{"id":"5b14ed01-cbb6-44fb-acfb-3b36f38a9746","arxiv_id":"2506.11569","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A new bihamiltonian structure built from the unconstrained Adler-Gelfand-Dickey bracket yields a logarithmic Dubrovin-Frobenius manifold, realized also on the permutation-group orbit space.","lead":"The paper constructs a new compatible Poisson bracket on the space of differential operators, whose dispersionless limit produces a logarithmic Dubrovin-Frobenius manifold, a geometric structure relevant to integrable systems and topological field theory. It then shows the same manifold arises from the orbit space of the permutation group's standard representation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rank-universal step Theorem 8.1 — that after the α-dependent change of coordinates B_2^Q becomes at most linear in s_{r-1}(x) — is only sketched ('Similar to Lemma 7.2'); if it fails for some r, the compatible bracket B_1^Q and the whole logarithmic Frobenius-manifold construction collapse.","rationale":"The reader's CONDITIONAL verdict is appropriately calibrated. The most load-bearing unproved assertion is Theorem 8.1, and its proof is a sketch citing Lemma 7.2, while the underlying coefficient lemmas are not fully derived. This is a verification gap in the central mechanism, not a difference of opinion with the community. The low-rank examples r=2,3,4 support the mechanism, so a flat REJECT would be too strong. I also observed that the printed potential in Theorem 9.2 conflicts with equation (9.13): the sum term has no t_{r-1}-dependence, so its t_{r-1}-derivative is zero; this strengthens the need for a complete proof of the general formulas, although it is probably a typo. A symbolic computation for r=5 or r=6, or a full re-derivation of Lemmas 6.3 and 6.5, would settle whether the central construction actually holds. Since my review does not move the reader's verdict, I recommend UNCHANGED.","tokens_in":25954,"tokens_out":18552,"duration_ms":163881,"concrete_test":"Perform a computer-algebra Drinfeld–Sokolov reduction for r=5 and r=6 using (5.7), transform to the z- and then s-coordinates of Sections 7–8, and check that every coefficient of (s_{r-1})^k with k≥2 vanishes in all entries of B_2^Q and that ∂_{s_{r-1}}Ω^{i,r-i+1}_2(s)=r-1. Independently re-derive the coefficient in Lemma 6.3 from (5.7) and (6.7) without the summarized step; any nonzero quadratic term or changed constant invalidates Theorem 8.1 and the subsequent construction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every subsequent claim rests on Theorem 8.1. The compatible bracket B_1^Q = Lie_{∂s_{r-1}(x)} B_2^Q exists only if B_2^Q is at most linear in s_{r-1}(x) in the coordinates s_r = ((r-1)/((r-1)+αr))(z_r + α z_1 z_{r-1}), with α as in (8.2). The proof of this theorem is an outline: the coefficient of (s_{r-1})^2 is said to vanish by the quadratic equation (8.2), and the rest is left to 'Similar to Lemma 7.2' and 'the remainder follows'. The input coefficients come from Lemmas 6.3 and 6.5, but those lemmas are themselves only summarized: equations (6.3)-(6.6) and (6.8) extract particular coefficients without a full derivation from (5.7). If one of these constants is wrong, or if a quadratic term in s_{r-1} survives in another entry through cross terms of the coordinate change, then Lie_{∂s_{r-1}}^2 B_2^Q ≠ 0, B_1^Q is not a Poisson bracket, and no flat pencil or Frobenius manifold follows. The explicit r=3,4 examples are genuine evidence but do not establish the general-rank statement. In addition, the potential displayed in Theorem 9.2 does not satisfy the relation ∂_{r-1}F = 1/2 Π_{ij}t_i t_j used in its proof: the displayed sum has zero derivative with respect to t_{r-1}; the sum appears to be missing a factor t_{r-1}. This is a real inconsistency in the stated central formula, though it looks