{"id":"bf78dfb5-3929-4a5a-962c-1ec237c3f2de","arxiv_id":"1908.03184","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Multiplier invariants of periodic points are regular functions on moduli spaces of endomorphisms of P^N, are finite-to-one on certain families, and admit explicit isospectral families.","lead":"This paper studies how the multiplier spectra of periodic points can act as coordinates on the moduli space of degree-d endomorphisms of projective space in dimension at least two. It constructs invariant functions, shows they are regular, and proves that in several families the multiplier map is finite-to-one, with isospectral families showing limits of this approach.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.13's degree count hinges on an unproved unique partition of the multiplier eigenvalues into coordinate spectra; if multiple partitions occur generically, the claimed ((d-2)!)^N fiber degree is an overcount.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing gap in Theorem 5.13: the assertion of a unique splitting of the unordered eigenvalue sets into coordinate multiplier spectra. My reading of the proof confirms that this uniqueness is stated without proof and is not a trivial consequence of the preceding lemmas. The factorization of Σ_1(f) gives the eigenvalue sets; the reconstruction of the N coordinate spectra is a nontrivial combinatorial step. The generic distinctness of eigenvalues may make the partition unique in many cases, but the proof does not establish this, and the special eigenvalues at infinity (0 and d) could create coincidences that break uniqueness. If multiple partitions occur on a Zariski-open set, the stated degree ((d−2)!)^N would be an underestimate of the true generic fiber cardinality. This concern does not require rejecting the paper: the map might still be finite-to-one with a larger degree, and the other families (triangular, monic quadratics) might remain correct. But because the headline Theorem C(1) is a precise numerical statement, the proof must close this gap before the claim is accepted as proved. The reader's verdict CONDITIONAL is appropriate; my stress-test does not move it, so the verdict is UNCHANGED. I also note the parallel independence issue in Theorem 5.17's interpolation argument, but the unique-partition gap is the most consequential because it directly affects the paper's sharpest quantitative claim.","tokens_in":22446,"tokens_out":9203,"duration_ms":100024,"concrete_test":"For N=2 and d=4 (where (d−2)!=2), choose a generic split polynomial F=(F_1,F_2) with distinct fixed point multipliers, e.g., F_1(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0 and F_2(y)=y^4+b_3y^3+b_2y^2+b_1y+b_0 with random rational coefficients. Compute Σ_1(F) either by the paper's elimination algorithm (Algorithm 4.3) or directly from the fixed point solutions, and factor it to obtain the multiset of unordered pairs of eigenvalues for all fixed points, including those at infinity. Enumerate all ways to partition this eigenvalue multiset into two 5-element multisets A and B such that the Cartesian-product family { {a,b}: a∈A_affine? plus the infinity fixed points } reproduces exactly the observed multiset of pairs. Count the number of distinct partitions up to swapping A and B.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim of Theorem 5.13 is that τ_{d,1}^N is generically ((d−2)!)^N-to-one on split polynomial endomorphisms. The proof factors Σ_1(f) into unordered eigenvalue sets, then asserts: 'based on which eigenvalues occur in which (unordered) sets, we can split the eigenvalues into multiplier spectra for each coordinate function in only one way.' This assertion is not proved, and it is not evident. For a split map F=(F_1,...,F_N), the affine fixed points are products of fixed points of the F_i, so the observed data consist of N-element sets {λ_1(p_1),...,λ_N(p_N)} one for each tuple (p_1,...,p_N). Recovering the N multiplier spectra A_i={λ_i(p): p∈Fix(F_i)} from this multiset of N-tuples is a factorization problem for a complete multipartite hypergraph. Even in the generic distinct-eigenvalue case, uniqueness up to permutation requires proof; with repeated eigenvalues, or with the special eigenvalues contributed by the fixed points at infinity (which the proof also assumes can be identified), the partition can be non-unique. If for a Zariski-open set of split maps there is more than one admissible partition, then the fiber contains more than ((d−2)!)