be verified.","rationale":"The reader correctly flagged both the abstract-only status and the mismatched body, and set UNVERDICTED with low confidence. The single load-bearing issue for the strongest claim is not a subtle gap inside a present proof; it is that the combinatorial proof is not present at all. No internal inconsistency of the hypergraph argument can be assessed until the right manuscript is attached. Therefore the verdict should stay UNVERDICTED; no upgrade or downgrade is justified. The concrete test is simply to obtain and check the real paper against the three preservation/isolation properties the abstract asserts.","tokens_in":12894,"tokens_out":533,"duration_ms":10591,"concrete_test":"Retrieve the actual arXiv:2605.02610 PDF (or source) and verify that its transformation section proves: (i) the combined shift/antilex compression preserves δ(F) ≥ binom(t,k−1), (ii) |∂_ℓ F| does not increase, and (iii) an isolated K^k_{t+1} appears for every t ≥ k−1 when |X| exceeds (1/4)(t+1)^2 binom(t−1,ℓ−2)+3t+1. If any of (i)–(iii) fails for some regime (e.g. t near k−1 or ℓ near k), the headline threshold claim weakens or collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The cacheable full manuscript under paper_id 2605.02610 is not “Shadows of Uniform Hypergraphs…”; it is the unrelated ML paper “Selective Prediction from Agreement” (arXiv 2605.02611). The abstract’s central claim—that a shifting + antilexicographic-compression transformation yields an extremal F containing an isolated K^k_{t+1} for |X| above the stated threshold—depends entirely on that combinatorial argument (preservation of δ(F) ≥ binom(t,k−1), non-increase of |∂_ℓ F|, and isolation of the clique for all t ≥ k−1 and k > ℓ ≥ 2). With no matching proof text, lemmas, or constructions present, that claim is uncheckable. The reader’s concern about the transformation is therefore not merely open; the supporting manuscript is absent.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The abstract of arXiv:2605.02610 claims a minimum-degree analogue of Kruskal–Katona for k-uniform families: if δ(F) ≥ binom(t,k−1), then there exists an extremal F (minimizing |∂_ℓ F|) that contains an isolated copy of K^k_{t+1} whenever |X| exceeds (1/4)(t+1)^2 binom(t−1,ℓ−2)+3t+1, for all t≥k−1 and k>ℓ≥2. The argument is said to rest on a new hypergraph transformation combining shifting with antilexicographic compression. For the special case k=3, ℓ=2 the abstract asserts an improved threshold over Füredi–Zhao (2022). The body supplied under this paper_id is not that manuscript; it is an unrelated machine-learning paper on selective prediction via Lipschitz version spaces (arXiv:2605.02611).","tokens_in":13132,"tokens_out":897,"duration_ms":11914,"significance":"If the combinatorial claim in the abstract is correct, the result would be a genuine advance in extremal hypergraph theory: a structural existence theorem for degree-constrained shadow minimizers in the general range k>ℓ≥2, together with a concrete improvement of the Füredi–Zhao threshold for triples. The announced technique (shifting plus antilexicographic compression that preserves minimum degree while not increasing the ℓ-shadow and isolating a clique) would also be of independent interest. None of this can be assessed from the text actually provided.","major_comments":[{"comment":"The full manuscript text attached under paper_id 2605.02610 is not “Shadows of Uniform Hypergraphs under a Minimum Degree Condition.” It is the unrelated paper “Selective Prediction from Agreement: A Lipschitz-Consistent Version Space Approach” (arXiv:2605.02611, cs.LG). Consequently every load-bearing claim of the abstract—the transformation that combines shifting with antilexicographic compression, preservation of δ(F)≥binom(t,k−1), non-increase of |∂_ℓ F|, isolation of K^k_{t+1}, and the stated threshold—is unsupported by any lemma, construction, or proof in the supplied body. The central result is uncheckable.","section":null},{"comment":"Even the special-case improvement for k=3, ℓ=2 (threshold ¼(t+1)^2+3t+1 versus Füredi–Zhao’s ¼(t+1)^2(t+2)) cannot be verified: no comparison argument, no extremal example, and no analysis of the critical range |X| near the new bound appear in the provided text.","section":null},{"comment":"The abstract’s weakest technical assumption—that the new compression preserves both the degree lower bound and shadow size while forcing isolation for every t≥k−1 and every k>ℓ≥2—is precisely the step that requires careful case analysis (especially near t=k−1 and when ℓ is close to k). With no proof text present, that