{"id":"05b9acc3-eb32-4a0c-8a3f-bc3baa84b2a6","arxiv_id":"2505.19326","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Martinet sub-Laplacian's quantum limits are described by four adapted two-microlocal measures whose concentration and invariance are governed by quartic and harmonic oscillators and a drift along the Reeb flow.","lead":"This paper proves a decomposition of quantum limit measures for eigenfunctions of a sub-Riemannian Laplacian on a flat toroidal cylinder, describing how oscillations concentrate and propagate at different semiclassical scales. A generalist reader might care because it provides precise phase-space rules for quantum dynamics in a degenerate geometry, with consequences for dispersion and observability in subelliptic evolution equations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5's projection formula doubles the regular part of μ1,k; the stated decomposition cannot hold as written and needs correction before the weak-drift conclusion is accepted.","rationale":"The reader's weakest assumption, non-degeneracy of the critical point of Λ_k, is a real and explicitly stated condition on an auxiliary spectral family; it correctly limits the unconditional validity of Theorems 5 and 6. I do not disagree with that. But I find a more direct internal inconsistency in the statement of Theorem 5: the summed projection identity is false as written, independent of the non-degeneracy issue. This matters because Theorem 6's rigidity conclusion (y-independence of μ1,k) is obtained from the two-microlocal decompositions of Theorem 5. The likely fix is minor — either write the identity for each j separately or use a single indicator excluding both critical points — but until corrected, the weak-drift theorems are not reliable as stated. I do not see a defect that invalidates the main construction of the four measures M1, M2, m^j_3, m4 or the invariance/concentration Theorems 2–4; those appear to be the substantial core of the paper. The proof of Theorem 6 also invokes a formal operator bG_h without full justification, which reinforces the need for a careful rewrite of Section 6. Overall the paper is a serious contribution, and the appropriate verdict remains conditional rather than accept or reject; the requested fixes are localized to the weak-drift part.","tokens_in":41866,"tokens_out":20636,"duration_ms":205768,"concrete_test":"Choose μ1,k = δ_{(y0,z0,η*,ζ*)} with η* distinct from both η^0 and η^1, and take a test symbol a(y,z,η,ζ,v) = φ(y,z,η,ζ) compactly supported near (η*,ζ*) and independent of v. Evaluate the displayed projection identity: the left side equals 1, while the right side equals 2 plus contributions from ν^j and Tr M^j (which vanish away from the critical points). The failure of the identity shows the theorem statement must be corrected; after correcting it, re-check that Theorem 6's conclusion 0 = [D_y^2, M^j_{1,k}] and ∂_y ν^j_{1,k} = 0 still implies y-independence of μ1,k.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 5, after the two-microlocal limit formula, the paper states: μ1,k = Σ_{j∈{0,1}} (1_{η≠η^j} μ1,k + ν^j_{1,k} + Tr M^j_{1,k}). This identity is inconsistent unless the two critical points coincide. If μ1,k has no atom at either η^0 or η^1, then both indicators equal 1 on the full mass of μ1,k away from the two points, so the right-hand side contains twice the regular part of μ1,k, plus the critical contributions. More concretely, testing with a compactly supported function a(y,z,η,ζ) that avoids both critical points and is independent of v: the left side gives ∫ a dμ1,k, while the right side gives 2∫ a dμ1,k plus the ν/TrM terms. The theorem should either state the j-wise decomposition μ1,k = 1_{η≠η^j} μ1,k + ν^j_{1,k} + Tr M^j_{1,k} for each fixed j, or replace the indicators by 1_{η∉{η0,η1}} in the summed formula. Because Theorem 6 derives the y-independence of μ1,k from these decompositions, the erroneous projection identity is load-bearing for the weak-drift part of the central claim. The proof of Theorem 6 also relies on a formal operator limit bG_h and a sketched normal-form argument, so this section is not yet at the same rigor as Theorems 1–4.