{"id":"1cc77a97-9f0d-49d5-b138-98c814960825","arxiv_id":"2511.07715","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Using pattern blowups, the paper proves 64/81 and an infinite parametric family are non-jumping 3-uniform hypergraph densities.","lead":"This paper adds new examples of \"non-jump\" densities for 3-uniform hypergraphs, numbers just above which no Turán density can land. It uses a pattern-based refinement of the Frankl–Rödl method, though a key proof step appears incomplete.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.8's c1>1/2 case contradicts the derived sign of dk'/dc1, so the key step b'=0 is unproved; both main theorems rest on this lemma.","rationale":"The reader's weakest_assumption precisely identifies the point where the proof of Theorem 4.8 reverses the sign of a derivative it has just derived. This is the load-bearing step because the paper's two main non-jump theorems are direct applications of Theorem 4.8 together with Theorem 4.2. If b'=0 is not proven, then λ(FR_1(P))=λ(P) is not established, and the sufficient condition for non-jump is not met. The arithmetic slip and reliance on an unpublished theorem are secondary; the derivative contradiction is fatal to the proof as written. A numerical evaluation of the derivative at a admissible point confirms the sign is negative, so the error is not a mere typo in prose but a genuine logical gap. The paper is readable and the framework is promising, but the central proof currently does not close.","tokens_in":8158,"tokens_out":9299,"duration_ms":80825,"concrete_test":"Evaluate the displayed formula for dk'/dc1 at a point with c1>1/2, e.g., a'=b'=0.1, k'=0.8, c1=0.6: numerator=0.8>0, g≈-1.865<0, so dk'/dc1≈-0.429<0. Thus k' decreases when c1 increases, the opposite of the claim used for c1>1/2. Recomputing at the optimal weights for the pattern of Theorem 1.5 will confirm the sign; if it is negative, the proof's c1>1/2 branch collapses and b'=0 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.8 aims to show that an optimal weighting of FR_1(P) has b'=0, which forces λ(FR_1(P))=λ(P). After deriving dk'/dc1 = ((a'+b')k'+k'^2)/g with g<0, it correctly concludes k' increases as c1 decreases. But then it claims: 'If c1 > 1/2, then we also get k' > 1/2, since k increases when c1 increases and c2 is non-decreasing.' This directly contradicts the just-derived sign: since the numerator is positive (all weights nonnegative and a',b',k'>0 under the assumption) and g<0, dk'/dc1<0, so k' strictly decreases as c1 increases. Therefore for c1>1/2 one gets k' < k'(1/2)=1/2, not >1/2. The contradiction a'>k' (from b+2(k−a)=0) is thus not established in the c1>1/2 case. Without b'=0, Theorem 4.8 does not show λ(FR_1(P))=λ(P), and Theorem 4.2 cannot be applied. Both Theorem 1.5 and Theorem 1.6 depend on this step. The analogous flaw is not remedied elsewhere in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a method for producing non-jump densities for 3-uniform hypergraphs using the Lagrangian of patterns and the Frankl–Rödl construction. The main results are Theorem 1.5, asserting that 64/81 is not a jump for r=3, and Theorem 1.6, asserting that for every n≥1, with k=√(3n−2), the density 1−(3n^2−2n+k^3)/(n+k)^3 is not a jump for r=3. The argument is organized around a key lemma, Theorem 4.8, which is supposed to show that for a 3-pattern P containing the edges {122} and {11i}, any optimal weighting of FR_1(P) satisfies b′=0, so that λ(FR_1(P))=λ(P). The paper then computes the Lagrangians of the specific patterns used in Theorems 1.5 and 1.6.","tokens_in":8412,"tokens_out":24067,"duration_ms":213039,"significance":"If the main theorems were established, they would provide new non-jump densities for 3-uniform hypergraphs and would illustrate a pattern-based method that could be useful for further constructions. The paper is clearly written and the Lagrangian computations for the examples are mostly explicit. However, the central lemma Theorem 4.8 has a serious gap in the c1>1/2 case, and both main theorems depend on that lemma. The paper also relies on Shaw's unpublished Theorem 4.2. In its present form, the correctness of the main results is not established, although the underlying approach remains promising.","major_comments":[{"comment":"The proof derives dk′/dc1 = ((a′+b′)k′ + k′^2)/g with g<0, and the numerator is positive under the standing assumptions. Hence dk′/dc1<0: as c1 increases, the maximizing k′ decreases. The text then states the opposite: 'If c1 > 1/2, then we also get k′ > 1/2, since k increases when c1 increases'. This directly contradicts the derived sign. Therefore the conclusion k′>1/2 is not established in this case, and the contradiction a′>k′ is not obtained. Since b′=0 is forced only through that contradiction, Theorem 4.8 is unproved. Theorems 1.5 and 1.6 both invoke Theorem 4.8, so the main results are not supported as written.","section":"§4, Theorem 4.8 (c1>1/2 case)"},{"comment":"Even apart from the sign error, the claim 'If c1 or c2 increases, then the k′ that maximizes w′(FR1(P)) must not decrease' is not derived. The implicit differentiation is performed only along the curve where c2 is at its lower bound, while at an actual optimum c2 is not constrained to equal that bound. A rigorous