{"id":"94a1865c-8198-4f2d-88d1-84e5ffac083e","arxiv_id":"2505.13014","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a compact Levi non-degenerate CR manifold, the semiclassical spectral projector of a Levi-elliptic Toeplitz operator is, modulo a negligible kernel, the sum of two oscillatory integrals with complex phases, with explicit leading coefficients.","lead":"This paper proves a full asymptotic expansion for spectral projections of semiclassical Toeplitz operators on compact Levi non-degenerate CR manifolds, a curved setting that generalizes boundaries of complex domains. The result gives explicit leading terms in terms of the Levi form and the operator's symbol, extending known strictly pseudoconvex theorems to indefinite Levi signature.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main expansion rests entirely on the quoted structure Theorem 1.5; its exact hypotheses must be verified, especially L^2-closed range in the exceptional case |n_- - n_+| = 1.","rationale":"The reader's weakest_assumption identifies the same load-bearing point: everything in the paper depends on the quoted structure theorem for spectral projections on lower energy forms. The paper's own Section 1.4 discussion of L^2-closed range and the exceptional case |n_- - n_+| = 1 makes this dependence concrete and gives a credible route to a false over-generalization if the quoted theorem carries extra hypotheses. I also noted a separate, apparently fixable typo in equation (2.2.7), where the definition of the symbol l_{I0,I0} cannot be in S^{-1} as written and presumably should include an inverse; this is not the central issue. The formula for C_P in Corollary 1.4 also appears to have a sign/absolute-value issue for p_{J0,J0}(alpha) < 0, but that is downstream of Theorem 1.2 and does not by itself change the main theorem. The recommended verdict remains CONDITIONAL: the central claim is plausible and the framework is coherent, but the decisive quoted input and the lengthy microlocal estimates need independent verification before unconditional acceptance.","tokens_in":937,"tokens_out":843,"duration_ms":283427,"concrete_test":"Retrieve Hsiao-Marinescu, J. Differential Geom. 107 (2017), Theorem 4.1 and compare its hypotheses verbatim with Theorem 1.5. In particular, determine whether it assumes L^2-closed range for the Kohn Laplacian, a condition Y(q)/Z(q), or |n_- - n_+| != 1 for the decomposition Pi_lambda^(q) = S_- + S_+. If such an assumption is present and is not satisfied by the hypotheses of Theorem 1.1, then Theorem 1.1 must be restricted accordingly. If Theorem 4.1 contains no such assumption, the concern is vacated. As a secondary check, test Theorem 2.6 or equation (1.1.11) on a compact 3D CR example with signature (1,0) and non-closed range, with a Levi-elliptic P; a counterexample would falsify the unrestricted statement.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central Theorem 1.1 is a direct consequence of the quoted structure Theorem 1.5, not independently proved in the paper. Section 1.4 shows the authors are aware that the two-FIO description of the true Szegő projection Pi^(q) is only available when the Kohn Laplacian has L^2-closed range, that closed range is automatic only when |n_- - n_+| != 1, and that a three-dimensional counterexample exists in the exceptional case [7]. Yet Theorem 1.1 imposes no closed-range or Y(q) hypothesis. If Theorem 1.5, quoted from [43, Theorem 4.1], actually contains such a hypothesis, or if the finite-energy projection Pi_lambda^(q) requires an additional condition when |n_- - n_+| = 1, then the two-phase oscillatory form of (1.1.11), the parametrix Theorem 2.6, and all subsequent expansions have no foundation in the exceptional case. The proof simply inherits this structure: Theorems 2.5, 2.17, and 3.5 all invoke Theorem 1.5. Because the paper does not re-derive the theorem and provides no machine-checkable artifact, this is the most load-bearing