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies semiclassical quantum limits of the Martinet sub-Laplacian Δ_M = -X_1^2 - X_2^2 on M = R × T^2, with X_1 = ∂_x and X_2 = ∂_y + x^2∂_z. The central claim is that the non-compact part ν_∞ of any quantum limit can be decomposed into four adapted two-microlocal semiclassical measures: an operator-valued measure M_1 associated with the singular set {x=0} and the quartic oscillator H_{η,ζ}, an operator-valued measure M_2 associated with the harmonic oscillator G_σ away from the singular set, and scalar measures m_3^j and m_4 in the subcritical regimes. The paper proves existence of these measures (Theorem 1), their invariance under the relevant oscillator flows (Theorem 2), their support on the level sets λ_k=1, ν_j=1, H_η^j=1, G=1 (Theorem 3), and drift-invariance properties in the y-direction or along the Reeb field (Theorem 4). Under an additional non-degeneracy assumption on critical points of the Montgomery eigenvalues Λ_k, Theorems 5 and 6 claim a refined two-microlocal decomposition of μ_{1,k} near the critical points and y-independence of μ_{1,k}. The proofs use Rothschild-Stein estimates, the Wigner equation, and normal-form/averaging arguments. The weak-drift results are explicitly conditional on a spectral conjecture for intermediate values of k.","tokens_in":2223,"tokens_out":2394,"duration_ms":103637,"significance":"If the main results are correct, the paper provides the first complete two-microlocal description of quantum limits for a step-two sub-Riemannian Laplacian with a non-regular distribution, going beyond the three-dimensional contact case treated in [4, 11]. The decomposition of ν_∞ into four adapted measures with explicit dynamics is a substantial and natural extension of earlier work on the Baouendi-Grushin operator and the Engel group. A particular strength is that the invariance properties are derived from the Wigner equation rather than imposed, and the concentration statements are sharp and falsifiable: they tie the measures to the Montgomery family of quartic oscillators and to the harmonic oscillator. The paper also correctly identifies the abnormal drift as a sub-principal effect and formulates precise conditional statements at critical points. However, the load-bearing parts of the weak-drift section (Theorems 5 and 6) are presented as sketches or with formal operators, and at least one projection identity in Theorem 5 is incorrect as stated. These issues are fixable, but they currently prevent the manuscript from being accepted in its present form.","major_comments":[{"comment":"The stated identity μ_{1,k} = Σ_{j∈{0,1}} (1_{η≠η^j} μ_{1,k} + ν^j_{1,k} + Tr M^j_{1,k}) cannot hold as written. For the k=1 case covered by [20], μ*=0 and hence η^0=η^1, so the two indicators coincide; testing with a compactly supported symbol avoiding η=0 gives twice the regular part of μ_{1,k} on the right. For a general non-degenerate μ*, the same doubling occurs unless μ_{1,k} has atoms at both η^0 and η^1. The correct statement is either the j-wise decomposition μ_{1,k} = 1_{η≠η^j} μ_{1,k} + ν^j_{1,k} + Tr M^j_{1,k} for each fixed j, or the summed formula with the indicator 1_{η∉{η^0,η^1}}. Since the conclusion 'μ_{1,k} is constant in y' in Theorem 6 uses this projection, the error is load-bearing and must be repaired.","section":"Section 6, projection formula after Eq. (40)"},{"comment":"The proof of Theorem 6 relies on the formal averaged operator ⟨w⟩_h, about which the paper explicitly says 'We do not give for the moment a more precise sense to the operator ⟨w⟩_h'. The subsequent cohomological equation (125), the definition of bG_h in (124), the localization operator bE_{h,τ}, and the O(h^3) commutator formula all presuppose a meaningful operator with uniform estimates on the spectral window λ_k(η,ζ)≈1. Without a precise definition, for example