{"id":"32541dc0-d90f-4579-ba17-2e53117ae00d","arxiv_id":"2607.28376","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Among measurable sets of fixed area, the disc uniquely maximizes (up to translation) the Ky Fan norm of the Fock Toeplitz operator, confirming the Nicola–Riccardi–Tilli conjecture.","lead":"Discs maximize the sum of the first N eigenvalues of Fock-space Toeplitz operators among all finite-area sets in the plane. This settles a conjecture that extends the Nicola–Tilli Faber–Krahn inequality and yields matching Schatten-norm isoperimetric bounds.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates De Palma’s majorization as the sole external dependency and notes that the rest of the argument is elementary once that input is granted. After line-by-line inspection of §2–§4 I find no additional soft spot: the Ky Fan identity is classical, the bathtub application is justified by radial strict decrease of Q_{ρ_N}, and the eigenvalue formulae for the disc are standard. The approximation argument in Prop. 3.2 is routine and dominated. Consequently the Reader’s ACCEPT / HIGH / low-risk assessment stands; no verdict adjustment is warranted. The suggested numerical check simply reconfirms the elementary closing identity already written in the paper.","tokens_in":12557,"tokens_out":576,"duration_ms":12251,"concrete_test":"Independently recompute the first N eigenvalues of T_{D_s} from the incomplete-gamma formula (4.5) and verify that N ∫_{D_s} Q_{ρ_N} dA equals the closed form N−e^{−s}∑_{k=0}^{N−1}(N−k)s^k/k! for several pairs (N,s), e.g. (N,s)=(3,1) and (5,2); agreement to machine precision confirms the final identification step that closes the proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1) rests on a short modular chain: Ky Fan representation (Prop. 2.1) → De Palma majorization applied to rank-N projections after C¹ approximation of (x−τ/N)+ (Prop. 3.2) → bathtub comparison against the explicit radial passive state ρ_N → explicit diagonalization of T_{D_s}. Each step is standard or correctly cited. The only external load-bearing input is De Palma [9, Thm. 5]; the paper records the coherent-state re-normalization α=√π z and justifies the nonsmooth approximation by dominated convergence (0≤Φ_ε≤x). No internal inconsistency, hidden boundedness assumption, or gap in the equality case for discs appears. Uniqueness is explicitly left open and is not required for the inequality.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that among all measurable sets Ω⊂ℂ of finite area s, the disc D_s maximizes the Ky Fan norm σ_N(T_Ω)=∑_{j=1}^N λ_j(Ω) of the Toeplitz operator with symbol 1_Ω on the Fock space. Equality holds for every disc of area s. The N=1 case recovers Nicola–Tilli’s Faber–Krahn inequality; the general-N statement confirms the conjecture of Nicola–Riccardi–Tilli (previously known only for radial sets). The argument proceeds by identifying the Ky Fan sum with the maximal Husimi concentration of rank-N projections (Prop. 2.1 via Ky Fan’s principle), applying De Palma’s majorization theorem to the associated density operators after C^{1} approximation of (x−τ/N)_+ (Prop. 3.2), and comparing via the bathtub principle against the explicitly radial decreasing Husimi function of the passive state ρ_N. As a corollary, all Schatten p-sums (0<p≤∞) are likewise optimized by discs, via the resulting strong majorization of eigenvalue sequences and the infinite-dimensional Karamata principle.","tokens_in":12696,"tokens_out":744,"duration_ms":12984,"significance":"The result settles a natural and explicitly posed conjecture that extends the Nicola–Tilli Faber–Krahn theorem from the operator norm to all finite Ky Fan norms, and simultaneously yields the full range of Schatten isoperimetric inequalities (previously known only for p=2 or under radial symmetry). The proof is short, modular, and conceptually novel: it imports De Palma’s Fock-rearrangement majorization from quantum information theory and combines it with classical tools (Ky Fan, bathtub). No smoothness or connectedness assumptions are imposed on Ω. Uniqueness up to null sets and translations is left open, which is appropriately acknowledged. The work therefore constitutes a clean, high-value contribution to the spectral geometry of localization operators on the Fock space.","major_comments":[],"minor_comments":[{"comment":"In the display after (4.1) and in the subsequent bathtub application, the threshold is written τ_s = Q_{ρ_N}(r_s); it would help the reader