{"id":"03048cd9-268c-4af0-89a7-75d9b7f3f7e1","arxiv_id":"2607.26551","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"This paper uses Random Duality Theory to give an alternative proof of Lehner's deterministic spectral edge formula for Kronecker-Gaussian matrices.","lead":"This paper gives a new proof of Lehner's formula for the spectral edges of Kronecker-Gaussian random matrices, using Random Duality Theory instead of spectral and free-probability methods. It aims to show that RDT's comparison and replicated-system machinery can recover known strong asymptotic freeness results exactly.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lower bound hinges on unproved matrix extension of RDT tightness principle (48)-(49); without it the equality lim Eλ_n = ρ_n is not established.","rationale":"The reader's weakest_assumption correctly identifies the RDT tightness principle as the load-bearing point. My read of the proof confirms this: the upper bound (Theorem 1 plus the Γ-minimization) is a valid Slepian-type comparison, and Theorem 5's verification of (92) appears algebraically sound. However, the step from (92) to the equality of interpolation limits (93)-(94) is purely an import. The paper states that the 'remaining parts of the [53,54] methodologies automatically extend', but for a matrix-valued overlap Q the usual scalar overlap order parameter becomes a k×k contraction, and the arguments of [53,54] do not obviously carry over. In particular, the strict inequality (92) controls the free-energy gap, but the equivalence in (49) is the statement that this gap implies tightness; this is a nontrivial theorem in the scalar case and cannot be assumed. The auxiliary issue of Γ's positive definiteness is real but secondary; it can be fixed by adding Γ≻0, as the optimum is positive definite when K is invertible. Because the lower bound is essential to (6), the paper should be accepted only conditionally, pending a self-contained derivation of (49) or a direct proof of (93). That is the same verdict as the reader, so no change.","tokens_in":19839,"tokens_out":17841,"duration_ms":144139,"concrete_test":"Independently re-derive the implication (92) ⇒ (93) for the matrix-valued case using only the published statements of Talagrand (Theorems 5.2 in [53] and 2.4 in [54]) and Gordon (Theorem 2 of [23]), and verify the extension of the overlap-order-parameter equations to noncommuting F_i. Specifically, check whether the positivity of the quadratic form in (42) for all Q∈Q, Q≠I, is sufficient to prove the strict superadditivity of the limiting free energy in the presence of a matrix-valued overlap. If the derivation cannot be completed without an additional assumption, the lower bound is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (6) is proved only if the interpolation limits coincide: lim 1/√n E D(1) = lim 1/√n E D(0), equations (93)-(94). The paper's only route to this equality is the imported equivalence (48)-(49), which asserts that a strictly positive gap between the two-replica and twice single-system free energies forces tightness of the interpolation. This principle is not derived here; it is taken from self-cited preprints [47,48,51] and Talagrand [53,54], and its extension to the matrix-valued Stiefel setting with noncommuting F_i is asserted in one sentence: 'the remaining parts of the [53,54] methodologies automatically extend'. That is precisely the point requiring proof: for matrix overlaps Q = (X(1))^T X(2), the overlap is a k×k contraction rather than a scalar, replica symmetry breaking could occur at the matrix level, and the second-moment control in (22) no longer suffices to rule out overlap dependence. If (48)-(49) fail for noncommuting F_i, the lower-bound argument in Section 3.5 collapses even though Theorem 5's inequality (92) may hold. The paper also leaves implicit that Γ in (32)-(37) must be positive definite; without this constraint the dual minimization is unbounded, and the substitution Γ→√n Γ in Theorem 4's proof is not justified for singular K.