{"id":"71261833-df44-4d3d-ad2f-13ae97e7f96b","arxiv_id":"hep-th/0411183","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Non-supersymmetric flux vacua correspond to eigenvectors of the superpotential Hessian, and in the generic random-matrix ensemble the count of metastable pure F-breaking vacua with near-zero cosmological constant scales as the twelfth power of the supersymmetry-breaking scale.","lead":"In the ensemble of type IIB flux compactifications, the authors reformulate the search for supersymmetry-breaking vacua as an eigenvector problem for the matrix of second derivatives of the superpotential, and connect its statistics to a known random-matrix ensemble. The headline consequence is that, for pure F-term breaking with near-zero cosmological constant, low supersymmetry-breaking scales are disfavored, scaling as the twelfth power of the breaking scale.","discovery_kind":"unclear","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The M*^12 scaling rests on CI-genericity of Z, which the authors acknowledge is unproven for the actual IIB ensemble and which their own tractable examples violate; the unresolved question is whether any realistic multi-modulus Calabi-Yau realizes the CI measure.","rationale":"The reader correctly identifies the CI genericity of Z as the weakest assumption behind the M*^12 headline. The structural result that non-supersymmetric F-breaking critical points are eigenvectors of D^2W with eigenvalue 2|W| is exact and independent of the ensemble, and the derivation of Eq. 3.4 is internally consistent once the typeset fractions in Eq. 3.2 are read correctly. The M*^12 scaling is explicitly ensemble-dependent: the paper says so in the abstract and in §3.1.3, and its own one-modulus and orbifold examples give different exponents. The D-term analysis in §4 is a separate and useful contribution. There is no circularity, no fitted parameter, and the self-citation to hep-th/0404116 is legitimate prior support. The concern is real but acknowledged, so it does not move the verdict; the paper is honest about the gap between the generic CI ensemble and the physically relevant IIB ensemble.","tokens_in":26196,"tokens_out":29250,"duration_ms":481606,"concrete_test":"Numerically sample the IIB flux ensemble for a multi-modulus Calabi-Yau (e.g. the mirror quintic with h^{2,1}=101, or a simpler two-modulus case), compute the distribution of the smallest eigenvalue lambda_1 of Z = D^2W at fixed moduli, and check whether it scales as d(lambda_1^2) for small lambda_1. If it does, the CI measure and the M*^12 scaling are supported; if it scales as d(lambda_1) or d(lambda_1^3), the exponent changes and the headline needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline result N ~ M*^12 (Eq. 3.9, Conclusions) is derived for the generic ensemble Eq. 2.12, in which Z = D^2W is a generic complex symmetric matrix in the CI class, so that a small eigenvalue is distributed as d(lambda^2) (Eq. 2.30). The authors state plainly (abstract:s1, §2.4, §3.1.3) that the actual IIB ensemble is a subset of CI and that they have seen no sign of the commuting-generator structure Eq. (2.33) elsewhere. But the paper's own tractable examples give different scalings: the one-modulus conifold gives F^3 dF (§3.1.1) because the see-saw structure forces det Z ~ 1, and T^6/Z2xZ2 has commuting generators producing degenerate eigenvalues (§3.1.2). The paper argues these are special, but does not demonstrate that a realistic multi-modulus Calabi-Yau (h^{2,1} >= 2) realizes the CI measure d(lambda^2). Since M*^12 is a product of four factors and the d(lambda^2) factor is one of them, a deviation in the eigenvalue measure would change the exponent. The authors acknowledge this as future work, so the concern is not a new finding but an unresolved gap between the headline and the physically relevant ensemble.","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The paper reformulates the search for nonsupersymmetric F-term flux vacua as an eigenvector problem for the matrix M = D^2W of second derivatives of the superpotential: critical points with F ≠ 0 are eigenvectors of M with eigenvalue 2|W| (Eqs. 2.13–2.19). It identifies the relevant ensemble of such matrices as the CI