via truncated time averages on the eigenspaces of H_h, the derivation of (41) and (42) is not complete. This is a load-bearing gap for the weak-drift part of the central claim.","section":"Section 6, proof of Theorem 6"},{"comment":"The proof of Theorem 5 is presented only as a sketch ('This proof is standard ... and we only sketch it'). The successive limits d→0, r→∞, R→∞, ε→0 and the extraction of subsequences leading to the new measures ν^j_{1,k} and M^j_{1,k} are not written in detail, and positivity of these measures is not established. Since these measures are new objects introduced in Theorem 5, the existence statement needs either a complete proof or a precise reduction to the arguments of Proposition 3.","section":"Section 6, proof of Theorem 5"},{"comment":"Equation (16) states ν_∞ = ∫_{Rη×R*_ζ} Tr dM_1 + ∫_{R*_σ} Tr dM_2 + Σ_j ∫ dm^j_3 + ∫ dm_4. As written, this compares a measure on M = R_x × T^2_{y,z} with expressions that, even after taking traces, are still measures on different spaces such as T^2_{y,z}×Rη×R*_ζ or M\\S×R*_σ. The equality should specify the relevant pushforwards or projections, for example in the weak form ∫ b dν_∞ = ∫_{T^2×Rη×R*_ζ} b(0,y,z) Tr dM_1 + ..., as is done informally in the proof at the end of Section 2. The statement of Theorem 1 should be corrected accordingly.","section":"Theorem 1, Eq. (16)"},{"comment":"Positivity of the scalar measures m^j_3 and m_4 is asserted but the proofs say 'the details are omitted' and 'positivity follows by similar arguments'. Since Theorem 1 claims these are positive Radon measures, a positivity argument should be supplied or explicitly reduced to the arguments already given for M_1 and M_2 in Propositions 3 and 4. Positivity is part of the definition of the objects whose existence is the central claim.","section":"Propositions 5 and 6"}],"minor_comments":[{"comment":"There are numerous typos and minor grammatical errors, for example 'Rotchschild' (p. 1), 'sepectrum' (p. 6), 'estabilshed' (p. 36), 'fruther' (p. 36), 'respcetively' (p. 41), 'adpatation' (p. 42), and 'Multyplier' (p. 26). A careful proofreading pass is recommended.","section":"Throughout"},{"comment":"In the proof of Proposition 3, the measure M_1 is first written as an element of M(T^2_{y,z} × Rξ × Rζ; ...), but the correct integration variable in the second factor is η, not ξ. This appears to be a typo, since the final statement uses Rη × R*_ζ.","section":"Proposition 3 proof"},{"comment":"In the final paragraph of Section 2, the displayed limit 'lim_{R→0}' should be 'lim_{R→+∞}'. The surrounding text and the definition of I^∞_{h,R} make the intended meaning clear.","section":"End of proof of Theorem 1"},{"comment":"The condition on the neighborhood B^j reads 'λ_k(η,ζ) ≠ 1, ∀k ≠ k', which is confusing because k is also the fixed index. This should be written as 'λ_l(η,ζ) ≠ 1 for all l ≠ k'.","section":"Theorem 5 statement"},{"comment":"The non-degeneracy assumption used in Theorems 5 and 6 is verified by [20] only for k=1 and for all sufficiently large k; for intermediate k, including k=2, it rests on the numerical conjecture [20, Conj. 1.6]. The manuscript states this explicitly, so this is not an error, but the abstract's phrase 'additional regularity properties' should make the conditional nature of the weak-drift results more prominent.","section":"Section 1.2.4 and Remark 6"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and represents a serious advance if the technical gaps are closed. I see no novelty or attribution concerns: the reliance on [4] and [7] is methodological and acknowledged. The main risk is the rigor gap in Section 6, especially the formal averaged operator in Theorem 6 and the incorrect projection formula in Theorem 5. These are local but load-bearing, so major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of this paper is strong. The author builds adapted two-microlocal semiclassical measures for the Martinet sub-Laplacian, a step-two structure with a singular set, and obtains a four-measure decomposition of the non-compact part of quantum limits. The quartic-oscillator invariance at the singular set, the harmonic-oscillator invariance away from it, and the abnormal Reeb drift are genuinely new and convincingly derived. Theorems 1–4 appear solid; the main calculations are detailed and the reliance on the author's earlier normal-form method is methodological rather than circular. The paper is honest about the non-degeneracy assumption in Section 1.2.4 and flags the numerical conjecture for k=2. For the core results alone, this paper deserves referee time.