{"id":"e834a648-36ab-4036-a9a8-632934a956f4","arxiv_id":"2607.18898","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New sufficient criterion for unliftability of degenerate geometric rank loci and a characterization of when the rank-1 locus causes the degeneracy.","lead":"This paper studies when a tensor has surprisingly large low-rank loci and proves a criterion for when such a tensor cannot be extended to a larger one, plus a classification of tensors whose rank-1 locus alone creates the degeneracy. It also gives new examples, including octonions and an SL3-invariant tensor.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7's key equality \\hat T_e Y = \\hat T_e Z \\cap E for scheme-theoretic hyperplane sections is asserted without proof for singular Z; Theorem 4 and the paper's central criterion rest on it.","rationale":"The reader's weakest assumption pinpoints the tangent-cone equality in Lemma 7. I agree this is the most load-bearing gap. The proof of Theorem 4 reduces to this lemma, and the lemma's proof contains an unproved equality that is not obvious for tangent cones. However, the gap may be repairable: at a point regular in Y, the equality for Zariski tangent spaces holds, and in the specific case X=σ_r Seg the relevant points are smooth in X. Still, as written, the proof is incomplete. Additionally, the induction in Theorem 10 does not explicitly preserve conciseness under the generic hyperplane restriction, which the theorem assumes; this is a secondary issue but needs a sentence. The computational examples in Section 5 are undocumented, weakening support for the claimed new examples, but this is less central. Overall, CONDITIONAL is appropriate.","tokens_in":128,"tokens_out":29316,"duration_ms":761661,"concrete_test":"Check the equality in Lemma 7 directly: in Macaulay2, take a singular projective variety X (e.g., X = V(x^2-yz, y^2-xz) ⊂ P^3), a hyperplane E' with Z = X∩E', and a hyperplane E such that Y = Z∩E is regular at a point e. Compute the initial ideals and compare \\hat T_e Y with \\hat T_e Z ∩ E. If they differ, the lemma is false; if they agree in all such examples, add a proof. Alternatively, verify the equality for X = σ_r Seg at a rank-r point with arbitrary lift E' by computing the tangent spaces directly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 7 claims U ⊆ ND_X(E). The proof's crucial step is the assertion that at e ∈ \\tilde Y_1, '\\hat T_e Y = \\hat T_e Z ∩ E due to the scheme-theoretic intersection Y = Z ∩ PE.' Here \\hat T denotes the affine tangent cone. For tangent cones, this equality is not a formal consequence of scheme-theoretic intersection; it is known to fail for general singular schemes. Even if regularity of e in Y forces e to be smooth in the relevant component of Z (so the equality would hold for Zariski tangent spaces), the paper does not prove this, and the notation explicitly refers to the tangent cone. Since Theorem 4 is derived from Lemma 7 via Corollary 8, an unproved equality at this point means the central unliftability criterion is not established. A separate gap appears in Theorem 10's induction: restricting to a general hyperplane A↠A' may lose conciseness, which is assumed by the theorem; this is likely repairable by a genericity argument but is also not written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies tensors of degenerate geometric rank, i.e., tensors T ∈ A⊗B⊗C for which some rank-r locus Y^r_T has unexpectedly large dimension. The main contributions are: (1) a sufficient criterion for a tensor (equivalently, a linear space E ⊂ Hom(B*,C)) to be r-unliftable, expressed through the set RND_r(E) of rank-r-neutral directions (Theorem 4, generalizing Draisma's criterion for spaces of bounded rank); (2) a structural theorem (Theorem 10) stating that if the rank-1 locus alone achieves degenerate geometric rank, then either the tensor's space of matrices contains a linear subspace of the Segre variety, or a quotient is GL(B)×GL(C)-equivalent to the space of symmetric 2×2 matrices, in which case the rank-2 locus also achieves the geometric rank; (3) several examples of nonlinearly degenerate geometric rank, including matrix multiplication tensors, octonions, minimal-border-rank tensors, and an SL_3-invariant tensor. The paper supplies a SageMath implementation of the liftability criterion.","tokens_in":13417,"tokens_out":21341,"duration_ms":197490,"significance":"If the results hold, they provide a