ensemble of Altland and Zirnbauer, derives a general density formula for nonsupersymmetric vacua (Eq. 3.4), and applies it to IIB flux compactifications. The headline result is that the number of metastable pure F-breaking vacua with Λ ≈ 0 and M_susy < M* scales as M*^12 (Eq. 3.9), a product of four factors: phase fixing (dF), the CI eigenvalue measure d(λ^2), the mass-matrix determinant (F^2), and metastability tuning (F^2). Two tractable examples (one-modulus conifold, T^6/Z2×Z2) are worked out, and a heuristic D-term analysis (§4) contrasts lifted supersymmetric vacua (N ~ 1/log M*) with F-term descendants.","tokens_in":26433,"tokens_out":14755,"duration_ms":258767,"significance":"The eigenvector reformulation is exact and makes manifest that nonsupersymmetric vacua are generic and comparable in number to supersymmetric ones; the CI connection is a genuine structural insight linking flux-vacuum statistics to random-matrix theory. The M*^12 scaling is a concrete, parameter-free-in-form prediction for the model ensemble, and the paper is careful to distinguish the model ensemble from the actual IIB ensemble. The D-term contrast is physically important. The derivations are checkable and the paper is well organized.","major_comments":[{"comment":"The M*^12 result is derived for the generic ensemble Eq. 2.12, in which Z is a generic CI matrix so that the small-eigenvalue measure is d(λ^2). The authors acknowledge (abstract:s1, §2.4, §3.1.3) that the actual IIB ensemble is a subset of CI and that their own tractable examples violate the genericity: the one-modulus conifold gives F^3 dF (§3.1.1) because the see-saw structure forces det Z ~ 1, and T^6/Z2×Z2 has commuting generators (Eq. 2.33) producing degenerate eigenvalues (§3.1.2). What remains unresolved is a quantitative statement of how the exponent depends on the eigenvalue measure: if the IIB measure is d(λ^p) with p ≠ 2, the exponent shifts from 12. A short paragraph stating the exponent as a function of p, and identifying which structural features of a Calabi-Yau (number of moduli, prepotential) would push p away from 2, would make the scope of the headline result precise. This is load-bearing because the d(λ^2) factor is one of the four powers.","section":"§3.1.3, Conclusions, Eq. 3.9","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The O(F^2) metastability suppression is derived assuming U is approximately uniformly distributed and independent of Z and F. In the IIB ensemble U is determined by Z and F (Eq. 3.11), and the explicit mass formula Eq. 3.13 shows the O(F) term is Re(e^{2iθ}D_I F_{JKL} Zbar^I Fbar^J Fbar^K Fbar^L). The paper checks the large-complex-structure limit (DF = 0) and the conifold limit, but does not give a general argument that the IIB correlations leave the O(F^2) suppression intact. Since this factor is one of the four powers in M*^12, a general argument (or a counterexample) would strengthen the claim. This is a correctness-risk concern, not a claim of error.","section":"Below Eq. 2.28","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The D-term analysis is explicitly heuristic (the text states there is no explicit ensemble of EFTs with D terms that looks simple and universal enough). The contrast between the lifted-susy result N ~ 1/log M* (Eq. 4.4, favoring low scale) and the F-term M*^12 (favoring high scale) is one of the paper's most physically interesting claims, but it rests on treating the D-term as a small perturbation with ǫ ~ |v|^{4/3} and on the cancellation of V''_D against H_N in the v-direction. The paper should state more explicitly which conclusions in §4.3 are robust to the choice of D-term ensemble and which depend on the antibrane/conifold specifics.","section":"§4, Eqs. 4.4, 4.10–4.12","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The factor written as df|f| appears to be a typo; the preceding line gives d|F_1|/|F_1|, so the measure should presumably be df/|f| (or the notation should be clarified). Please check the power of |F| carried into Eq. 3.4.","section":"Eq. 3.