\n\nThat said, the stress-test note is correct. The projection identity in Theorem 5, μ1,k = Σ_{j∈{0,1}} (1_{η≠η^j} μ1,k + ν^j_{1,k} + Tr M^j_{1,k}), doubles the regular part of μ1,k away from the two critical points. Since ν^j and Tr M^j are supported at η=η^j, the right-hand side contains twice the full mass of μ1,k outside the two points. The correct identity should either be stated for each j separately as μ1,k = 1_{η≠η^j} μ1,k + ν^j_{1,k} + Tr M^j_{1,k}, or the summed version should read μ1,k = 1_{η∉{η^0,η^1}} μ1,k + Σ_j (ν^j_{1,k} + Tr M^j_{1,k}). As written, the theorem is false. This is not a minor typo: the proof of Theorem 6 derives y-independence of μ1,k from this projection, so the weak-drift conclusion does not follow without correction. The sketched proof of Theorem 5, the formal operator bG_h in Theorem 6, and the omitted positivity proofs for m^j_3 and m4 are secondary issues but should be tightened in a revision.\n\nMy overall take: the main quantum-limit results are an important step forward and the paper is worth serious refereeing, but the weak-drift section needs a real fix, not just cosmetic editing. I would recommend acceptance contingent on correcting the projection formula and re-checking the proof of Theorem 6.","headline":"First adapted two-microlocal description for the Martinet sub-Laplacian is a real contribution, but the weak-drift projection identity in Theorem 5 is wrong as written and needs correcting before the y-independence conclusion can stand.","tokens_in":42697,"tokens_out":6255,"would_cite":false,"duration_ms":54529,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P20","35H20","58J51","81Q20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves every quantum limit of the Martinet sub-Laplacian decomposes into four adapted two-microlocal measures, with quartic- and harmonic-oscillator dynamics and drift-invariance laws at critical points.","keywords":["Martinet sub-Laplacian","quantum limits","two-microlocal semiclassical measures","quartic oscillator","Montgomery family","Rothschild-Stein estimates","sub-Riemannian geometry","drift invariance"],"falsifier":"A decisive calculation is the second eigenvalue $\\Lambda_2(\\mu)$ of $D_x^2 + (\\mu + x^2)^2$: the paper leaves the non-degeneracy of its critical point as a numerical conjecture. If $\\Lambda_2''(\\mu^*) = 0$ at the critical point $\\mu^*$, the decomposition of $\\mu_{1,2}$ into $\\nu^j_{1,2}$ and $M^j_{1,2}$ at the points $(\\eta^j,\\zeta^j)$ with $\\partial_\\eta\\lambda_2 = 0$, and the resulting $y$-independence of $\\mu_{1,2}$, do not follow. A dynamical check: simulate the Schrödinger flow $e^{it\\Delta_M}$ on a spectrally truncated cylinder with data on the $k$-th eigenspace and test whether the long-time spatial density on the singular set is $y$-independent.","tokens_in":41663,"feed_emoji":"⚛️","tokens_out":14878,"duration_ms":115347,"temperature":0.7,"pith_summary":"The paper's goal is a complete asymptotic description of the quantum limits of the Martinet sub-Laplacian on $M = \\mathbb{R} \\times \\mathbb{T}^2$, the step-two subelliptic operator $-\\partial_x^2 - (\\partial_y + x^2\\partial_z)^2$ whose bracket structure degenerates on the singular set $x = 0$. The central claim is that the noncompact part of every accumulation point of eigenfunction densities decomposes into four positive two-microlocal measures, one for each oscillation scale fixed by the Rothschild–Stein estimates. The