substantial generalization of Draisma's liftability framework to arbitrary rank loci and give the first structural dichotomy for tensors whose degeneracy is caused by the rank-1 locus. The explicit examples and the accompanying code are useful contributions; in particular, the unliftability of matrix multiplication tensors and the octonion structure tensor are concrete and checkable. The paper is written in a clear style and engages honestly with computational verification. The main theorems are significant for the geometric-rank community and for the broader study of tensor degenerations.","major_comments":[{"comment":"The proof of U ⊆ ND_X(E) hinges on the equality \\hat T_e Y = \\hat T_e Z ∩ E, asserted 'due to the scheme-theoretic intersection Y = Z∩PE'. This equality is not a formal consequence of scheme-theoretic intersection for tangent cones when e is singular in Z, and the proof only assumes e ∈ Y_reg, not e ∈ Z_reg. Indeed, the argument explicitly allows dim \\hat T_e Z > dim \\hat T_e Y. A counterexample to this tangent-cone equality for a singular variety and a hyperplane section would invalidate Lemma 7, and hence Theorem 4. The authors need either to prove this equality under the stated hypotheses (e.g., by showing e is regular in Z or by a transversality statement for tangent cones) or to modify the definition of RND_r(E) so that the argument goes through. As written, the central unliftability criterion is not established.","section":"Section 3, Lemma 7"},{"comment":"The induction applies the theorem to the restriction T' induced by A ↠ A', where (A')* = T_A^{-1}(T_A(A*)∩H) for a general hyperplane H ⊂ B⊗C. The paper assumes throughout that all tensors are concise, but T' may lose conciseness: the flattenings T'_B and T'_C can fail to be injective even when T is concise. The proof does not show that a general H can be chosen to preserve conciseness, nor does it state a version of Theorem 10 for non-concise tensors. This is a load-bearing omission because the inductive hypothesis requires the theorem to apply to T'. The gap appears repairable by a genericity argument, but it must be supplied.","section":"Section 4, Theorem 10 proof, induction step"}],"minor_comments":[{"comment":"The title line reads 'ON DEGENERA TE GEOMETRIC RANK' due to a missing space; likely a LaTeX typo.","section":"Title/Abstract"},{"comment":"The projective dimension of PA* is m−1, but the text sometimes writes dimPA* where the vector-space dimension m is meant. This is a source of the '+' signs in §4; please unify notation.","section":"Section 2.1"},{"comment":"The set RND_r(E) is a union of intersections; it is not asserted to be a linear subspace. This is fine, but in the remark after Theorem 4 the statement 'RND_r(E) is a linear space of dimension dim E + 1' should be justified, since the union may be nonlinear in general.","section":"Section 3, Definition 3"},{"comment":"The verification that the chosen permutation matrices A suffice to force the intersection to equal E is sketched rather than proved in full. A short argument for the general n case would improve readability.","section":"Section 5.1, Example 15"},{"comment":"In the displayed matrices, several entries are written as '0x' without a space (e.g., '0x 0 0' and '0x 0 0 0'); the intended multiplication sign is clear but should be fixed.","section":"Section 5.3"}],"recommendation":"major_revision","confidential_remarks":"This is a promising paper with interesting results and useful examples. However, the proof of Lemma 7 contains a serious gap in the tangent-cone equality that is load-bearing for Theorem 4. The authors should be asked to either provide a rigorous proof of the equality under the stated assumptions, or to reformulate the definitions so the argument becomes valid, or to give a counterexample that would delimit the theorem's scope. The conciseness issue in Theorem 10's induction is more clearly repairable but also needs to be addressed in the revision. I do not think rejection is warranted at this stage."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth reading. It does two new things: it generalizes Draisma's unliftability criterion from linear spaces of bounded rank to arbitrary rank loci, via the notion of RND_r, and it proves a clean classification of tensors whose rank-1 locus alone achieves geometric rank. The examples—matrix multiplication, octonions, and the SL3-invariant E6—are concrete and should be useful for people hunting for nonlinear flattenings or border subrank bounds.