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Reference [4] gives arXiv:hep-th/030704, which appears truncated; the Ashok–Douglas paper is hep-th/0307041. Reference [1] reads A, Altland and should read A. Altland.","section":"References [1], [4]","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The statement that a rough fraction 1/n of all nonsupersymmetric critical points are metastable sits somewhat uneasily with the later O(F^2) metastability suppression (Eq. 2.28); a sentence reconciling the two regimes would help the reader.","section":"§2.3, below Eq. 2.24","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The symbol M is used both for the matrix Eq. 2.15 and for M_susy/M_planck; a notational distinction would reduce ambiguity.","section":"§2.4","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claim that (m^-_1)^2 = -8|F|^2/3 < 0 for pure F-breaking with Λ ≈ 0 could be spelled out; it is a useful check on the general formula and on the statement that this example has no such metastable vacua.","section":"§3.1.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The discussion of the Dine et al. W = 0 peak is a disagreement with a specific proposal; it would be clearer to state explicitly which of their assumptions the paper is rejecting and which part of the argument is being accepted.","section":"§3.2.2","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a follow-up to the authors' own hep-th/0404116 and imports the CI ensemble from condensed matter; the novelty is the application, which is substantial and clearly explained. The D-term section is the weakest part and could be shortened or more explicitly flagged as exploratory. The paper is otherwise suitable for the journal."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"The thing worth your time here is section 2: the F-breaking critical point equations are rewritten as an eigenvector condition on the matrix of second superpotential derivatives, with eigenvalue 2|W| (Eqs. 2.13–2.19). That is exact, it is clean, and it immediately explains why metastable nonsupersymmetric vacua are generic rather than fine-tuned — you can only use the lowest eigenvalue, so roughly 1/n of critical points survive. The identification of the relevant random-matrix class as Altland-Zirnbauer CI (Eq. 2.29) is a genuinely nice observation, and the degeneracy-counting argument in §2.4 is careful. Section 4 on D-terms is a separate, useful contribution: lifted susy vacua give a 1/log M* distribution favoring low scales, while perturbed F-breaking vacua keep the F-scaling. That contrast is the most physically interesting thing in the paper and it is stated plainly. The paper is also unusually candid about its own weak points, which I appreciate. The headline N ~ M*^12 (Eq. 3.9) is a product of four order-of-magnitude factors, and one of them — the d(lambda^2) eigenvalue measure — assumes Z behaves as a generic CI matrix. The authors say outright that the actual IIB ensemble is only a subset of CI, and their own tractable examples give different answers: the one-modulus conifold gives F^3 dF because the see-saw forces det Z ~ 1 (§3.1.1), and T^6/Z2xZ2 has commuting generators producing degenerate eigenvalues (§3.1.2). They argue both are special, and I believe them, but they do not show that a realistic multi-modulus CY realizes the CI measure. So the exponent is conditional, not derived. That is a real gap, but it is a gap the paper names rather than hides, and the structural result does not depend on it. The other soft spot is the metastability tuning argument in §2.3, where the O(F^2) suppression from Eq. 2.28 rests on an assumption of uniform distribution of one component of U — plausible, not proved. Neither of these is fatal to the paper's actual contribution. Who should read it: anyone working on flux-vacuum statistics or on whether the landscape favors low-scale susy. The structural reformulation will be cited regardless of whether M*^12 survives. This deserves referees.","headline":"The eigenvector reformulation is exact and useful; the M*^12 scaling is an honest order-of-magnitude estimate whose exponent depends on an ensemble assumption the authors themselves flag as unproven.","tokens_in":27142,"tokens_out":716,"would_cite":true,"duration_ms":11633,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["11.25.Mj","11.30.Pb"],"model":"selfhosted/deepseek-v4.1-flash","headline":"Nonsupersymmetric