assumption remains an unexamined axiom rather than a theorem.","section":null}],"minor_comments":[{"comment":"Metadata inconsistency: the CACHEABLE PAPER SOURCE CONTEXT header lists paper_id 2605.02610 and the hypergraph title, while the body is arXiv:2605.02611. This should be corrected before any further review cycle.","section":null},{"comment":"The abstract alone is well written and clearly positions the result against Kruskal–Katona and Füredi–Zhao; once the correct manuscript is supplied, only ordinary presentation checks will be needed on that front.","section":null}],"recommendation":"uncertain","confidential_remarks":"The submission package appears to have swapped manuscripts (combinatorics abstract + ML full text). I cannot produce a scientific evaluation of 2605.02610 until the correct PDF/source is provided. Please re-queue with the matching full text; I am happy to re-referee promptly. Do not treat the present report as a judgment on the mathematical content of the hypergraph paper."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing you need to know: the cacheable full text under 2605.02610 is not Liu–Lu–Zhang. It is Khosravani’s selective-prediction / Lipschitz version-space paper (2605.02611). So this is effectively abstract-only for the hypergraph claim, and the central proof cannot be read.\n\nFrom the abstract alone, the contribution is clear and useful to specialists. Füredi–Zhao (2022) showed that for 3-uniform hypergraphs and 2-shadows, every extremal family under a minimum-degree condition eventually contains an isolated K_{t+1}^3 once the ground set is larger than (1/4)(t+1)^2(t+2). This note claims a general k>ℓ≥2 version: there exists an extremal F (minimizing |∂_ℓ F| subject to δ(F)≥binom(t,k−1)) that contains an isolated K^k_{t+1} once |X| exceeds (1/4)(t+1)^2 binom(t−1,ℓ−2)+3t+1. For the classical (3,2) case the new threshold is (1/4)(t+1)^2+3t+1, strictly smaller for t≥3. The advertised tool is a transformation that mixes shifting with antilexicographic compression. That is honest incremental progress inside the Kruskal–Katona / degree-shadow program, not a reorganization of the field.\n\nWhat we cannot check is exactly what the abstract hangs on: whether that transformation preserves the degree lower bound, does not increase the shadow, and forces isolation of the clique for all t≥k−1 and all k>ℓ≥2 (including the awkward regimes near t=k−1 or ℓ close to k). The reader’s worry about degree loss or non-isolated edges is therefore not a nitpick; it is the whole argument, and the supporting lemmas are missing from the supplied text.\n\nCitation pattern in the abstract is appropriate (Kruskal–Katona, Füredi–Zhao). There is no free-parameter fitting or circular construction visible from the claim statement.\n\nWho it is for: people who already work on shadows, shifting, and degree versions of extremal set theory. A serious editor in math.CO should send the real manuscript to referees if it matches this abstract; the claimed threshold improvement and the general (k,ℓ) extension are enough to deserve referee time. I would not bring the current package to reading group, and I would not cite until the correct combinatorial text is in hand. Engage only after the right PDF is attached.","headline":"We only have the abstract for the hypergraph paper; the attached full text is an unrelated ML manuscript, so the claimed compression argument is uncheckable.","tokens_in":13698,"tokens_out":651,"would_cite":false,"duration_ms":11230,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05D05"],"pacs":[],"model":"grok-4.5","headline":"Under a minimum-degree condition, an extremal k-uniform hypergraph can isolate a full clique once the ground set is large enough.","keywords":["Kruskal–Katona theorem","shadows","uniform hypergraphs","minimum degree","extremal set theory","shifting","antilexicographic compression","isolated clique"],"falsifier":"Exhibit a single triple (k,ℓ,t) with t ≥ k−1 and ground-set size just above the stated threshold for which every minimum-shadow hypergraph of minimum degree binom(t,k−1) has no isolated K^k_{t+1}, or show that the transformation itself decreases degree or increases shadow size on that instance.","tokens_in":13799,"feed_emoji":"🔷","tokens_out":1003,"duration_ms":14127,"temperature":0.7,"pith_summary":"The classical Kruskal–Katona theorem says that if a family of k-subsets is large enough, its ℓ-shadow cannot be smaller than that of a complete clique. This paper studies the minimum-degree version: when every vertex has degree at least that of a clique of order t, how small can the ℓ-shadow