{"id":"e376f703-886b-4bb0-8b1d-1035b1de4bc6","arxiv_id":"2608.02594","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An argmax principle over high pseudo-moments yields degree-O(sqrt(n/eps)) SoS algorithms for Best Separable State, multiplicative 2->4 norm approximation, and a shorter proof of the known sphere-polynomial convergence bound.","lead":"This paper introduces an argmax principle for sum-of-squares relaxations on the sphere: the maximizer of a high-moment polynomial built from the relaxation is used as a rounding target. It yields sharper SoS degree bounds for Best Separable State, a multiplicative 2-to-4 norm approximation, and simpler proofs for spherical polynomial optimization.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 overclaims p→q norm results for rational p; body only proves even integer p.","rationale":"The reader's verdict is CONDITIONAL, citing the overclaim in Theorem 1.3 as the main issue, while listing exact-arithmetic assumptions as the weakest assumption. I agree that the p→q overclaim is the most concrete and load-bearing concern: it is a direct mismatch between the introduction and the body, and the proofs genuinely depend on p being even. The exact-arithmetic issue is a standard caveat in SoS papers and is explicitly acknowledged, so it is less decisive. Since my concern supports the reader's CONDITIONAL verdict without changing it, I set verdict_should_be to UNCHANGED. My agreement_with_reader is partial because the reader's formal 'weakest_assumption' field points to the SDP solver precision, whereas I find the p→q scope overclaim more central; however, the reader's rationale does identify the same theorem overclaim, so we are in substantive agreement.","tokens_in":44432,"tokens_out":21668,"duration_ms":253277,"concrete_test":"Fix p=3, q=4, and construct a degree-D_{3,4} pseudo-expectation as a point mass at x=(−2,0,0,…,0). Check whether it satisfies the constraints (5.4a) and (5.4b) for a suitable threshold τ, while the true ℓ_3→ℓ_4 norm of a corresponding matrix A (e.g., A with a row equal to a vector dominated by this x) is bounded away from the claimed relation. If the relaxation admits a pseudo-expectation that violates the ℓ_3-ball bound, the proof of Theorem 5.2 relies on g_p(x) and cannot be extended to odd p. Alternatively, inspect Lemma 5.1 for p=3 and verify that its SoS certificate fails because 1−s^{2j} requires x_i^3 to be a polynomial, which is not valid on all of R.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The introduction's Theorem 1.3 (and the abstract's 'family of p→q norms with even q') advertises approximation of matrix p→q norms for any rational 1<p<q with even q. However, Section 5 explicitly restricts to even integers 2≤p<q, and the proofs rely critically on p being even. In particular, Lemma 5.1 uses the polynomial g_p(x)=1−Σ_j x_j^p as an SoS certificate for the ℓ_p-ball constraint (5.4a). For p odd or non-integer, x_j^p is not a polynomial, or the inequality g_p(x)≥0 does not enforce Σ_j |x_j|^p ≤ 1. For example, with p=3, x=(−2,0) satisfies g_3(x)=1−(−8)=9>0 but has ℓ_3-norm 2>1. Thus the relaxation itself changes and the proof of Theorem 5.2 cannot be extended as stated. This overclaim is not merely a cosmetic gap: it broadens one of the paper's three headline applications beyond what is proven. The other two results (BSS and degree-d polynomial optimization) appear internally consistent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an 'argmax principle' for analyzing sum-of-squares relaxations on the sphere: from a feasible pseudo-expectation Ẽ, form a high-moment polynomial (e.g., Φ_k(u)=Ẽ⟨x,u⟩^{2k}, or a fold-based analogue), then use a maximizer of this polynomial as the rounding object; first/second-order or global optimality conditions convert into reweighed moment inequalities that drive the convergence analysis. Applications: (1) perfect-completeness Best Separable State, where a degree-O(√(n/ε)) SoS is shown to distinguish h_sep=1 from h_sep≤1−ε through the rank-one-subspace reduction, with an ETH-based tightness discussion for inverse-linear gaps; (2) the projector 2→4 norm, where degree-O(√n/ε) SoS gives a (1+O(ε))-multiplicative approximation via a threshold relaxation and binary search, plus an existential rounding proof and a claimed deterministic finite-search rounding; (3) degree-d polynomial optimization, where a fold-selection argument with pseudo-Hölder and spherical-moment identities recovers the BGG+17 bound O_d((n/k)^{d/2−1}). The paper also advertises an extension to p→q norms for rational p and even q (Theorem 1.3), and works throughout in the exact real-arithmetic SDP model.","tokens_in":44685,"tokens_out":36542,"duration_ms":244688,"significance":"The argmax principle is genuinely unifying, and the paper is rich in verifiable content: explicit derivative identities (3.3)–(3.6), the reweighing/block-PSD residual bound in Theorem 3.6, argmax concentration Lemma 4.4, the Bernstein-grid discretization, and the pseudo-Hölder/polarization machinery of Section 6 all come with explicit constants and complete proofs. If the main theorems stand after the fixes below, the contributions are substantial: an essentially tight BSS degree with a proof simpler than BKS17; a (claimed) first multiplicative (1+ε)-approximation for the projector 2→4 norm in subexponential time; and a short proof of the BGG+17 convergence rate. The paper introduces no new relaxation, involves no fitting, and is free of circularity. However, the advertised rational-p extension is not proved in the body, and a numerical inequality in the finite-search rounding proof is false; the stated scope therefore needs