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a proof, based on Random Duality Theory (RDT), of Lehner's deterministic spectral-edge formula for Kronecker–Gaussian matrices of the form H = A0⊗I + n^{-1/2} Σ Ai⊗G_i. The main claim is that lim_{n→∞} Eλ_n(H) = ρ_n, where ρ_n is the deterministic SDP value appearing in (5)–(6) and (38). The upper bound lim Eλ_n(H) ≤ ρ_n is obtained through a Slepian/Gordon comparison of Gaussian processes followed by a Lagrangian dual calculation. The lower bound is attempted through a replicated-system argument, Section 3.5, whose key step is an RDT 'tightness' equivalence stated in (48)–(49). The paper concludes that this also reproves the strong asymptotic freeness edge results of [25,43].","tokens_in":20159,"tokens_out":30137,"duration_ms":235443,"significance":"If the proof were complete, the paper would provide a genuinely different route to a known and important result: Lehner's formula for the spectral edges of Kronecker–Gaussian matrices would follow from RDT rather than from free probability or spectral methods. The upper-bound half is essentially rigorous and the algebra in Theorem 5 is internally coherent. The paper contains no fitted parameters and the final formula is explicit and falsifiable. However, the lower-bound half, which is the load-bearing part, depends on a strong RDT tightness principle that is imported from self-cited preprints and asserted to extend to matrix-valued overlaps without proof. As written, the central equality is therefore not established.","major_comments":[{"comment":"The lower-bound argument rests on the equivalence min_{Q∈Q} (lim 2·n^{-1/2} E D(0) − lim n^{-1/2} E D^(2)(t)) > 0 ⇔ lim n^{-1/2} E D(1) = lim n^{-1/2} E D(0). This is asserted, not proved. The text says that 'the remaining parts of the [53,54] methodologies automatically extend', but for k>1 the overlap Q = (X^(1))^T X^(2) is a k×k contraction rather than a scalar, replica symmetry breaking could occur at matrix level, and the second-moment control in (22) does not by itself rule out overlap dependence. Since (93)–(94), and hence the final equality lim Eλ_n(H) = ρ_n, follow only through this equivalence, the central claim is unsupported as written. A self-contained proof of (49) in this matrix-valued Stiefel setting, or a precise reduction to the scalar case, is required.","section":"§3.5, Eqs. (48)–(49)"},{"comment":"The dual minimization is over Γ = Γ^T with no positive-definiteness restriction. For the scalar analogue, min_{γ≠0} (a/γ + γ) is unbounded below, and for matrices the same phenomenon occurs when Γ is allowed to be indefinite. The optimality condition (37) and the substitution Γ → √n Γ in Theorem 4's proof presuppose Γ ≻ 0 and invertible. The domain must be explicitly restricted to Γ ≻ 0 (or Γ ⪰ 0 with a limiting argument), and the existence of the minimizer should be stated.","section":"§3.3, Eqs. (32)–(37)"},{"comment":"The main theorem as stated is not dimensionally consistent. H ∈ R^{nk×nk} has nk eigenvalues, but (4) lists λ_1 ≤ ... ≤ λ_n, and (6) writes lim_{n→∞} λ_n(H) = ρ_n. With λ_i defined as the i-th smallest eigenvalue, λ_n(H) is the n-th smallest among nk eigenvalues, not the maximum. The object in (7) is max over S^{nk}, i.e., the largest eigenvalue λ_{nk}(H). The statements in (6) and throughout should use the correct eigenvalue index, or λ_n(H) should be explicitly redefined as the maximal eigenvalue.","section":"§2.1, Eqs. (4)–(7) and (6)"},{"comment":"The displayed definition of ρ_n and ρ_1 is not a well-defined matrix expression as written: A0⊗I is nk×nk, while Z and A_i Z^{-1} A_i (with A_i being k×k) cannot be added to it. The later SDP formula in (38), taken from [18], is clear, but (5) needs to be corrected to that representation or to the intended Schur-complement form so that the central object is defined precisely.","section":"§2.1, Eq. (5)"},{"comment":"The limit n→∞ is interchanged with the minimization over Γ (and Λ) without justification. The law of large numbers gives convergence of (1/n)Σ (F_i