{"id":"d9ebfdf3-e883-44ae-82af-2f6746b588b9","arxiv_id":"2507.09679","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The automorphism group of each cyclic quotient X_{d,e} of the affine plane has either one or two conjugacy classes of Borel subgroups, depending on whether e^2 is congruent to 1 modulo d.","lead":"This mathematics paper classifies the maximal solvable connected symmetry subgroups of cyclic quotients of the affine plane, showing they come in one or two conjugacy classes depending on an arithmetic congruence. For affine geometers, it extends the classical Borel theorem to a family of infinite-dimensional automorphism groups and reveals a sharp dichotomy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The internal hinge is Cor 5.5(b): its Serre bounded-length step is unproved for general amalgams, and Theorem 7.10 inherits that gap.","rationale":"The reader identified the imported amalgam theorem (Theorem 7.5) as the weakest assumption, and that is a legitimate concern because the present paper does not reprove it. But importing a prior published theorem is standard mathematical practice, and the paper [1] is by the same authors and appears to supply the needed structural input. The more load-bearing internal point is Corollary 5.5(b): it is used directly in Theorem 5.8, which underpins Corollary 5.9 and hence Theorem 7.10, yet its proof applies Serre's bounded-length theorem without establishing bounded length. This is a genuine proof gap, not merely a citation or typo issue. It is likely fixable via the divisibility of connected commutative algebraic groups, and the main claim may well be correct, but as written the argument has an unsupported step. The reader's verdict of CONDITIONAL is therefore appropriate, and no change to that verdict is needed.","tokens_in":34448,"tokens_out":23072,"duration_ms":258735,"concrete_test":"Check the step for the actual groups N_{d,e}/G_{d,e}. Either (1) prove directly that every algebraic subgroup of these amalgams has bounded reduced-word length, using degree bounds such as Lemma 7.3(d), so that Serre's Theorem 2.14 legitimately applies; or (2) prove the divisibility alternative: every element of a connected commutative algebraic subgroup of an amalgam over k has n-th roots for all n, hence cannot act loxodromically on the Bass-Serre tree. If either route succeeds, the gap is an expositional fix and the conditional verdict stands; if both fail, Theorem 5.8 and the classification are in jeopardy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 5.5(b) asserts that every connected commutative subgroup G contained in an amalgam A *_{A∩B} B of ind-subgroups of Aut(X) acts elliptically on the Bass-Serre tree. The proof chooses an algebraic curve C through 1 and f, applies Cantat–Regeta–Xie to obtain a commutative algebraic subgroup ⟨C⟩, and then invokes Serre's Theorem 2.14 as though ⟨C⟩ automatically had bounded length. Bounded length is not proved, and algebraicity does not formally imply it for an arbitrary amalgam of ind-subgroups; for Aut(A2) it follows from degree bounds, but Corollary 5.5(b) is stated in full generality. Theorem 5.8 uses this corollary to force the penultimate derived subgroup of a loxodromic solvable connected H to be elliptic, and Corollary 5.9 plus Theorem 7.10 inherit that dependence. The statement is probably true: a connected commutative algebraic group over an algebraically closed field of characteristic 0 is divisible, while a loxodromic element of a simplicial tree has positive integer translation length and cannot have n-th roots for arbitrarily large n. But that replacement argument is not what is written. As printed, the proof of the one-versus-two Borel dichotomy contains an unsupported step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Borel subgroups, defined as maximal connected solvable ind-subgroups, of the automorphism groups of cyclic quotient affine toric surfaces X_{d,e} = A^2/µ_d. The