^N maps, so Theorem 5.13's stated degree is wrong, even if finite-to-one-ness survives. The proof of Theorem 5.17 has a related unproven independence claim in its interpolation step; however, the explicit numerical degree in Theorem 5.13 makes the partition uniqueness the most load-bearing unsupported step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines multiplier invariants for endomorphisms of projective space of degree d, collected as coefficients σ_{i,j}^{(n)} of a polynomial Σ_n(f) built from characteristic polynomials of multiplier matrices at periodic points. It proves these are regular functions on the moduli space M_d^N, derives some relations among them, gives an elimination-theoretic algorithm for computing them without finding periodic points, and then addresses the extent to which multiplier data determine the conjugacy class. The main results are a conjecture generalizing McMullen's theorem, several constructions of isospectral families (symmetric products, Cartesian products, Segre products), and three finite-to-one statements: for split polynomial endomorphisms (a generic degree ((d−2)!)^N is claimed), for triangular polynomial endomorphisms, and for monic quadratic endomorphisms of P^2. The paper also contains an explicit computational description of the multiplier map on the monic quadratic family.","tokens_in":22709,"tokens_out":8921,"duration_ms":94070,"significance":"If the finite-to-one results are correct, the paper makes a meaningful step toward a higher-dimensional analogue of McMullen's theorem and provides useful tools for studying the moduli space M_d^N. The construction of the invariants is natural, the isospectral families via symmetric and Cartesian products are elegant, and the explicit computation on monic quadratics gives a rare concrete data point. The paper is also honest about its own gaps: the proof of Theorem 5.17 explicitly states that the independence of the interpolation equations remains a question, and the proof of Theorem 5.13 asserts a partition-uniqueness step without proof. These gaps are load-bearing for the stated degree and finite-to-one claims, so the paper's central new assertions require additional work before they can be accepted.","major_comments":[{"comment":"The claimed degree ((d−2)!)^N depends on the unsupported assertion that, from the unordered N-tuples of eigenvalues of the multiplier matrices, one can split the eigenvalues into the multiplier spectra of the coordinate polynomials in only one way. The data form a complete multipartite hypergraph with vertex sets the coordinate spectra, and generic uniqueness of the factorization is not evident; with repeated eigenvalues or with the special eigenvalues contributed by fixed points at infinity, the partition can fail to be unique. If multiple partitions occur on a Zariski-open set, the fiber contains more than ((d−2)!)^N maps and the theorem's degree is an overcount, even if finite-to-one-ness survives. This step must be proved or the statement weakened to an upper bound.","section":"§5.2.1, Theorem 5.13"},{"comment":"The interpolation argument assumes that the linear equations determining the coefficients of F_k from their values at the fixed points of (F_1,...,F_{k−1}) are generically independent. The proof shows that a dependency would place the fixed points on a degree-j hypersurface and that this is a closed condition, but it does not prove that the union of these closed conditions is a proper subset of the moduli space; the example of the powering map avoids one such condition, but that does not rule out another closed condition covering the entire space. In addition, the earlier step of selecting, among the total eigenvalue data, the finite subsets that could be the multiplier spectrum of F_1 is again a partition-uniqueness problem that is not addressed.","section":"§5.2.2, Theorem 5.17"},{"comment":"The proof that σ_{i,j}^{(n)} is regular on Hom_d^N is too terse. The assertion that the only possible poles come from partial derivatives of the dehomogenized map, and that these yield only powers of the resultant, needs a careful argument: the fixed points themselves are algebraic over the coefficient field and their coordinates can have denominators, and one must show these cancel in the symmetric functions. This is load-bearing because regularity on M_d^N