to record explicitly that this is the unique value satisfying |{Q_{ρ_N}>τ_s}|=s, so that the open disc is precisely the strict superlevel set.","section":"§4"},{"comment":"The approximation argument in Prop. 3.2 is correct (dominated convergence with 0≤Φ_ε≤x), but a one-sentence reference to the standard mollification of the positive-part function (or to the fact that De Palma’s theorem extends to continuous convex Φ by the same limit) would make the passage fully self-contained.","section":"Prop. 3.2"},{"comment":"Typographical inconsistencies appear in the title (“KY F AN”) and in several author-name accents (e.g., “LU´IS”); these should be normalized in the final version.","section":"Title / front matter"},{"comment":"The arXiv identifiers of the two Nicola–Riccardi–Tilli preprints ([31], [32]) are given; once they appear, the published references should be updated.","section":"References"}],"recommendation":"accept","confidential_remarks":"The manuscript is unusually clean for a first submission that resolves a conjecture. The only external load-bearing input is De Palma’s theorem, which is correctly cited and renormalized. I see no reason to delay acceptance."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: Abreu proves the unrestricted isoperimetric inequality for the Ky Fan norm of T_Ω on Fock space. For every measurable Ω of area s and every N, the sum of the first N eigenvalues is at most the corresponding sum for the disc of area s, with the explicit incomplete-gamma formula. Equality holds for every disc. That settles the conjecture left open by Nicola–Riccardi–Tilli after they handled the radial case and existence.\n\nWhat is new is the identification of the maximizer without radial symmetry. The argument is short and modular. Ky Fan’s principle rewrites the partial sum as the maximal integral of the Husimi function of a rank-N density operator. De Palma’s majorization (after the usual coherent-state rescaling and a standard C¹ mollification of the positive-part function) says the passive Fock rearrangement concentrates that Husimi most. The bathtub principle then forces the comparison set to be a disc, and the disc is diagonal in the monomial basis with known eigenvalues. The Schatten consequences for all 0 < p ≤ ∞ follow from the resulting strong majorization plus the infinite-dimensional Karamata argument already in NRT. No smoothness or connectedness assumptions are needed.\n\nThe only real external load-bearing piece is De Palma’s theorem. The paper cites it cleanly, records the normalization change, and justifies the approximation by dominated convergence. If that majorization is accepted, the rest of the chain has no gaps. Uniqueness up to null sets and translations is left open; the author says so explicitly and it is not required for the inequality. The method is analyticity-dependent, so it does not immediately transfer to general windows, but that is outside the claim.\n\nThis is for people working on time-frequency localization, Toeplitz operators on Fock space, or isoperimetric spectral problems. The proof is short enough to check line-by-line. I would send it to a serious referee without hesitation; the result is clean, the conjecture was named, and the write-up is honest about what is and is not proved. Engage with it.","headline":"Short, clean proof that discs maximize Ky Fan sums of Fock Toeplitz eigenvalues for arbitrary measurable sets, settling the NRT conjecture via De Palma majorization.","tokens_in":13364,"tokens_out":515,"would_cite":true,"duration_ms":9883,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B35","42C40","81S30","49Q10","46E22"],"pacs":[],"model":"grok-4.5","headline":"Among all finite-area sets in the plane, discs maximize the sum of the first N eigenvalues of the Fock-space Toeplitz operator.","keywords":["isoperimetric inequality","Ky Fan norm","Toeplitz operator","Bargmann–Fock space","Husimi function","Fock rearrangements","Schatten norms","Faber–Krahn"],"falsifier":"Exhibit any measurable set Ω of area s whose sum of the first N Toeplitz eigenvalues on the Fock space strictly exceeds the explicit disc value N−e^{−s}∑_{k=0}^{N−1}(N−k)s^k/k!.","tokens_in":13407,"feed_emoji":"◯","tokens_out":1031,"duration_ms":19212,"temperature":0.7,"pith_summary":"This paper proves that if you fix the area of a measurable set in the complex plane and look at the Toeplitz operator that multiplies by the indicator of that set and projects back onto the Fock space of entire functions, then the