repairable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for each rank r, a local Poisson bracket B^Q_1 compatible with the second unconstrained Adler-Gelfand-Dickey bracket B^Q_2 on the Slodowy slice Q of gl_r. The construction uses Drinfeld-Sokolov reduction, introduces coordinates s_i depending on a parameter α solving equation (8.2), and asserts that in these coordinates B^Q_2 is at most linear in s_{r-1}(x). The bracket B^Q_1 is then defined as the Lie derivative Lie_{∂s_{r-1}}B^Q_2, and the leading terms of the bihamiltonian structure are claimed to form a flat pencil of metrics, hence a logarithmic Dubrovin-Frobenius manifold with explicit potential (9.6). The paper further relates the resulting structure to the orbit space of the standard representation of the permutation group S_r via invariant theory. Explicit verifications are given for r=2,3,4.","tokens_in":26368,"tokens_out":10893,"duration_ms":92048,"significance":"If the rank-universal statements hold, the paper provides a new family of logarithmic Dubrovin-Frobenius manifolds arising from unconstrained AGD brackets and connects them to the constrained-KP and B_n orbit-space examples of [2], [24], and [25]. The concrete computations for gl_2, gl_3, and gl_4 (Examples 8.3, 9.3, 9.4, 10.2), the Miura-type link to invariant theory in §10, and the central-invariant calculations in §11 are genuine strengths. However, the proofs of the general-r statements are partly delegated to 'similar' computations, and the central potential formula contains an inconsistency, so the general-r claims are not yet fully established.","major_comments":[{"comment":"The proof that B^Q_2 is at most linear in s_{r-1}(x) is not supplied for general r. The text says 'Similar to Lemma 7.2' and 'the remainder follows', and the input Lemmas 6.3 and 6.5 are themselves only partially derived: Lemma 6.3 computes the coefficient of (u_{r-1})^2 in Ω^{rr}_2 but does not show the vanishing of quadratic terms in all other entries after the change (8.1), and Lemma 6.5 extracts the constants F^k without displaying the full expansion (6.8). Since the definition of B^Q_1 = Lie_{∂s_{r-1}}B^Q_2 and its compatibility (Theorem 8.2) depend on this linearity, the general-r construction requires a complete, verifiable proof rather than an outline.","section":"§8, Theorem 8.1"},{"comment":"The displayed potential F does not satisfy the relation ∂_{r-1}F = (1/2)Π_{ij}t^it^j used in the proof (Eq. (9.13)). For r≥4, the sum Σ_{i≠2,r-1} t_i t_{r+1-i} is annihilated by ∂_{r-1}, so ∂_{r-1}∂_i∂_{r+1-i}F = 0 for i≠2,r-1, whereas Π_{i,r+1-i} = 1/(r-1) requires this third derivative to equal 1/(r-1). The sum term appears to be missing a factor t_{r-1}; the explicit potentials (9.18) and (10.5) for r=3,4 are consistent with the corrected form. The term as written also has the wrong quasihomogeneous degree for the claimed Euler identity.","section":"§9, Eq. (9.6)"},{"comment":"The proof states 'with the distinction that here we have Ω^{11}_2 = r−1, whereas in [31], Ω^{11}_2 = r(r−1)'. This contradicts the value Ω^{11}_2 = r established in Proposition 5.3, Lemma 7.2, Theorem 8.1, and Proposition 9.1. The constants in equations (9.7)–(9.16) depend on this value, so the discrepancy must be resolved and the subsequent coefficients re-checked.","section":"§9, Theorem 9.2 proof"},{"comment":"The associativity of the would-be Frobenius algebra (the WDVV equations, Eq. (9.16)) is asserted with 'Detailed computations confirm' but no computation or general argument is presented. Since the adaptation of [31] is nontrivial and the value of Ω^{11}_2 differs from the reference, the verification of (9.16) for all r is a load-bearing step and should be either included explicitly or replaced by a precise structural argument.","section":"§9, after Eq. (9.15)"}],"minor_comments":[{"comment":"The sentence 'Similar