{"id":"a2013c9b-58ad-4a85-bd2c-2a37e4cc8f29","arxiv_id":"1908.03713","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The set of algebraic curvature operators with sectional curvature at least k is a spectrahedron only up to dimension 3, a spectrahedral shadow in dimension 4, and not a spectrahedral shadow from dimension 5 onward, with algorithms for membership testing.","lead":"Curvature operators describe how a space bends at a point; this paper maps exactly when the set of operators with a sectional curvature lower bound can be described by simple matrix inequalities (spectrahedra) or their projections. The answer is yes in low dimensions, only with projections in dimension four, and impossible in dimension five and up, which also gives new counterexamples to a conjecture in convex algebraic geometry.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The n≥5 non-shadow proof rests on an unverified coefficient-in-L condition in Proposition 4.1; if it fails, Theorem A(1) has no known proof.","rationale":"The reader and I identify the same weakness. I read the n=4 material and the Claim in Section 4 carefully; the algebra there is plausible. The Claim's use of the determinantal ideal and injectivity of ψ is sound, and the non-SOS conclusion follows. However, Theorem A(1) requires more than a nonnegative non-SOS quadratic form: the translated homogenization coefficients must remain in L for every translation. This is exactly the hypothesis of Theorem 2.14, and it is not a consequence of non-SOS alone. The paper's one-sentence justification is too compressed for the step on which the main counterexample depends. The external inputs—[BSV16], [Zol79], and [Sch18b]—can be accepted on authority, but the reduction to the explicit 6-variable setting is original and needs verification. My proposed symbolic computation is cheap and decisive: it checks the coefficient-in-L condition directly for the actual form RZol and the actual space L. If the test passes, I see no obstacle to the central claim; if it fails, Theorem A(1) is unproven. This matches the reader's CONDITIONAL verdict, so no adjustment is needed.","tokens_in":27596,"tokens_out":20884,"duration_ms":235764,"concrete_test":"Use exact arithmetic in Macaulay2 or similar to construct a basis of L=ψ(R[Gr2(5)]_2) in the coordinates a=x12,b=x13,c=x14,d=x25,e=x35,f=x45; generate f^h(t,x-y) from RZol; and express each coefficient of t^j, j=0..4, in that basis, treating y12,...,y45 as indeterminates. Since the membership condition is polynomial in y, a symbolic computation proves it for all y. If every coefficient lies in L, the application of Theorem 2.14 is supported; if any coefficient does not, Proposition 4.1 collapses. For robustness, repeat for several P∈P_Gr2(5)\\Σ_Gr2(5) produced by SOS software.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing point is the final step of Proposition 4.1, where Theorem 2.14 is applied to L=ψ(R[Gr2(5)]_2) and f=ψ(P), with P∈P_Gr2(5)\\Σ_Gr2(5). For the criterion to fire, every coefficient of f^h(t,x12-y12,...,x45-y45), as a polynomial in t, must lie in L for every y∈R^6. The text disposes of this condition in one sentence: 'This and multilinearity of the determinant imply...'. No basis of L is exhibited and no coefficient is written out, so the only connection between the Zoltek/BSV16 form and Scheiderer's theorem is asserted rather than verified. If some t-coefficient lies outside L for some y, Theorem 2.14 is inapplicable, and Theorem A(1) loses its only proof. The Claim that f is non-SOS is internally coherent, but it only proves f∉Σ; it does not by itself establish the stronger 'badness' property that Scheiderer's machinery needs. This is not an accusation of error; it is a precise location of the unsecured load.