{"id":"70a1e14f-90b2-465d-8d5e-df8db01e1f47","arxiv_id":"2509.24354","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Pattern-colorable spectral extremal hypergraphs are asymptotically regular, yielding a reduction theorem that determines the alpha-spectral radius and edge extremal hypergraphs for expansions of color-critical graphs.","lead":"This paper proves a regularity property for hypergraphs that maximize a generalized spectral radius under coloring constraints, and uses it to solve a class of spectral extremal problems. The result pinpoints the extremal hypergraphs that avoid a family of forbidden patterns, including r-expansions of color-critical graphs.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5's edge-extremal equality is a non sequitur: taking α→∞ in Theorem 5.1 proves only the upper bound, not the 'Moreover' characterization, so Corollaries 5.3–5.4 are unsupported.","rationale":"I read the full manuscript. The main spectral reduction (Theorem 4.2) is internally plausible; the induction in its proof appears sound once Theorem 3.10 is granted. The regularity theorem 3.10 relies on Cooper–Desai–Sahay's principal-ratio bound; an initial worry about Col(P) being clonal evaporates, since for G_{u→v} one can give u the color of v, so this import is consistent. The degree-stability input from [12] remains an external dependency, exactly as the reader notes; I agree that Theorem 4.8 is conditional on it. However, the more pressing flaw I see is in Section 5: the passage from the α-spectral inequality to an exact edge-extremal characterization is a non sequitur. This is an internal proof gap, not an external dependency, and it affects the advertised edge-Turán uniqueness results. Therefore I keep the reader's CONDITIONAL verdict but for a slightly different reason; I would not reject the paper because the main spectral theorem may be correct and the edge result may be repairable.","tokens_in":22459,"tokens_out":28391,"duration_ms":206944,"concrete_test":"Analytic check: attempt to derive the missing equality in Theorem 5.1 by showing e(G) = ex(Col(P), n) forces λ^(α)(G) = λ^(α)(Col(P), n) for some α > 1. This is the only route from the paper's tools, and it is false in general: two graphs with equal edge counts can have distinct α-spectral radii for every finite α. Decisive special case: r = 2, F = K_{l+1}, P = K_l^2; run the proof of Theorem 5.1 without importing the classical Simonovits stability/symmetrization argument. If the 'Moreover' cannot be derived from the spectral inequality alone, the theorem as stated is unproved. Equivalently, check whether Corollary 5.4's uniqueness is independently established anywhere in §5; it is not.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised edge-Turán characterization is not proved. In Theorem 5.1 the proof establishes e(G) ≤ ex(Col(P), n) by letting α→∞ in λ^(α)(G) ≤ λ^(α)(Col(P), n). That limiting argument is valid for the inequality, but it cannot yield the stated equality case: if e(G) = ex(Col(P), n), the two sides approach the same limit r!·ex(Col(P), n), and equality of limits does not imply λ^(α)(G) = λ^(α)(Col(P), n) for any finite α. Theorem 4.2's equality clause requires exactly that finite-α spectral equality, so it cannot be invoked. Consequently Corollary 5.3 (\"equality holds iff G = T_l^r(n)\") and Corollary 5.4 do not follow from the material in the paper. A separate argument—for example, showing that an F-free graph with ex(Col(P), n) edges has minimum degree above the degree-stability threshold after deleting o(n) vertices, and then extending a P-coloring—is needed but absent. The asymptotic/upper-bound part of the spectral reduction is unaffected, but the paper's abstract promises \"every F-free edge extremal hypergraph must be P-colorable,\" which is exactly the unproved part.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a reduction framework for spectral Turán-type problems for the α-spectral radius of r-uniform hypergraphs. It defines r-patterns P and the family Col(P) of P-colorable hypergraphs, proves asymptotic regularity of spectral extremal hypergraphs in Col(P) (Theorem 3.10), and then proves that if a forbidden family F is degree-stable with respect to Col(P), every F-free r-graph G satisfies λ^(α)(G) ≤ λ^(α)(Col(P), n) (Theorem 4.2). The paper applies this to r-expansions of color-critical graphs (Theorem 4.8), using a degree-stability result imported from [12], and derives edge-Turán results by taking α → ∞ in Section 5. A further theorem (Theorem 3.12) shows that spectral extremal k-chromatic r-graphs are asymptotically balanced, giving partial information toward a conjecture of Kang–Nikiforov–Yuan.","tokens_in":22766,"tokens_out":11834,"duration_ms":86523,"significance":"If