\n\nThe main theorems appear correct. Theorem 4 is a natural extension, and the proof follows Draisma's structure. The one spot that needs attention is Lemma 7: the equality \\hat T_e Y = \\hat T_e Z ∩ E is asserted 'due to the scheme-theoretic intersection.' That is not automatic for tangent cones. But in the situation of the lemma, e is regular in Y, and a standard argument shows that a smooth hyperplane section forces the ambient component to be smooth at e (otherwise the tangent space of the section would be too large). Then the equality holds. The paper does not include that argument, which is an omission, but it is repairable and I don't think the proof is broken.\n\nTheorem 10's induction also skips the detail that the restricted tensor T' may lose conciseness. A genericity choice of the hyperplane should preserve it, but the text doesn't say so. Another minor gap.\n\nThe computational claims (octonions unliftable, E6 4-unliftable) are asserted without details in the text. There is a GitHub link, so a referee can check, but as written the reader has to trust the computation. That should be fixed before publication.\n\nOtherwise, the examples are consistent and the Section 5 discussion is honest about the fact that minimal border rank examples always have a linear locus achieving the rank jointly. The paper's claim to have identified nonlinear degeneracy is fine—matrix multiplication and octonions do show that.\n\nBottom line: this deserves a serious referee. The gaps are expository, not fatal. I would send it to review, and I'd cite it if I worked on geometric rank or tensor flattenings.","headline":"A genuine generalization of Draisma's unliftability criterion to general rank loci, plus a clean classification of rank-1-induced degeneracy; the proofs are mostly sound, with a few spots that need clarification rather than major rework.","tokens_in":13929,"tokens_out":11927,"would_cite":true,"duration_ms":98438,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","14N07","14M12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Rank-neutral directions reveal when degenerate tensors cannot lift","keywords":["tensor","geometric rank","rank loci","liftability","degenerate geometric rank","secant varieties","rank-neutral directions","border subrank"],"falsifier":"Find a triple $(X,E,E')$ with $E' = E + Cv$, where $Y = X \\cap PE$ is regular at a point $e$ but $Z = X \\cap PE'$ is singular at $e$, and compute $\\hat{T}_e Y$ and $\\hat{T}_e Z \\cap E$. If they differ, Lemma 7 collapses, and with it the general unliftability criterion of Theorem 4.","tokens_in":13077,"feed_emoji":"🧮","tokens_out":4532,"duration_ms":41328,"temperature":0.7,"texified_at":"2026-08-05T21:32:39.550977+00:00","pith_summary":"The paper studies tensors whose geometric rank is lower than expected—tensors in which some locus of matrices of rank at most $r$ has surprisingly large dimension. It proves a sufficient criterion for when such a tensor cannot be extended to a larger tensor without enlarging that rank locus: if the tensor's space of matrices equals its set of rank-neutral directions, it is 'unliftable'. It also characterizes tensors whose degeneracy is caused entirely by the rank-1 locus: either the space contains a linear subspace of the Segre variety, or it is equivalent to $2\\times 2$ symmetric matrices, in which case the rank-2 locus also achieves the geometric rank. These results are accompanied by new examples of degenerate geometric rank arising from nonlinear rank loci, including matrix multiplication and the octonions.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":5958,"prompt_tokens":769,"completion_tokens":5189,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":769,"completion_tokens_details":{"reasoning_tokens":4411}},"feed_headline":"Rank-neutral directions reveal when degenerate tensors cannot lift","feed_subtitle":"A new liftability criterion and a dichotomy for rank-1 degeneracy narrow the search for exceptional tensors.","key_machinery":"The argument rides on the set of rank-neutral directions $RND_r(E)$: for each maximal component $Y_i$ of the rank-$r$ locus, intersect the spaces $E + \\hat{T}_e \\sigma_r \\mathrm{Seg}$ over all regular points $e \\in Y_i$ where $\\operatorname{rank}(e)=r$. The tangent-cone formula $\\hat{T}_{[e]}\\sigma_r \\mathrm{Seg} = \\{M \\mid M(\\ker e) \\subseteq \\operatorname{im} e\\}$ turns the