unverified point in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies generalized Toeplitz operators T_{P,λ}^{(q)} = Π_λ^{(q)} P Π_λ^{(q)} on compact Levi non-degenerate CR manifolds, where Π_λ^{(q)} is the spectral projection of the Kohn Laplacian on (0,q)-forms for the interval [0,λ]. For q = n_- and P satisfying the Levi-ellipticity conditions (1.1.3)-(1.1.4), the main theorem claims that for any χ ∈ C_c^∞(R\\{0}) the Schwartz kernel of χ(k^{-1}T_{P,λ}^{(q)}) is, modulo O(k^{-∞}), the sum of two semi-classical oscillatory integrals with complex phases φ_- and φ_+, with a complete asymptotic expansion in powers k^{n+1-j} and explicit leading coefficients (Theorems 1.1 and 1.2). A Szegő-type limit theorem for the scaled spectral measure and two examples involving free and locally free circle actions are also given. The proof is built on the quoted structure theorem of Hsiao-Marinescu for lower-energy Szegő projections (Theorem 1.5), followed by a parametrix construction for (z - T_{P,λ}^{(q)}) and a Helffer-Sjöstrand resolvent calculation.","tokens_in":54725,"tokens_out":20120,"duration_ms":228870,"significance":"If the result is correct, it substantially extends semi-classical Toeplitz spectral asymptotics from the strictly pseudoconvex setting to Levi non-degenerate CR manifolds of arbitrary constant signature, with explicit parameter-free leading coefficients and concrete applications to CR circle actions. The paper gives precise theorem statements, a complete proof architecture in the style of Melin-Sjöstrand and Boutet de Monvel, and worked examples in §3.2-§3.3. The main results are, however, conditional on the quoted structure theorem and on a dense sequence of microlocal estimates; the two load-bearing points identified below require explicit verification or repair before the central claim can be regarded as established.","major_comments":[{"comment":"The main expansion is logically inherited from the quoted structure theorem Theorem 1.5, yet the exceptional signature case is left ambiguous. The text states that the two-FIO description of the true Szegő projection Π^{(q)} is available only when □_b^{(q)} has L^2-closed range, that closed range is automatic only when |n_- - n_+| ≠ 1, and that a three-dimensional counterexample exists in the exceptional case; Theorem 1.1, however, imposes no closed-range or Y(q) hypothesis. Since Theorems 2.5, 2.17 and 3.5 all invoke Theorem 1.5, the authors should either quote the exact hypotheses of [43, Theorem 4.1] and verify explicitly that it applies to Π_λ^{(q)} for every compact non-degenerate CR manifold when |n_- - n_+| = 1, or add the missing hypothesis to Theorem 1.1 and restrict the corollaries accordingly. A single sentence citing the precise statement of [43, Theorem 4.1] may suffice if the theorem is unconditional, but as written the paper's own §1.4 discussion creates a genuine ambiguity that blocks acceptance.","section":"§1.4, Theorem 1.1"},{"comment":"The proof uses the identity p_{J0,J0}(-α) = -p_{J0,J0}(α) > 0 in the change of variables for the C^{+,+} term. This identity is not a consequence of formal self-adjointness and first-order classicality of P. For example, a formally self-adjoint first-order operator with principal symbol -|ξ| satisfies p(α)<0 but p(-α)=p(α)<0, so it meets (1.1.4) without making p(-α) positive. The subsequent definition of Ψ_+ ∈ Ph(p^{-1}_{J0,J0}(-α)α) requires the phase coefficient Λ to be positive, and the leading coefficients (2.3.15)-(2.3.16) depend on the same change of variables. The proof should either derive the parity identity from the assumptions on P and on the CR structure, or reformulate the parametrix using p_{J0,J0}(α) directly, e.g. by the change of variables t = -β p_{J0,J0}(α_w) in (2.3.37), which requires no assumption on p(-α). This is load-bearing because Theorem 2.17 is the core parametrix underlying the spectral expansion.","section":"§2.3, proof of Theorem 