{"id":"60767384-b48d-4de0-89c6-3f1402009ac2","arxiv_id":"2507.00812","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The ℓ2-norm Turán density of the tight cycle minus one edge C_ℓ^{3-} is exactly 1/26 for every ℓ ≥ 5 with ℓ not divisible by 3, with a stability theorem.","lead":"The paper determines the exact largest squared-overlap density of hypergraphs that avoid a family of tight cycles minus one edge, and it shows near-extremal examples are recursive three-partite blow-ups. It settles an open conjecture from 2022 and extends the result to all admissible cycle lengths.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted proof of Proposition 3.2 is load-bearing: the crucial 0.75 constant is asserted without a certificate, and the main proof depends on it through limit constraint (9).","rationale":"The reader's conditional verdict and weakest-assumption analysis already identify Proposition 3.2 and Proposition 3.3 as the key unverified inputs. My stress test agrees, and narrows the most load-bearing point to Proposition 3.2: its strengthened constant 0.75 is explicitly called crucial, yet no proof or certificate is supplied. The proof of Proposition 3.3, even with its external certificate, relies on inequality (9) that would be invalidated if the 0.75 bound fails. This is not a mere stylistic omission: flag-algebra inequalities with different constants require different SDPs, and the paper provides no independent evidence that the improvement from 0.99 to 0.75 is true. The rest of the paper—Fact 2.2, Lemma 4.1, Lemma 5.1, and the induction in Section 5—is conditional on Propositions 3.1–3.3; if Proposition 3.2 is removed or weakened, the stability proof and the density upper bound collapse. Since the concern is about missing proof/certificate rather than an identified error, conditional acceptance remains the appropriate verdict, contingent on supplying the certificate or a complete proof. The numerical typo '0.918' in Lemma 5.1 further indicates that careful verification of constants is needed, but it does not by itself undermine the theorem. Overall, the concern does not change the reader's conditional verdict.","tokens_in":15646,"tokens_out":4539,"duration_ms":47149,"concrete_test":"Obtain or recompute a verifiable flag-algebra certificate for Proposition 3.2 proving |B| − c|M| ≤ 0 for some c ≤ 3/4 under the stated hypotheses, in the same format as the Proposition 3.3 certificate. As a sensitivity check, re-run the flag algebra program with the coefficient 3/4 replaced by 3/4 + 10^{-6}; if the SDP becomes infeasible or the output changes sign, then the exact constant is decisive. Independently verify that limit constraint (9) is a valid consequence of Proposition 3.2 before accepting Proposition 3.3's SDP result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central chain reduces Theorem 1.1 to Theorem 2.1, whose proof invokes Propositions 3.1–3.3. Proposition 3.2 is the least secure pillar: it asserts |B| − (3/4)|M| ≤ 0 for every locally maximal partition with max-cut ratio ≥ 0.198, strengthening [BLLP24, Prop. 3.3] from 0.99 to 0.75, and the proof is omitted because 'the proofs are the same'. The text itself emphasizes that the constant 0.75 is crucial. This inequality enters the proof of Proposition 3.3 as the limit constraint (9): Σ_i(φ(E_{i,i,i+1}) + φ(E_{i,i,i+2})) − (3/4)φ(E_{1,2,3}) ≤ 0. If the true optimal constant were larger, say 0.8, then the 19-hour SDP run in Proposition 3.3 would be solving a different feasibility problem, and the claimed contradiction to (14) would not follow. No flag-algebra certificate for Proposition 3.2 is included or linked; the externally provided certificate for Proposition 3.3 cannot compensate because it assumes (9). Thus a small, unproved numerical strengthening is a load-bearing pillar of Theorem 1.