G_i)(F_i G_i)^T to K, but the minimizer depends on n, and the parameter domain is unbounded. Before writing the displayed limits, one needs uniform concentration over Γ (and Λ), or an epsilon-net argument. Without this, the upper bound is not fully rigorous.","section":"§3.3 and §3.5.2, Eqs. (33)–(34), (95)–(96)"}],"minor_comments":[{"comment":"In the sentence following (23), '(X^(1))^T X^(1) = 0' should be '= I'.","section":"§3.2, Eq. (23)"},{"comment":"The set Q_F is garbled: the condition '>0∈R^{n×k}' is not meaningful, and the set should be defined as a subset of R^{k×k} (or with a precise trace inequality).","section":"§3.5, Eq. (42)"},{"comment":"The second labeled line for θ̄_i,2 should be θ̄_i,4; the current label duplicates the second term.","section":"§3.5.1, Eq. (61)"},{"comment":"The notation ∥(I−Q)F_i∥²_F is not literally the same as the preceding trace tr((I−Q^T)F_i(I−Q)F_i); the latter equals ∥F_i^{1/2}(I−Q)F_i^{1/2}∥²_F. Please clarify the notation.","section":"§3.5.3, Eq. (107)"},{"comment":"The sentence 'we assume that A_i's are such that inf can be replaced by max' is vague; specify the nondegeneracy condition that guarantees attainment.","section":"§2.1, after Eq. (5)"},{"comment":"There are several typographical issues, e.g., 'eignevalue' after (4), 'pne obtains' in §3.5.1, and the limiting notation in (41) and (65) without explicit existence statements.","section":"Throughout"},{"comment":"The claim of 'effectively reproving the key asymptotic freeness results obtained via spectral methods in [25,43]' is stronger than what is shown: the paper treats the specific Kronecker–Gaussian sum (2) and its spectral edge, not general polynomials in independent Gaussian matrices.","section":"Abstract and §4"}],"recommendation":"major_revision","confidential_remarks":"The decisive lower-bound input is imported from the author's own preprints [47,48,51] and the equivalence (49) is the heart of the proof. I would ask the authors to provide a self-contained proof of (49) in the matrix-valued setting, or to cite a published version with explicit theorem numbers. The paper would also benefit from correcting the dimension and eigenvalue-index issues before further review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a serious attempt to reprove Lehner's spectral-edge formula for Kronecker-Gaussian matrices using Random Duality Theory, but the central equality is not actually proved. The upper bound is rigorous and elegant; the lower bound rests on a matrix-valued extension of a replica tightness principle that the paper asserts rather than proves.\n\nWhat is genuinely new: the paper recasts λ_n(H) as a Gaussian optimization over the Stiefel manifold, proves the upper bound via a Slepian comparison in Theorem 1, and reduces the missing lower bound to a concrete algebraic inequality. Theorem 5 is the real work: it checks the strict inequality (92) between the two-replica and twice-single-system free energies. The algebra there is intricate and, as far as I can see, internally consistent. It is also honest about the status of the formula: equations (5)-(6) are Lehner's, and the paper credits [31] and the strong asymptotic freeness literature.\n\nWhere it falls short: the jump from (92) to lim Eξ = lim EL is mediated by (48)-(49), an equivalence imported from [47,48,51,53]. The paper says the remaining parts of [53,54] 'automatically extend' to the matrix setting. That is the crux. For scalar overlaps the Talagrand/Parisi machinery is known; here Q=(X1)^T X2 is a k×k contraction, the F_i do not commute, and the second-moment argument in (22) does not control overlap dependence. If (48)-(49) fail, the lower bound in Section 3.5 collapses even though Theorem 5 may hold. The Γ issue is smaller but real: the dual minimization in (32)-(37) is unbounded below unless Γ is restricted to be positive definite, and the scaling Γ→√n Γ needs that constraint. Also, the conclusion claims to 'reestablish' the full strong asymptotic freeness results of [25,43]; what is actually handled is the Kronecker-Gaussian spectral edge with positive semidefinite A_i, which is a special case.