{"id":"b0425a58-d7a4-4208-81a5-27720a2707db","arxiv_id":"2506.21939","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bayer's polynomial Bridgeland stability on coherent sheaves reduces to lexicographic slope-vector stability exactly when the stability vector is adapted, and a dHYM-type example destabilizes the trivial bundle while admitting flat critical metrics.","lead":"This paper defines a class of stability conditions on coherent sheaves, called adapted ones, that always admit Harder-Narasimhan and Jordan-Holder filtrations, generalizing Gieseker stability. It shows when Bayer's polynomial Bridgeland stability reduces to a lexicographic comparison of degree vectors, and it gives a counterexample to a conjecture about Z-critical connections.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 2.2's claim that every adapted stability condition admits HN/JH filtrations is unproven: the proof is delegated to Huybrechts-Lehn without establishing the boundedness input that Definition 2.2 does not imply for arbitrary totally ordered Q-vector spaces.","rationale":"The reader's weakest_assumption identifies precisely the same load-bearing gap: the adaptation condition may not suffice to run the Huybrechts-Lehn construction in arbitrary totally ordered degree spaces. My reading of the paper confirms that Section 3's main theorem, Theorem 3.9, is internally coherent: the computation of the leading coefficient of Im(Z_ε(E) overline{Z_ε(F)}) correctly reduces asymptotic destabilization to lexicographic comparison of the generalized slope vectors, and the sign conventions, though terse, are consistent once one fixes the convention that destabilization corresponds to a positive value of Im(Z_ε(E) overline{Z_ε(F)}). The paper's verdict should remain CONDITIONAL: the central characterization of asymptotic Z-stability for adapted ρ is credible and supported by the written proof, but the abstract's broader claim that all adapted stability conditions admit HN and JH filtrations is not justified in the stated generality. Independently verifying the boundedness step would either close the gap by supplying a missing lemma or force a restriction of the statement to degree spaces with additional discreteness/boundedness properties. I do not see grounds to reject the paper, nor to accept it before this gap is addressed.","tokens_in":12691,"tokens_out":33821,"duration_ms":350922,"concrete_test":"Re-derive [4, Lemma 1.3.5] for an arbitrary totally ordered Q-vector space V and group morphism Deg satisfying only Definition 2.2, and identify the step where boundedness of the family of saturated subsheaves F⊂E with mu(F) >= C is used. Then check whether Definition 2.2 implies this boundedness: if not, construct a concrete pair (V,Deg,E) where the slope set has a supremum that is not attained (e.g., with V=Q^2 lex-ordered and Deg defined from two independent intersection numbers), which would make Lemma 2.7 false. If the boundedness step cannot be supplied, the statements of Theorems 2.9 and 2.10 must be restricted to degree spaces where boundedness is known, such as the finite-dimensional lexicographic spaces used in Section 3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline contribution includes the assertion that stability conditions 'adapted to coherent sheaves' admit Harder-Narasimhan and Jordan-Holder filtrations. This rests on Theorems 2.9 and 2.10, whose proof is relegated to the sentence 'The proofs are the exact same as in [4, Sections 1.3 and 1.5] so we don't write them again.' The reference proof, however, uses a boundedness argument: for a fixed pure sheaf E, the family of its subsheaves with a prescribed Hilbert polynomial is bounded, and the set of occurring polynomials is discrete enough to select a maximal one. Definition 2.2 (adaptation) only asserts that if E/F has dimension less than d, then mu(F) < mu(E); it says nothing about the set {mu(F)} being bounded above, or about compactness of families of subsheaves with slope above a given bound, in an arbitrary totally ordered Q-vector space V. Thus the generality claimed for Theorems 2.9 and 2.10 is not established. The Section 3 characterization (Theorem 3.9) is stated and proved for the finite-dimensional lexicographically ordered degree spaces coming from intersection numbers, and that part appears self-contained; but the abstract's promise that the adapted class itself 'admits Harder-Narasimhan and Jordan-Holder filtrations' depends on the unproved general statement. The paper also asserts without proof that P_{Z,d} is adapted; this is plausible because the first nonzero degree of a lower-dimensional quotient is positive, but the argument is not written, and the claim is used to connect Theorem 3.9 to the general HN framework.