valid, the spectral reduction is genuinely useful: it converts spectral Turán problems for degree-stable forbidden families into problems about pattern-colorable families, and Theorem 3.10 provides quantitative regularity information (eigenvector lower bounds and minimum-degree lower bounds) for a broad class of hypergraphs. The paper also includes a self-contained appendix for some foundational lemmas and clearly identifies Col(P) as hereditary and multiplicative. However, the advertised equality characterizations — both in the spectral theorem and in the edge-Turán application — are not fully justified by the proofs as written, and the headline concrete application depends on an unproved external table from an arXiv preprint. These issues affect load-bearing claims, though they appear to be local and potentially repairable.","major_comments":[{"comment":"The limiting argument in the proof of Theorem 5.1 proves only the upper bound e(G) ≤ ex(Col(P), n). From r!e(G)=lim_{α→∞} λ^(α)(G) = lim_{α→∞} λ^(α)(Col(P), n) one cannot infer λ^(α)(G) = λ^(α)(Col(P), n) for any finite α, so the 'Moreover' clause cannot be obtained by invoking Theorem 4.2's equality case. Consequently the equality characterizations in Corollaries 5.3 and 5.4 are unsupported. A separate argument — for example, showing that an edge-extremal F-free graph has high minimum degree after deleting o(n) vertices, and then applying degree stability — is needed but absent.","section":"Theorem 5.1, Section 5"},{"comment":"The induction in the proof shows only that the particular chosen maximizer H_n is P-colorable for all n ≥ n1. It does not show that an arbitrary F-free G satisfying λ^(α)(G)=λ^(α)(Mon(F), n) is P-colorable. Thus the assertion 'equality holds only if G is P-colorable' is not established. This equality clause is used in Corollaries 4.7 and 4.8 via Lemma 4.6, and also in Corollary 4.4. To complete the proof one must prove that every spectral extremizer in Mon(F), not merely one member of a chosen sequence, has the eigenvector/min-degree properties; this is missing.","section":"Theorem 4.2"},{"comment":"The proof of Theorem 4.8 invokes 'As established in Table 1 of [12]' for the degree-stability of the r-expansion of an (l+1)-color critical graph with respect to Col(K_l^r). This is a load-bearing external input from an arXiv preprint and is not reproduced or proved in the manuscript. If that table is unavailable or incorrect, the applications Theorems 4.8 and Corollary 5.4 do not follow from the present work. The authors should state the exact stability theorem used and either prove it or make the citation self-contained.","section":"Theorem 4.8 / §4"}],"minor_comments":[{"comment":"The proof of Theorem 3.10(2) cites [4, Theorem 4.6] for the statement that any hereditary and multiplicative family is clonal. The theorem is not quoted. Please state the precise result from [4] or give a proof in the appendix.","section":"Lemma 3.6 / Theorem 3.10(2)"},{"comment":"'Principle eigenvector' should be 'principal eigenvector'.","section":"Corollary 4.4"},{"comment":"There are several reference typos: [30] lists pages '183-179' (should be '179-183'); [8] lists year '1996' (should be '1966'); [37] gives 'N. Sergey, Y. Liana' — the authors are S. Norin and L. Yepremyan.","section":"References"},{"comment":"The construction H_n∘k is introduced only inline and would benefit from a formal definition and a brief explanation of why it is edge-maximal and P-colorable.","section":"Theorem 3.10(1)"}],"recommendation":"major_revision","confidential_remarks":"The core spectral upper-bound reduction is interesting and appears largely sound, but the equality claims are overstated: Theorem 4.2 only proves that a chosen maximizer is P-colorable, and Section 5's edge-equality claim is a non sequitur from the limiting argument. The concrete application to color-critical expansions also depends on an external arXiv table ([12]) that should be verified by the editor or made self-contained. I recommend major revision rather than rejection, because the missing arguments appear to be local and likely repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things before you spend time on this one. First, the core idea is genuinely useful: Theorem 3.10 gives asymptotic regularity for spectral extremal hypergraphs in arbitrary pattern-colorable families, and the reduction in Theorem 4.2 — from degree-stable forbidden families to pattern-colorable spectral problems — is a clean template that goes beyond the earlier K^r_{l+1} and F_{r,l+1} cases. The analytic machinery (eigenvector lower bounds, the permanence inequality, the use of Cooper–Desai–Sahay) is competently executed, and the parts I checked internally are coherent. Second, the edge-extremal characterization in Section 5 is not proved. Taking α→∞ in Theorem 4.2 gives the upper bound e(G) ≤ ex(Col(P), n), but equality of limits does not imply equality of λ^(α) at any finite α, so the \"Moreover\" clause in Theorem 5.1, and with it Corollaries 5.3 and 5.4, are unsupported. The abstract promises that every edge-extremal hypergraph must be P-colorable; that is exactly the unproved part.