{"id":"6bc5e8b7-286e-4617-9360-83138a028b04","arxiv_id":"2608.09866","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every nonzero q with |q|<=1, the sharp spectral constant of the scaled q-numerical range is max{1, 2|q|/(1+sqrt(1-|q|^2))}.","lead":"This paper proves an exact identity relating scaled q-numerical ranges to ordinary numerical ranges of matrices under similarity, and uses it to pin down the sharp spectral constant for every q. It settles a conjecture from earlier work and gives a transfer theorem that may apply to other generalized numerical ranges.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Upper bound of Theorem 4.1 depends entirely on unverified preprints [8]/[1] for Crouzeix's 2-spectral-set theorem; if that theorem fails, the claimed sharp constant does not follow.","rationale":"The reader's weakest_assumption names exactly the same point: the upper bound in Theorem 4.1 cites [8]/[1] without reproof. I agree that this is the single most load-bearing assumption. The internal proofs of the similarity formula (Theorem 2.2), the transfer Theorem 3.3, and the lower-bound computation are self-contained and appear correct; my re-derivation of the lower bound for the nilpotent matrix N matches the claimed constant. The only place where the sharp value 2|q|/(1+sqrt(1-|q|^2)) can enter is C=2 in Theorem 3.3, and that value is supplied entirely by the external theorem. Because both references are preprints and the result is a recent proof of a long-standing conjecture, the conditional nature should be explicit. I would therefore mark the paper CONDITIONAL rather than ACCEPT: the mathematical reduction is sound, but the headline theorem currently inherits its sharpness from an unverified external proof. If [8]/[1] are confirmed, the verdict becomes ACCEPT with no changes.","tokens_in":6699,"tokens_out":19801,"duration_ms":180105,"concrete_test":"Obtain an independent line-by-line verification of the proof in [8] (or [1]) of the theorem 'W(B) is a 2-spectral set for every square matrix B', for all matrix sizes n and all B. If the proof is complete and the constant is exactly 2, then Theorem 4.1's upper bound follows as written; if a gap is found, recompute the upper bound in Theorem 3.3 with the corrected constant and compare it with the lower bound from the nilpotent example.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing step is the upper bound in Theorem 4.1: it applies Theorem 3.3 with K(B)=W(B), C=2, and gamma=chi(|q|). Theorem 3.3 only outputs max{1,C/gamma}, so the value 2 enters precisely through the assertion, cited to the 2026 preprints [8] and [1], that W(B) is a 2-spectral set for every B in M_n(C). No part of the present manuscript proves this fact. If the true universal constant for W(B) were C>2, the argument would give only C_q <= max{1,C/chi(|q|)}, which is strictly larger than the claimed value for |q| sufficiently close to 1; the rank-one lower bound would then no longer match. The proof is a reduction, not a self-contained proof of the sharp constant. The central claim is therefore conditional on correctness and completeness of [8]/[1].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies scaled q-numerical ranges Ω_q(A)=q^{-1}W_q(A). Its central results are: (i) Theorem 2.2, an exact similarity representation Ω_{η(γ)}(A)=union_{κ(S)≤γ} W(S^{-1}AS); (ii) Theorem 3.3, an abstract transfer principle that turns a C-spectral-set bound for W(B) into a spectral-set bound for Ω_q(A); and (iii) Theorem 4.1, the sharp spectral constant C_q = max{1, 2|q|/(1+√(1-|q|^2))} for every n≥2 and 0<|q|≤1. The upper bound is obtained by applying Theorem 3.3 with K(B)=W(B) and C=2, relying on the recently announced proof of Crouzeix's conjecture; the lower bound uses a rank-one nilpotent matrix and a cited formula for its q-numerical radius.","tokens_in":6891,"tokens_out":11557,"duration_ms":100880,"significance":"If the external 2-spectral-set theorem is accepted, the paper settles the conjecture from [11] and