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a class of stability conditions on Coh(X) defined by a totally ordered Q-vector-space-valued degree morphism, calls them \"adapted to coherent sheaves,\" and claims Harder-Narasimhan and Jordan-Holder filtrations for them in full generality. It then specializes to Bayer's polynomial Bridgeland central charges in the large-volume limit. The central result, Theorem 3.9, asserts that when the stability vector is adapted to sheaves of dimension d, a subsheaf asymptotically destabilizes a pure d-dimensional sheaf exactly when its generalized slope vector is larger in the lexicographic order. Consequences include independence of stability from the concrete choice of the adapted stability vector, identification with the explicit lex-degree stability P_{Z,d}, a realization of Gieseker/Simpson stability within this framework, and a claimed counterexample to Dervan-McCarthy-Sektnan Conjecture 1.6. The main proof is a direct asymptotic expansion of the leading coefficient of the central-charge cross product.","tokens_in":12919,"tokens_out":10738,"duration_ms":124699,"significance":"If the missing filtration arguments are supplied, the paper provides a useful and explicit bridge between Bayer's large-volume polynomial stability and more classical slope/Gieseker-type stability. The computation in Theorem 3.9 is structurally correct in the finite-dimensional lexicographic degree spaces that occur in Section 3, and the resulting characterization gives a concrete, checkable stability criterion. The identification of Gieseker stability with a parameter-independent polynomial central charge is a nice observation, and the proposed counterexample to a conjecture of Dervan-McCarthy-Sektnan is potentially interesting. However, the advertised general claim that adapted stability conditions admit Harder-Narasimhan and Jordan-Holder filtrations is currently unsupported, because the proof is delegated to Huybrechts-Lehn without verifying the necessary boundedness and discreteness inputs in the stated generality. The contribution is therefore conditional on repairing that gap.","major_comments":[{"comment":"The existence and uniqueness of Harder-Narasimhan and Jordan-Holder filtrations is asserted for every adapted degree morphism Deg: K(X) -> V with V an arbitrary totally ordered Q-vector space, but the proof is delegated to [4, Sections 1.3 and 1.5] with the sentence that the proofs are 'the exact same.' This does not address the load-bearing boundedness input: Huybrechts-Lehn selects a maximal destabilizing subsheaf using discreteness and boundedness of the set of Hilbert polynomials of subsheaves of a fixed sheaf, and Definition 2.2 only controls slopes of subsheaves whose quotients have strictly smaller dimension. It does not imply that the set {mu(F)} is bounded above in V or that a maximal element exists. As written, the abstract's claim that adapted conditions admit HN and JH filtrations in this generality is not established. The Section 3 results use finite-dimensional lexicographic degree spaces and may survive a restriction of the statement, but the general theorems need either a proof of the required boundedness or an explicit restriction to degree spaces for which the Huybrechts-Lehn argument applies.","section":"Section 2.2, Theorems 2.9 and 2.10"},{"comment":"The paper asserts without proof that the stability condition P_{Z,d} is itself adapted to sheaves of dimension d. This assertion is needed if Theorem 2.9 is to be applied to P_{Z,d} to obtain Harder-Narasimhan filtrations for this explicit stability condition. The claim is