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies, from the viewpoint of convex algebraic geometry, the convex semialgebraic sets Rsec≥k(n) of algebraic curvature operators whose sectional curvature is bounded below (or above) by k. The main result, Theorem A, asserts a complete classification: for n≥5 these sets are not spectrahedral shadows; for n=4 they are spectrahedral shadows but not spectrahedra; for n≤3 they are spectrahedra. The proof uses the identification R[Gr_2(n)]_2 ≅ Sym^2_b(∧^2R^n), the Blekherman–Smith–Velasco/Zoltek non-SOS quadratic form, and a criterion extracted from Scheiderer's work. Theorem B provides O(n)-invariant nested inner approximations by spectrahedral shadows and outer approximations by spectrahedra; Theorem C characterizes Rsec≥0(4) as an algebraic interior whose minimal defining polynomial is the discriminant disc_x(det(R+x*)). The paper also gives SDP-based algorithms for n≥5 and Sturm-based algorithms for n=4, plus a semi-Riemannian analogue in Appendix A.","tokens_in":27806,"tokens_out":31308,"duration_ms":302008,"significance":"The results, if correct, are significant. They supply a new geometric family of counterexamples to the Helton–Nie conjecture, give a concrete and reusable form of Scheiderer's non-shadow criterion, and turn the dimension-four case into an explicit algebraic-interior description with efficient membership tests. The proof structure is sound: Propositions 3.2 and 3.5 establish the algebraic-interior structure; Appendix B proves the needed irreducibility of the discriminant of symmetric matrices; and the n≥5 argument reduces via Corollary 4.2 to Scheiderer's criterion. I found no circularity: Theorem A(1) does not depend on the authors' earlier work, and the only use of [BM] is for the outer approximations in Section 5, which is external. The paper also includes checkable algorithmic consequences and a reproducible Macaulay2 verification in Example 5.3.","major_comments":[],"minor_comments":[{"comment":"The stated component identification is false as written: for every k, p_k((k+1)Id) = disc_x(det(Id+x*)) = disc_x((1+x)^3(1-x)^3) = 0, so (k+1)Id cannot belong to the set {p_k>0}, let alone to C_k. The intended condition is presumably that (k+1)Id lies in the closure of C_k, or equivalently that C_k is the component whose closure is Rsec≥k(4). Please correct this sentence.","section":"Theorem C, final sentence"},{"comment":"The verification of the coefficient-in-L hypothesis of Theorem 2.14 is compressed into the sentence 'This and multilinearity of the determinant imply...'. The claim is correct, but because this is the load-bearing connection between the Zoltek form and Scheiderer's criterion, please spell out the argument: write Γ_t(x-y)=Γ_t(x)-Γ_t(y), expand each 2×2 minor by row multilinearity, and observe that every t-coefficient of the resulting product is a y-dependent linear combination of products of two t-coefficients of minors of Γ_t(x), each of which lies in L by definition of ψ.","section":"Section 4, Proposition 4.1"},{"comment":"The isomorphism in the induction step should read Gr_k(n+1) ≅ Gr_{n+1-k}(n+1), not Gr_{n-k}(n+1). The subsequent range reduction remains valid after this correction.","section":"Section 4, Corollary 4.2"},{"comment":"The proof of Theorem 2.14 delegates the key implication to 'the exact same reasoning as in [Sch18b, Ex. 4.20, Rem. 4.21]'. Since the theorem is advertised as a conveniently applicable criterion, a fuller derivation, or an explicit statement of the cited results, would make the paper more self-contained and easier to verify.","section":"Section 2.6, Theorem 2.14"},{"comment":"In line 5, the condition that σ_i has a root in (a_j,a_{j+1}) is tested for each i; the implementation via Sturm's root-isolating partitions should be described explicitly so that the decidability of this test is transparent.","section":"Section 5.4, Algorithm 3"}],"recommendation":"minor_revision","confidential_remarks":"I checked the coefficient-in-L step in Proposition 4.1 that the stress test highlighted; after expanding by row multilinearity, the condition is indeed satisfied, so I do not regard it as a gap. The only mathematical error I found is the final component-identification sentence of Theorem C, which is a local typo but should be corrected. The Gr_{n-k} typo in Corollary 4.2 is also local. The paper is otherwise coherent and suitable for the journal after these corrections."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take.