is one of the paper's foundational claims (Theorem A).","section":"§2, Theorem 2.4(1)"},{"comment":"The hypersurface equation for the image of τ_{2,1}^2 restricted to monic polynomials is asserted after a Sage computation, but no code, script, or certificate is provided. Since Corollary 5.20 and the claim that the image is a hypersurface rest entirely on this calculation, the computation should be reproducible (for example, by including the Sage code and the elimination Groebner basis) or verified by an independent method.","section":"§5.2.3, Theorem 5.19"},{"comment":"The stated number of points of period n is D_n = (d^{n(N+1)}−1)/(d−1), but the number of fixed points of f^n on P^N is ((d^n)^{N+1}−1)/(d^n−1). For N=1, d=2, n=2 the formula gives 15, while f^2 has 5 fixed points counted with multiplicity. This error propagates into the indexing of Σ_n in equations (1)–(2) and into the proof of Theorem 3.1 for n>1.","section":"§2, paragraph after Definition 2.1"}],"minor_comments":[{"comment":"The example states that the invariants for F=(x^2+c, y^2+d) are generated by σ_{2,2}=8(c+d)+60 and σ_{2,3}=16(c+d)+24, both of which depend only on c+d; this cannot determine the pair (c,d) up to permutation. Please provide the full set of invariants or correct the assertion.","section":"Example 5.14"},{"comment":"The proof of independence of {σ_{1,j},...,σ_{j,j}} is terse; the claim that each σ_{b,j} contains a partition not found in σ_{a,j} for a<b≤j requires a more formal statement about partitions of j into at most i parts.","section":"§3, Theorem 3.1"},{"comment":"The algorithm is described as a way to compute Σ_1(f), but the treatment of multiplicities is not fully justified; the remark that Groebner bases lose multiplicity information and the claimed fix via Chow forms need a precise correctness statement.","section":"§4, Algorithm 4.3"},{"comment":"The phrase 'Fujimura's results summarized in [6]' is confusing because earlier in the paper Fujimura–Nishizawa [7] is cited for the polynomial multiplier coordinate; please clarify which reference is meant.","section":"§5.2.1"},{"comment":"There are numerous typos and OCR artifacts (e.g., 'endomorph ism' in the abstract, 'indeterminant' for 'indeterminate', stray spacing in 'Latt` es'), which should be corrected in a final version.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is a 2019 arXiv preprint; before sending to referees the editor may wish to check whether the author has released a revised version addressing the partition-uniqueness and independence gaps. The computational claims in Theorem 5.19 should be supported by code. The inconsistency in Example 5.14 is small but suggests that some Sage computations may not have been double-checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my read. The paper does something real. It defines multiplier invariants for endomorphisms of P^N, proves they are regular functions on the moduli space, gives relations and a computational method, constructs several isospectral families, and proves finite-to-one results for split and triangular polynomial families and monic quadratics. The isospectral constructions (symmetric products of Lattès maps, cartesian products, Segre embeddings) are clean and genuinely useful. Theorem 5.19's explicit hypersurface for monic quadratics is a concrete computational contribution. The conjectured McMullen analogue is natural and well-motivated.\n\nThe soft spots are exactly where the stress-test note lands. In Theorem 5.13, the proof asserts without proof that the unordered eigenvalue sets split uniquely into the coordinate multiplier spectra. That is not obvious; it is a factoring problem for a complete multipartite hypergraph, and if multiple partitions occur generically, the ((d−2)!)^N degree is an overcount. Finite-to-one-ness may survive, but the stated degree is load-bearing and unsupported. Theorem 5.17 has a structurally similar gap: the interpolation equations are assumed generically independent, with the closed-condition argument sketched rather than fully proved. Plausible, but not complete. Theorem 5.9's proof is one line, probably fine but terse. The computational theorem has no shipped code, but the explicit polynomial is checkable and the computations look reproducible. Theorem 3.1's proof is hard to follow but seems to work.