{"id":"56eae9b2-f778-4e87-a9b8-4c426f452587","arxiv_id":"2607.28616","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new transference method proves sharp real-exponent degree-power bounds for t-intersecting, intersecting, and bounded-matching-uniform families, with complete equality classifications.","lead":"This paper builds a reusable convex-majorization framework that turns sharp two-moment estimates into sharp bounds for real-power degree sums in extremal set systems, and applies it to three exact extremal problems. The results show that in the sharp Erdős–Ko–Rado range, a designated star family maximizes the codegree power sum for every real exponent p≥2, with all equality families listed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 inherits its entire real-p codegree bound from the Wu–Zhang tight-pair estimate (Lemma 2.3); if that black-box bound has a hidden range restriction at n=(t+1)(k−t+1), the headline application and its equality classification need rework.","rationale":"The reader's weakest_assumption and my stress-test converge on the same point: the paper's first application is conditional on an external second-moment estimate. My independent reading of the internal proofs—the divided-difference argument, the monotonicity claims in Lemma 2.6, the Johnson eigenspace classification, and the codegree-excess proof—uncovered no specific error. The transference layer is modular and mathematically sound as far as I can see; the known risks are exactly the stated black boxes (Wu–Zhang, Wilson/AK boundary, Frankl). Such reliance is standard, and the paper identifies it explicitly. The proposed boundary verification would settle the residual risk, but on the evidence currently available the verdict should not change.","tokens_in":20929,"tokens_out":34662,"duration_ms":429983,"concrete_test":"Verify [21, Theorem 1.4] at the Wilson boundary: independently re-derive M2(F)≤(k−t)(n−k)C(n−t,k−t) for n=(t+1)(k−t+1) from the Ahlswede–Khachatrian classification, checking both S_T and A_Z; alternatively exhaustively enumerate all t-intersecting families for (t,k,n)=(2,3,6) and (3,4,8) and compare their M2 with the bound. If the boundary estimate holds and its equality cases match the two families in Theorem 1.1, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I rechecked the analytic engine—Lemma 2.5's Hermite remainder for 2<p<3, Lemma 2.6's two-point replacement and monotonicity, the Johnson-scheme equality classification in Proposition 4.1, and the shifted-slice proof of (CE)—and found no internal error. The genuinely load-bearing point is Lemma 2.3: the second-moment bound M2(F)≤(k−t)(n−k)C(n−t,k−t) for t-intersecting families in the Wilson range is imported verbatim from [21, Theorem 1.4], including the boundary n=(t+1)(k−t+1). The transference framework only converts this quadratic statistic into real-p bounds; it cannot supply or repair the statistic. The p=2 equality classification in Theorem 1.1 also uses equality in that same bound, so a hidden strictness restriction or an incorrect boundary equality statement would propagate into Theorem 1.1's upper bound and into cases (ii)–(iii). This is an explicit, clearly acknowledged external dependence (Remark 1.2), not an internal inconsistency; the framework itself is independent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a convex-transference framework for bounding degree power sums of uniform set families. The framework takes as input sharp low-order information (first and second moment bounds, or first moment plus an excess-mass bound) and transfers it to sharp bounds on ℓ_{r,p} for every real p, retaining equality information. Three applications are given: (1) Theorem 1.1 extends the Wu–Zhang quadratic theorem for t-intersecting families to every real p ≥ 2 throughout the sharp Erdős–Ko–Rado range, with equality classifications including boundary cases; (2) Theorem 1.5 handles all r-degree levels for intersecting families at n ≥ 2k; (3) Theorem 1.7 gives the sharp codegree-power bound for families with matching number at most s in Frankl's linear range. The analytic engine consists of three majorant certificates: an endpoint secant, a two-point Hermite envelope, and a one-knot hinge envelope. The equality analyses use the EKR/Wilson/Ahlswede–Khachatrian boundary classification, Bey's inequality with a Johnson-scheme spectral decomposition, and Frankl's matching theorem.","tokens_in":21258,"tokens_out":32001,"duration_ms":316196,"significance":"If the external inputs are correct, this is a significant and reusable contribution. The transfer principles in Lemmas 2.4–2.7 are genuinely modular: they are parameter-free in the combinatorial sense, being constructed solely from the target degree distribution, and they convert quadratic or excess-mass estimates into real-power bounds without requiring higher-moment counts. The applications are exact and include complete equality classifications, including