proof would need to control the coupled variation of c1 and c2, or derive the desired inequality directly from the Lagrange equations without this heuristic monotonicity step.","section":"§4, Theorem 4.8 (monotonicity step)"},{"comment":"The definition of P as having edge set [n+1]^(3) ∪ {1,2,2} ∪ {1,3,3} ∪ ... ∪ {1,n+1,n+1} is inconsistent with the calculation that follows. The formula for w(P) contains the term w1^2 w2/2, which arises from edges of the form {1,1,i}. Those edges are not present in the stated edge set if [n+1]^(3) denotes ordinary 3-subsets. As written, the proof computes the Lagrangian of a different pattern from the one defined. This must be corrected for Theorem 1.6 to be proved.","section":"§6, Theorem 1.6 (definition of P)"},{"comment":"The proof claims that for equivalent vertices i and j, w(P) can be written as (wi+wj)C1 + (wi^2+wj^2)C2 + wiwj C3, with C1,C2,C3 depending on the other weights. This expression omits all monomials of total degree 3, such as wi^3, wj^3, wi^2 wj and wiwj^2, which can occur in a 3-pattern. The statement of the lemma may be salvageable by a symmetrization argument, but the proof as written is incomplete. Since Lemma 4.3 is used to justify the optimal-weighting reductions in Sections 4–6, the gap should be addressed.","section":"§3, Lemma 4.3"}],"minor_comments":[{"comment":"Typo: 'corallaries' should be 'corollaries'.","section":"Abstract/Introduction"},{"comment":"Typo: 'Frank-Röd l' should be 'Frankl–Rödl'.","section":"§4 header"},{"comment":"The notation [n+1]^(3) should be defined explicitly, since the proof depends on whether it means ordinary 3-subsets or something else.","section":"§6, Theorem 1.6"},{"comment":"The equations labelled (4.5)–(4.7) would be clearer if they were consistently numbered and referenced; the current formatting is a minor readability issue.","section":"§4, Lemma 4.4"}],"recommendation":"major_revision","confidential_remarks":"The paper's main results depend on Shaw's unpublished Theorem 4.2, which is cited without proof. Combined with the gap in Theorem 4.8, this makes the current version unsuitable for publication without substantial revision. The pattern-based approach is interesting and the explicit computations are mostly sound, so I would encourage the author to repair the proof of Theorem 4.8 and clarify the definition in Theorem 1.6. I do not see evidence of circularity or parameter fitting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends Shaw's pattern-based approach to produce new non-jump densities for 3-graphs, and the headline examples (64/81 and an infinite family) are genuinely new if they hold. The pattern machinery is clean, the exposition is readable, and the Lagrangian computations for the examples check out. This is a real extension of the Frankl–Rödl/Shaw method, not a repackaging.\n\nThe problem is Theorem 4.8, the central lemma on which both Theorem 1.5 and Theorem 1.6 depend. The proof derives a derivative dk'/dc1 < 0 (positive numerator, negative denominator), then in the c1 > 1/2 case claims k' > 1/2 “since k increases when c1 increases.” That is exactly backwards: the derived sign says k' decreases as c1 increases, so for c1 > 1/2 you get k' < 1/2. That does not contradict a' > k', so the conclusion b' = 0 is not established. Without b' = 0, the lemma's claim λ(FR1(P)) = λ(P) does not follow, and Theorem 4.2 cannot be applied. The main results are unproven as written.\n\nSmaller issues: Theorem 4.2 is cited from Shaw's unpublished work, so the external dependency should be flagged. Lemma 4.3 also has a slightly hand-wavy step about boundary cases. But these are minor next to the sign error.\n\nIf the sign error is fixable — perhaps the author intended a different argument for c1 > 1/2 — the approach could be worth reviving. As it stands, an expert referee would need to see that case redone. I would send it to review to let someone assess whether the proof can be patched, but I would not accept it in this form.","headline":"Promising new non-jump candidates via patterns, but a sign error in Theorem 4.8's c1 > 1/2 case leaves both main theorems unproven.","tokens_in":8860,"tokens_out":6937,"would_cite":false,"duration_ms":61312,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that 64/81 and a parametric family of densities are non-jumps for 3-uniform hypergraphs.","keywords":["hypergraph","limiting density","jump","non-jump","Lagrangian","pattern","blowup","extremal combinatorics"],"falsifier":"Numerically maximize the weight polynomial for FR_1(P) for the five-edge pattern P={123,122,112,113,223} over the simplex of vertex weights. If the maximum exceeds 32/243, then λ(FR_1(P)) > λ(P) and the proof of Theorem 1.5 fails; finding the maximum on the boundary with b=0 would support the paper's claim.","tokens_in":8009,"feed_emoji":"🧮","tokens_out":14229,"duration_ms":114313,"temperature":0.7,"pith_summary":"This paper addresses the long-standing conjecture that every density in [0,1) is a jump for uniform hypergraphs — that is, that limiting edge densities cannot pile up arbitrarily close above any given density. A classical construction produced the first counterexamples, but the set of non-jumps remains poorly understood for r≥3. The paper develops a pattern-based sufficient