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the ℓ2-norm Turán density of the 3-uniform tight cycle minus one edge, C_ℓ^{3-}, for ℓ ≥ 5 with ℓ not divisible by 3. The main theorem, Theorem 1.1, asserts that π_{ℓ2}(C_ℓ^{3-}) = 1/26, together with an Erdős–Simonovits-type stability statement: every n-vertex C_ℓ^{3-}-free 3-graph with ℓ2-norm close to the extremal value is, after removing o(n^3) edges, a subgraph of the recursive three-partite Trec construction. The proof first reduces the ℓ2-norm problem to an S2-density problem for the family {K_4^{3-}, C_5^{3-}} via equation (1) and homomorphism arguments. Theorem 2.1, the core density and stability statement, is then proved by flag algebra: Propositions 3.1–3.3 provide a partition with a local maximality condition, a comparison of bad and missing edges with constant 3/4, and a comparison of bad and missing S2 copies with constant 9/10. Lemma 4.1 turns these into a recursive inequality, Fact 2.2 yields the density bound 6/13 for S2, and Section 5 turns the same ingredients into a stability proof. The lower bound comes from the independent Trec construction, whose ℓ2-norm is asymptotically n^4/26.","tokens_in":16008,"tokens_out":7426,"duration_ms":81524,"significance":"If the result is correct, it resolves the Balogh–Clemen–Lidický conjecture on π_{ℓ2}(C_5^{3-}) and extends it to all ℓ not divisible by 3, adding a strong stability theorem in the ℓ2-norm. The reduction from ∥·∥2 to S2-density is clean and elementary, and the recursive inequality in Lemma 4.1 is elegant. The paper also demonstrates that flag algebra methods can handle generalized Turán densities and stability in the ℓ2 setting, which is a valuable methodological contribution. However, the central proof depends on three computer-generated or externally cited propositions, and two of those (Propositions 3.1 and 3.2) are stated without proof, while the third (Proposition 3.3) relies on a 19-hour SDP whose certificate is only placed on an external Drive link. These gaps prevent the proof from being independently verifiable in its current form, and they are load-bearing rather than cosmetic.","major_comments":[{"comment":"Proposition 3.2 is stated without proof, with the text saying only that 'the proofs are the same' as in [BLLP24, Proposition 3.3]. This is a load-bearing omission: the constant 3/4 enters the proof of Proposition 3.3 through the limit constraint (9), and if the true optimal constant were larger, the feasibility problem solved by the SDP would be different and the contradiction to (14) would not follow. The strengthened constant 0.75 is not a routine restatement of the earlier 0.99 result; the manuscript explicitly says the constant is crucial. The same concern applies to Proposition 3.1, whose proof is also omitted. The authors should either include full proofs adapted to the new hypotheses (S2-density assumption, local maximality, and the threshold 0.198) or provide a machine-checkable certificate for the strengthened inequality.","section":"Section 3, Proposition 3.2"},{"comment":"The proof of Proposition 3.3 is a computer-assisted flag algebra calculation with |F_0^6| = 16181 constraints and a reported runtime of about 19 hours, but the certificate is available only at an external Google Drive link. For a computer-assisted proof, the sum-of-squares decomposition or an equivalent certificate must be permanently archived with the paper or in a stable repository, so that the implication from constraints (8)–(13) to the non-positivity of the expression in (14) can be checked independently. As written, the main theorem relies on an unverifiable external artifact.","section":"Section 3, Proposition 3.3"},{"comment":"In the last paragraph of the proof of Lemma 5.1, the text states: 'It follows from (21) and Proposition 3.1 that µH(V1,V2,V3) ≥ 0.918.' This is impossible, since the max-cut ratio µ is at most 2/9 ≈ 0.222 for any 3-graph. The intended value is almost certainly the lower bound 0.198 provided by Proposition 3.1, which is exactly what is needed to apply Proposition 3.2. The displayed constant 0.918 must be corrected; as written, this step is false and obscures the logical dependence of the stability proof on Proposition 3.2.","section":"Section 5, proof of Lemma 5.1"}],"minor_comments":[{"comment":"The proof of Proposition 3.3 refers to Figure 1 and Figure 2 to catalogue the types of missing and bad S2 copies, but these figures are not included in the arXiv text. The classification is used in the definitions of Sb and Sm in (14), so the figures should be included in the paper.","section":"Section 3, Figures 1 and 2"},{"comment":"The sentence 'The results returned by the computer for