provides a clean mechanism that may be useful beyond q-numerical ranges. The similarity identity in Theorem 2.2 is elegant and proved in detail, and the internal proofs in Sections 2 and 3 are careful and correct. However, the advertised sharp constant is conditional on unproved external preprints ([8], [1]) for the upper bound and on [6] for the lower-bound formula; the unconditional contribution is the reduction, not the final number. This does not diminish the value of the mechanism, but the conditional nature should be stated precisely.","major_comments":[{"comment":"The upper bound C_q ≤ max{1, 2|q|/(1+√(1-|q|^2))} is obtained by feeding the assertion 'W(B) is a 2-spectral set for every square matrix B' into Theorem 3.3, with that assertion cited only to the preprints [8] and [1]. No part of the manuscript proves or sketches this fact, and if the true universal constant for W(B) were C>2, the argument would only give C_q ≤ max{1, C/χ(|q|)}, which would not match the lower bound for |q| close to 1. The central theorem is therefore conditional. Please either state Theorem 4.1 explicitly as conditional on Crouzeix's conjecture, cite a peer-reviewed version of the 2-spectral-set theorem, or include a verification of the cited result.","section":"Section 4, Theorem 4.1 upper bound"},{"comment":"The sharpness example N=[[0,2],[0,0]] uses the formula max_{z∈Ω_q(N)}|z|=(1+√(1-|q|^2))/|q| from [6, Theorem 2.1]. Since [6] is also an arXiv preprint, the lower bound is not fully proved within the manuscript. This is a smaller external dependency than the upper bound, but for a claimed sharp constant both sides should be justified either by a proof in the paper or by a reference that the reader can verify.","section":"Section 4, Theorem 4.1 lower bound"}],"minor_comments":[{"comment":"As written, the equality Ω_q(A)=Ω_{|q|}(A)=Ω_{|q|}(A)^\\circ identifies the set with its interior. For a non-scalar A, Ω_q(A) is a closed convex set with nonempty interior and is not equal to its interior; the intended argument is that Theorem 3.3 is applied to D=Ω_{|q|}(A)^\\circ and the spectral constant for the interior is the same as for its closure. Please rephrase.","section":"Section 4, proof of Theorem 4.1"},{"comment":"The sentence about Jin's preprint [8] should be clarified: does [8] already prove (1.3) directly, and does it use the same similarity mechanism? The authors should state precisely how their result differs, especially since the abstract presents the sharp inequality as a main novelty.","section":"Introduction, final paragraph"},{"comment":"The strict interior inclusion for non-scalar matrices is cited from [9, Theorem 2.5]; adding a few words about why that theorem applies to the scaled ranges would help the reader.","section":"Section 2, Proposition 2.1(2)"},{"comment":"The full-text header renders the title with an extra space ('CONST ANTS'), which is presumably a typesetting artifact; the metadata title is correct.","section":"Title/header"}],"recommendation":"major_revision","confidential_remarks":"The paper is a reduction of the sharp spectral constant for scaled q-numerical ranges to the Crouzeix 2-spectral-set theorem, together with an elegant similarity formula and a transfer principle. The main theorem's upper bound depends essentially on [8] and [1], both preprints; the lower bound also depends on [6]. If the journal accepts citations to unrefereed preprints, the mathematical content is sound. The authors' own note that [8] also proves (1.3) raises a novelty question; the editor may wish to verify how much of Theorem 4.1 is already in [8]. The internal proofs in Sections 2 and 3 are clean and correct."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this is a solid paper that resolves the sharp spectral constant for scaled q-numerical ranges, Conjecture 4.2 of the authors' own earlier paper, and does it with a genuinely reusable mechanism. The constant itself was also proved by Jin in a concurrent preprint, but the method here is the real contribution.