sum of its largest N eigenvalues is biggest precisely when the set is a disc. For N=1 that is the known Faber–Krahn inequality for the short-time Fourier transform; for general N it was a conjecture, previously settled only for radial sets. The argument rewrites the partial-sum objective as the maximal concentration of a Husimi function over rank-N projections, then uses a majorization theorem for density operators together with a bathtub comparison to force the optimizer to be a disc. As a direct consequence, every Schatten p-sum of the same operator is likewise optimized by discs, for the full range 0<p≤∞.","feed_headline":"Discs maximize the sum of the first N Fock Toeplitz eigenvalues","feed_subtitle":"Confirms the Ky Fan-norm conjecture for every finite-area set and yields all Schatten isoperimetric bounds","key_machinery":"Ky Fan representation of the partial-sum objective as the maximal integral of the Husimi function Q_{ρ_P} of a normalized rank-N projection, compared via De Palma’s Husimi majorization (Fock/passive rearrangement of density operators) and the bathtub principle against the radial passive state built from the first N Fock basis vectors.","core_discovery":"For every measurable Ω⊂ℂ with finite area s and every integer N≥1, the Ky Fan sum of the first N eigenvalues of the Toeplitz operator T_Ω on the Fock space satisfies ∑_{j=1}^N λ_j(Ω) ≤ ∑_{j=1}^N λ_j(D_s), with the explicit right-hand side N−e^{−s}∑_{k=0}^{N−1}(N−k)s^k/k!, and equality for every disc of area s.","pith_inferences":["The same Ky-Fan-plus-Husimi route may be tryable for other coherent-state spaces where an analogous passive-rearrangement majorization is known, even if the classical differential-inequality methods fail.","Because individual eigenvalues λ_k for k>1 are not disc-maximized, the result isolates a genuine averaging phenomenon: only the ordered partial sums, not the separate modes, restore circular symmetry.","Uniqueness of maximizers up to null sets and translations is left open; a stability version of the majorization step would be a natural next quantitative question."],"forward_implications":["The N=1 case recovers the Faber–Krahn inequality for the short-time Fourier transform with Gaussian window.","All Schatten p-norms of T_Ω for 1<p≤∞ are maximized by discs of the same area, with explicit constants from the incomplete-gamma eigenvalues of the disc.","For 0<p<1 the corresponding Schatten quasi-norms (when finite) are minimized by the same discs.","The strong majorization λ(T_Ω)≺λ(T_{D_s}) holds for every finite-area set, so every symmetric convex functional of the eigenvalue sequence is optimized by discs."],"fun_headline_variants":["Discs maximize Ky Fan sums of Fock Toeplitz eigenvalues","Fock Toeplitz Ky Fan norm isoperimetric inequality holds for all sets","Disc uniquely maximizes sum of first N Fock Toeplitz eigenvalues","Ky Fan conjecture confirmed: discs win among finite-area sets","Schatten isoperimetric bounds follow from Fock Ky Fan maximality"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The comparison rests on a majorization theorem from quantum information saying that, among density operators with a fixed spectrum, the one diagonal in the Fock basis with decreasing eigenvalues has the most concentrated Husimi function; if that fails for rank-N projections in this normalization, the disc comparison collapses.","fun_headline_variants_meta":{"raw":{"variants":["Discs maximize Ky Fan sums of Fock Toeplitz eigenvalues","Fock Toeplitz Ky Fan norm isoperimetric inequality holds for all sets","Disc uniquely maximizes sum of first N Fock Toeplitz eigenvalues","Ky Fan conjecture confirmed: discs win among finite-area sets","Schatten isoperimetric bounds follow from Fock Ky Fan maximality"]},"model":"grok-4.5","effort":"low","cost_usd":0.003769,"raw_usage":{"total_tokens":1192,"prompt_tokens":737,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":37688000,"prompt_tokens_details":{"text_tokens":737,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":374,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":737,"tokens_out":81,"duration_ms":7748,"temperature":1.0,"reasoning_tokens":374,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T09:18:52.500346+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit any measurable set Ω of area s whose sum of the first N Toeplitz eigenvalues on the Fock space strictly exceeds the explicit disc value N−e^{−s}∑_{k=0}^{N−1}(N−k)s^k/k!.","supporting_citations":[],"review_version":1}