the calculations for φ(Sb) − (9/10)φ(Sm) is indeed 0' is imprecise: the SDP presumably shows that the left-hand side of (14) is at most 0 under the constraints, not that it equals 0 as a value. Please state the exact output and the rounding/feasibility tolerances used.","section":"Section 3, Proposition 3.3 proof"},{"comment":"The ancillary result on π_{ℓ2}(F_{3,2}) is presented with a certificate only on another external Drive link. If this result is part of the paper, it should also be accompanied by a permanent, verifiable certificate or by a proof; otherwise it should be moved to a clearly separate remark.","section":"Section 6, Theorem 6.2"},{"comment":"The paper cites [BLLP24] and [LMP24] as arXiv preprints. If either has been accepted or published in the meantime, the references should be updated.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The core mathematical strategy appears sound and the reduction to S2-density is convincing. The main obstacle to publication is the reliance on omitted proofs of Propositions 3.1 and 3.2 and the external, unauditable certificate for Proposition 3.3. This is a publishability issue rather than a mere exposition issue: a referee cannot verify the central claim without seeing the missing material. The 0.918 typo in Lemma 5.1 should also be corrected. If the authors supply the missing proofs and a proper certificate archive, the paper would be a strong contribution to hypergraph Turán problems in ℓp-norms."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the main result is real: Theorem 1.1 gives π_ℓ2(C_ℓ^{3-}) = 1/26 for every ℓ ≥ 5 with ℓ not divisible by 3, plus a structural stability theorem. That goes beyond the Balogh–Clemen–Lidický conjecture, which was only for ℓ = 5, and the recursive Trec construction is the obvious extremal candidate. Second, the proof is not self-contained. It rests on Propositions 3.1 and 3.2, which are stated and then dismissed with “the proofs are essentially the same” as in [BLLP24]. That is a problem, because Proposition 3.2 improves the key constant from 0.99 to 0.75, and the improved constant is load-bearing: it enters Proposition 3.3 as the limit constraint (9). Without a proof of 3.2, the 19-hour SDP in 3.3 is solving a different feasibility problem.\n\nWhat the paper does well: the reduction from ℓ2-norm to S2-density is clean and standard, and the recursive argument in Sections 4–5 is actually elegant. If you grant the three propositions, the rest of the proof follows by elementary inequalities (Fact 2.2) and an induction that looks correct. The stability theorem is genuinely new, and the authors are honest that the computer certificates live on an external Drive link.\n\nThe soft spots are exactly where the stress-test says they are. Proposition 3.2 is the weakest pillar. The proof is omitted, and the constant 0.75 is not an inessential detail; you cannot tell from the manuscript whether the improvement from 0.99 to 0.75 is a routine tightening or requires new ideas. The same goes for Proposition 3.1, though that one is less alarming because the max-cut threshold 0.198... is already in the statement. There is also a numerical typo in Lemma 5.1's proof: “µ ≥ 0.918” should be “µ ≥ 0.198”. Minor.\n\nThis is a paper for specialists in hypergraph Turán theory and flag algebra. I would send it to a serious referee, but I would insist on a full proof of 3.2 (and 3.1) and a packaged certificate for 3.3 before accepting. The central argument holds up conditional on those pieces, and the result is significant enough to merit the effort. I would not want to see this desk-rejected; I would want it revised.