\n\nWhat's new: Theorem 2.2 is an exact similarity formula: Omega_{eta(gamma)}(A) = union_{kappa(S)<=gamma} W(S^{-1}AS) with eta(gamma)=2/(gamma+gamma^{-1}). It is a nice geometric fact, and the proof via the Kantorovich inequality is short and correct. Theorem 3.2 is an extremal-pair amplification: if a convex domain D has spectral constant M_D(A)>1, then you can find a similarity with condition number gamma that multiplies that constant by gamma. Theorem 3.3 turns that into an abstract transfer principle: if every matrix has a C-spectral set K(B), and all K(S^{-1}AS) sit compactly inside D for kappa(S)<gamma, then M_D(A) <= max{1, C/gamma}. These are clean, correct, and widely applicable beyond q-numerical ranges.\n\nThe soft spot is exactly what the stress test flags: the upper bound in Theorem 4.1 assumes that W(B) is a 2-spectral set for every B, citing two 2026 preprints, [8] and [1]. The paper does not prove this. If those preprints are wrong, the upper bound falls to max{1, C/chi(|q|)} with an unknown C>2, and the lower bound would no longer match. That is a real dependency, but it is explicitly disclosed, and the authors also note Jin's independent verification of (1.3). A referee can check the cited preprints. The lower bound uses the rank-one formula from [6], another preprint, but that one is older and less likely to be problematic.\n\nMy take: the transfer mechanism is the intellectual content, and it is well executed. The proof of the sharp constant is conditional on the Crouzeix conjecture resolution, but that resolution now has two independent proofs, so the dependency is about as safe as it gets for a brand-new result. The paper deserves a serious referee and should be accepted, with a referee note to verify the external theorems. I would use Theorem 3.3 in my own work, and I would bring this to reading group.\n\nRecommendation: send it to a good operator theory journal; accept after a standard check of [8] and [1].","headline":"A clean, reusable transfer theorem for spectral constants that settles the scaled q-numerical range conjecture, with the caveat that the upper bound leans on the just-announced proof of Crouzeix's conjecture.","tokens_in":7409,"tokens_out":4405,"would_cite":true,"duration_ms":35872,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A12","47A25","15A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the optimal spectral constant for every scaled $q$-numerical range is $\\max\\{1, 2|q|/(1+\\sqrt{1-|q|^2})\\}$, with equality attained by a rank-one matrix.","keywords":["q-numerical range","scaled q-numerical range","numerical range","spectral set","optimal spectral constant","condition number","polynomial functional calculus","operator norm bound"],"falsifier":"Compute, for a fixed nonnormal $A$ and fixed $|q|<1$, the supremum of $\\|p(A)\\|/\\max_{z\\in\\Omega_q(A)}|p(z)|$ over polynomials; any value above $\\max\\{1, 2|q|/(1+\\sqrt{1-|q|^2})\\}$ would disprove the theorem. The paper's own sharpness example is the $2\\times2$ matrix $N=\\begin{pmatrix}0&2\\\\0&0\\end{pmatrix}$, which attains the claimed constant, so a counterexample would have to be found elsewhere.","tokens_in":6529,"feed_emoji":"📐","tokens_out":13129,"duration_ms":105250,"temperature":0.7,"pith_summary":"This paper settles the question of the optimal spectral constant for scaled $q$-numerical ranges. For every matrix $A$, every polynomial $p$, and every $0<|q|\\leq 1$, the norm $\\|p(A)\\|$ is bounded by the maximum of $|p|$ on the scaled range $\\Omega_q(A)=q^{-1}W_q(A)$, and the best possible constant is $\\max\\{1, 2|q|/(1+\\sqrt{1-|q|^2})\\}$. This confirms a conjecture from earlier work and is the first exact value for these sets. The proof is driven by a new geometric identity that writes $\\Omega_{\\eta(\\gamma)}(A)$ as a union of ordinary numerical ranges of similarities of $A$; that identity transfers sharp spectral-set estimates from the classical numerical range to every scaled $q$-numerical range. Knowing