\n\nNone of these are fatal. The framework is sound and the special cases are genuinely new. The gaps are addressable in revision: either prove the partition uniqueness for 5.13, or weaken the statement to finite-to-one and drop the degree, and expand the independence argument in 5.17. The self-citation for symmetric products is to a published external result, so that is not a problem.\n\nThis paper deserves a serious referee. It is a real step for higher-dimensional moduli in arithmetic dynamics, and even if the degree count needs correction, the isospectral families and finite-to-one results would still be a solid contribution. I would send it to review, with a note that the referee should focus on Section 5.2.1.","headline":"A genuine extension of multiplier invariants to higher-dimensional moduli spaces with real new results, but the headline degree count in Theorem 5.13 rests on an unproved uniqueness claim that needs fixing before the quantitative statement is trusted.","tokens_in":23316,"tokens_out":1821,"would_cite":true,"duration_ms":19778,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37P45","37P05","37A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Higher-dimensional dynamical maps can be told apart, up to finitely many choices, by the eigenvalues of the derivative at periodic points.","keywords":["multiplier invariants","moduli space of endomorphisms","projective space dynamics","isospectral families","split polynomial endomorphisms","triangular polynomial endomorphisms","Lattès maps","fixed point multipliers"],"falsifier":"Compute $\\Sigma_1(F)$ for every monic split polynomial $F=(x^3+a x+b,\\, y^3+c y+e)$ in two variables with distinct fixed points. Factor $\\Sigma_1(F)$ and enumerate all ways to split the nine pairs of eigenvalues into two sets of three pairs that could be the fixed-point spectra of cubic polynomials. If any two different splits produce the same coefficient list $\\sigma^{(1)}_{i,j}$, or if one split supports more than $((3-2)!)^2=1$ pair of coordinate polynomials, the claimed generic degree is wrong.","tokens_in":22165,"feed_emoji":"🔄","tokens_out":14028,"duration_ms":128362,"temperature":0.7,"pith_summary":"The paper aims to show that fixed-point multiplier data—the eigenvalues of the derivative at periodic points—can serve as (almost) separating invariants for dynamical systems on higher-dimensional projective space, not just on the Riemann sphere. It defines regular functions $\\sigma^{(n)}_{i,j}$ on the moduli space $\\mathcal{M}^N_d$, proves they satisfy relations, and gives an elimination-theoretic method to compute them. The main results show that the fixed-point multiplier map $\\tau^N_{d,1}$ is generically $((d-2)!)^N$-to-one on split polynomial endomorphisms and generically finite-to-one on triangular polynomial endomorphisms and on a family of monic quadratic maps of $\\mathbb{P}^2$. It also constructs several isospectral families whose multiplier invariants are constant. These results support a proposed higher-dimensional analogue of the classical one-dimensional theorem that multiplier maps are quasi-finite away from Lattès maps.","feed_headline":"Fixed-point multipliers separate higher-dimensional maps","feed_subtitle":"For split, triangular, and quadratic P^2 maps, periodic-point eigenvalues are enough to identify maps up to finitely many choices.","key_machinery":"The central object is the $n$-multiplier spectrum. For each point $P$ of exact period $n$, form the characteristic polynomial $\\gamma_{f^n,P}(t)$ of the multiplier matrix $d(f^n)_P$; collect all of them into $\\Sigma_n(f)=\\prod_{P\\in\\mathrm{Per}_n(f)}(w-\\gamma_{f^n,P}(t))$, and define $\\sigma^{(n)}_{i,j}$ as its coefficients. These coefficients are symmetric functions of the eigenvalues and are invariant under $\\mathrm{PGL}_{N+1}$ conjugation, and Theorem 2.4 shows they lie in the ring of regular functions $\\mathbb{Q}[\\mathcal{M}^N_d]$. The proof mechanism for the finite-to-one results is separation: since split maps have diagonal multiplier matrices, the eigenvalues from $\\Sigma_1(f)$ can be assigned to coordinate polynomials, reducing the problem to the one-dimensional polynomial case; for triangular maps the same assignment feeds a multivariate Lagrange interpolation step; and for monic quadratics explicit elimination computes the image hypersurface. The fixed-point index identity $\\sum_{P\\in\\mathrm{Fix}(f)}\\gamma_{f,P}(t)/\\gamma_{f,P}(1)=(t^{N+1}-d^{N+1})/(t-d)$ supplies relations among the invariants.","core_discovery":"The central discovery is that the multiplier spectrum—the unordered collection of characteristic polynomials of the derivative maps at periodic points—defines a system of regular conjugation-invariant functions on the moduli space $\\mathcal{M}^N_d$, and that this system controls conjugacy classes in several natural families. Concretely, for split polynomial endomorphisms (each coordinate polynomial in a single variable), the fixed-point multiplier map $\\tau^N_{d,1}$ is generically $((d-2)!)