computations by evaluating the one form A with dz1 = du2' appears to contain a typo: since z1 = u1, the correct differential is dz1 = du1, not dz1 = du2.","section":"§7, proof of Lemma 7.2"},{"comment":"The sentence 'Thus, Ω^{ij}_2(z) = Ω^{ij}_2(z) = Σ_k ...' repeats the symbol Ω^{ij}_2; the first occurrence should presumably be a different notation (e.g., the pullback metric) or be deleted.","section":"§10, proof of Theorem 10.1"},{"comment":"The claim that Ω^{ij}_2(z) 'is identical to the metric defined on the orbits space by the inverse of the Hessian of z2' is not demonstrated; a short verification would improve clarity.","section":"§10, proof of Theorem 10.1"},{"comment":"Reference [24] contains an extraneous fragment '138, pp. 154-167 (2019)' after the journal citation; this appears to be a duplicate or misplaced bibliographic entry.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper overlaps with the author's earlier work and with [31], but the construction from unconstrained AGD brackets is new. The editor may wish to insist that the revised version contain full proofs of Theorem 8.1 and of the WDVV verification, rather than references to 'similar' computations, before the general-rank claims are accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the low-rank part of this paper is a genuine and mostly checkable construction; the general-rank part is a plausible conjecture supported by r=3,4, not by a complete proof. I would send it to a referee, but I would not bet on the general theorem as written.\n\nWhat's new: starting from the unconstrained second AGD bracket, Dinar produces a new local Poisson bracket B_1^Q = Lie_{\\partial s_{r-1}} B_2^Q, shows the pair is bihamiltonian after a specific coordinate change, and derives a logarithmic Dubrovin-Frobenius manifold. That is not in the older literature. The r=3 and r=4 explicit potentials are worked out, and the equivalence with the constrained-KP/B_n orbit-space potentials is shown by explicit maps. The connection to the permutation-group orbit space via Miura transformation is a nice added angle. The low-rank computations look internally consistent; I checked the r=3 bracket and the t-coordinate change, and they line up.\n\nThe soft spot is exactly where the reader's report puts it. Theorem 8.1 is load-bearing: if B_2^Q is not at most linear in s_{r-1}, then Lie-deriving it twice in the s_{r-1} direction does not vanish, and B_1^Q is not a Poisson bracket. The proof says 'similar to Lemma 7.2' and 'the remainder follows' after one quadratic expression. The input Lemmas 6.3 and 6.5 themselves summarize coefficient extraction from (5.7). For a general rank statement, this is too thin. The r=3,4 examples are evidence, but not proof.\n\nThere is also a real-looking typo in the main potential (9.6): the sum over i\\neq 2,r-1 of t_i t_{r+1-i} does not depend on t_{r-1}, so (9.13) cannot hold as written. I suspect the formula is missing a t_{r-1} factor or the sum should be over a different range; either way the displayed central formula needs correction. The WDVV check is also delegated to 'detailed computations,' which is not enough for a general theorem, even if the low-rank cases are fine.\n\nThe paper is not a fraud; it is a serious construction with real low-rank content. But the general-r claim is not yet supported at the level of proof. A good referee could probably make it work or find the place it breaks. Send it to peer review.","headline":"Low-rank constructions are solid and new; the general-rank proof rests on a sketched key theorem that needs full computation before the main claim is accepted.","tokens_in":26891,"tokens_out":1852,"would_cite":false,"duration_ms":17537,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37K25","37K30","53D45","17B80","53D17","13A50","17B68","17B08"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the