flux vacua are counted by eigenvectors of a Hessian, and low-scale breaking is suppressed","keywords":["flux vacua","supersymmetry breaking","string landscape","random matrix ensembles","Calabi-Yau compactification","supergravity","eigenvalue degeneracy","cosmological constant tuning"],"falsifier":"Compute the eigenvalue distribution of $Z_{IJ} = F_{IJK} 0^K$ in an explicit multi-modulus Calabi-Yau compactification and check whether the smallest eigenvalue follows $d(0^2)$ as the CI ensemble predicts, or whether see-saw or commuting-generator structure pushes small eigenvalues away; a measured exponent other than 12 in the small-$F$ vacuum count would settle it.","tokens_in":27468,"feed_emoji":"🌌","tokens_out":3560,"duration_ms":73829,"temperature":0.7,"pith_summary":"This paper asks how many string-theory flux vacua break supersymmetry at a given scale, and finds that the answer is governed less by the details of any particular compactification than by the statistics of a single matrix: the matrix of second derivatives of the superpotential. The authors show that a critical point with non-zero $F$-terms is exactly an eigenvector of that matrix, with the eigenvalue fixed to twice the superpotential, so counting vacua becomes counting eigenvectors. Because the vacua of interest require a small eigenvalue and a small superpotential, the answer depends on how a generic complex symmetric matrix distributes its smallest eigenvalue — an ensemble that turns out to be the CI class of Altland and Zirnbauer, borrowed from mesoscopic physics. Carrying the calculation through, including the mass-matrix determinant that normalizes the delta function and the fine-tuning needed for stability, gives the paper's headline count: the number of metastable pure $F$-term vacua with breaking scale below $M_*$ and cosmological constant near zero grows as $M_*^{12}$. That scaling runs against the intuition that many independent hidden sectors should pile up at high breaking scales, and it means that tuning the vacuum energy to zero is itself a strong selector.","feed_headline":"Tuning the cosmological constant favors high-scale supersymmetry","feed_subtitle":"Requiring near-zero vacuum energy costs a factor of the breaking scale to the twelfth, so low-scale flux vacua are scarce.","key_machinery":"The load-bearing object is the matrix $M$ of Eq. (2.15), a hermitian matrix built from the symmetric matrix $Z_{AB} = D_A D_B W$ and the phase of $W$, obeying the symmetry conditions $M = -0_3 M 0_3 = -0_2 M^* 0_2$. Its eigenvalues come in $+0,-0$ pairs, and its eigenvectors, written as $0^+_a$ and $0^-_a$, are approximate mass eigenstates of the bosonic mass matrix when $|F|$ is small. Solving $dV=0$ becomes the eigenvalue condition $2|W| = 0_1$, with $F$ proportional to the corresponding eigenvector. The symmetry conditions place $M$ in the CI ensemble of Altland and Zirnbauer, whose measure the authors write explicitly as $[dU] 0_a d(0_a^2) 0_{a<b} |0_a^2 - 0_b^2|$; the factor $0_a d(0_a^2)$ is what converts a small eigenvalue into a factor of $0^2$ in the vacuum count, and the Vandermonde-like product is what makes degenerate eigenvalues cost codimension. The paper then uses this to argue that higher branches with $k$-fold degenerate eigenvalues sit at real codimension $k(k-1)/2$ and so do not contribute to the leading large-$L$ asymptotics, except in special cases where the matrices $0Z/0|Z_i|$ commute after a similarity transformation.","core_discovery":"The central claim is that the distribution of nonsupersymmetric flux vacua is, to a good approximation, the distribution of supersymmetric vacua multiplied by computable correction factors, and that those factors can be read off from the eigenvalue statistics of $Z_{AB} = D_A D_B W$. The derivation runs in three steps. First, the equations $dV=0$ are rewritten as a linear problem $N(W,Z) F = 0$, whose solutions are eigenvectors of a hermitian matrix $M$ built from $Z$; a non-supersymmetric vacuum sits at $2|W| = |Z|$'s smallest eigenvalue, and the phase of $F$ is fixed, so the measure is $dF$ rather than $d^2F$. Second, the density of such vacua is written as $|det H|$ times a correction factor involving the two squared