^N$-to-one: the only ambiguity left after reading the multipliers is an arbitrary permutation of the $N$ coordinate polynomials, each of which is recoverable up to the familiar $(d-2)!$ choices from one-variable polynomial dynamics. For triangular polynomial endomorphisms and for monic quadratic maps of $\\mathbb{P}^2$ of the form $f=[x^2+a_1xz+a_2yz-a_1z^2 : y^2+b_1xz+b_2yz-b_1z^2 : z^2]$, the same map is generically finite-to-one; for the monic quadratics the image is an explicit hypersurface in $\\mathbb{A}^5$. The paper also shows that Lattès-type constructions—symmetric products, cartesian products, and Segre embeddings of a Lattès family with the power map—give isospectral families whose multiplier invariants are constant for all periods.","pith_inferences":["A concrete test of the split-polynomial degree formula: enumerate all monic split degree-3 polynomials in two variables and compare fibers of the map sending the polynomial to the coefficients of $\\Sigma_1(f)$; finding a fiber with two different valid partitions of the eigenvalue pairs would lower the generic degree below $((d-2)!)^N$.","The generic degree of $\\tau^2_{2,1}$ on the monic quadratic family remains open; partial computations in the paper suggest degree 8 on an open set and degree 12 on closed subsets, so a direct fiber-count would settle it.","The isospectral property for Segre embeddings seems to persist for more general isospectral factors; the paper notes computations suggest this but were beyond available machine power, so a proof for arbitrary isospectral pairs is a natural extension.","If the uniqueness-of-partition assumption fails, the corrected statement would likely involve a combinatorial factor counting compatible partitions of the multiplier spectrum, computable explicitly for small $N$ and $d$."],"forward_implications":["If the theorems are correct, the multiplier invariants give an explicit, computable coordinate system on large parts of $\\mathcal{M}^N_d$, so conjugacy classes can be compared by finite multiplier data rather than by searching for conjugacies.","For split polynomial endomorphisms, including period-2 multiplier invariants should make $\\tau^N_{d,n}$ generically one-to-one, assuming the one-dimensional conjecture on period-2 multipliers; the paper states this as a corollary of that conjecture.","The monic quadratic family of $\\mathbb{P}^2$ shows that $\\tau^2_{2,1}$ has image of codimension one in $\\mathbb{A}^5$, with an explicit hypersurface equation that can be used to test membership and compute fibers.","Isospectral families provide higher-dimensional analogues of Lattès maps: symmetric products, cartesian products, and Segre images of Lattès families have constant multiplier invariants, so any quasi-finiteness statement must exclude them just as dimension 1 excludes Lattès maps.","The conjecture that $\\tau^N_{d,n}$ is quasi-finite for large $n$ reduces the classification problem to the finite ambiguity encoded by the multiplier spectra, making the moduli space more accessible to arithmetic and computational study."],"supporting_citations":[{"why":"The dimension-one theorem that the multiplier map is finite-to-one away from Lattès maps; the result being generalized.","marker":"[20]"},{"why":"Provides the explicit isomorphism of the degree-2 one-dimensional moduli space with affine space via fixed-point multiplier symmetric functions.","marker":"[21]"},{"why":"Supplies the fixed-point index relation in projective space used to derive relations among the multiplier invariants.","marker":"[30]"},{"why":"Gives the one-dimensional polynomial fiber count (d-2)! that is applied coordinate-by-coordinate in the split-polynomial theorem.","marker":"[7]"},{"why":"Provides an algorithm for computing fibers of the one-dimensional polynomial multiplier map, applied componentwise to split maps.","marker":"[29]"},{"why":"Establishes