\n\nThe paper is serious and mostly rigorous. Its real contribution is Theorem A: the sectional-curvature bound sets are spectrahedra for n≤3, spectrahedral shadows but not spectrahedra for n=4, and not even spectrahedral shadows for n≥5. The last part is a new family of Helton-Nie counterexamples, and the authors correctly note the conjecture was already dead via Scheiderer and Fawzi. The point is that these are natural convex bodies from Riemannian geometry. The n=4 minimal defining polynomial in Theorem C and the O(n)-invariant inner/outer approximations in Theorem B are also new and useful.\n\nI want to address the specific worry about Proposition 4.1. The stress-test concern is that the coefficient-in-L condition is asserted without proof, and if it failed, the n≥5 non-shadow conclusion would lose its only proof. I read the algebra carefully and the condition holds. Once you write the Plücker coordinates of the two-row matrix Γ(t) as t^k times the six affine coordinates or the 2×2 minors of Γ′, the translated polynomial f^h(t,x-y) expands by multilinearity into linear combinations, with y-coefficients, of products of these same functions. Those products span L by definition. So Scheiderer's criterion applies. The step is compressed—it deserves a few explicit lines or a footnote—but it is not an error.\n\nTwo minor caveats. Proposition 5.7 uses [BM, Thm. A] for the outer approximations without stating the exact statement; that is normal use of prior work, but the dependency should be explicit. And the SDP-based membership algorithms assume exact feasibility, while interior-point methods return numerical approximations; the algorithms are rigorous in an oracle model, not as floating-point routines. The authors are honest that the n≥5 algorithm may fail to halt on a measure-zero bad set.\n\nWho is this for? People in convex algebraic geometry, and differential geometers who care about curvature conditions as semialgebraic sets. It deserves serious peer review. The referee should push for a few more lines in Proposition 4.1 and a sentence about the SDP exactness caveat; otherwise it is ready.","headline":"New curvature-operator counterexamples to Helton-Nie; the proof is sound, with one terse step that deserves expansion.","tokens_in":28378,"tokens_out":13210,"would_cite":true,"duration_ms":130677,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14P10","53B20","53C21","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"For curvature operators, semidefinite descriptions exist only in low dimensions.","keywords":["curvature operators","sectional curvature bounds","spectrahedra","spectrahedral shadows","semidefinite programming","Grassmannian","sums of squares","Helton-Nie conjecture"],"falsifier":"Take the specific nonnegative non-sum-of-squares quadratic form on $\\mathrm{Gr}_2(5)$ that the paper uses, write its homogenization in the affine chart, and check whether every coefficient of the shifted polynomial $f^h(t, x-y)$ belongs to the subspace $L$ spanned by the affine-chart images of quadratic forms; a single violation would invalidate the $n\\ge 5$ theorem. Alternatively, produce an explicit spectrahedral shadow description of $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(5)$; if one exists, the theorem's first part is false.","tokens_in":27376,"feed_emoji":"📐","tokens_out":6315,"duration_ms":52553,"temperature":0.7,"pith_summary":"This paper determines exactly which dimensions admit a semidefinite description of the set of algebraic curvature operators satisfying a sectional curvature bound. In dimensions at most three these sets are spectrahedra; in dimension four they are projections of spectrahedra but not spectrahedra themselves; and in dimensions five and higher they are neither, which supplies a new family of counterexamples to the Helton–Nie conjecture that every convex semialgebraic set is a spectrahedral shadow. The paper also gives efficient membership tests for dimension four via Sturm's root counting, and for higher dimensions it constructs nested inner and outer semidefinite approximations that converge to the true set.","feed_headline":"Curvature cones defeat semidefinite representation in high dimensions.","feed_subtitle":"In