second unconstrained Adler-Gelfand-Dickey bracket admits a compatible local Poisson bracket whose dispersionless limit defines a logarithmic Dubrovin-Frobenius manifold, and that the same manifold arises on the…","keywords":["Adler-Gelfand-Dickey bracket","Dubrovin-Frobenius manifold","logarithmic Frobenius manifold","bihamiltonian structure","flat pencil of metrics","Drinfeld-Sokolov reduction","permutation group orbit space","WDVV equations"],"falsifier":"Compute for $r=5$ the coefficient of $(s_{r-1}(x))^2$ in the entry $\\{s_r(x),s_r(y)\\}$ after the coordinate change (8.1), using the explicit formulas (5.7), (6.2), and Lemma 6.3; if this coefficient is nonzero, Theorem 8.1 fails and the constructed Frobenius manifold does not exist. A cheaper partial check is to compute the central invariants of $(B^Q_2,B^Q_1)$ for $r=5$ and compare them with the topological-type values found for $r=2,3,4$.","tokens_in":25754,"feed_emoji":"🌀","tokens_out":9865,"duration_ms":95216,"temperature":0.7,"pith_summary":"This paper studies the second unconstrained Adler-Gelfand-Dickey bracket, a Poisson bracket on the space of $r$-th order scalar differential operators $L=D^r+v_1D^{r-1}+\\cdots+v_r$. On a Slodowy slice $Q$ of the regular nilpotent element of $\\mathfrak{gl}_r$, it constructs a new local Poisson bracket $B^Q_1=\\mathrm{Lie}_{\\partial s_{r-1}(x)}B^Q_2$ that is compatible with $B^Q_2$, so the pair forms a bihamiltonian structure. The key step is a coordinate change chosen so that $B^Q_2$ depends at most linearly on $s_{r-1}(x)$; then the Lie derivative automatically gives a compatible bracket. The pair admits a dispersionless limit whose leading term is a flat pencil of metrics, and away from the zeros of $\\det\\Omega^{ij}_2$ and $t_1=0$ this pencil defines a logarithmic Dubrovin-Frobenius manifold, meaning a manifold whose tangent spaces carry a compatible Frobenius algebra structure encoded by a potential satisfying the WDVV equations. The potential has the explicit form $F=\\frac{1}{(r-1)(2+4\\delta_{r,3})}(t_{r-1})^2t_2+\\frac{1}{2(r-1)}\\sum_{i\\neq 2,r-1}t_it_{r+1-i}+\\frac{1}{2(r-1)}(t_r)^2\\log t_r+G$ with $G$ a quasihomogeneous polynomial of degree $2r$, and the same structure is realized on the orbit space of the standard representation of the permutation group $S_r$.","feed_headline":"New compatible bracket builds logarithmic Frobenius manifolds","feed_subtitle":"This produces explicit WDVV potentials from r-th order scalar differential operators.","key_machinery":"The central mechanism is Drinfeld-Sokolov reduction of the Lie-Poisson bracket on $L(\\mathfrak{gl}_r)$ to the Slodowy slice $Q=L_2+L(\\mathfrak{gl}_r^f)$, followed by two coordinate changes: the invariant coordinates $z_i=\\frac{1}{i}\\mathrm{Tr}(g^i)$ and the modified coordinates $s_i$ of Theorem 8.1. The quadratic equation for $\\alpha$ is chosen to cancel the $(s_{r-1})^2$ term in the entry $\\Omega^{rr}_2$, making $B^Q_2$ at most linear in $s_{r-1}(x)$; the criterion $\\mathrm{Lie}^2_X\\{\\cdot,\\cdot\\}=0$ then turns the Lie derivative into a compatible bracket. The flat-coordinate and WDVV-verification machinery of the cited construction is adapted to the present setting, with the entry $\\Omega^{11}_2=r-1$ instead of $r(r-1)$, yielding the logarithmic potential.","core_discovery":"On the Slodowy slice $Q=L_2+L(\\mathfrak{gl}_r^f)$ associated with the regular nilpotent element $L_2$ of $\\mathfrak{gl}_r$, the paper establishes that after the quasihomogeneous coordinate change $s_i=z_i$ for $i\\neq r$ and $s_r=\\frac{r-1}{(r-1)+\\alpha r}(z_r+\\alpha z_1z_{r-1})$, with $\\alpha$ solving $r\\alpha^2+2(r-1)\\alpha+(r-1)=0$, the second Adler-Gelfand-Dickey bracket $B^Q_2$ becomes at most linear in $s_{r-1}(x)$. Consequently $B^Q_1=\\mathrm{Lie}_{\\partial s_{r-1}(x)}B^Q_2$ is a nontrivial local Poisson bracket compatible with $B^Q_2$. The bihamiltonian structure $(B^Q_2,B^Q_1)$ admits a dispersionless limit whose leading term is a flat pencil of metrics, and on the open dense set where $\\det\\Omega^{ij}_2\\neq 0$ and $t_1\\neq 0$ it defines a logarithmic Dubrovin-Frobenius manifold with the explicit potential displayed in Theorem 9.2. The same manifold is shown to arise naturally on the orbit space of the standard representation of the permutation group $S_r$, through the Hessian metric of the degree-two invariant and the same coordinate change.","pith_inferences":["Extending beyond the paper: the natural next test is to compute, for $r=5$, both the claimed linearity and the central invariants of $(B^Q_2,B^Q_1)$; the paper only verifies $r=2,3,4$.","Extending beyond the paper: the logarithmic term $(t_r)^2\\log t_r$ suggests that the flat pencil degenerates or becomes non-semisimple at the divisor $t_r=0$, a locus the paper does not analyze.","Extending beyond the paper: the realization on the permutation-group orbit space suggests a general recipe for other finite reflection groups: start with the Hessian metric of the degree-two invariant, apply the same $\\alpha$-coordinate change, and look for logarithmic Frobenius manifolds; the paper does not pursue this beyond $S_r$."],"forward_implications":["The second unconstrained Adler-Gelfand-Dickey bracket admits a previously unknown compatible local Poisson bracket, so the pair forms a bihamiltonian structure on the space of $r$-th order scalar differential operators.","The bihamiltonian structure has a dispersionless limit whose leading term is a flat pencil of metrics, and on an open dense subset it defines a logarithmic Dubrovin-Frobenius manifold with an explicit potential.","The same logarithmic Dubrovin-Frobenius manifold can be constructed on the orbit space of the standard representation of the permutation group by a Dubrovin-Saito invariant-theoretic procedure.","For $r=2,3,4$ the bihamiltonian structures are of topological type, with central invariants $-\\frac{1}{24}$ for $r=2$ and $-\\frac{1}{8}$ for $r=3,4$, and for $r=3,4$ they are equivalent to the constrained-KP bihamiltonian structures.","The paper conjectures that the equivalence with constrained-KP structures and the topological-type property extend to all $r>2$."],"supporting_citations":[{"why":"Supplies the Drinfeld-Sokolov reduction through which $B^Q_2$ is defined as the reduced Lie-Poisson bracket on the Slodowy slice.","marker":"[12]"},{"why":"Supplies the theory of Dubrovin-Frobenius manifolds and the intersection-form formula used to pass from flat pencils of metrics to Frobenius manifolds.","marker":"[14]"},{"why":"Provides the existence of quasihomogeneous flat coordinates of the form $t_i=s_i+$ nonlinear terms, used in Proposition 9.1.","marker":"[15]"},{"why":"Establishes the bridge between flat pencils of metrics and Dubrovin-Frobenius manifolds that Theorem 9.2 relies on.","marker":"[16]"},{"why":"Supplies the flat-coordinate and WDVV-verification machinery adapted in Theorem 9.2, specifically the case $\\Omega^{11}_2=r(r-1)$ that the paper modifies to $r-1$.","marker":"[31]"},{"why":"Gives an alternative Dubrovin-Saito construction of logarithmic Dubrovin-Frobenius manifolds on orbit spaces of $B_n$, used as a cross-check and in the Section 11 equivalence.","marker":"[2]"},{"why":"Provides the central-invariant formula used to test whether the bihamiltonian structures are of topological type.","marker":"[18]"},{"why":"Supplies the constrained-KP bihamiltonian structures whose associated logarithmic Frobenius manifolds are compared and shown equivalent for $r=3,4$.","marker":"[24]"},{"why":"Gives the criterion $\\mathrm{Lie}^2_X\\{\\cdot,\\cdot\\}=0$ used to conclude that the Lie derivative $B^Q_1$ is compatible with $B^Q_2$.","marker":"[28]"}],"fun_headline_variants":["Poisson bracket builds logarithmic