\n\nThe main theorem also has a more subtle gap. In the proof of Theorem 4.2, the induction assumes λ^(α)(H_n) = λ^(α)(G_n) once H_n ∈ Col(P)_n, but the base case only establishes H_{n1} ∈ Col(P)_{n1}, not equality of the two spectral radii. Without an argument that the Col(P)-extremal G_{n1} is F-free (which is true in the applications but not stated in the theorem), the equality step does not follow. This may be patchable, but as written the proof is incomplete. I would also flag the heavy reliance on the external preprint [12] for the degree-stability of r-expansions of color-critical graphs; that is a legitimate dependence, but the paper imports it as fact, so the headline application is conditional on verification of that table.\n\nThe paper is written for people working on spectral Turán problems for hypergraphs. If the Section 5 claims are weakened to upper bounds, and the induction base in Theorem 4.2 is repaired, it is a solid contribution. As it stands, it deserves a serious referee — the tools are real and the reduction idea is worth engaging with — but it needs substantial revision before the advertised results are fully supported.","headline":"The reduction idea is good and the regularity theorem is a real step forward, but the edge-extremal equality claims don't follow from the α→∞ argument, and the main induction has a base-case gap.","tokens_in":23253,"tokens_out":6365,"would_cite":true,"duration_ms":54109,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C50","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that degree-stable forbidden families of hypergraphs have spectral Turán problems reducible to pattern-colorable families, and uses this to identify the unique extremal hypergraph for r-expansions of color-critical graphs.","keywords":["spectral Turán-type problems","α-spectral radius","hypergraph","degree stability","r-pattern","P-colorable hypergraph","r-expansion","color-critical graph"],"falsifier":"Compute the α-spectral radius for α=2 of the 3-uniform expansion of a 5-cycle, C_5^(3), on n=9 vertices and compare it with λ^(2)(T_2^3(9)); if any C_5^(3)-free 3-graph on 9 vertices has larger value, Theorem 4.8 fails. More systematically, a computational search over all F^(r)-free r-graphs on small n for a given color-critical F would reveal whether T_l^r(n) is the unique maximizer; the first counterexample would localize the failure of the imported degree-stability table.","tokens_in":22341,"feed_emoji":"🧮","tokens_out":8880,"duration_ms":81485,"temperature":0.7,"pith_summary":"The paper develops a general reduction for spectral Turán-type problems in r-uniform hypergraphs. Its central theorem states that if a forbidden family F is degree-stable with respect to the family Col(P) of P-colorable r-graphs for some r-pattern P, then for all sufficiently large n every n-vertex F-free r-graph G satisfies λ^(α)(G) ≤ λ^(α)(Col(P), n), and equality forces G to be P-colorable. This reduces the spectral extremal problem for F to the spectral extremal problem inside Col(P). The reduction is powered by a new asymptotic-regularity theorem showing that spectral extremal hypergraphs in Col(P) have minimum eigenvector component and minimum degree close to uniform bounds. As the headline application, the authors prove that for the r-expansion F^(r) of an (l+1)-color critical graph, the balanced complete l-partite r-graph T_l^r(n) is the unique maximizer of the α-spectral radius among F^(r)-free r-graphs, and the same machinery yields the exact edge Turán number e(T_l^r(n)).","feed_headline":"Degree stability resolves hypergraph spectral Turán problems","feed_subtitle":"For r-expansions of color-critical graphs, the balanced complete l-partite hypergraph uniquely maximizes α-spectral radius.","key_machinery":"The argument rests on three pieces. (i) Theorem 3.10, an asymptotic-regularity theorem for spectral extremal hypergraphs in Col(P): for α>1, the minimum component of a principal eigenvector satisfies (x_min)^α ≥ (1/n)(1−O(1/n)), and the minimum degree satisfies δ(G_n) ≥ π(Col(P))(1−O(1/n)) binom(n,r−1). (ii) The paper's degree-stability notion (Definition 4.1): a family F is degree-stable with respect to Col(P) if every F-free r-graph with minimum degree