masses in the supersymmetry-breaking direction, where $H$ is the same matrix whose determinant weights the supersymmetric count. Third, evaluating the correction factors in the generic ensemble gives $N(M_{susy} < M_*, 0 < M_*^{12}$ for pure $F$ breaking with $M_*$ small. The authors are explicit that this is a sum of four effects of comparable size — the fixed phase, the $d(0^2)$ distribution of the smallest eigenvalue, a factor $F^2$ from $det V''$, and a further $F^2$ from the metastability tuning — and that no single one of them dominates. They also treat the case where a small $D$-term lifts supersymmetric vacua, finding a distribution that instead favors low breaking scales, going roughly as $1/0 M_*$.","pith_inferences":["If the same eigenvalue-repulsion logic applies to the Kähler sector once it is properly stabilized, then the $M_*^{12}$ suppression may be partly offset by the extra measure factors that sector contributes, which the paper explicitly sets aside.","The result suggests a sharper statement than the paper makes: the tuning of the cosmological constant, rather than the dynamics of supersymmetry breaking, is the dominant selector among flux vacua, which would make anthropic-style weighting of $0$ less necessary than usually assumed.","The paper's own counterexamples — the one-modulus conifold with its see-saw determinant, and the orbifold with commuting generators — suggest a practical test: survey the eigenvalue spacing of $Z$ across many explicitly constructible Calabi-Yau moduli spaces and see whether the CI prediction holds or whether commuting structure is common.","If explicit enumeration of quantized fluxes at moderate $L$ confirms the continuous approximation for small $F$, that would validate the whole approach; if it instead cuts off small $F$ as the authors suspect, the suppression would be even stronger than $M_*^{12}$."],"forward_implications":["The naive expectation that independent hidden sectors add their supersymmetry breaking and push the distribution toward high scales is not borne out: the different $F$-terms are independently distributed only if the sectors are completely decoupled.","A low supersymmetry breaking scale is disfavored for pure $F$ breaking once the cosmological constant is tuned near zero, by a factor of order $M_*^8$ relative to the untuned count.","Supersymmetry breaking by a small $D$-term, such as an antibrane near a conifold, produces a distribution falling only as $1/0 M_*$, so the two mechanisms point in opposite directions and their relative abundance is what decides the question.","The universal $-2|W|^2$ term in the bosonic mass matrix destabilizes vacua with light fermions and $|D|, |W| 00 |F|$, removing a large class of otherwise realistic-looking vacua.","Because the mass-matrix determinant enters as a weight, the general expectation that bosons much lighter than the natural scale of the potential are disfavored is made quantitative."],"supporting_citations":[{"why":"Supplies the joint flux distribution Eq. (2.11), the $det H$ density for supersymmetric vacua, and the special-geometry relations for $Z_{IJ}$ and $U_{IJK}$ that the whole calculation is built on.","marker":"[11]"},{"why":"Defines the CI symmetry class and its measure, which the paper adopts as the generic ensemble governing eigenvalue degeneracies and the small-eigenvalue distribution.","marker":"[1]"},{"why":"Establishes the programme of counting string vacua statistically and the large-$L$ volume argument for replacing sums over quantized fluxes by continuous integrals.","marker":"[18]"},{"why":"Provides the geometric picture of the region in flux space supporting vacua, which the paper extends to the nonsupersymmetric case in section 3.2.1.","marker":"[4]"},{"why":"The suggestion of power-law growth in the number of $F$-breaking vacua with the breaking scale, which is the claim the paper's $M_*^{12}$ result argues against.","marker":"[22]"},{"why":"Source for the supergravity fermion mass matrix, used to identify the eigenvector at $2|W|$ as the goldstino and to connect the matrix $M$ to physical masses.","marker":"[37]"},{"why":"Supplies the concrete $D$-term example, an antibrane at the bottom of a warped throat near a conifold, used