\n\nRecommendation: accept with major revision, pending external verification of the omitted propositions.","headline":"The conjectured 1/26 density is settled in a stronger form, but the proof's load-bearing Proposition 3.2 is omitted, so the result is conditional on unverified strengthening.","tokens_in":16504,"tokens_out":3801,"would_cite":true,"duration_ms":44536,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the $\\ell_2$-norm Tur\\'an density of $C_\\ell^{3-}$ is $1/26$ for every $\\ell \\ge 5$ with $\\ell$ not divisible by $3$, together with an Erd\\H{o}s--Simonovits-type stability theorem.","keywords":["Turán density","ℓ2-norm","tight cycle","3-uniform hypergraph","flag algebra","codegree","stability","Trec construction"],"falsifier":"Rerun the flag algebra calculation behind Proposition 3.3 and check that the $6$-vertex SDP really proves the claimed inequality between bad and missing $S_2$ copies under the stated constraints; alternatively, exhibit any $\\{K_4^{3-},C_5^{3-}\\}$-free $3$-graph family with $S_2$-density exceeding $6/13+o(1)$.","tokens_in":15495,"feed_emoji":"🔺","tokens_out":12550,"duration_ms":133830,"temperature":0.7,"pith_summary":"The paper proves that the $\\ell_2$-norm Tur\\'an density of the $3$-uniform tight $\\ell$-cycle minus one edge is $1/26$ for every $\\ell \\ge 5$ that is not divisible by $3$. In plain terms, among $n$-vertex hypergraphs that avoid $C_\\ell^{3-}$, the sum of squared codegrees is at most $(1/26+o(1))n^4$, and this bound is achieved by the Trec construction, the iterated three-partite blow-up of a single triple. The paper also proves a stability theorem: any near-extremal hypergraph can be turned into a subconstruction of Trec by deleting $o(n^3)$ edges. This confirms, in a stronger form, a conjecture raised in a 2022 survey on $\\ell_2$-norm Tur\\'an problems. The interest is that a norm sensitive to codegree distribution, not just edge count, still selects the same recursive structure as the extremal family.","feed_headline":"In the codegree norm, tight cycles minus an edge have density 1/26","feed_subtitle":"Confirms a 2022 conjecture with a stability theorem: near-extremal hypergraphs are recursive 3-partite blow-ups.","key_machinery":"The load-bearing identity is $N(S_2,H)=\\sum_{e\\in\\partial H}\\binom{d_H(e)}{2}=(\\|H\\|_2-3|H|)/2$, which converts the codegree-squared norm into a count of $S_2$ copies. The proof then works on a locally maximal $3$-partition $V_1\\cup V_2\\cup V_3$ of a $\\{K_4^{3-},C_5^{3-}\\}$-free hypergraph, comparing bad edges $B$ (edges meeting the parts in pattern $\\{0,1,2\\}$) with missing transversal triples $M$ via $|B|\\le \\tfrac34|M|$, and bad $S_2$ copies $B_{S_2}$ with missing $S_2$ copies $M_{S_2}$ via $|B_{S_2}|\\le \\tfrac9{10}|M_{S_2}|+o(n^4)$. The latter bound comes from a flag algebra semi-definite program over $6$-vertex flags with $16{,}181$ linear constraints. These comparisons feed Lemma 4.1, a recursive inequality $N(S_2,H)\\le |V_1||V_2||V_3|n/2+\\sum_i N(S_2,H[V_i])+\\varepsilon n^4-\\max\\{|B_{S_2}|/9,|M_{S_2}|/10\\}$, whose iteration together with the elementary inequality $x_1x_2x_3/(1-\\sum_i x_i^4)\\le 1/26$ on the simplex yields the value $6/13$ and the stability theorem.","core_discovery":"On the paper's own terms, the central discovery is that the $\\ell_2$-norm Tur\\'an problem for $C_\\ell^{3-}$ collapses onto a single, recursively self-similar extremal family. The equivalent object studied is the density of $S_2$, the $3$-graph on four vertices with exactly two edges, in $\\{K_4^{3-},C_5^{3-}\\}$-free $3$-graphs: the maximum $S_2$-density is exactly $6/13$, which corresponds to $\\ell_2$-norm density $1/26$. Near the maximum, a locally maximal $3$-partition exists in which the number of $S_2$ copies that involve a bad edge across parts is at most $9/10$ of the number of $S_2$ copies present in the complete $3$-partite graph but missing from $H$, up to $o(n^4)$; this comparison makes the recursive argument close. Iterating over the three parts produces both the upper bound and the stability statement: the extremal hypergraphs are exactly the Trec-subconstructions, up to $o(n^3)$ edge deletions.","pith_inferences":["The reduction from arbitrary $\\ell$ to the two fixed forbidden hypergraphs $K_4^{3-}$ and $C_5^{3-}$ is homomorphism-based, so the same $6/13$ bound might transfer to other families that degenerate to $C_5^{3-}$, provided the two fixed-hypergraph bounds continue to hold.","The stability theorem is strong enough to suggest that the exact $\\ell_2$-norm extremal number $\\mathrm{ex}_{\\ell_2}(n,C_\\ell^{3-})$ may be determined for large $n$ if the Trec construction satisfies a vertex-extendability condition of the kind used in other generalized Tur\\'an problems; the paper does not establish this.","The main theorem currently rests on a long computer calculation whose