highest-frequency regime, concentrated on the singular set, is governed by the quartic Montgomery oscillator $\\hat{H}_{\\eta,\\zeta} = D_x^2 + (\\eta + x^2\\zeta)^2$, while the regime away from the singular set is governed by the harmonic oscillator $\\hat{G}_\\sigma = \\sigma^2 D_w^2 + w^2$; two scalar measures cover the subcritical scales. The paper proves that these measures concentrate on the level sets $\\lambda_k = 1$ and $\\nu_j = 1$ and satisfy drift-invariance laws, including Reeb-flow invariance away from the singular set and an abnormal drift, with a further two-microlocalization at non-degenerate critical points yielding $y$-independence of the singular-regime spectral measures. If correct, quantum-limit questions for this non-regular step-two operator reduce to the spectral theory of one quartic oscillator family, and the obstructions to dispersion are located exactly at its critical points.","feed_headline":"Every Martinet eigenfunction limit splits into four measures","feed_subtitle":"New two-microlocal measures reveal the hidden noncompact mass and the quartic-oscillator dynamics behind it.","key_machinery":"The load-bearing object is a new family of adapted two-microlocal semiclassical measures, extracted from rescaled Wigner distributions $\\langle \\mathrm{Op}_h^w(a_{h,R})\\psi_h, \\psi_h\\rangle$ whose symbols read $a(x,y,z,w,\\xi,h\\sigma,h^2\\zeta)$ with $w = \\eta + x^2\\zeta$ and $\\sigma = 2x\\zeta$; the cut-off $\\breve\\chi(\\zeta/R)$ discards the compact part of phase space. The construction uses dilation–contraction symplectic changes of variables that rescale $x$ by $h$, $(y,z)$ by $h^{1/2}$, and $(\\xi,\\eta,\\zeta)$ by $h^{-1}$, $h^{-1/2}$, $h^{-1/2}$, matching the Rothschild–Stein estimates $hX_j \\lesssim 1$, $h^2X_3 \\lesssim 1$, $h^3X_4 \\lesssim 1$ so that each regime converges to its own operator-valued measure. The effective quantum models are the Montgomery quartic oscillator $\\hat{H}_\\mu = D_x^2 + (\\mu + x^2)^2$ with simple eigenvalues $\\Lambda_k(\\mu)$, governing the singular set, and the harmonic oscillator $\\hat{G}_\\sigma = \\sigma^2D_w^2 + w^2$ with eigenvalues $|\\sigma|(2j+1)$, governing the regular region. Drift invariance comes from a normal-form and averaging procedure in complex coordinates $Z = \\xi + iw$ that cancels the fast oscillation and isolates the subprincipal transport term, which is the Reeb flow away from the singular set and the abnormal $\\partial_y$-drift with coefficient $\\partial_\\eta\\lambda_k$ (or the elliptic-integral average $\\Upsilon^j(\\eta)$) near it.","core_discovery":"The paper proves that for the Martinet sub-Laplacian on the flat toroidal cylinder the noncompact part of any quantum limit splits as $\\nu_\\infty = \\int \\operatorname{Tr} dM_1 + \\int \\operatorname{Tr} dM_2 + \\sum_{j\\in\\{0,1\\}}\\int dm_3^j + \\int dm_4$, where $M_1$, $M_2$, $m_3^j$, $m_4$ are positive Radon measures obtained as limits of rescaled Wigner distributions at four different semiclassical scales (Theorem 1). The high-oscillation dynamics is explicit: $M_1$ commutes with the quartic oscillator $\\hat{H}_{\\eta,\\zeta} = D_x^2 + (\\eta + x^2\\zeta)^2$ on $L^2(\\mathbb{R}_x)$, $M_2$ commutes with the harmonic oscillator $\\hat{G}_\\sigma = \\sigma^2 D_w^2 + w^2$, and the scalar measures $m_3^j$ and $m_4$ are invariant under the classical flows of $H^j_\\eta(\\varsigma,\\xi) = \\xi^2 + (\\eta + (-1)^j\\varsigma^2)^2$ and $G(w,\\xi) = \\xi^2 + w^2$ (Theorem 2). Concentration holds on $\\lambda_k(\\eta,\\zeta) = 1$ and $\\nu_j(\\sigma) = (2j+1)|\\sigma| = 1$ (Theorem 3). At the subprincipal scale the paper obtains the drift laws $0 = \\partial_\\eta\\lambda_k\\,\\partial_y\\mu_{1,k}$, $0 = (2j+1)Z\\mu_{2,j}$, $0 = \\Upsilon^j(\\eta)\\partial_y m_3^j$, $0 = Zm_4$, with $Z = X_3 - x^{-1}X_2$ the Reeb field and $\\Upsilon^j$ an average computed by complete elliptic integrals (Theorem 4). Under the additional assumption that the critical point of $\\Lambda_k$ is non-degenerate, a further