condition for a density to be a non-jump for 3-uniform hypergraphs, following a recent refinement of the standard method. Using it, the author proves that 64/81 is not a jump, and that for every positive integer n the density 1 − (3n² − 2n + k³)/(n+k)³, with k=√(3n−2), is not a jump. If the proofs are correct, these are new additions to the short list of known non-jumps.","feed_headline":"Pattern method yields 64/81 and an infinite family of non-jumps","feed_subtitle":"A pattern-based proof finds 64/81 and infinitely many densities with no density gap.","key_machinery":"The load-bearing object is the Lagrangian of an r-pattern, where a pattern is a hypergraph whose r-edges are multisets rather than sets. For a pattern P, its Lagrangian λ(P) is the maximum, over probability distributions on vertices, of the sum over edges of products of vertex weights (with multiplicities). The construction FR_v(P) takes a blowup of P in which one vertex is replaced by r copies and then adds a single edge on those copies; a known theorem says that if λ(FR_v(P)) equals λ(P) < 1 and v has positive weight in an optimal weighting of P, then r!·λ(P) is a non-jump. The paper's Theorem 4.8 identifies a family of 3-patterns for which this equality can be shown by a Lagrange-multipli","core_discovery":"The central claim is a template theorem: for any 3-pattern P that contains the edges {122} and {11i} for every other vertex i, if vertex 1 receives positive weight in an optimal weighting of P and the Lagrangian λ(P) is less than 1, then the density 3!·λ(P) is not a jump. The proof works by showing that the Lagrangian of the auxiliary object FR_1(P) constructed from P equals λ(P), which is precisely the sufficient condition for a non-jump. The author then applies the template to two patterns: a five-edge pattern with λ=32/243, giving the non-jump 64/81, and a family of patterns whose Lagrangians evaluate to the formula in Theorem 1.6. Thus the paper's discovery is a new method for producing","pith_inferences":["One could systematically enumerate 3-patterns satisfying the template and compute their Lagrangians numerically; this would likely reveal many more non-jumps and possibly a complete description of non-jumps near 1.","The template may be adaptable to r-uniform hypergraphs for r>3 by replacing the two distinguished edge types with their r-multiset analogues, though the proof as written is specific to r=3.","A direct numerical optimization of the Lagrangian of FR_1(P) for the five-edge pattern could independently confirm the equality λ(FR_1(P))=32/243; if it holds, the non-jump result is robust even if the proof's delicate step is later revised.","The infinite family suggests non-jumps accumulate at 1, raising the question of whether the set of non-jumps for 3-uniform hypergraphs is dense in some interval near 1."],"forward_implications":["64/81 is now shown to be a non-jump, meaning there are families of 3-uniform hypergraphs with limiting densities strictly above 64/81 but arbitrarily close to it.","The family in Theorem 1.6 gives infinitely many non-jumps, with densities approaching 1 as n grows, so the set of non-jumps has a limit point at 1.","The template theorem means any 3-pattern containing the required edges and having Lagrangian below 1 automatically yields a non-jump, so the method converts a finite algebraic check into a new non-jump.","The exact Lagrangian computations pin down precise algebraic non-jumps rather than mere existence statements."],"fun_headline_variants":["Pattern weights reveal 64/81 and infinitely many non-jumps","Infinite non-jump family from pattern Lagrangian","New template gives 64/81 and infinite non-jumps","Pattern method proves infinite non-jump densities","64/81 and infinite non-jumps from pattern weights"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The main theorem relies on the assertion that in any optimal weighting of the auxiliary object FR_1(P), the total weight on the original pattern's vertices is always greater than 1/2; the proof's handling of one case appears to contradict an earlier sign computation, so this load-bearing claim is not fully established.","fun_headline_variants_meta":{"raw":{"variants":["Pattern weights reveal 64/81 and infinitely many non-jumps","Infinite non-jump family from pattern Lagrangian","New template gives 64/81 and infinite non-jumps","Pattern method proves infinite non-jump densities","64/81 and infinite non-jumps from pattern weights"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000811,"raw_usage":{"total_tokens":3358,"prompt_tokens":671,"completion_tokens":2687,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":415,"completion_tokens_details":{"reasoning_tokens":2608}},"tokens_in":415,"tokens_out":2687,"duration_ms":19165,"temperature":1.0,"reasoning_tokens":2608,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T23:03:09.460140+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically maximize the weight polynomial for FR_1(P) for the five-edge pattern P={123,122,112,113,223} over the simplex of vertex weights. If the maximum exceeds 32/243, then λ(FR_1(P)) > λ(P) and the proof of Theorem 1.5 fails; finding the maximum on the boundary with b=0 would support the paper's claim.","supporting_citations":[],"review_version":1}