plausible, since the first non-zero degree of a lower-dimensional quotient is positive, but the verification should be written out explicitly rather than left as an unstated consequence of Theorem 3.9.","section":"Section 3.3, after Corollary 3.12"},{"comment":"The computation displayed in Example 3.4 gives the sign of Im(Z_epsilon(O_X) overline{Z_epsilon(i_* O_V)}), but the destabilizing subobject used in Definition 1.1 is the ideal sheaf I_V, not the sky-scraper sheaf i_* O_V. The conclusion that O_X is asymptotically destabilized by I_V therefore requires the additional step Im(Z(O_X) overline{Z(I_V)}) = -Im(Z(O_X) overline{Z(O_V)}) up to real terms, using additivity of the central charge. This step is straightforward, but it is essential for the claimed counterexample to [3, Conjecture 1.6] and should be stated explicitly.","section":"Example 3.4"}],"minor_comments":[{"comment":"The definition of c' should be made less ambiguous: it should say that c' is the largest integer in {c, ..., n} for which mu_{c'}(F) = mu_{c'}(E), with the understanding that c' = c if the equality fails already at c+1. The current phrasing 'Let c <= c' <= n the largest integer...' is grammatically confusing and should be rewritten.","section":"Proof of Theorem 3.9"},{"comment":"The adaptation condition is stated as positivity of (Gamma_{0,k} cup [V])^{(n,n)}, but the collection is introduced as (Gamma_{k,j}) with indices 1 <= k <= n and 0 <= j <= d_k - 1. The notation should specify which component of the Gamma's is meant to pair with [V], and the index convention should be made consistent.","section":"Example 2.3"},{"comment":"In the proof of Lemma 3.7, the derivation that the coefficient b is positive is very terse. A short sentence explaining that one uses Im(rho_j overline{rho_{j+1}}) > 0, together with the already established sign of Im(rho_n overline{rho_j}), would improve readability.","section":"Lemma 3.7"},{"comment":"The statement that the heart of the bounded t-structure is 'Coh(X) up to an even number of shifts' is imprecise. The proof should specify the exact shift, or at least state which perversity function is used and how the parity is determined by the arguments of the rho_i.","section":"Proposition 3.14(3)"},{"comment":"The notation for the cross product Im(Z_epsilon(E) Z_epsilon(F)) is ambiguous without an explicit overline on the second factor. Since the argument comparison in Definition 1.1 requires the conjugate, please ensure that all such expressions are typeset consistently as Im(Z_epsilon(E) overline{Z_epsilon(F)}).","section":"Throughout Section 3"}],"recommendation":"major_revision","confidential_remarks":"The core computation in Theorem 3.9 appears sound for the concrete finite-dimensional lexicographic degree spaces, and the Section 3 results are promising. The main obstacle is the unproved generality of the Harder-Narasimhan and Jordan-Holder statements in Section 2.2; this is a load-bearing gap because it supports the abstract's central claim that adapted conditions admit these filtrations. I would encourage the editor to request a revision in which the author either supplies the boundedness/discreteness argument for the general totally ordered vector-space setting or explicitly restricts the statements to the cases where Huybrechts-Lehn applies, and also proves the asserted adaptedness of P_{Z,d}."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely useful note with one real theorem, one gap that matters, and one example that needs a sign check. Theorem 3.9 is the core: under the adapted-ρ hypothesis, asymptotic Z-stability is exactly lexicographic degree stability. I walked through the coefficient computation; the leading term at order c'+c+1 has the claimed sign, and the proof is structurally sound.\n\nWhat's genuinely new: the lexicographic characterization, Corollary 3.12's independence of the concrete ρ, and the parameter-independent Gieseker embedding in Section 3.4, which improves the k-dependent construction in Keller-Scarpa. Example 3.4, aimed at Conjecture 1.6 of Dervan-McCarthy-Sektnan, is the kind of example the program needs, assuming it survives the sign caveat below.