one-dimensional multiplier-spectrum results and the period-2 injectivity conjecture whose analogue extends to split polynomials.","marker":"[14]"},{"why":"Shows multipliers of symmetric products depend only on the multipliers of the original map, making Lattès symmetric products isospectral.","marker":"[8]"},{"why":"Proves the invariant ring of rational maps equals the regular functions on the one-dimensional moduli space, the model for the regularity theorem.","marker":"[26]"}],"fun_headline_variants":["Multiplier spectra: finite ambiguity in P^N moduli","Periodic-point eigenvalues fix split maps up to swaps","Isospectral Lattès families evade multiplier invariants","Multiplier map: finite-to-one for split, triangular, quadratic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, given the full list of eigenvalue pairs of a split polynomial, there is only one way to assign those pairs to the $N$ coordinate polynomials so that they form valid multiplier spectra; if more than one assignment works, the claimed degree $((d-2)!)^N$ would be an overcount, and a similar independence assumption is needed for the triangular interpolation step.","fun_headline_variants_meta":{"raw":{"variants":["Multiplier spectra: finite ambiguity in P^N moduli","Periodic-point eigenvalues fix split maps up to swaps","Isospectral Lattès families evade multiplier invariants","Multiplier map: finite-to-one for split, triangular, quadratic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000398,"raw_usage":{"total_tokens":2117,"prompt_tokens":1012,"completion_tokens":1105,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":628,"completion_tokens_details":{"reasoning_tokens":1036}},"tokens_in":628,"tokens_out":1105,"duration_ms":13450,"temperature":1.0,"reasoning_tokens":1036,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:23:02.317378+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\Sigma_1(F)$ for every monic split polynomial $F=(x^3+a x+b,\\, y^3+c y+e)$ in two variables with distinct fixed points. Factor $\\Sigma_1(F)$ and enumerate all ways to split the nine pairs of eigenvalues into two sets of three pairs that could be the fixed-point spectra of cubic polynomials. If any two different splits produce the same coefficient list $\\sigma^{(1)}_{i,j}$, or if one split supports more than $((3-2)!)^2=1$ pair of coordinate polynomials, the claimed generic degree is wrong.","supporting_citations":[{"cited_title":"Families of rational maps and iterati ve root-finding algorithms","cited_arxiv_id":null,"evidence_quote":"The dimension-one theorem that the multiplier map is finite-to-one away from Lattès maps; the result being generalized."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit isomorphism of the degree-2 one-dimensional moduli space with affine space via fixed-point multiplier symmetric functions."},{"cited_title":"Complex dynamics on projective spaces – in dex formula for fixed points","cited_arxiv_id":null,"evidence_quote":"Supplies the fixed-point index relation in projective space used to derive relations among the multiplier invariants."},{"cited_title":"The real multiplier coordinate space of the quartic polynom ials, pages 61–69","cited_arxiv_id":null,"evidence_quote":"Gives the one-dimensional polynomial fiber count (d-2)! that is applied coordinate-by-coordinate in the split-polynomial theorem."},{"cited_title":"The moduli space of polynomial maps and their fixed-point multipliers","cited_arxiv_id":null,"evidence_quote":"Provides an algorithm for computing fibers of the one-dimensional polynomial multiplier map, applied componentwise to split maps."},{"cited_title":"Multiplier spectra a nd the moduli space of degree 3 morphisms on P1","cited_arxiv_id":null,"evidence_quote":"Establishes one-dimensional multiplier-spectrum results and the period-2 injectivity conjecture whose analogue extends to split polynomials."},{"cited_title":"Symmetrization of Rational Maps: Arithmetic Properties and Families of Latt\\`es Maps of $\\mathbb P^k$","cited_arxiv_id":"1603.04887","evidence_quote":"Shows multipliers of symmetric products depend only on the multipliers of the original map, making Lattès symmetric products isospectral."},{"cited_title":"Silverman","cited_arxiv_id":null,"evidence_quote":"Proves the invariant ring of rational maps equals the regular functions on the one-dimensional moduli space, the model for the regularity theorem."}],"review_version":1}