dimensions 5 and up no exact semidefinite formula exists, but converging algorithms still certify or refute the bound.","key_machinery":"The machinery turns curvature operators into quadratic forms on the Grassmannian $\\mathrm{Gr}_2(n)$: an algebraic curvature operator $R$ is identified with the quadratic form $q_R$ on Plücker coordinates, and the condition $\\mathrm{sec}_R\\ge 0$ becomes nonnegativity on the real Grassmannian. Under this identification, $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(n)$ is exactly the cone $P_{\\mathrm{Gr}_2(n)}$ of nonnegative quadratic forms, while sums of squares in the homogeneous coordinate ring correspond to strongly nonnegative curvature. Two named objects carry the argument: the Finsler–Thorpe trick says that in dimension four, $\\mathrm{sec}_R\\ge 0$ iff $R+x*$ is positive semidefinite for some real $x$, exhibiting $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(4)$ as a spectrahedral shadow; and the discriminant $p_k(R)=\\mathrm{disc}_x(\\det(R-k\\,\\mathrm{Id}+x*))$ gives the minimal defining polynomial of the algebraic interior in dimension four. For $n\\ge 5$, the proof applies a criterion that detects when such a cone is not a spectrahedral shadow, using a quadratic form on $\\mathrm{Gr}_2(5)$ that is nonnegative but not a sum of squares, and a translation-invariance property of its homogenization.","core_discovery":"The central discovery, stated as Theorem A, is a complete dimensional dichotomy. For every real k and every n, the convex semialgebraic set $\\mathcal{R}_{\\mathrm{sec}\\ge k}(n)$ of algebraic curvature operators with $\\mathrm{sec}\\ge k$ (and its mirror $\\mathcal{R}_{\\mathrm{sec}\\le k}(n)$) is a spectrahedron when $n\\le 3$, a spectrahedral shadow but not a spectrahedron when $n=4$, and not even a spectrahedral shadow when $n\\ge 5$. The proof for $n=4$ rests on the Finsler–Thorpe trick: $\\mathrm{sec}_R\\ge 0$ in dimension four is equivalent to $R+x*$ being positive semidefinite for some real $x$, where $*$ is the Hodge star, so $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(4)$ is the linear projection of a spectrahedron. The proof for $n\\ge 5$ identifies $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(n)$ with the cone of nonnegative quadratic forms on the Grassmannian $\\mathrm{Gr}_2(n)$, extracts a criterion from recent work on spectrahedral shadows to show such a cone is not a spectrahedral shadow, and exhibits the required non-sum-of-squares input using a known explicit quadratic form. Theorem C adds that in dimension four the set is an algebraic interior whose minimal defining polynomial is the discriminant of $\\det(R+x*)$, and Theorem B provides nested inner approximations by spectrahedral shadows and outer approximations by spectrahedra whose unions and intersections exhaust the set.","pith_inferences":["Since $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(n)$ for $n\\ge 5$ is not a spectrahedral shadow, any exact semidefinite representation would have to use an infinite-dimensional or non-polynomial lifting; a concrete test would be to see whether the boundary of the cone has positive curvature in the sense of convex algebraic geometry that obstructs such lifts.","The construction of the bad example suggests a general recipe: any Grassmannian $\\mathrm{Gr}_k(n)$ with $2\\le k\\le n-2$ and $n\\ge 5$ yields a non-spectrahedral-shadow cone of nonnegative quadratic forms, so the phenomenon is not special to $\\mathrm{sec}\\ge 0$ but is shared by positivity conditions defined by other $O(n)$-representations.","A natural extension would be to replace the Grassmannian by other homogeneous varieties and ask whether the corresponding nonnegative quadratic forms are spectrahedral shadows exactly when sums of squares coincide; the paper's criterion might be the right tool to test this.","The algorithms' numerical robustness could be benchmarked against random curvature operators in dimensions 5 and 6; if the bad set, though of measure zero, is approached by typical inputs, the stopping criterion might require many iterations in practice."],"forward_implications":["In dimension 