to model $0 V_D$ and its scaling with the modulus.","marker":"[26]"},{"why":"The proposal that small-$|W|$ vacua are enhanced, which the paper examines in section 3.2.2 and translates into a hypothetical $|W|^{-2}$ factor in the initial distribution.","marker":"[15]"}],"fun_headline_variants":["Flux vacua counts follow eigenvector statistics of the Hessian","Nonsupersymmetric flux vacua inherit CI random-matrix eigenvalue statistics","CI ensemble sets the distribution of nonsupersymmetric flux vacua","Small D-terms can favor low-scale supersymmetry breaking in flux vacua"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The $M_*^{12}$ scaling assumes that the matrix of second superpotential derivatives behaves like a generic member of the CI random-matrix ensemble, so that a small eigenvalue is distributed as $d(0^2)$ with no special structure — an assumption the paper's own tractable examples violate.","fun_headline_variants_meta":{"raw":{"variants":["Flux vacua counts follow eigenvector statistics of the Hessian","Nonsupersymmetric flux vacua inherit CI random-matrix eigenvalue statistics","CI ensemble sets the distribution of nonsupersymmetric flux vacua","Small D-terms can favor low-scale supersymmetry breaking in flux vacua"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":1579,"prompt_tokens":994,"completion_tokens":585,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":516}},"tokens_in":994,"tokens_out":585,"duration_ms":10451,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":516,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-29T16:31:30.602360+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"Compute the eigenvalue distribution of $Z_{IJ} = F_{IJK} 0^K$ in an explicit multi-modulus Calabi-Yau compactification and check whether the smallest eigenvalue follows $d(0^2)$ as the CI ensemble predicts, or whether see-saw or commuting-generator structure pushes small eigenvalues away; a measured exponent other than 12 in the small-$F$ vacuum count would settle it.","supporting_citations":[{"cited_title":"Distributions of flux vacua","cited_arxiv_id":"hep-th/0404116","evidence_quote":"Supplies the joint flux distribution Eq. (2.11), the $det H$ density for supersymmetric vacua, and the special-geometry relations for $Z_{IJ}$ and $U_{IJK}$ that the whole calculation is built on."},{"cited_title":"Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures","cited_arxiv_id":"cond-mat/9602137","evidence_quote":"Defines the CI symmetry class and its measure, which the paper adopts as the generic ensemble governing eigenvalue degeneracies and the small-eigenvalue distribution."},{"cited_title":"The statistics of string/M theory vacua","cited_arxiv_id":"hep-th/0303194","evidence_quote":"Establishes the programme of counting string vacua statistically and the large-$L$ volume argument for replacing sums over quantized fluxes by continuous integrals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the geometric picture of the region in flux space supporting vacua, which the paper extends to the nonsupersymmetric case in section 3.2.1."},{"cited_title":"Statistical analysis of the supersymmetry breaking scale","cited_arxiv_id":"hep-th/0405279","evidence_quote":"The suggestion of power-law growth in the number of $F$-breaking vacua with the breaking scale, which is the claim the paper's $M_*^{12}$ result argues against."},{"cited_title":"Supersymmetry and supergravity,","cited_arxiv_id":null,"evidence_quote":"Source for the supergravity fermion mass matrix, used to identify the eigenvector at $2|W|$ as the goldstino and to connect the matrix $M$ to physical masses."},{"cited_title":"Hierarchies from Fluxes in String Compactifications","cited_arxiv_id":"hep-th/0105097","evidence_quote":"Supplies the concrete $D$-term example, an antibrane at the bottom of a warped throat near a conifold, used to model $0 V_D$ and its scaling with the modulus."},{"cited_title":"Low Energy Supersymmetry From the Landscape","cited_arxiv_id":"hep-th/0407043","evidence_quote":"The proposal that small-$|W|$ vacua are enhanced, which the paper examines in section 3.2.2 and translates into a hypothetical $|W|^{-2}$ factor in the initial distribution."}],"review_version":1}