the previously overlooked boundary families in the p = 2 case of Theorem 1.1. The paper also answers Zhou–Yuan's problem in the sharp star range and removes the integrality restriction on p for the matching-number problem. The main caveat is the dependence of Theorem 1.1 on the Wu–Zhang tight-pair bound; this is clearly acknowledged and does not affect the framework itself, but the exact statement of that external theorem should be verified and made precise.","major_comments":[{"comment":"The headline theorem's upper bound and its p=2 equality classification import the Wu–Zhang tight-pair estimate M2(F) ≤ (k−t)(n−k) C(n−t,k−t) verbatim, including the boundary n=(t+1)(k−t+1). This is load-bearing because the analytic transfer cannot repair a false second-moment input. The manuscript cites [21, Theorem 1.4] but does not reproduce its exact hypotheses. Since the authors later note in Remark 1.2 that the equality statement of [21] needs supplementation, the referee requests a precise statement of [21, Theorem 1.4] — or a verification sketch — confirming that the inequality holds with equality at the stated boundary and that there is no hidden strictness restriction. This is a verification request rather than an internal error.","section":"Lemma 2.3 / Theorem 1.1"}],"minor_comments":[{"comment":"The assertion that shifting does not increase the matching number is invoked without proof. It is standard, but a reference or a one-sentence justification would improve self-containedness.","section":"Lemma 5.2"},{"comment":"The claim that Φ_{q−1} is monotone under inclusion is used to compare the q≥2 contributions. This is true because the excess is computed pointwise, but the manuscript could spell this out.","section":"Section 5.2, proof of Lemma 5.5"},{"comment":"There is a small typo in the equality statement: 'Fis isomorphic' should read 'F is isomorphic'.","section":"Theorem 1.7"},{"comment":"Lemma 2.6 is dense and its parameters are somewhat intricate. A short table translating the abstract parameters (N, L, D, M, a, c, ξ, η) into the quantities used in Theorems 1.5 and 1.7 would aid readability.","section":"Section 2, Lemma 2.6"}],"recommendation":"minor_revision","confidential_remarks":"The paper appears technically sound after my reading; I found no internal error in the central analytic lemmas or in the three applications. The main risk is the black-box Wu–Zhang input, which the authors cite rather than prove. Since this is exactly the kind of recent, sharp external theorem that a careful referee must ask authors to state precisely, I recommend minor revision rather than outright acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. It delivers: a modular convex-transference architecture (endpoint secant, Hermite envelope, hinge envelope) that converts sharp low-order combinatorial statistics into real-exponent degree-power bounds while preserving equality. The three applications are real theorems. Theorem 1.1 extends the Wu–Zhang quadratic theorem to all real p≥2 at the exact Wilson threshold and answers Zhou–Yuan's problem in that range. Theorem 1.5 covers every degree level, and Theorem 1.7 removes the integrality restriction on p and gives an explicit linear range for bounded matching number. I checked the central analytic lemmas — the two-point replacement in Lemma 2.6, the Hermite remainder for 2<p<3, the monotonicity claim, and the shifted-slice proof of Lemma 5.5 — and they are internally consistent. The Johnson-scheme classification at n=2k is careful. The paper earns its claims.\n\nThe honest soft spot is the one the authors flag in Remark 1.2: Theorem 1.1's upper bound and its p=2 equality cases import the Wu–Zhang tight-pair estimate (Lemma 2.3) wholesale, including the boundary n=(t+1)(k−t+1). The transform framework cannot repair a false second-moment input. So the referee's job is not to recheck the whole proof but to verify that Lemma 2.3 holds with equality at the boundary as stated. If [21, Theorem 1.4] has a hidden range restriction there, then Theorem 1.1's equality classification needs rework. The authors are transparent about this dependence, even correcting an equality oversight in Wu–Zhang. That is how it should be. The other external inputs (Bey, Frankl, AK, Wilson) are standard, and the subquadratic counterexample in Section 6 explains why p≥2 is necessary.\n\nFor whom: anyone working on degree-power extremal problems or convex-majorant methods. It deserves a serious referee and, after the external-input check, acceptance. I'd bring it to the reading group.","headline":"A genuinely reusable convex-transfer framework for degree powers, with three sharp applications; the main risk is the external Wu–Zhang second-moment input, not the internal proof.","tokens_in":21704,"tokens_out":2901,"would_cite":true,"duration_ms":40572,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-04T03:20:57.024053+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":2}