2.17, Eq. (2.3.37)"},{"comment":"The leading coefficient formulas (3.1.35)-(3.1.36) and (1.1.20)-(1.1.21) are stated without an independent derivation: they are presented as consequences of Theorem 2.17 and the quoted leading symbol computations. In particular, the sign and the argument of χ in the A^+_0 term are only consistent with the rest of the proof under the parity identity discussed in the previous comment. The authors should give the short calculation showing how (3.1.36) follows from Theorem 2.17 within their chosen phase normalization, or, if they adopt the alternative change of variables, update the statements of Theorem 1.2 and (3.1.36) accordingly. This is necessary because Theorem 1.2 is one of the two headline results and its explicit form is used in Corollary 1.4.","section":"§3.1, Theorem 3.5 and Theorem 1.2"}],"minor_comments":[{"comment":"The cross-references to '§4.1', '§4.2' and '§4.3' in the introduction should be to '§3.1', '§3.2' and '§3.3', respectively.","section":"§1.1"},{"comment":"There are small typos: 'Our main result conclude' should be 'Our main result concludes', and 'microloal' should be 'microlocal'.","section":"Abstract and §2.2"},{"comment":"The phrase 'proplery supported H¨ormander symbol' should read 'properly supported Hörmander symbol'.","section":"§1.1, around (1.1.7)"},{"comment":"The condition on r^+_1 in the statement of Theorem 2.17 appears to contain a typo: it should presumably be 'r^+_1 = 0 when n_- ≠ n_+', consistent with (2.3.34), (2.3.38) and all other statements, rather than 'n_- = n_+' as printed.","section":"§2.3, Theorem 2.17"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious and competent extension of the author's previous work with Herrmann, Hsiao and Marinescu, and the main structural outline is plausible. My recommended revision is driven by two concrete gaps rather than by skepticism about the overall method: the exceptional signature case in the quoted structure theorem, and the unproved parity identity in the parametrix proof. The second issue in particular is likely repairable within the manuscript's scope, but it affects the central derivation and cannot be waived as a presentation detail."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuine extension of the semiclassical Toeplitz spectral program from strictly pseudoconvex to all compact Levi non-degenerate CR manifolds with constant signature. The main theorem — full off-diagonal expansion as two oscillatory integrals plus explicit leading term — is new, and the Levi-ellipticity condition is a sensible generalization. Worth a serious referee.\n\nWhat it does well: the paper is honestly built on two pillars, Hsiao–Marinescu's structure theorem for lower-energy Szegő projections and the semiclassical FIO machinery from [32]. It doesn't pretend to re-derive those. The parametrix construction (Theorems 2.6, 2.17, 2.18) is intricate and looks coherent; the Helffer–Sjöstrand remainder estimates in Section 3.1 are the right kind of work for this problem. The leading coefficient formulas (1.1.20)–(1.1.21) are explicit and plausible. The examples with free and locally free circle actions ground the abstract result.\n\nSoft spots, in order of size. First, the load-bearing structure: Theorem 1.1's two-phase expansion for q = n_- rests entirely on Theorem 1.5, quoted from [43]. The paper notes in Section 1.4 that L²-closed range of the Kohn Laplacian is automatic only when |n_- - n_+| ≠ 1, and that the exceptional case has counterexamples. Yet Theorem 1.1 carries no closed-range or Y(q) hypothesis. If Theorem 1.5 as stated in [43] does include such a hypothesis — or if the finite-energy projection Π_λ^(q) requires an extra condition in the exceptional case — then the two-phase form has no foundation there. The authors owe the reader one explicit paragraph reconciling the quoted theorem's hypotheses with the exceptional case. This is not a fatal flaw on the evidence here, but it is the first thing a referee should check.