main theorem, Theorem 1.1, asserts a dichotomy: when e^2 ≡ 1 mod d the Borel subgroups form one conjugacy class, and when e^2 ≢ 1 mod d they form exactly two conjugacy classes represented by the two triangular factors of an amalgam. The proof combines Bass–Serre theory for amalgams with the authors' earlier description of Aut(X_{d,e}) as a quotient of a normalizer in Aut(A^2). The paper also gives a criterion, Theorem 1.2, for maximal solvable subgroups to be Borel subgroups in terms of absence of finite-index proper subgroups, and it classifies maximal solvable subgroups of Aut(A^2) and Aut_0(A^2).","tokens_in":34600,"tokens_out":17791,"duration_ms":191065,"significance":"If the main classification is correct, it is a valuable contribution to the structure theory of automorphism groups of affine surfaces: it confirms and extends, for cyclic quotients of the affine plane, the phenomenon already known for Aut(A^2), and it gives a clean one-versus-two conjugacy dichotomy controlled by the congruence e^2 ≡ 1 mod d. The paper uses a well-chosen mix of Bass–Serre theory, algebraic-group arguments, and the Cantat–Regeta–Xie algebraicity theorem. The free-subgroup argument in Proposition 3.1 and the root-of-unity computation in Lemma 7.12 are explicit and rigorous. However, several load-bearing steps in the treatment of the e^2 ≡ 1 case are either unsupported or actually false as printed, so the paper needs substantial revision before the central claims can be accepted.","major_comments":[{"comment":"For e > 1 with e^2 ≡ 1 mod d, equation (12) gives N^+_{d,e} = T and Notation 7.2 gives N_{d,e} = ⟨T, τ⟩. Since T is a normal subgroup of ⟨T, τ⟩ with quotient of order 2, the whole group N_{d,e} is solvable. Hence T/G_{d,e} is a proper subgroup of the solvable group N_{d,e}/G_{d,e}, so the claim in Proposition 7.8 that N^±_{d,e}/G_{d,e} is maximal among the solvable subgroups of Aut(X_{d,e}) is false in this case. Consequently Theorem 1.1(c), which states that every Borel subgroup is maximal among the solvable subgroups, also fails for this family. The proof's appeal to Proposition 3.1 is not available here because the amalgam in (17) has one factor equal to the amalgamated subgroup and is not proper.","section":"§7.2, Proposition 7.8 and Theorem 1.1(c)"},{"comment":"The proof of Corollary 5.5(b) asserts that the commutative algebraic subgroup ⟨C⟩ is elliptic by invoking Serre's Theorem 2.14, but it does not prove that ⟨C⟩ has bounded length. Bounded length is not a formal consequence of algebraicity for an arbitrary amalgam of ind-subgroups; for Aut(A^2) it follows from degree bounds, but Corollary 5.5(b) is stated in full generality and no such bound is supplied. This step is load-bearing because Theorem 5.8 uses Corollary 5.5(b) to conclude that the penultimate derived subgroup of a loxodromic solvable connected subgroup consists of elliptic elements, and Corollary 5.9 and Theorem 7.10 inherit the gap. In the specific setting of X_{d,e} the missing fact could be supplied by the analogue of Theorem 6.3 imported from [1, Theorem 4.15], but that replacement is not made in the manuscript.","section":"§5, Corollary 5.5(b) and Theorem 5.8"},{"comment":"Corollary 5.9 is applied to the amalgams in (17), but for e = 1 and for e > 1 with e^2 ≡ 1 mod d the expression (17) has the form A *_{A} B and is not a proper amalgam. The associated Bass–Serre tree, as defined in §2, is degenerate or not a tree in the relevant sense, so the torsionally unboundedness argument and Corollary 5.9 do not apply to these cases. Furthermore, in the e > 1 case the sentence 'the inclusion T ⊂ N^+_{d,e} is strict' in the proof of Theorem 7.10 is incorrect, since (12) gives equality. The e^2 ≡ 1 cases require a separate proof, using for example the algebraic group structure of PGL(2,k) when e = 1 and of the solvable extension T ⋊ ⟨τ⟩ when e > 1.","section":"§7.10, proof of Theorem 7.10 and Corollary 7.14"}],"minor_comments":[{"comment":"The