within ε of the Turán density is P-colorable. (iii) A one-vertex extension lemma (Claim 1 in Theorem 3.10) that propagates the eigenvector lower bound from n to n+1, enabling induction that the spectral extremizer of the F-free family lies i","core_discovery":"The core discovery is that degree stability functions as a bridge from spectral Turán problems to pattern-coloring problems. Theorem 4.2 states: if F is degree-stable with respect to Col(P), then for sufficiently large n, every n-vertex F-free r-graph G obeys λ^(α)(G) ≤ λ^(α)(Col(P), n), with equality only when G is P-colorable. The proof shows the spectral extremal F-free hypergraph inherits a minimum-degree lower bound sufficient to trigger the degree-stability hypothesis, thereby forcing membership in Col(P). The advertised application, Theorem 4.8, states that for an (l+1)-color critical graph F and α≥1, the maximum λ^(α) among n-vertex F^(r)-free r-graphs is exactly λ^(α)(T_l^r(n)); for","pith_inferences":["The reduction likely extends to other families for which degree stability is known or can be proved, such as cancellative 3-graphs or Fano-plane-free hypergraphs, yielding new spectral Turán results.","The O(1/n) regularity bounds in Theorem 3.10 suggest the extremal spectral radius has a well-defined second-order term; determining it might give sharper asymptotic formulas for λ^(α)(M on(F), n).","If the authors' Problem 5.6 has an affirmative answer, then the spectral and edge extremal sets for Col(P) coincide, which combined with Theorem 4.2 would characterize the extremal F-free hypergraphs completely.","A natural test of the color-critical hypothesis: check whether the uniqueness of T_l^r(n) persists when F is only (l+1)-chromatic but not color-critical; a counterexample would show the critical-coloring condition is essential."],"forward_implications":["For any degree-stable F, the spectral Turán problem for F collapses to the spectral extremal problem in Col(P): λ^(α)(M on(F), n) = λ^(α)(Col(P), n) for all large n.","For F the r-expansion of an (l+1)-color critical graph, the unique α-spectral extremizer (α>1) is T_l^r(n), and Corollary 4.5 gives λ^(α)(M on(F), n) = π(Col(P)) n^{r−r/α} − O(1) n^{r−r/α−1}.","The same reduction yields edge Turán results: every F-free n-vertex r-graph has at most ex(Col(P), n) edges, and equality forces P-colorability; for r-expansions this gives the exact Turán number e(T_l^r(n)).","Theorem 3.12 shows spectral extremal k-chromatic r-graphs are asymptotically balanced, partially addressing a conjecture on k-chromatic r-graphs.","The method is a template: any degree-stable family automatically inherits both spectral and edge extremal theorems with the same extremal structure."],"fun_headline_variants":["Stability forces extremal hypergraphs to be pattern-colored","Degree stability unlocks hypergraph spectral Turán limits","Spectral extremal hypergraphs: the degree stability bridge","How degree stability solves α-spectral Turán problems","Hypergraph spectral Turán: the power of degree stability"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof imports, without proof, the fact that the r-expansion of any (l+1)-color critical graph is degree-stable with respect to Col(K_l^r), as tabulated in an external preprint; if that degree-stability assertion is false for some expansion, the conclusion that the extremal hypergraph is P-colorable — and hence that T_l^r(n) is extremal — no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Stability forces extremal hypergraphs to be pattern-colored","Degree stability unlocks hypergraph spectral Turán limits","Spectral extremal hypergraphs: the degree stability bridge","How degree stability solves α-spectral Turán problems","Hypergraph spectral Turán: the power of degree stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1333,"prompt_tokens":1001,"completion_tokens":332,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":745,"completion_tokens_details":{"reasoning_tokens":254}},"tokens_in":745,"tokens_out":332,"duration_ms":5598,"temperature":1.0,"reasoning_tokens":254,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T14:39:44.424792+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the α-spectral radius for α=2 of the 3-uniform expansion of a 5-cycle, C_5^(3), on n=9 vertices and compare it with λ^(2)(T_2^3(9)); if any C_5^(3)-free 3-graph on 9 vertices has larger value, Theorem 4.8 fails. More systematically, a computational search over all F^(r)-free r-graphs on small n for a given color-critical F would reveal whether T_l^r(n) is the unique maximizer; the first counterexample would localize the failure of the imported degree-stability table.","supporting_citations":[],"review_version":1}