\n\nSecond, several internal references are wrong: the Helffer–Sjöstrand formula lives in Section 3.1, not Section 4.1, and the circle-action examples are in Sections 3.2–3.3, not Section 4.2–4.3. The abstract also omits the O(k^{-∞}) remainder. These are trivial to fix.\n\nThird, the proof is dense and relies on unverified stationary-phase computations. That is normal for this literature; no machine-checked artifact is provided. I could not spot a circular step. The paper extends a known program rather than introducing a new framework, which is fine.\n\nVerdict: for a reader working in CR geometry, microlocal analysis, or Berezin–Toeplitz quantization, this is a useful and substantial paper. I would send it to peer review with one request: sort out the closed-range/exceptional case in Theorem 1.1.","headline":"Real extension of the semiclassical Toeplitz program to Levi non-degenerate CR manifolds; needs one clarification on the quoted Szegő structure theorem's hypotheses in the exceptional signature case.","tokens_in":55364,"tokens_out":2602,"would_cite":false,"duration_ms":25886,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32V05","32A25","35P20","58J40"],"pacs":[],"model":"deepseek-v4-flash","headline":"On any compact Levi non-degenerate CR manifold, the spectral function of a Levi-elliptic Toeplitz operator has a complete asymptotic expansion into two oscillatory integrals with complex phases.","keywords":["CR manifolds","Levi non-degenerate","Toeplitz operators","semiclassical spectral asymptotics","Szegő projection","Kohn Laplacian","Fourier integral operators","lower energy forms"],"falsifier":"Take a compact CR manifold with a transversal CR circle action and a first-order $P$ satisfying (1.1.3)--(1.1.4), compute the left side of (1.1.11) from the explicitly known Fourier components of the Szegő projection, and compare the $O(k^{n+1})$ diagonal coefficient with $\\int_0^\\infty t^n\\chi(t)\\,dt\\,|\\det L_x|/(2\\pi^{n+1})\\,(v(x)/m(x))$; a mismatch at a single point, or a surviving $k^{n}$ term from the wrong phase, would falsify the leading coefficient of Theorem 1.2.","tokens_in":54270,"feed_emoji":"📐","tokens_out":11352,"duration_ms":106129,"temperature":0.7,"pith_summary":"The paper establishes that, on any compact CR manifold whose Levi form is non-degenerate with constant signature $(n_-,n_+)$, the spectral calculus of a generalized Toeplitz operator built from a formally self-adjoint first-order pseudodifferential operator has a complete semiclassical asymptotic expansion. At the critical form degree $q=n_-$, and under the Levi-ellipticity conditions (1.1.3)--(1.1.4), the Schwartz kernel of $\\chi(k^{-1}T_{P,\\lambda}^{(q)})$ is, up to $O(k^{-\\infty})$, the sum of two semiclassical oscillatory integrals whose phases are the two complex Szegő phases $\\varphi_-$ and $\\varphi_+$. The leading diagonal coefficient is computed explicitly from the Levi determinant, the contact-form geometry, and the principal symbol of $P$. For forms of other degrees the same operator is smoothing and hence asymptotically invisible. If the central claim is right, this gives the CR counterpart of Bergman kernel expansions for high powers in mixed-curvature complex geometry, together with a Szegő-type Weyl law for these operators.","feed_headline":"Two oscillatory phases govern Toeplitz spectra on CR manifolds","feed_subtitle":"Full semiclassical expansion with explicit leading coefficient at the critical form degree.","key_machinery":"The load-bearing object is the microlocal decomposition of the spectral projection onto lower-energy $(0,q)$-forms (quoted Theorem 1.5): at $q=n_-$, $\\Pi_\\lambda^{(q)}\\equiv S_-+S_+$ modulo smoothing, where $S_\\pm$ are Fourier integral operators with complex phases $\\varphi_\\pm$ satisfying $\\operatorname{Im}\\varphi_\\pm\\ge0$, $\\varphi_\\pm(x,x)=0$, and $d_x\\varphi_\\pm(x,x)=-d_y\\varphi_\\pm(x,x)=\\mp\\alpha(x)$, with $S_+=0$ when $n_-\\neq n_+$. On this base the paper builds a parametrix for $z-T_{P,\\lambda}^{(q)}$ inside the same complex-phase calculus and feeds it into the almost-analytic functional calculus for $\\chi(k^{-1}T_{P,\\lambda}^{(q)})$. Complex stationary phase then converts the resolvent-type Toeplitz kernels into semiclassical oscillatory integrals whose phases are rescaled by the principal-symbol factors $p_{I_0,I_0}(-\\alpha)$ and $p_{J_0,J_0}(\\alpha)$; the Levi determinant enters through the leading Szegő symbols (1.1.15)--(1.1.16).","core_discovery":"Theorem 1.1 asserts that for $q=n_-$ the kernel $\\chi(k^{-1}T_{P,\\lambda}^{(q)})(x,y)$ equals, modulo $O(k^{-\\infty})$ on each coordinate patch, the sum of $\\int_0^\\infty e^{ikt\\varphi_-(x,y)}A_-(x,y,t;k)\\,dt$ and $\\int_0^\\infty e^{ikt\\varphi_+(x,y)}A_+(x,y,t;k)\\,dt$, with $A_\\mp(x,y,t;k)\\sim\\sum_{j\\ge0}A^\\mp_j(x,y,t)k^{n+1-j}$ and $A_+$ identically zero unless $n_-=n_+$. At the diagonal the leading symbol is $A^-_0(x,x,t)=t^n\\chi(p_{I_0,I_0}(-\\alpha_x)t)\\,|\\det L_x|/(2\\pi^{n+1})\\,(v(x)/m(x))\\,\\tau_x^{n_-}$, with the mirror formula $A^+_0(x,x,t)=t^n\\chi(p_{J_0,J_0}(\\alpha_x)t)\\,|\\det L_x|/(2\\pi^{n+1})\\,(v(x)/m(x))\\,\\tau_x^{n_+}$ when $n_-=n_+$. This is a full asymptotic expansion of the spectral projector, not merely a counting law: every coefficient $A^\\mp_j$ has a well-defined symbol class, and the phases carry the off-diagonal structure.","pith_inferences":["The two-phase structure suggests that the positive and negative spectral ends are governed by two distinct effective Hamiltonians $p_{I_0,I_0}(-\\alpha)$ and $p_{J_0,J_0}(\\alpha)$; a natural continuation would be a two-sided quantum ergodicity statement for eigenfunctions on each end, which the paper does not address.","The $O(k^{-\\infty})$ vanishing outside $q\\in\\{n_-,n_+\\}$ can be read as a spectral refinement of the classical coarse vanishing theorem for high powers of line bundles: in the CR picture, low-energy Toeplitz modes of other form degrees carry no leading semiclassical trace at all.","For CR manifolds with transversal circle action, the parametrix should reproduce known orbifold Bergman kernel expansions on each orbit-type stratum; the paper works out the globally free and locally free cases in Section 3, but a systematic treatment of the singular strata as an independent derivation would make the two-phase form directly checkable."],"forward_implications":["For $q\\notin\\{n_-,n_+\\}$, the operator $\\chi(k^{-1}T_{P,\\lambda}^{(q)})$ is smoothing for large $k$, so the semiclassical spectral function vanishes to infinite order and the Toeplitz operator is compact.","For $q=n_-$, the kernel admits a full off-diagonal expansion in powers $k^{n+1-j}$; away from the diagonal the positive imaginary parts of the phases make the kernel rapidly decreasing in $k$.","On the diagonal, the leading coefficient is explicit in local geometry: it is proportional to $|\\det L_x|$, the volume ratio $v(x)/m(x)$, the projector $\\tau_x^{n_-}$, and the cutoff $\\chi$ evaluated at $p_{I_0,I_0}(-\\alpha_x)t$; an analogous positive-phase term appears when $n_-=n_+$.","The scaled spectral measures $k^{-n-1}\\sum_j\\delta(t-k^{-1}\\lambda_j)$ converge weakly to the absolutely continuous measure $C_P^{(q)}t^n\\,dt$ of Corollary 1.4, giving a Szegő-type Weyl law.","The spectrum of $T_{P,\\lambda}^{(q)}$ at $q=n_-$ is discrete with accumulation only at $\\pm\\infty$, so the cutoff $\\chi$ cleanly separates the positive and negative spectral sides, which the two-phase expansion captures separately."],"supporting_citations":[{"why":"supplies the microlocal structure theorem for the spectral projection on lower energy forms, the quoted decomposition $\\Pi_\\lambda^{(q)}\\equiv S_-+S_+$ modulo smoothing on which Theorem 1.1 rests.","marker":"[43]"},{"why":"provides the calculus of Fourier integral operators with complex phase and the complex stationary phase