Frobenius manifolds","Compatible bracket yields log Frobenius manifolds","Permutation orbits give logarithmic Frobenius manifolds","Bihamiltonian structure from AGD brackets yields Frobenius manifolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the assertion, sketched rather than fully proved in Theorem 8.1, that after the coordinate change (8.1) the second Adler-Gelfand-Dickey bracket is at most linear in $s_{r-1}(x)$, and on the assumption that the flat-coordinate and WDVV machinery of the cited construction remains valid when $\\Omega^{11}_2=r-1$ instead of $r(r-1)$; if either premise fails, the compatible bracket, the flat pencil, and the Frobenius manifold need not exist.","fun_headline_variants_meta":{"raw":{"variants":["Poisson bracket builds logarithmic Frobenius manifolds","Compatible bracket yields log Frobenius manifolds","Permutation orbits give logarithmic Frobenius manifolds","Bihamiltonian structure from AGD brackets yields Frobenius manifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00047,"raw_usage":{"total_tokens":2321,"prompt_tokens":911,"completion_tokens":1410,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":1342}},"tokens_in":527,"tokens_out":1410,"duration_ms":11910,"temperature":1.0,"reasoning_tokens":1342,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:04:41.967381+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute for $r=5$ the coefficient of $(s_{r-1}(x))^2$ in the entry $\\{s_r(x),s_r(y)\\}$ after the coordinate change (8.1), using the explicit formulas (5.7), (6.2), and Lemma 6.3; if this coefficient is nonzero, Theorem 8.1 fails and the constructed Frobenius manifold does not exist. A cheaper partial check is to compute the central invariants of $(B^Q_2,B^Q_1)$ for $r=5$ and compare them with the topological-type values found for $r=2,3,4$.","supporting_citations":[{"cited_title":"G., Sokolov, V","cited_arxiv_id":null,"evidence_quote":"Supplies the Drinfeld-Sokolov reduction through which $B^Q_2$ is defined as the reduced Lie-Poisson bracket on the Slodowy slice."},{"cited_title":"Integrable systems and quantum groups (Montecatini Terme, 1993), 120-348, Lecture Notes in Math., 1620, Springer, Berlin, (1996)","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of Dubrovin-Frobenius manifolds and the intersection-form formula used to pass from flat pencils of metrics to Frobenius manifolds."},{"cited_title":"Surveys in differential geometry IV: integrable systems, 181-211 (1998)","cited_arxiv_id":null,"evidence_quote":"Provides the existence of quasihomogeneous flat coordinates of the form $t_i=s_i+$ nonlinear terms, used in Proposition 9.1."},{"cited_title":"Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 47-72, World Sci","cited_arxiv_id":null,"evidence_quote":"Establishes the bridge between flat pencils of metrics and Dubrovin-Frobenius manifolds that Theorem 9.2 relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the flat-coordinate and WDVV-verification machinery adapted in Theorem 9.2, specifically the case $\\Omega^{11}_2=r(r-1)$ that the paper modifies to $r-1$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives an alternative Dubrovin-Saito construction of logarithmic Dubrovin-Frobenius manifolds on orbit spaces of $B_n$, used as a cross-check and in the Section 11 equivalence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the central-invariant formula used to test whether the bihamiltonian structures are of topological type."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the constrained-KP bihamiltonian structures whose associated logarithmic Frobenius manifolds are compared and shown equivalent for $r=3,4$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the criterion $\\mathrm{Lie}^2_X\\{\\cdot,\\cdot\\}=0$ used to conclude that the Lie derivative $B^Q_1$ is compatible with $B^Q_2$."}],"review_version":1}