proof cites 'Theorem 4.13(a)–(d)', but no Theorem 4.13 appears in Section 4; from the content of the cited statements, the reference should be Theorem 6.6(a)–(d).","section":"§7.15, proof of Theorem 7.15"},{"comment":"The notation N^+_{d,e} *_{N^+_{d,e}} N_{d,e} is confusing because the two appearances of N^+_{d,e} are equal, so the amalgam is not proper; the later notation in Remark 7.6.2, N^+_{d,e} *_{T} ⟨T,τ⟩, clarifies the intended edge group for e > 1 but should be made consistent with (17).","section":"§7.2, equation (17)"},{"comment":"In Claim 2 of Theorem 5.8, the phrase 'every elliptic f ∈ H is a torsion element' is then used to assert that all elements of H^{(n-1)} are torsion; this is not fully spelled out, since H^{(n-1)} consists of products of commutators, and one must first know that each such commutator is elliptic, which is exactly what Corollary 5.5(b) is being used to provide.","section":"§5, Theorem 5.8 proof"}],"recommendation":"major_revision","confidential_remarks":"The false maximality claim in Proposition 7.8 for the e > 1, e^2 ≡ 1 case is a substantive mathematical error in a central statement, not merely a presentation slip. The likely fix is to restrict the maximality assertions to maximality among connected solvable subgroups, or to exclude the solvable disconnected case from Theorem 1.1(c). The conjugacy dichotomy itself appears plausible, but the proof as written does not cover the non-proper amalgam cases in (17). The paper deserves a chance after these points are repaired."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colloquially: this paper proves that Aut(X_{d,e}) has one or two conjugacy classes of Borel subgroups depending on whether e^2 is congruent to 1 modulo d. That dichotomy is new, and the main argument is sound. The Bass-Serre reduction is well executed: Proposition 3.1 gives a hands-on free-subgroup argument, Lemma 7.12 does the root-of-unity computation that makes the torsionality come out, and Theorem 5.8 cleanly forces connected solvable subgroups to be elliptic. The classification of maximal solvable subgroups of Aut(A2) in Theorem 1.5 is largely an assembly of known pieces, but that assembly is useful and the paper is honest about what is new.\n\nThe reliance on the authors' earlier [1] for the amalgam structure of Aut(X_{d,e}) is legitimate, not circular: the target classification is not an input there. There are no fitted parameters and no invented entities.\n\nTwo soft spots, both correctable. First, the proof of Theorem 7.15 cites \"Theorem 4.13(a)-(d)\", which does not exist in the paper. The intended reference is evidently Theorem 6.6 (Lamy's classification of subgroups of Aut(A2)). This is a cross-reference slip, but it sits in a load-bearing proof and should be fixed.\n\nSecond, Corollary 5.5(b) applies Serre's bounded-length theorem (Theorem 2.14) to a commutative algebraic subgroup <C> of a general amalgam of ind-subgroups of Aut(X), without proving that <C> has bounded length. The stress-test note is right that this is unsupported as written. For Aut(A2) one gets bounded length from degree bounds, but the corollary is stated in full generality. The likely repair is a divisibility argument: a connected commutative algebraic group over an algebraically closed field of characteristic zero is divisible, while a loxodromic element of a tree has positive integer translation length and cannot have n-th roots for arbitrarily large n. That would make the step work, but it is not what is printed. Theorem 5.8 and the final one-versus-two dichotomy inherit this dependency, so the authors should expand the proof.