two-microlocalization near each point $(\\eta^j,\\zeta^j)$ with $\\partial_\\eta\\lambda_k = 0$ decomposes $\\mu_{1,k}$ into pieces satisfying $\\partial_y\\nu^j_{1,k} = 0$ and $[D_y^2, M^j_{1,k}] = 0$, so that $\\mu_{1,k}$ is independent of $y$ (Theorems 5 and 6).","pith_inferences":["If some eigenvalue $\\Lambda_k(\\mu)$ ever had a degenerate critical point, the $y$-independence of $\\mu_{1,k}$ would fail, but a slower, higher-order drift along $y$ would likely persist; testing the $k = 2$ case numerically (left open by the paper) is the direct way to probe this.","The same four-measure architecture plausibly transfers to other step-two sub-Riemannian structures with a codimension-one singular locus, with a Montgomery-type quartic family in the transverse variable as the universal singular model; the paper does not claim this.","A numerical Schrödinger simulation on a truncated cylinder, with initial data concentrated on the $k$-th spectral band of the fiber operator, could measure the drift speed $\\partial_\\eta\\lambda_k$ against the paper's prediction and test the $y$-independence at critical points."],"forward_implications":["Quantum-limit questions for the Martinet sub-Laplacian reduce to spectral data of one quartic family: the singular-regime measures are fibered over the simple eigenvalues $\\lambda_k(\\eta,\\zeta)$, so finer knowledge of $\\Lambda_k(\\mu)$ translates directly into finer control of eigenfunction limits.","The obstructions to dispersion are located exactly at the critical points of $\\Lambda_k(\\mu)$: at the two points $(\\eta^j,\\zeta^j)$ on the level set $\\lambda_k = 1$ where $\\partial_\\eta\\lambda_k$ vanishes the $y$-drift stops, and by Theorems 5 and 6 the singular-regime measure becomes $y$-independent there.","Away from the singular set the noncompact part of a quantum limit is invariant under the Reeb flow $Z$, the step-two analogue of the Reeb invariance theorem known in the three-dimensional contact case.","On the compact torus version of the operator, with $V(x) = \\sin^2(x/2)$, the same decomposition holds, so complete quantum limits exist on a compact Martinet-type manifold.","The abnormal drift term $\\partial_\\eta\\lambda_k\\,\\partial_y\\mu_{1,k}$ is the quantum counterpart of the abnormal propagation along Martinet singular geodesics previously found at the classical level."],"supporting_citations":[{"why":"Supplies the Rothschild–Stein sub-elliptic estimates that fix the three oscillation scales ($h$, $h^2$, $h^3$) around which the two-microlocal measures are built.","marker":"[30]"},{"why":"Provides the spectral structure of the quartic oscillator; Proposition 1 of the paper is [6, Corollary B.4], giving the eigenvalues $\\lambda_k(\\eta,\\zeta) = |\\zeta|^{2/3}\\Lambda_k(\\eta\\operatorname{sgn}(\\zeta)/|\\zeta|^{1/3})$.","marker":"[6]"},{"why":"Provides the critical-point theory of the Montgomery eigenvalues $\\Lambda_k(\\mu)$ (uniqueness and non-degeneracy for $k = 1$ and all large $k$, with the $k = 2$ conjecture) on which Theorems 5 and 6 and the dispersion obstruction rest.","marker":"[20]"},{"why":"The three-dimensional contact quantum-limit theorem whose Reeb-flow invariance is the model for the drift invariance in the regular regime.","marker":"[11]"},{"why":"Source of the two-microlocal measure technique for perturbed contact sub-Laplacians and of the normal-form and averaging procedure in complex coordinates reused in Section 5.2.","marker":"[4]"},{"why":"Introduced the two-microlocalization near critical points of the dispersion relation that Theorems 5 and 6 adapt to the singular-set regime.","marker":"[25]"},{"why":"Identified the abnormal propagation along Martinet singular geodesics that the paper's drift term $\\partial_\\eta\\lambda_k\\partial_y\\mu_{1,k}$ is the quantum manifestation of.","marker":"[10]"},{"why":"The Engel-group analogue whose weak-invariance refinement is the template for Theorems 5 and 6.","marker":"[7]"}],"fun_headline_variants":["Four-measure split for Martinet quantum limits","Martinet