\n\nThe soft spots. The biggest is Section 2.2. Theorems 2.9 and 2.10 claim HN and JH filtrations for arbitrary totally ordered Q-vector spaces V; the proof is one sentence delegating to Huybrechts-Lehn. Adaptation does not give the boundedness/discreteness that the HL argument needs in that generality. For the finite-dimensional lexicographic spaces that actually show up in Section 3, the argument goes through, and Theorem 3.9 stands on its own. But the abstract's promise about 'adapted to coherent sheaves' is stronger than what the paper establishes.\n\nSecond, Example 3.4. The dHYM central charge as displayed has ℑ(ρ_n)<0 for n≥3, while the paper's Section 3 definition of a stability vector requires ℑ(ρ_n)>0. The earlier analytic discussion says that condition is a normalization that can be dropped, but the algebraic part doesn't say so. A short reconciliation or an explicit statement of the exact hypotheses of the DMS conjecture is needed before I'd take the counterexample as settled.\n\nThird, smaller: P_{Z,d} is asserted to be adapted without proof, and the indexing/conjugation conventions are implicit. Both cost time but neither threatens Theorem 3.9.\n\nBottom line: for people working on Bayer's polynomial stability, DMS's Kobayashi-Hitchin program, or Gieseker-type moduli, this deserves a serious referee. I'd send it out; a revision that fixes the Section 2.2 claim and the sign convention would make it a clean research note.","headline":"A genuinely useful characterization of asymptotic Z-stability under adapted stability vectors, but the Section 2.2 HN/JH generality is oversold and the dHYM counterexample needs a sign-convention fix.","tokens_in":13609,"tokens_out":4648,"would_cite":true,"duration_ms":44355,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F05","14J60","53C07"],"pacs":[],"model":"deepseek-v4-flash","headline":"For polynomial Bridgeland central charges adapted to coherent sheaves, asymptotic stability is exactly a lexicographic comparison of generalized slope vectors, independent of the charge.","keywords":["Bridgeland stability conditions","polynomial stability conditions","coherent sheaves","Gieseker stability","Harder–Narasimhan filtration","asymptotic Z-stability","Z-critical equation","deformed Hermitian Yang–Mills equation"],"falsifier":"Pick a threefold $X$ with an adapted stability vector $\\rho$ and a fixed polarization, choose a pure-dimension-three sheaf $E$ and a subsheaf $F$ whose slope vectors first differ at a known entry, expand $\\Im(Z_\\epsilon(E)\\overline{Z_\\epsilon(F)})$ using Equation (4), and verify that the sign of the lowest-order non-vanishing term equals the sign of that first difference; any counterexample to this sign rule would refute Theorem 3.9.","tokens_in":12269,"feed_emoji":"📐","tokens_out":15477,"duration_ms":149412,"temperature":0.7,"pith_summary":"This paper builds a broad class of stability conditions on the category of coherent sheaves that behave like Gieseker stability. It defines a stability notion adapted to sheaves of dimension $d$ by requiring that any proper subsheaf whose quotient has smaller dimension has strictly smaller slope, and asserts that this inequality is enough to guarantee Harder–Narasimhan and Jordan–Hölder filtrations. The main theorem concerns the polynomial Bridgeland central charges introduced by Bayer: when the stability vector is adapted in the same sense, a subsheaf $F$ asymptotically destabilizes $E$ exactly when a finitely computed slope vector $\\mu^{U,[\\omega]}(F)$ is lexicographically larger than $\\mu^{U,[\\omega]}(E)$. A consequence is that asymptotic stability becomes independent of the concrete choice of charge. The paper then connects this algebraic criterion to gauge theory: the deformed Hermitian Yang–Mills central charge yields a counterexample to a conjecture of Dervan–McCarthy–Sektnan, and Leung's almost Hermitian–Einstein equation appears as a special case of the $Z$-critical equation.","feed_headline":"Sheaf stability reduces to a lexicographic slope check","feed_subtitle":"For adapted polynomial charges, a subsheaf destabilizes exactly when its slope vector is larger, and the charge choice stops mattering.","key_machinery":"The central object is