formula used to reduce Toeplitz products to two-phase oscillatory integrals.","marker":"[53]"},{"why":"establishes the singularity theory of the Szegő and Bergman kernels from which the phases and leading symbols are taken.","marker":"[10]"},{"why":"introduces the semiclassical spectral asymptotics method for Toeplitz operators on CR manifolds that the paper extends to Levi-elliptic order-one operators.","marker":"[32]"},{"why":"gives the microlocal description of Szegő projections on non-degenerate CR manifolds, including the two-phase form at $q=n_-$.","marker":"[37]"},{"why":"supplies the Levi-ellipticity condition and the Toeplitz parametrix idea for the $(0,0)$-form case that the paper adapts to higher form degrees.","marker":"[26]"}],"fun_headline_variants":["Two oscillatory phases give full Toeplitz spectral asymptotics","CR Toeplitz spectra: exact two-phase semiclassical expansion","Critical form degree yields double-phase kernel asymptotics","Full spectral projector expansion on Levi non-degenerate CR"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole two-phase expansion rests on the quoted structure theorem that the spectral projection onto lower-energy $(0,n_-)$-forms is, modulo smoothing, exactly the sum of two complex-phase oscillatory integral operators $S_-+S_+$ and is smoothing for other degrees; if that decomposition is false or requires additional hypotheses such as $L^2$-closed range, the main expansion loses its stated two-phase form.","fun_headline_variants_meta":{"raw":{"variants":["Two oscillatory phases give full Toeplitz spectral asymptotics","CR Toeplitz spectra: exact two-phase semiclassical expansion","Critical form degree yields double-phase kernel asymptotics","Full spectral projector expansion on Levi non-degenerate CR"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000716,"raw_usage":{"total_tokens":3282,"prompt_tokens":1074,"completion_tokens":2208,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":690,"completion_tokens_details":{"reasoning_tokens":2139}},"tokens_in":690,"tokens_out":2208,"duration_ms":17846,"temperature":1.0,"reasoning_tokens":2139,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:21:58.811887+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a compact CR manifold with a transversal CR circle action and a first-order $P$ satisfying (1.1.3)--(1.1.4), compute the left side of (1.1.11) from the explicitly known Fourier components of the Szegő projection, and compare the $O(k^{n+1})$ diagonal coefficient with $\\int_0^\\infty t^n\\chi(t)\\,dt\\,|\\det L_x|/(2\\pi^{n+1})\\,(v(x)/m(x))$; a mismatch at a single point, or a surviving $k^{n}$ term from the wrong phase, would falsify the leading coefficient of Theorem 1.2.","supporting_citations":[{"cited_title":"Hsiao and G","cited_arxiv_id":null,"evidence_quote":"supplies the microlocal structure theorem for the spectral projection on lower energy forms, the quoted decomposition $\\Pi_\\lambda^{(q)}\\equiv S_-+S_+$ modulo smoothing on which Theorem 1.1 rests."},{"cited_title":"Melin and J","cited_arxiv_id":null,"evidence_quote":"provides the calculus of Fourier integral operators with complex phase and the complex stationary phase formula used to reduce Toeplitz products to two-phase oscillatory integrals."},{"cited_title":"Boutet de Monvel and J","cited_arxiv_id":null,"evidence_quote":"establishes the singularity theory of the Szegő and Bergman kernels from which the phases and leading symbols are taken."},{"cited_title":"Hsiao, Projections in several complex variables, M´em","cited_arxiv_id":null,"evidence_quote":"gives the microlocal description of Szegő projections on non-degenerate CR manifolds, including the two-phase form at $q=n_-$."},{"cited_title":"Galasso and C.-Y","cited_arxiv_id":null,"evidence_quote":"supplies the Levi-ellipticity condition and the Toeplitz parametrix idea for the $(0,0)$-form case that the paper adapts to higher form degrees."}],"review_version":1}