correction.","major_comments":[{"comment":"Theorem 1.3 advertises p→q-norm approximation for every rational 1<p<q with even q, but the body proves only even integers: Section 5 opens with 'fixed even integers 2≤p<q' and disclaims 'the mixed regime 1<p<2<q'. Evenness is load-bearing: Lemma 5.1 encodes the ℓ_p ball by g_p(x)=1−Σ_j x_j^p, a valid SoS certificate only for even p. For p=3, x=(−2,0) satisfies g_3(x)=9>0 but ∥x∥_3=2>1, so (5.4a) does not enforce the intended constraint; for non-integer rational p, x_j^p is not a polynomial and (5.4) is undefined. Note p=3/2, q=4 is exactly the disclaimed regime. Fix: restrict Theorem 1.3 and the abstract to even integers 2≤p<q (matching Theorem 5.2), or supply a different relaxation for rational p.","section":"Section 1.1 Theorem 1.3 vs. Section 5 / Theorem 5.2 / Lemma 5.1"},{"comment":"The asserted bound 4Δ_α/τ^{3/4} ≤ (ε/16)τ^{1/4} is false. With (4.32), α=ε²τ²/216 (when the min is active), and (4.27), Δ_α=2√α/(1−α), so 4Δ_α/τ^{3/4} = (8ετ/(6√6))/((1−α)τ^{3/4}) ≥ (4/(3√6))ετ^{1/4} ≈ 0.544ετ^{1/4} > (ε/16)τ^{1/4}; the τ-dependence cancels, so no regime makes the inequality true. Consequently the derivation of τ^{1/4}−a ≤ (ε/4)τ^{1/4} collapses. Corollary 4.6 and Theorem 4.1 do not use Algorithm 4.2 and are unaffected, but the advertised deterministic finite-search rounding needs a corrected parameter, e.g., α=Θ(ε²τ^{3/2}) with a small constant, which still gives polynomial 1/α for τ≥1/n.","section":"Section 4.6, Theorem 4.14, Eq. (4.34)"}],"minor_comments":[{"comment":"The paper works in the exact real-arithmetic SDP model and assumes pseudo-expectations are exactly feasible with positive reweighing denominators. This is a standard convention, and it is disclosed, but the guarantees (e.g., (3.6), (4.16), (5.19)) have no slack for solver error; a stability analysis would strengthen the algorithmic claims.","section":"Section 2.3; Algorithms 3.1, 4.1, 5.1"},{"comment":"Notation: w denotes both the reweighing weight (Sections 3–4) and the p∗-unit direction maximizing Φ_k in Section 5; s similarly denotes both ⟨x,u⟩ and ⟨w,x⟩. Distinct symbols would considerably improve readability.","section":"Section 5"},{"comment":"Once Theorem 1.3 is restricted to even p (see Major 1), the parameter λ(p)=max{p,2} is never used and could be removed from the statement.","section":"Theorem 1.3"},{"comment":"The line 'Choose a universal constant 2√2 δ0 < 1 − 1/√2' is garbled (presumably 2√2·δ0 < 1 − 1/√2); please fix the typesetting. The same passage would benefit from a pointer to the exact trust-region assumption labels of [BAC19].","section":"Appendix A, proof of Theorem 3.8"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the introduction is considerably more expansive than the body; the rational-p scope of Theorem 1.3 directly contradicts the body's own scope statement, so it is an internal inconsistency rather than a merely optimistic conjecture. The (4.34) slip is a parameter error in a secondary algorithmic-rounding result; neither issue affects the BSS theorem, the degree-d polynomial theorem, or the existential 2→4 approximation. I did not re-verify every constant in the appendices, but the main identities check out. The paper fits cs.CC well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: you should send this to review. The argmax principle — maximize a high-moment polynomial built from the pseudo-expectation, then use the maximizer's optimality conditions — is genuinely new and it does real work. It gives a cleaner BSS analysis, improving the degree from Barak–Kothari–Steurer's O~(sqrt(n)/eps^2) to O(sqrt(n/eps)), matching ETH in the inverse-linear gap regime. It upgrades the BBH+12 decision algorithm for the 2→4 norm to a multiplicative (1+eps)-approximation at comparable running time; that's a new result. The third application recovers the BGG+17 convergence rate with a shorter proof; not a new bound, but the proof is neat.\n\nI went through the main derivations — derivative identities, reweighing, block-PSD residual bound, fourth-moment concentration, Bernstein-grid finite search. They're internally consistent. No fitting, no circularity; the bounds are derived directly.\n\nThe real problem is the p→q statement. Theorem 1.3 in the introduction claims rational 1<p<q with even q. Section 5 proves only even integers 2≤p<q. This is not cosmetic: Lemma 5.1 builds the SoS certificate for the ℓ_p ball from g_p(x)=1−sum x_j^p, which only enforces the ball when p is even. The stress-test example p=3 is correct: g_3(x)>0 does not imply ||x||_3≤1, so the relaxation changes and the proof cannot extend as stated. The abstract's \"even q\" is fine; the theorem statement is the issue. Fix: restrict Theorem 1.3 to even p, or supply the general p proof.\n\nMinor caveats: exact real arithmetic throughout, no numerical stability analysis — standard in this literature, so minor. BSS rounding leans on the Boumal–Absil–Cartis Riemannian trust-region theorem as a black box; acceptable, though check Appendix A's constants if you care about explicit rates.\n\nBottom line: the BSS and 2→4 results are significant and the core math holds up. The p→q overclaim is fixable and should be fixed. I'd send this to a serious referee; it deserves engagement. Good reading-group material if you want to show how a simple principle unifies several SoS analyses.","headline":"Argmax principle is genuinely new and the BSS/2→4 results are solid, but Theorem 1.3 overclaims rational p→q — the body only proves even integer p.","tokens_in":45175,"tokens_out":3368,"would_cite":true,"duration_ms":23109,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-04T03:26:24.758266+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}