\n\nThe central claim is probably true, and the issues are fixable. This paper deserves a serious referee. I would accept it for peer review and ask for those two corrections. I would cite it if I worked on automorphism groups of affine varieties; it is also a good candidate for a reading group, since the tree arguments are instructive.","headline":"A clean arithmetic dichotomy for Borel subgroups of Aut(X_{d,e}), presented with a couple of fixable gaps in the written proof.","tokens_in":35258,"tokens_out":3299,"would_cite":true,"duration_ms":32582,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J50","20E08","14M25","14R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For each affine toric surface $X_{d,e}$, the automorphism group has one or two conjugacy classes of Borel subgroups, according as $e^2 \\equiv 1 \\pmod d$.","keywords":["affine toric surface","automorphism group","Bass–Serre tree","Borel subgroup","maximal solvable subgroup","cyclic quotient","triangular subgroup","amalgam"],"falsifier":"Take $d=5$, $e=2$, so that $e^2 \\not\\equiv 1 \\pmod 5$. If one could exhibit an element of $\\operatorname{Aut}(X_{5,2})$ of infinite order whose fixed subtree in the Bass–Serre tree contains a geodesic segment of length 3, Lemma 7.12 and the uniqueness of the two Borel classes would fail; conversely, finding a connected solvable subgroup of $\\operatorname{Aut}(X_{5,2})$ not conjugate into either $N^+_{5,2}/G_{5,2}$ or $N^-_{5,2}/G_{5,2}$ would contradict Theorem 7.10(b).","tokens_in":34141,"feed_emoji":"🔺","tokens_out":9106,"duration_ms":89873,"temperature":0.7,"pith_summary":"This paper classifies the Borel subgroups—the maximal connected solvable subgroups—of the automorphism groups of the cyclic quotients $X_{d,e}$ of the affine plane, where the group of $d$-th roots of unity acts by $(x,y) \\mapsto (\\zeta^e x, \\zeta y)$. The answer is a clean dichotomy: if $e^2 \\equiv 1 \\pmod d$, all Borel subgroups are conjugate to a single triangular subgroup; if $e^2 \\not\\equiv 1 \\pmod d$, there are exactly two conjugacy classes, represented by the two triangular factors of an amalgam decomposition. In both cases every connected solvable subgroup is conjugate into one of these triangular subgroups, and every Borel subgroup is maximal among all solvable subgroups, not just connected ones. The proof imports the known amalgam structure of $\\operatorname{Aut}(X_{d,e})$ and analyzes it through Bass–Serre trees, showing that connected solvable subgroups can only live in the amalgam factors.","feed_headline":"One congruence decides the Borel subgroup count","feed_subtitle":"For each affine toric surface X_{d,e}, the automorphism group has one or two conjugacy classes of maximal solvable subgroups.","key_machinery":"The load-bearing machinery is the Bass–Serre tree $T_{d,e}$ of the amalgam decomposition $\\operatorname{Aut}(X_{d,e}) \\cong N_{d,e}/G_{d,e} = A \\,*_C\\, B$ supplied by the authors' earlier theorem. A group acting on a tree is called torsionally unbounded when every element with unbounded fixed subtree is a torsion element; the paper proves $N_{d,e}$ and $\\operatorname{Aut}(X_{d,e})$ have this property. For such groups, Theorem 5.8 and Corollary 5.9 force every connected solvable subgroup to be elliptic—conjugate into one of the factors—so Borel subgroups must be conjugate to Borel subgroups of $N^+_{d,e}$ or $N^-_{d,e}$. The decisive local calculation is Lemma 7.12: when $e^2 \\not\\equiv 1 \\pmod d$, any element fixing a geodesic segment of length 3 in $T_{d,e}$ is conjugate to a diagonal matrix with primitive root eigenvalues, hence is torsion.","core_discovery":"On the authors' terms, the central discovery is Theorem 1.1 and Theorem 7.10: for the affine toric surface $X_{d,e}=\\mathbb{A}^2/G_{d,e}$, the group $\\operatorname{Aut}(X_{d,e})$ has a unique conjugacy class of Borel subgroups when $e^2 \\equiv 1 \\pmod d$, represented by $N^+_{d,e}/G_{d,e}$ (equivalently $N^-_{d,e}/G_{d,e}$, since the twist conjugates them), and exactly two conjugacy classes when $e^2 \\not\\equiv 1 \\pmod d$, represented by the two factors $N^+_{d,e}/G_{d,e}$ and $N^-_{d,e}/G_{d,e}$ of the amalgam $\\operatorname{Aut}(X_{d,e}) = N^+_{d,e}/G_{d,e} \\,*_{T/G_{d,e}}\\, N^-_{d,e}/G_{d,e}$. Every solvable connected subgroup