\n\nNet: the paper is not ready as a proof. But the RDT approach to spectral edges is interesting, the upper bound is clean, and Theorem 5 is a substantial algebraic contribution. If the author can supply a self-contained proof of the matrix-valued tightness principle, the paper would be a genuine alternative route. I would send it to a serious referee, expecting major revision, rather than desk reject.","headline":"A careful RDT reproof of a known spectral-edge formula; the upper bound is solid, but the lower bound relies on an unproved matrix-valued replica tightness principle, so the main equality is not established.","tokens_in":20624,"tokens_out":12234,"would_cite":false,"duration_ms":96497,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","46L54","15B52","60G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The asymptotic largest eigenvalue of a Kronecker–Gaussian matrix is exactly a deterministic max-over-positive-semidefinite value, and a comparison-based random-duality argument proves it without spectral methods.","keywords":["Kronecker–Gaussian matrices","strong asymptotic freeness","random duality theory","spectral edge","Gaussian comparison principle","replica overlap tightness","semicircular free operators","matrix concentration"],"falsifier":"Evaluate condition (92) for a concrete instance: fix $k,l$, positive-semidefinite $A_i$, an admissible overlap $Q\\in\\mathcal{Q}$, and $t\\in(0,1)$, and compute $\\min_{\\Gamma=\\Gamma^T,\\Lambda=\\Lambda^T}\\bar L^{(2)}$ and $2\\min_{\\Gamma=\\Gamma^T}\\bar L$ from (90) and (35). If the former is not strictly smaller than the latter, the proof's tightness condition fails. A sharper check: find any admissible $Q$ and $t$ for which the $\\Lambda$-derivative in (106) vanishes at $(\\tilde\\Gamma/2,0)$, since Theorem 5's contradiction argument requires that derivative to be nonzero.","tokens_in":19619,"feed_emoji":"🎲","tokens_out":13371,"duration_ms":104846,"temperature":0.7,"pith_summary":"The paper takes on a known deterministic formula: for a Kronecker–Gaussian matrix $H=A_0\\otimes I+\\frac{1}{\\sqrt{n}}\\sum_{i=1}^l A_i\\otimes \\bar G_i$ with symmetric positive-semidefinite $A_i$ and symmetric Gaussian $\\bar G_i$, the limiting largest eigenvalue should be $\\rho_n=\\max_{S\\succeq 0,\\ \\mathrm{tr}(S)=1}\\left(\\mathrm{tr}(A_0S)+2\\,\\mathrm{tr}\\sqrt{\\sum_{i=1}^l (A_iS)^2}\\right)$. The paper's objective is to prove this equality without free probability or spectral methods, using only comparison principles from Random Duality Theory. A sympathetic reading is that RDT's random-dual upper bound and its two-replica-cannot-double lower-bound principle are enough to pin the edge exactly, thereby rederiving the strong asymptotic freeness edge results of [25,43] as a byproduct.","feed_headline":"RDT proves the exact spectral edge of Kronecker–Gaussian matrices","feed_subtitle":"A comparison-based proof reaches the deterministic edge formula without free probability or spectral methods","key_machinery":"The machinery is the four-step RDT protocol adapted to the matrix-valued setting. First, $\\lambda_n(H)$ is rewritten as $\\max_{\\mathrm{tr}(R^TR)=1}(\\mathrm{tr}(F_0)+\\frac{\\sqrt{2}}{\\sqrt{n}}\\xi)$ with $F_i=R A_i R^T$ and $\\xi=\\max_{X^TX=I}\\sum_i \\mathrm{tr}(X^TG_iXF_i)$. Second, a Gaussian-process comparison inequality bounds $\\xi$ by the random dual $L=\\max_{X^TX=I}\\sum_i\\sqrt{2}\\,\\mathrm{tr}(G_i^{(1)}XF_i)$, giving the upper bound. Third, the dual is handled by the Lagrangian identity $L=\\sqrt{2}\\min_{\\Gamma=\\Gamma^T}\\left(\\frac14\\mathrm{tr}(\\sum_i F_iF_i^T\\Gamma^{-1})+\\mathrm{tr}(\\Gamma)\\right)$, whose large-$n$ limit is $\\min_{\\Gamma}(\\mathrm{tr}(K\\Gamma^{-1})+\\mathrm{tr}(\\Gamma))$ with $K=\\sum_i F_iF_i^T$ and optimizer $\\tilde\\Gamma=\\sqrt{K}$. Fourth, the missing lower bound is reduced to condition (92): $\\min_{\\Gamma,\\Lambda}\\bar L^{(2)}<2\\min_\\Gamma\\bar L$ over admissible