criterion into an explicit matrix computation. The proof of Theorem 4 generalizes Draisma's liftability criterion to arbitrary projective varieties $X$, with Lemma 7 as the load-bearing step. Theorem 10 uses Hopf's theorem on the span of an irreducible curve in the Segre variety, plus an induction cutting with a general hyperplane.","core_discovery":"The central results are Theorem 4 and Theorem 10. Theorem 4 states that if $E = RND_r(E)$, then the space $E$ is r-unliftable: it cannot be contained in a larger space $E'$ of matrices with the same codimension of the rank-$r$ locus. The set $RND_r(E)$ collects directions whose addition to $E$ does not force the rank locus to shrink in codimension, computed via tangent cones of the secant variety $\\sigma_r \\mathrm{Seg}$. Theorem 10 characterizes degenerate geometric rank caused by the rank-1 locus: for a concise degenerate tensor $T$, if $Y^1_T$ achieves $GR(T)$, then either $PT_A(A^*)$ contains a linear subspace of the Segre variety, or some quotient of $T$ is $\\operatorname{GL}(B)\\times \\operatorname{GL}(C)$-equivalent to the space of symmetric $2\\times 2$ matrices, and th","pith_inferences":["If the tangent-cone equality in Lemma 7 fails in some case, the Rank-Neutral-Direction criterion would need refinement; a search for such a counterexample could reveal a sharper condition.","The characterization suggests that 'purely nonlinear' degenerate geometric rank may not exist for rank 1, but the paper's own examples show it does for higher ranks; extending Theorem 10 to r>1 is a natural test.","The SageMath implementation could be used to scan known tensor families for unliftability, potentially preparing the ground for new border subrank lower bounds.","The connection to Hopf's theorem hints that span-of-curve inequalities in the Segre variety may yield further bounds for secant varieties, beyond the geometric rank setting."],"forward_implications":["A tensor satisfying E = RND_r(E) cannot be extended to any larger tensor with the same codimension of the rank-r locus, giving a practical unliftability certificate.","Degenerate geometric rank caused solely by the rank-1 locus is severely constrained: only two exceptional configurations exist, which simplifies classification efforts.","The examples of matrix multiplication and octonions show that the criterion works beyond linear spaces of bounded rank, suggesting a new tool for subrank upper bounds.","If the theorem's conclusion holds generally, tensors of degenerate geometric rank must either contain a linear bounded-rank space or exhibit a symmetric 2×2 block; nonlinear rank loci alone cannot be the sole source.","The computationally verified unliftability of several minimal-border-rank tensors provides a new batch of 'maximal' degenerate tensors."],"fun_headline_variants":["Unliftable tensors exposed by rank-neutral directions","Rank-1 degeneracy explains why tensors resist lifting","A liftability test: when rank-neutral directions fail","New examples: nonlinear loci trigger degenerate rank"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of Theorem 4 assumes that, for points $e$ that are regular in the intersection $Y = X \\cap PE$ but possibly singular in $Z = X \\cap PE'$, the tangent cone of $Y$ equals the intersection of the tangent cone of $Z$ with $E$; this equality is asserted without proof and is not automatic for singular varieties.","fun_headline_variants_meta":{"raw":{"variants":["Unliftable tensors exposed by rank-neutral directions","Rank-1 degeneracy explains why tensors resist lifting","A liftability test: when rank-neutral directions fail","New examples: nonlinear loci trigger degenerate rank"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000127,"raw_usage":{"total_tokens":880,"prompt_tokens":603,"completion_tokens":277,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":347,"completion_tokens_details":{"reasoning_tokens":215}},"tokens_in":347,"tokens_out":277,"duration_ms":3521,"temperature":1.0,"reasoning_tokens":215,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T14:00:57.583107+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a triple $(X,E,E')$ with $E' = E + Cv$, where $Y = X \\cap PE$ is regular at a point $e$ but $Z = X \\cap PE'$ is singular at $e$, and compute $\\hat{T}_e Y$ and $\\hat{T}_e Z \\cap E$. If they differ, Lemma 7 collapses, and with it the general unliftability criterion of Theorem 4.","supporting_citations":[],"review_version":1}