of $\\operatorname{Aut}(X_{d,e})$ is conjugate into one of these triangular subgroups, and every Borel subgroup is maximal among the solvable subgroups. Theorem 1.2 adds that, outside the abelian exceptions, a maximal solvable subgroup of $\\operatorname{Aut}(X_{d,e})$ is a Borel subgroup exactly when it contains no proper subgroup of finite index.","pith_inferences":["Across all normal affine toric surfaces, the number of conjugacy classes of Borel subgroups is always 1 or 2: the remaining cases $\\mathbb{A}^2$, $(\\mathbb{A}^1_*)^2$, and $\\mathbb{A}^1 \\times \\mathbb{A}^1_*$ each have a single Borel subgroup, so the dichotomy for $X_{d,e}$ completes a full census.","The torsionally-unbounded criterion suggests a reusable mechanism: any automorphism group admitting an amalgam decomposition in which elliptic non-torsion elements have bounded fixed trees will have Borel subgroups inherited from the amalgam factors; this could be tested on other amalgams arising from affine surfaces such as Gizatullin surfaces.","The finite-index condition gives an intrinsic, subgroup-theoretic way to recognize Borel subgroups in these ind-groups, and could be checked without comparing conjugacy classes, potentially extending Theorem 7.15 to other automorphism groups of affine varieties."],"forward_implications":["For $X_{d,e}$, the automorphism group has one or two conjugacy classes of Borel subgroups according as $e^2 \\equiv 1 \\pmod d$ or not.","Every connected solvable subgroup of $\\operatorname{Aut}(X_{d,e})$ is triangulable: it is conjugate into one of the triangular factors, an analogue of the classical Borel and Lie–Kolchin theorems for these infinite-dimensional groups.","Borel subgroups of $\\operatorname{Aut}(X_{d,e})$ are maximal among all solvable subgroups, not merely among connected solvable ones, and each coincides with its normalizer.","A non-abelian maximal solvable subgroup is a Borel subgroup precisely when it has no proper subgroup of finite index.","Every abstract group automorphism of $\\operatorname{Aut}(X_{d,e})$ preserves the set of Borel subgroups."],"supporting_citations":[{"why":"Supplies the amalgam decomposition $\\operatorname{Aut}(X_{d,e}) \\cong N_{d,e}/G_{d,e}$ as $N^+_{d,e}/G \\,*_{T/G}\\, N^-_{d,e}/G$ or the one-factor amalgam when $e^2 \\equiv 1 \\pmod d$, the structural theorem on which the whole proof rests.","marker":"[1]"},{"why":"Provides Theorem 5.4, the algebraicity of connected commutative subgroups generated by commuting families, used to show connected solvable torsion subgroups and parabolic subgroups are trivial.","marker":"[7]"},{"why":"Supplies the Bass–Serre techniques and classification results for $\\operatorname{Aut}(\\mathbb{A}^2)$ that the paper adapts to $N_{d,e}$.","marker":"[33]"},{"why":"Provides the classification of elliptic, parabolic, and loxodromic subgroups of $\\operatorname{Aut}(\\mathbb{A}^2)$, including unbounded fixed subtrees and centralizers of torsion elements.","marker":"[34]"},{"why":"Supplies Serre's bounded-length theorem and the Bass–Serre tree formalism used to force algebraic subgroups into amalgam factors.","marker":"[45]"},{"why":"Source for the analogue of Steinberg's characterization: a non-abelian maximal solvable subgroup is a Borel subgroup iff it has no proper finite-index subgroup.","marker":"[4]"},{"why":"Supplies the Borel-subgroup results for $\\operatorname{Aut}(\\mathbb{A}^2)$ and the maximality of the triangular subgroup used for $N^\\pm_{d,e}$.","marker":"[16]"},{"why":"Supplies the ind-group and nested-subgroup facts, including that connected nested subgroups have no proper finite-index subgroup.","marker":"[30]"},{"why":"Supplies the pseudo-divisibility and algebraicity criteria for automorphisms used in the torsionally-unbounded arguments.","marker":"[35]"}],"fun_headline_variants":["One congruence decides Borel subgroup count","Toric surface automorphisms: one or two Borel classes","e^2 ≡ 1 mod d splits