sub-Laplacian limits: a four-measure decomposition","Two-microlocal measures crack Martinet eigenfunctions","Quartic oscillators behind Martinet eigenfunction limits","Splitting into four scales: Martinet quantum limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption, stated in Section 1.2.4, is that the critical point of $\\Lambda_k(\\mu)$ used in the refined analysis is non-degenerate: the paper proves this for $k = 1$ and for all sufficiently large $k$ via [20, Th. 1.2 and 1.5], and only conjectures it numerically for $k = 2$, so if non-degeneracy fails for some intermediate $k$ the $y$-independence conclusion for $\\mu_{1,k}$ does not follow from the proof.","fun_headline_variants_meta":{"raw":{"variants":["Four-measure split for Martinet quantum limits","Martinet sub-Laplacian limits: a four-measure decomposition","Two-microlocal measures crack Martinet eigenfunctions","Quartic oscillators behind Martinet eigenfunction limits","Splitting into four scales: Martinet quantum limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000483,"raw_usage":{"total_tokens":2514,"prompt_tokens":1202,"completion_tokens":1312,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":818,"completion_tokens_details":{"reasoning_tokens":1232}},"tokens_in":818,"tokens_out":1312,"duration_ms":10717,"temperature":1.0,"reasoning_tokens":1232,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:16:39.359619+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive calculation is the second eigenvalue $\\Lambda_2(\\mu)$ of $D_x^2 + (\\mu + x^2)^2$: the paper leaves the non-degeneracy of its critical point as a numerical conjecture. If $\\Lambda_2''(\\mu^*) = 0$ at the critical point $\\mu^*$, the decomposition of $\\mu_{1,2}$ into $\\nu^j_{1,2}$ and $M^j_{1,2}$ at the points $(\\eta^j,\\zeta^j)$ with $\\partial_\\eta\\lambda_2 = 0$, and the resulting $y$-independence of $\\mu_{1,2}$, do not follow. A dynamical check: simulate the Schrödinger flow $e^{it\\Delta_M}$ on a spectrally truncated cylinder with data on the $k$-th eigenspace and test whether the long-time spatial density on the singular set is $y$-independent.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Rothschild–Stein sub-elliptic estimates that fix the three oscillation scales ($h$, $h^2$, $h^3$) around which the two-microlocal measures are built."},{"cited_title":"Bahouri, D","cited_arxiv_id":null,"evidence_quote":"Provides the spectral structure of the quartic oscillator; Proposition 1 of the paper is [6, Corollary B.4], giving the eigenvalues $\\lambda_k(\\eta,\\zeta) = |\\zeta|^{2/3}\\Lambda_k(\\eta\\operatorname{sgn}(\\zeta)/|\\zeta|^{1/3})$."},{"cited_title":"Helffer and M","cited_arxiv_id":null,"evidence_quote":"Provides the critical-point theory of the Montgomery eigenvalues $\\Lambda_k(\\mu)$ (uniqueness and non-degeneracy for $k = 1$ and all large $k$, with the $k = 2$ conjecture) on which Theorems 5 and 6 and the dispersion obstruction rest."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The three-dimensional contact quantum-limit theorem whose Reeb-flow invariance is the model for the drift invariance in the regular regime."},{"cited_title":"Arnaiz and G","cited_arxiv_id":null,"evidence_quote":"Source of the two-microlocal measure technique for perturbed contact sub-Laplacians and of the normal-form and averaging procedure in complex coordinates reused in Section 5.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the two-microlocalization near critical points of the dispersion relation that Theorems 5 and 6 adapt to the singular-set regime."},{"cited_title":"Colin de Verdière and C","cited_arxiv_id":null,"evidence_quote":"Identified the abnormal propagation along Martinet singular geodesics that the paper's drift term $\\partial_\\eta\\lambda_k\\partial_y\\mu_{1,k}$ is the quantum manifestation of."},{"cited_title":"Benedetto","cited_arxiv_id":null,"evidence_quote":"The Engel-group analogue whose weak-invariance refinement is the template for Theorems 5 and 6."}],"review_version":1}