the optimal constant makes the functional-calculus bound dimension-free and parameter-tight.","feed_headline":"Optimal constant found for scaled q-numerical ranges","feed_subtitle":"The least C in the polynomial bound is max of 1 and 2|q|/(1+sqrt(1-|q|^2)).","key_machinery":"The engine of the paper is Theorem 2.2, the exact similarity formula $\\Omega_{\\eta(\\gamma)}(A)=\\bigcup_{\\kappa(S)\\le\\gamma} W(S^{-1}AS)$ with $\\eta(\\gamma)=2/(\\gamma+\\gamma^{-1})$. One inclusion uses the classical inequality $\\langle Pu,u\\rangle\\langle P^{-1}u,u\\rangle\\le (M^2+m^2)^2/(4M^2m^2)$ for a positive matrix $P$ with extremal singular values $M,m$, which gives the condition-number bound; the reverse inclusion constructs an explicit positive matrix with eigenvalues $\\gamma$ and $\\gamma^{-1}$. The formula is used through the inverse relation $\\chi(r)=(1+\\sqrt{1-r^2})/r$: similarities with $\\kappa(S)<\\chi(r)$ send $W(S^{-1}AS)$ compactly inside $\\Omega_r(A)$, which is precisely the containment hypothesis needed by the paper's abstract transfer theorem for spectral constants.","core_discovery":"The central discovery is an exact similarity formula and the sharp inequality that follows from it. For every $\\gamma\\ge 1$, $\\Omega_{\\eta(\\gamma)}(A)=\\bigcup_{\\kappa(S)\\le\\gamma} W(S^{-1}AS)$, where $\\eta(\\gamma)=2/(\\gamma+\\gamma^{-1})$ and $\\kappa(S)=\\|S\\|\\,\\|S^{-1}\\|$. Using this formula, the paper proves that for every $0<|q|\\le 1$, every $n\\ge 2$, every $A\\in M_n(\\mathbb C)$, and every polynomial $p$, $\\|p(A)\\|\\le \\max\\{1, 2|q|/(1+\\sqrt{1-|q|^2})\\}\\max_{z\\in\\Omega_q(A)}|p(z)|$, and that the constant is optimal. The upper bound transfers the theorem that ordinary numerical ranges are $2$-spectral sets through the similarity formula, while the lower bound is a direct computation on a rank-one matrix.","pith_inferences":["The lower-bound example is only $2\\times2$, which suggests that the extremal matrices for this sharp constant may always be low-dimensional; a testable extension is to check whether every $q$ admits an extremizer of size at most $2$.","Because the constant collapses to $1$ as $|q|\\to0$, one could investigate whether for small $q$ the scaled range behaves like a genuine spectral set with additional structure, such as a completely bounded version of the inequality.","The identity with condition-number-bounded similarities points to a computational route for the $q$-numerical radius: maximize $|z|$ over $W(S^{-1}AS)$ with $\\kappa(S)\\le\\chi(r)$, which may be more tractable than directly solving the constrained bilinear form."],"forward_implications":["For every $n\\ge2$, $A\\in M_n(\\mathbb C)$, and polynomial $p$, the bound $\\|p(A)\\|\\le \\max\\{1, 2|q|/(1+\\sqrt{1-|q|^2})\\}\\max_{z\\in\\Omega_q(A)}|p(z)|$ holds, and no smaller universal constant exists.","As $|q|$ runs from $0$ to $1$, the optimal constant runs from $1$ to $2$, so the estimate interpolates between a trivial spectral-set bound and the classical $2$-spectral-set bound for ordinary numerical ranges.","The exact similarity formula gives a compact-containment criterion: whenever $\\kappa(S)<\\chi(|q|)$, the numerical range $W(S^{-1}AS)$ lies strictly inside $\\Omega_q(A)$, so polynomial bounds on $\\Omega_q(A)$ automatically control all sufficiently well-conditioned similarities.","The abstract transfer theorem applies to any family of $C$-spectral sets, so the same mechanism can turn a spectral-set proof for one class of sets into sharp constants for a parameterized family."],"supporting_citations":[{"why":"Supplies the external theorem that every ordinary numerical range $W(B)$ is a $2$-spectral set, which the upper bound in Theorem 4.1 requires.","marker":"[8]"},{"why":"Gives a second proof of the same $2$-spectral-set theorem, cited by the authors as independent support for that assumption.","marker":"[1]"},{"why":"Formulates the numerical-range spectral-set conjecture and provides the extremal-function theorem used in Lemma 