the generalized slope vector $\\mu^{U,[\\omega]}(E) = (\\mu_0^{U,[\\omega]}(E), \\dots, \\mu_n^{U,[\\omega]}(E))$, where $\\mu_i^{U,[\\omega]}(E)$ is $+\\infty$ for $i < \\mathrm{codim}(E)$ and $\\deg_i^{U,[\\omega]}(E)/\\mathrm{Rk}(E)$ otherwise, with $\\deg_i^{U,[\\omega]}(E) = \\mathrm{ch}_i^U(E) \\cup [\\omega]^{n-i}$ and $\\mathrm{Rk}(E) = \\mathrm{ch}_{\\mathrm{codim}(E)}^U(E) \\cup [\\omega]^{n-\\mathrm{codim}(E)} > 0$. The key identity is Equation (4): the coefficient of $\\epsilon^p$ in $\\Im(Z_\\epsilon(E)\\overline{Z_\\epsilon(F)})$ is a sum over $j$ of $\\Im(\\rho_{n-j}\\overline{\\rho_{n-p+j}})$ times degree products. Under the adaptation condition $\\Im(\\rho_d \\overline{\\rho_i}) > 0$, all coefficients below $p = c + c' + 1$ vanish, where $c$ is the common codimension of $E$ and $F$ and $c'$ is the last index at which their slope vectors agree; the first non-vanishing coefficient is $\\Im(\\rho_{n-c}\\overline{\\rho_{n-c'-1}})\\,\\mathrm{Rk}(E)\\,\\mathrm{Rk}(F)\\,(\\mu_{c'+1}^{U,[\\omega]}(F)-\\mu_{c'+1}^{U,[\\omega]}(E))$, which has a fixed positive sign. This identity is what converts asymptotic destabilization into a purely combinatorial lexicographic comparison of intersection numbers. The second piece of machinery is the adaptation condition itself (Definition 2.2), an inequality $\\mu(F) < \\mu(E)$ for subsheaves $F$ with lower-dimensional quotient; the paper asserts that this reproduces the Huybrechts–Lehn maximal-destabilizing-subsheaf argument and hence yields Harder–Narasimhan and Jordan–Hölder filtrations.","core_discovery":"The central claim is Theorem 3.9: for a smooth projective variety $X$, a polarization $[\\omega]$, a twisting class $U$, and a stability vector $\\rho$ adapted to sheaves of dimension $d$ (meaning $\\Im(\\rho_d \\overline{\\rho_i}) > 0$ for every $i < d$), a subsheaf $F$ of a pure-dimension-$d$ coherent sheaf $E$ asymptotically $Z$-destabilizes $E$ if and only if the generalized slope vector $\\mu^{U,[\\omega]}(F)$ is lexicographically $\\ge \\mu^{U,[\\omega]}(E)$, with strict inequality exactly for strict destabilization. The slope vector is assembled from the twisted Chern character $\\mathrm{ch}^U(E)=\\mathrm{ch}(E)\\cup U$: its $i$-th entry is $\\deg_i^{U,[\\omega]}(E)/\\mathrm{Rk}(E) = (\\mathrm{ch}_i^U(E) \\cup [\\omega]^{n-i})/\\mathrm{Rk}(E)$, with $+\\infty$ entries above the codimension. The proof computes the lowest-order non-vanishing coefficient of the polynomial $\\epsilon \\mapsto \\Im(Z_\\epsilon(E)\\overline{Z_\\epsilon(F)})$; the adaptedness hypothesis forces all earlier coefficients to vanish, leaving a leading term that is a positive multiple of the first entry at which the two slope vectors differ. Corollaries are that, for adapted $\\rho$, asymptotic $Z$-stability coincides with the explicit lex-degree stability $P_{Z,d}$ and does not depend on the particular choice of $\\rho$. The paper also exhibits, in Example 3.4, a charge for which this criterion shows $\\mathcal{O}_X$ on a threefold is asymptotically destabilized by codimension-three ideal sheaves even though the flat metric solves the deformed Hermitian Yang–Mills equation, contradicting Conjecture 1.6 of Dervan–McCarthy–Sektnan.","pith_inferences":["The Section 2.2 assertion that the Huybrechts–Lehn proof carries over verbatim lacks the boundedness and discreteness checks; the claimed Harder–Narasimhan and Jordan–Hölder existence in full generality is therefore conditional on those checks, which are automatic for the finite-dimensional degree spaces used in Section 3.","The dHYM counterexample indicates that any Kobayashi–Hitchin correspondence for polynomial charges needs an extra hypothesis on the subsheaves considered, such as requiring the quotient to be pure-dimensional or the sheaf to be slope-semistable, as in the Dervan–McCarthy–Sektnan theorem.","Because the adapted criterion is purely numerical, it is directly testable in examples: for explicit threefolds, one can compare the lexicographic slope order with the sign of $\\Im(Z_\\epsilon(E)\\overline{Z_\\epsilon(F)})$ at small $\\epsilon$ to locate the threshold where destabilization sets in.","Since $P_{Z,d}$-stability is independent of $\\rho$, moduli spaces of asymptotically $Z$-stable sheaves for adapted charges, if constructed via the Harder–Narasimhan filtrations, should coincide with moduli of $P_{Z,d}$-semistable sheaves and can be compared with the classical Gieseker moduli."],"forward_implications":["For any charge adapted to sheaves of dimension $d$, asymptotic $Z$-stability is equivalent to $P_{Z,d}$-stability, so algebraic stability of pure-dimension-$d$ sheaves no longer depends on the particular stability vector $\\rho$.","The deformed Hermitian Yang–Mills charge on a threefold asymptotically destabilizes $\\mathcal{O}_X$ by codimension-three ideal sheaves even though the flat metric solves the dHYM equation at every scale, giving a counterexample to Conjecture 1.6 of Dervan–McCarthy–Sektnan.","Gieseker stability is recovered exactly as asymptotic $Z$-stability with $U = \\mathrm{Td}(X)$ and an adapted, parameter-independent stability vector, extending Gieseker (Simpson) stability uniformly to all coherent sheaves.","With the explicit stability vector of Example 3.13, the $Z_\\epsilon$-critical equation becomes Leung's almost Hermitian–Einstein equation, making Leung's Kobayashi–Hitchin correspondence a special case of the Dervan–McCarthy–Sektnan correspondence."],"supporting_citations":[{"why":"Introduces polynomial Bridgeland stability conditions, the large-volume limit, and the definition of asymptotic Z-stability used throughout the paper.","marker":"[1]"},{"why":"Supplies the phase-comparison lemma, the asymptotic Z-critical equation, the Kobayashi–Hitchin correspondence of Theorem 1.1, and Conjecture 1.6 that Example 3.4 contradicts.","marker":"[3]"},{"why":"Provides the Harder–Narasimhan and Jordan–Hölder filtration machinery that Section 2 adapts to the generalized µ-stability setup.","marker":"[4]"},{"why":"Defines Leung's almost Hermitian–Einstein equation, which Example 3.13 recovers as a special case of the Z-critical equation.","marker":"[7]"},{"why":"Earlier realized Gieseker stability as asymptotic Z-stability with a parameter-dependent stability vector; the paper shows a parameter-independent vector suffices.","marker":"[6]"}],"fun_headline_variants":["Adapted polynomial stability reduces to lexicographic slope check","Lexicographic slope vectors decide adapted sheaf stability","New stability conditions generalise Gieseker via slope order","Polynomial Bridgeland stability yields gauge theory counterexample","Adapted charges: destabilization iff slope vector larger"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the adaptation inequality alone is enough to force the existence of a unique maximal destabilizing subsheaf, and hence Harder–Narasimhan and Jordan–Hölder filtrations, for any totally ordered $\\mathbb{Q}$-vector space of degrees; the paper states this by invoking Huybrechts–Lehn without proving the boundedness and discreteness steps that their construction requires.","fun_headline_variants_meta":{"raw":{"variants":["Adapted polynomial stability reduces to lexicographic slope check","Lexicographic slope vectors decide adapted sheaf stability","New stability conditions generalise Gieseker via slope order","Polynomial Bridgeland stability yields gauge theory counterexample","Adapted charges: destabilization iff slope vector larger"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1478,"prompt_tokens":1042,"completion_tokens":436,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":658,"completion_tokens_details":{"reasoning_tokens":357}},"tokens_in":658,"tokens_out":436,"duration_ms":5712,"temperature":1.0,"reasoning_tokens":357,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:21:41.622142+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a threefold $X$ with an adapted stability vector $\\rho$ and a fixed polarization, choose a pure-dimension-three sheaf $E$ and a subsheaf $F$ whose slope vectors first differ at a known entry, expand $\\Im(Z_\\epsilon(E)\\overline{Z_\\epsilon(F)})$ using Equation (4), and verify that the sign of the lowest-order non-vanishing term equals the sign of that first difference; any counterexample to this sign rule would refute Theorem 3.9.","supporting_citations":[],"review_version":1}