still be? An extremal family is one that realises that minimum. The authors prove that for every k > ℓ ≥ 2 and t ≥ k−1, once the ground set exceeds a quadratic threshold in t, there exists an extremal family that contains an isolated copy of the complete k-uniform clique on t+1 vertices. The argument improves an earlier threshold of Füredi and Zhao in the special case k=3, ℓ=2. The result matters because it shows that the degree condition alone forces a rigid local structure—an isolated clique—rather than a more diffuse configuration, and it supplies a concrete size bound that is often substantially smaller than previous ones.","feed_headline":"Min-degree extremal hypergraphs isolate a clique past a quadratic size","feed_subtitle":"A shifting-plus-compression map improves the earlier Füredi–Zhao threshold for every t ≥ 3.","key_machinery":"A hypergraph transformation that combines shifting operations with antilexicographic compression. It preserves both the minimum-degree lower bound and the shadow size while forcing an isolated clique into an extremal example.","core_discovery":"For every integer t ≥ k−1 and every k > ℓ ≥ 2, if the ground set is larger than (1/4)(t+1)^2 binom(t−1,ℓ−2) + 3t + 1, then there exists a k-uniform hypergraph that minimises the size of the ℓ-shadow subject to minimum degree at least binom(t,k−1) and that contains an isolated copy of K^k_{t+1}. In the case k=3, ℓ=2 the new threshold is (1/4)(t+1)^2 + 3t + 1, improving the earlier bound of Füredi and Zhao for all t ≥ 3.","pith_inferences":["The quadratic dependence on t, multiplied only by a binomial in ℓ, hints that the obstruction is essentially a two-dimensional covering argument rather than a higher-dimensional packing.","If the compression step can be made fully constructive, one could algorithmically produce extremal examples rather than merely prove their existence.","The same isolated-clique phenomenon may hold under weaker average-degree or codegree conditions once an analogous compression is found."],"forward_implications":["Extremal examples under the minimum-degree condition can be taken to contain a pure isolated clique once the ground set is only quadratically large in t.","The improved numerical threshold for triples (k=3,ℓ=2) immediately tightens all subsequent quantitative statements that relied on the Füredi–Zhao bound.","The same transformation may be reusable for other degree-constrained shadow problems beyond the single-parameter minimum-degree setting.","Existence of an isolated-clique extremal example suggests that the global minimum shadow is realised by a disjoint union of a clique and a residual structure of controlled size."],"fun_headline_variants":["Min-degree extremal hypergraphs isolate a clique past quadratic threshold","Improved size forces isolated cliques in min-degree extremal hypergraphs","Extremal k-graphs isolate K_{t+1}^k under degree condition for large n","Shifting-compression yields isolated cliques in degree-extremal shadows","Degree-extremal hypergraphs isolate cliques past better quadratic bound"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The whole argument rests on the claim that this shifting-plus-antilexicographic-compression map never lowers minimum degree or enlarges the shadow while it isolates a clique, for every regime t ≥ k−1 and k > ℓ ≥ 2.","fun_headline_variants_meta":{"raw":{"variants":["Min-degree extremal hypergraphs isolate a clique past quadratic threshold","Improved size forces isolated cliques in min-degree extremal hypergraphs","Extremal k-graphs isolate K_{t+1}^k under degree condition for large n","Shifting-compression yields isolated cliques in degree-extremal shadows","Degree-extremal hypergraphs isolate cliques past better quadratic bound"]},"model":"grok-4.5","effort":"low","cost_usd":0.006802,"raw_usage":{"total_tokens":1893,"prompt_tokens":1043,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":68020000,"prompt_tokens_details":{"text_tokens":1043,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":768,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1043,"tokens_out":82,"duration_ms":80571,"temperature":1.0,"reasoning_tokens":768,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T17:43:16.383712+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single triple (k,ℓ,t) with t ≥ k−1 and ground-set size just above the stated threshold for which every minimum-shadow hypergraph of minimum degree binom(t,k−1) has no isolated K^k_{t+1}, or show that the transformation itself decreases degree or increases shadow size on that instance.","supporting_citations":[],"review_version":2}