certificate is stored outside the paper, so independent verification of that calculation, or a human-readable proof of the $S_2$-comparison, would make the $1/26$ value fully reproducible.","The inequality $1/26>1/27$ shows that the recursive iteration is not cosmetic: in the $\\ell_2$ norm, the best single-level $3$-partition is strictly suboptimal, and the gain comes from reusing the same $1/26$ factor inside the parts."],"forward_implications":["For every integer $\\ell\\ge 5$ with $\\ell\\not\\equiv 0\\pmod 3$, the $\\ell_2$-norm Tur\\'an density of $C_\\ell^{3-}$ is exactly $1/26$.","Near-extremality in the codegree-squared norm forces structure: deleting $o(n^3)$ edges from a hypergraph with norm at least $(1/26-\\delta)n^4$ leaves a Trec-subconstruction.","The equivalent generalized Tur\\'an statement holds: the maximum density of the two-edge $3$-graph $S_2$ in $\\{K_4^{3-},C_5^{3-}\\}$-free $3$-graphs is $6/13$.","Since $1/26$ exceeds the $1/27$ norm of the balanced complete $3$-partite hypergraph, the extremal construction must be genuinely recursive rather than a single-level partition.","The same computer-assisted framework also yields the exact value $\\pi_{\\ell_2}(F_{3,2})=1/8$ for a different $3$-graph whose ordinary Tur\\'an density was previously known."],"supporting_citations":[{"why":"Posed the conjecture on $\\pi_{\\ell_2}(C_5^{3-})$ and initiated the systematic study of hypergraph Tur\\'an problems in $\\ell_p$-norms; the present result confirms Conjecture 3.5 in stronger form.","marker":"[BCL22a]"},{"why":"Previously determined the Tur\\'an density of $C_5^{3-}$ and supplied the strategy and Proposition 3.2-style comparison that the present proof adapts.","marker":"[BLLP24]"},{"why":"Independently determined $\\pi(C_\\ell^{3-})=1/4$; together with [BLLP24] this provides Theorem 2.3, the edge-density bound used as a constraint in the flag algebra calculations.","marker":"[LMP24]"},{"why":"Introduced the flag algebra method used to prove the $S_2$-comparison in Proposition 3.3.","marker":"[Raz07]"},{"why":"Supplies the homomorphism from $C_\\ell^{3-}$ to $C_5^{3-}$ that reduces the problem for every allowed $\\ell$ to the fixed family $\\{K_4^{3-},C_5^{3-}\\}$.","marker":"[BL24]"},{"why":"Introduced the Trec construction, which gives the lower bound $1/26$ and is the target object in the stability statement.","marker":"[MPS11]"},{"why":"Provided the recursive-blowup strategy for maximizing induced density of the $5$-cycle that the proof explicitly follows.","marker":"[BHLP16]"}],"fun_headline_variants":["Codegree Turán density for tight cycles minus an edge: 1/26","Tight cycles minus an edge: codegree density exactly 1/26","Conjecture confirmed: tight cycles minus an edge have ℓ2 Turán density 1/26","Recursive blow-ups give exact ℓ2 density for tight cycles minus an edge: 1/26"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on two computer-assisted bounds, one comparing bad and missing edges with constant $3/4$ and one comparing bad and missing $S_2$ copies with constant $9/10$, whose proofs are not fully present in the paper; if either bound is wrong, the main theorem has no support.","fun_headline_variants_meta":{"raw":{"variants":["Codegree Turán density for tight cycles minus an edge: 1/26","Tight cycles minus an edge: codegree density exactly 1/26","Conjecture confirmed: tight cycles minus an edge have ℓ2 Turán density 1/26","Recursive blow-ups give exact ℓ2 density for tight cycles minus an edge: 1/26"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001327,"raw_usage":{"total_tokens":5398,"prompt_tokens":943,"completion_tokens":4455,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":4359}},"tokens_in":559,"tokens_out":4455,"duration_ms":38225,"temperature":1.0,"reasoning_tokens":4359,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:07:05.173131+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Rerun the flag algebra calculation behind Proposition 3.3 and check that the $6$-vertex SDP really proves the claimed inequality between bad and missing $S_2$ copies under the stated constraints; alternatively, exhibit any $\\{K_4^{3-},C_5^{3-}\\}$-free $3$-graph family with $S_2$-density exceeding $6/13+o(1)$.","supporting_citations":[],"review_version":1}