3.1.","marker":"[3]"},{"why":"Supplies the orthogonality result combined with the extremal function in Lemma 3.1.","marker":"[2]"},{"why":"Gives the condition-number inequality used to prove one inclusion in the exact similarity formula.","marker":"[7]"},{"why":"Provides the convexity, strict nesting, and spectral-containment facts for $q$-numerical ranges used in Proposition 2.1.","marker":"[9]"},{"why":"Introduced $\\Omega_q(A)$, proved preliminary spectral-set estimates, and stated the conjecture that the sharp constant settles.","marker":"[11]"},{"why":"Provides the rank-one $q$-numerical-radius formula used to compute the sharpness example in the lower bound.","marker":"[6]"}],"fun_headline_variants":["Sharp constant proves optimal bound for q-numerical ranges","Exact similarity formula yields sharp matrix polynomial inequality","Optimal spectral constant for q-scaled numerical ranges","Matrix norm bound made sharp for scaled q-numerical ranges","New inequality settles constant for q-numerical range polynomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper bound in Theorem 4.1 is proved only on top of the external theorem, cited as [8] and [1], that every matrix's ordinary numerical range is a $2$-spectral set; if that theorem failed, the upper bound would fail, although the lower-bound example would still stand.","fun_headline_variants_meta":{"raw":{"variants":["Sharp constant proves optimal bound for q-numerical ranges","Exact similarity formula yields sharp matrix polynomial inequality","Optimal spectral constant for q-scaled numerical ranges","Matrix norm bound made sharp for scaled q-numerical ranges","New inequality settles constant for q-numerical range polynomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000178,"raw_usage":{"total_tokens":1292,"prompt_tokens":936,"completion_tokens":356,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":276}},"tokens_in":552,"tokens_out":356,"duration_ms":3889,"temperature":1.0,"reasoning_tokens":276,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:16:08.343639+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a fixed nonnormal $A$ and fixed $|q|<1$, the supremum of $\\|p(A)\\|/\\max_{z\\in\\Omega_q(A)}|p(z)|$ over polynomials; any value above $\\max\\{1, 2|q|/(1+\\sqrt{1-|q|^2})\\}$ would disprove the theorem. The paper's own sharpness example is the $2\\times2$ matrix $N=\\begin{pmatrix}0&2\\\\0&0\\end{pmatrix}$, which attains the claimed constant, so a counterexample would have to be found elsewhere.","supporting_citations":[{"cited_title":"A solution to Crouzeix's conjecture","cited_arxiv_id":"2608.03841","evidence_quote":"Gives a second proof of the same $2$-spectral-set theorem, cited by the authors as independent support for that assumption."},{"cited_title":"Crouzeix,Bounds for analytical functions of matrices, Integral Equations Operator Theory48(2004), no","cited_arxiv_id":null,"evidence_quote":"Formulates the numerical-range spectral-set conjecture and provides the extremal-function theorem used in Lemma 3.1."},{"cited_title":"Bickel, P","cited_arxiv_id":null,"evidence_quote":"Supplies the orthogonality result combined with the extremal function in Lemma 3.1."},{"cited_title":"Greub and W","cited_arxiv_id":null,"evidence_quote":"Gives the condition-number inequality used to prove one inclusion in the exact similarity formula."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the convexity, strict nesting, and spectral-containment facts for $q$-numerical ranges used in Proposition 2.1."},{"cited_title":"O’Loughlin and J","cited_arxiv_id":null,"evidence_quote":"Introduced $\\Omega_q(A)$, proved preliminary spectral-set estimates, and stated the conjecture that the sharp constant settles."},{"cited_title":"$q$-numerical radius of rank-one operators and the generalized Buzano inequality","cited_arxiv_id":"2503.05036","evidence_quote":"Provides the rank-one $q$-numerical-radius formula used to compute the sharpness example in the lower bound."}],"review_version":1}