4, membership in $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(4)$ (and strict variants) can be decided exactly by Sturm's root-counting algorithm, without semidefinite programming.","For $n\\ge 5$, Algorithm 1 terminates correctly on every input outside a measure-zero bad set, so sectional curvature bounds can be tested in practice up to numerical precision.","The inner approximations $I_m$ and outer approximations $O_m$ are $O(n)$-invariant and geometric, so they define coordinate-free curvature conditions; in particular $I_0$ is strongly nonnegative curvature and $O_0$ is nonnegative Ricci curvature.","Because the approximations do not stabilize for $n\\ge 5$, no finite truncation of the Lasserre-type hierarchy or of the Weitzenböck-formula hierarchy can exactly capture $\\mathrm{sec}\\ge 0$ in high dimensions.","The same dimensional dichotomy holds for semi-Riemannian curvature operators with the natural analogue of sectional curvature bounds."],"supporting_citations":[{"why":"Supplies the criterion used to prove that the nonnegative quadratic forms on $\\mathrm{Gr}_2(5)$ are not a spectrahedral shadow.","marker":"[Sch18b]"},{"why":"Establishes that $P_X = \\Sigma_X$ exactly for varieties of minimal degree, locating the threshold $n=4$ vs $n\\ge 5$ for $\\mathrm{Gr}_2(n)$.","marker":"[BSV16]"},{"why":"Provides the explicit nonnegative, non-sum-of-squares quadratic form that drives the counterexample construction in dimensions $\\ge 5$.","marker":"[Zol79]"},{"why":"Gives the Thorpe trick characterizing $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(4)$ as the projection of a spectrahedron.","marker":"[Tho72]"},{"why":"Finsler's lemma, which the paper identifies as the underlying classical result behind the Thorpe trick.","marker":"[Fin36]"},{"why":"Provides the Weitzenböck-formula curvature terms used to build the outer approximations $O_m$.","marker":"[BM]"},{"why":"The Lasserre hierarchy idea adapted to build the inner approximations $I_m$.","marker":"[Las01]"},{"why":"Formulates the Helton–Nie conjecture that the paper disproves for these sets.","marker":"[HN09]"}],"fun_headline_variants":["High-dimensional curvature sets escape semidefinite shadows","Dimension 5+ kills spectrahedral shadow for curvature","Curvature cones in n≥5 reject semidefinite form","No spectrahedral shadow for curvature bounds in high dims","Curvature operators: semidefinite failure above dimension 4"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For dimensions five and up, the whole non-representability argument rests on a single technical criterion and on one explicitly supplied polynomial that must satisfy a certain translation-coefficient property; if that criterion does not apply here, or the polynomial fails that property, the paper's counterexample claim for $n\\ge 5$ has no proof.","fun_headline_variants_meta":{"raw":{"variants":["High-dimensional curvature sets escape semidefinite shadows","Dimension 5+ kills spectrahedral shadow for curvature","Curvature cones in n≥5 reject semidefinite form","No spectrahedral shadow for curvature bounds in high dims","Curvature operators: semidefinite failure above dimension 4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000861,"raw_usage":{"total_tokens":3754,"prompt_tokens":984,"completion_tokens":2770,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":2688}},"tokens_in":600,"tokens_out":2770,"duration_ms":19639,"temperature":1.0,"reasoning_tokens":2688,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:08:03.022507+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the specific nonnegative non-sum-of-squares quadratic form on $\\mathrm{Gr}_2(5)$ that the paper uses, write its homogenization in the affine chart, and check whether every coefficient of the shifted polynomial $f^h(t, x-y)$ belongs to the subspace $L$ spanned by the affine-chart images of quadratic forms; a single violation would invalidate the $n\\ge 5$ theorem. Alternatively, produce an explicit spectrahedral shadow description of $\\mathcal{R}_{\\mathrm{sec}\\ge 0}(5)$; if one exists, the theorem's first part is false.","supporting_citations":[],"review_version":1}