overlaps $Q$, which the paper proves by contradiction using the $\\Lambda$-derivative at $(\\tilde\\Gamma/2,0)$. The load-bearing object is the two-replica overlap $Q$ and the strict inequality that certifies the two-replica-cannot-double principle.","core_discovery":"The central claim is that $\\lim_{n\\to\\infty}\\mathbb{E}\\lambda_n(H)=\\rho_n$ (and consequently $\\lim_{n\\to\\infty}\\mathbb{E}\\lambda_1(H)=\\rho_1$ by Gaussian sign symmetry), where $\\rho_n$ is the deterministic maximum over $S\\succeq 0$, $\\mathrm{tr}(S)=1$, of $\\mathrm{tr}(A_0S)+2\\,\\mathrm{tr}\\sqrt{\\sum_i(A_iS)^2}$, matching the formula of [31] for the semicircular free counterpart. The proof is a sandwich: the upper bound comes from comparing the Gaussian process defining $\\lambda_n(H)$ with a simpler random-dual process and minimizing a Lagrangian over a dual variable $\\Gamma$; the matching lower bound comes from the RDT tightness principle that a two-copy system with any nontrivial overlap $Q$ has strictly less than twice the single-copy free energy, which forces the interpolated upper and lower limits in (41) to coincide. The paper therefore claims to reconfirm the spectral-edge formula of [31] and to reprove the strong-asymptotic-freeness edge statements of [25,43] without spectral theory.","pith_inferences":["If the overlap-tightness principle tolerates noncommuting $F_i$ and indefinite $A_i$, the comparison route could prove spectral-edge formulas for ensembles outside the reach of current free-probability tools, such as rectangular or weakly dependent Gaussian models; that extension is not in the paper.","Condition (92) is a finite-dimensional inequality for fixed $k,l$ and concrete $A_i$, so it could be certified by numerical optimization for specific instances, giving a practical check of the lower bound even where the analytic contradiction argument is hard to run.","The proof implicitly restricts the dual variable $\\Gamma$ to the positive-definite cone in (32)–(37); a fully explicit statement of that restriction and a justification of the square-root optimizer $\\tilde\\Gamma=\\sqrt{K}$ inside the cone would close a small rigor gap the paper leaves open.","The author flags the indefinite-$A_i$ case as future work; if RDT handles it, the same two-sided comparison would reprove broader polynomial-norm convergence results that currently rely on free probability."],"forward_implications":["The exact spectral edge of Kronecker–Gaussian matrices is established without free probability, Stieltjes transforms, or matrix Dyson equations; the comparison argument alone carries the proof.","The strong asymptotic freeness edge statements of [25,43] follow as corollaries of the RDT comparison route, not as inputs.","By Gaussian symmetry the same result covers the smallest eigenvalue $\\lambda_1(H)$, so both edges of the spectrum are pinned deterministically.","The deterministic edge $\\rho_n$ is a convex SDP, so the theorem turns an asymptotic spectral statement into a finite-dimensional optimization that can be computed in principle.","The proof extends the replica-overlap tightness principle to the $k$-fold matrix-valued setting, which the paper presents as a generic mechanism for other Gaussian-process comparisons."],"supporting_citations":[{"why":"supplies the deterministic spectral-edge formula for the semicircular free counterpart that the paper sets out to confirm.","marker":"[31]"},{"why":"established the strong asymptotic freeness edge result that the paper reproves through RDT.","marker":"[25]"},{"why":"established the real/symplectic variant of the strong asymptotic freeness result that the paper also reproves.","marker":"[43]"},{"why":"supplies the comparison upper-bound route and the SDP reformulation of $\\rho_n$ used in (38).","marker":"[18]"},{"why":"provides the Gaussian comparison theorem (Theorem 2 in the paper) used to bound the random dual.","marker":"[23]"},{"why":"supplies the generic lifted-process comparison principles behind the two-replica tightness condition.","marker":"[47]"},{"why":"supplies