Borel conjugacy","Borel subgroups of toric automorphism groups classified","Two species of affine toric surfaces for Borel classes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire classification depends on the amalgam decomposition $\\operatorname{Aut}(X_{d,e}) \\cong N_{d,e}/G_{d,e}$ imported from the authors' previous paper—and, in one step, on applying Serre's bounded-length theorem to algebraic subgroups without an explicit bounded-length proof—so if that decomposition were false, the Bass–Serre tree arguments would no longer apply.","fun_headline_variants_meta":{"raw":{"variants":["One congruence decides Borel subgroup count","Toric surface automorphisms: one or two Borel classes","e^2 ≡ 1 mod d splits Borel conjugacy","Borel subgroups of toric automorphism groups classified","Two species of affine toric surfaces for Borel classes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1300,"prompt_tokens":967,"completion_tokens":333,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":253}},"tokens_in":583,"tokens_out":333,"duration_ms":4022,"temperature":1.0,"reasoning_tokens":253,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:51:24.528112+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $d=5$, $e=2$, so that $e^2 \\not\\equiv 1 \\pmod 5$. If one could exhibit an element of $\\operatorname{Aut}(X_{5,2})$ of infinite order whose fixed subtree in the Bass–Serre tree contains a geodesic segment of length 3, Lemma 7.12 and the uniqueness of the two Borel classes would fail; conversely, finding a connected solvable subgroup of $\\operatorname{Aut}(X_{5,2})$ not conjugate into either $N^+_{5,2}/G_{5,2}$ or $N^-_{5,2}/G_{5,2}$ would contradict Theorem 7.10(b).","supporting_citations":[{"cited_title":"Arzhantsev and M","cited_arxiv_id":null,"evidence_quote":"Supplies the amalgam decomposition $\\operatorname{Aut}(X_{d,e}) \\cong N_{d,e}/G_{d,e}$ as $N^+_{d,e}/G \\,*_{T/G}\\, N^-_{d,e}/G$ or the one-factor amalgam when $e^2 \\equiv 1 \\pmod d$, the structural theorem on which the whole proof rests."},{"cited_title":"Cantat, A","cited_arxiv_id":null,"evidence_quote":"Provides Theorem 5.4, the algebraicity of connected commutative subgroups generated by commuting families, used to show connected solvable torsion subgroups and parabolic subgroups are trivial."},{"cited_title":"Lamy, L’alternative de Tits pour Aut[C2], J","cited_arxiv_id":null,"evidence_quote":"Supplies the Bass–Serre techniques and classification results for $\\operatorname{Aut}(\\mathbb{A}^2)$ that the paper adapts to $N_{d,e}$."},{"cited_title":"Lamy, The Cremona Group , preliminary version, July 5, 2025","cited_arxiv_id":null,"evidence_quote":"Provides the classification of elliptic, parabolic, and loxodromic subgroups of $\\operatorname{Aut}(\\mathbb{A}^2)$, including unbounded fixed subtrees and centralizers of torsion elements."},{"cited_title":"Serre, Trees, Springer Monographs in Mathematics","cited_arxiv_id":null,"evidence_quote":"Supplies Serre's bounded-length theorem and the Bass–Serre tree formalism used to force algebraic subgroups into amalgam factors."},{"cited_title":"Berest, A","cited_arxiv_id":null,"evidence_quote":"Source for the analogue of Steinberg's characterization: a non-abelian maximal solvable subgroup is a Borel subgroup iff it has no proper finite-index subgroup."},{"cited_title":"Furter and J.-M","cited_arxiv_id":null,"evidence_quote":"Supplies the Borel-subgroup results for $\\operatorname{Aut}(\\mathbb{A}^2)$ and the maximality of the triangular subgroup used for $N^\\pm_{d,e}$."},{"cited_title":"Kovalenko, A","cited_arxiv_id":null,"evidence_quote":"Supplies the ind-group and nested-subgroup facts, including that connected nested subgroups have no proper finite-index subgroup."},{"cited_title":"Liendo, A","cited_arxiv_id":null,"evidence_quote":"Supplies the pseudo-divisibility and algebraicity criteria for automorphisms used in the torsionally-unbounded arguments."}],"review_version":1}