the companion comparison principle used to justify the two-replica-cannot-double condition.","marker":"[48]"},{"why":"extends the tightness principle to sample-covariance-type settings, the template the paper follows for matrix-valued processes.","marker":"[51]"},{"why":"defines the cannot-double principle for single-partite mean-field systems that the paper extends to the $k$-fold matrix case.","marker":"[53]"},{"why":"contains the single-partite theorem whose $k$-fold matrix analogue the paper claims to establish via condition (92).","marker":"[54]"}],"fun_headline_variants":["RDT nails the spectral edge of Kronecker-Gaussian matrices","Spectral edge of Kronecker-Gaussian matrices proven via RDT","RDT confirms Lehner's formula without spectral theory","New proof of Lehner's formula for Kronecker-Gaussian matrices","RDT recovers exact spectral edge for Kronecker-Gaussian matrices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that for these matrix-valued Gaussian processes, a two-copy system with any non-trivial overlap cannot achieve twice the optimal value of a single copy, and that this failure forces the two comparison limits to coincide—a principle the paper takes from earlier work and extends to the $k$-fold matrix setting rather than deriving from a more basic argument.","fun_headline_variants_meta":{"raw":{"variants":["RDT nails the spectral edge of Kronecker-Gaussian matrices","Spectral edge of Kronecker-Gaussian matrices proven via RDT","RDT confirms Lehner's formula without spectral theory","New proof of Lehner's formula for Kronecker-Gaussian matrices","RDT recovers exact spectral edge for Kronecker-Gaussian matrices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1400,"prompt_tokens":900,"completion_tokens":500,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":409}},"tokens_in":516,"tokens_out":500,"duration_ms":4457,"temperature":1.0,"reasoning_tokens":409,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:25:15.651464+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate condition (92) for a concrete instance: fix $k,l$, positive-semidefinite $A_i$, an admissible overlap $Q\\in\\mathcal{Q}$, and $t\\in(0,1)$, and compute $\\min_{\\Gamma=\\Gamma^T,\\Lambda=\\Lambda^T}\\bar L^{(2)}$ and $2\\min_{\\Gamma=\\Gamma^T}\\bar L$ from (90) and (35). If the former is not strictly smaller than the latter, the proof's tightness condition fails. A sharper check: find any admissible $Q$ and $t$ for which the $\\Lambda$-derivative in (106) vanishes at $(\\tilde\\Gamma/2,0)$, since Theorem 5's contradiction argument requires that derivative to be nonzero.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the deterministic spectral-edge formula for the semicircular free counterpart that the paper sets out to confirm."},{"cited_title":"Haagerup and S","cited_arxiv_id":null,"evidence_quote":"established the strong asymptotic freeness edge result that the paper reproves through RDT."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"established the real/symplectic variant of the strong asymptotic freeness result that the paper also reproves."},{"cited_title":"Collins, A","cited_arxiv_id":null,"evidence_quote":"supplies the comparison upper-bound route and the SDP reformulation of $\\rho_n$ used in (38)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the Gaussian comparison theorem (Theorem 2 in the paper) used to bound the random dual."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the generic lifted-process comparison principles behind the two-replica tightness condition."},{"cited_title":"Precise sample covariance spectral norm error -- an RDT view","cited_arxiv_id":"2607.14460","evidence_quote":"extends the tightness principle to sample-covariance-type settings, the template the paper follows for matrix-valued processes."},{"cited_title":"Talagrand","cited_arxiv_id":null,"evidence_quote":"contains the single-partite theorem whose $k$-fold matrix analogue the paper claims to establish via condition (92)."}],"review_version":2}