{"id":"af86f566-261b-4d4f-a741-f38f46d1de55","arxiv_id":"2411.15789","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Sublevel sets of asymptotic tensor rank are Zariski-closed, making the parameter well-ordered in value, complete over the complex numbers, and computable from above.","lead":"This paper proves that the sublevel sets of asymptotic tensor rank, the tensors whose rank stays below a threshold in the limit of Kronecker powers, are defined by polynomial equations. This structural result implies discreteness of the values of asymptotic rank from above, including a gap above the matrix multiplication exponent, and yields a non-uniform algorithm to decide rank-at-most-r.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Computability-from-above is unsupported: Theorem 1.2 shows existence of defining polynomials, but no procedure to compute them is given, so the algorithm of Theorem 1.1 is not established.","rationale":"The stress-test pass finds the core mathematics of Theorem 1.2 convincing. The admissible functional axioms are indeed satisfied by tensor rank: subadditivity and submultiplicativity under the grouped tensor product are classical, and the remaining axioms are immediate. The double-blocking proof of Theorem 2.2 checks out, including the polynomial bound p(n) ≤ dim Sym^n V and the Fekete-based asymptotic argument. Consequently the sublevel sets are Zariski-closed, and the well-orderedness and completeness corollaries follow. The load-bearing concern is the computational claim. The abstract promises an efficient algorithm that evaluates a finite list of polynomials, but the paper explicitly says the polynomials are not exhibited. Existence of a finite generating set follows from Noetherianity, but no method is given to compute it; without that, Theorem 1.1 is not an algorithm. This is a significant overclaim but does not undermine the main theorem. The reader's CONDITIONAL verdict is therefore appropriate. We also note the BDR22 attribution issue flagged by the reader, but it is secondary. The proposed test checks whether the proof can be made constructive; we suspect it cannot as written.","tokens_in":20558,"tokens_out":25716,"duration_ms":219549,"concrete_test":"Instantiability test: take V = Qbar^2 ⊗ Qbar^2 ⊗ Qbar^2 over the algebraic closure of Q, and r = 2. Attempt to follow the proof of Theorem 2.2 to produce an explicit finite list of polynomials that define {T ∈ V : eR(T) ≤ 2}. If the proof yields no construction or effective search procedure with a termination guarantee (without an oracle for eR), then Theorem 1.1 is not an algorithm in the standard sense, and the computability-from-above claim should be withdrawn or restricted to an existence statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central mathematical theorem (Theorem 1.2) is sound: tensor rank satisfies the admissible functional axioms, and the proof of Theorem 2.2 correctly shows eF[A]=eF[\\bar A], making sublevel sets Zariski-closed. However, the advertised claim 'computable from above' (Theorem 1.1 and abstract) requires, for each format and r, an explicit finite list of polynomials whose vanishing decides eR(T)≤r. The paper states it does not exhibit these polynomials, and the proof of Theorem 2.2 is purely existential: it uses Zariski closure and Noetherianity to assert a finite generating set exists, without constructing it. Over a computable field, one cannot simply search for the polynomials, because there is no termination test without an oracle for the predicate 'eR(T)≤r' being decided. Thus the algorithm of Theorem 1.1 cannot be instantiated from the proof. This does not refute Theorem 1.2, but it means the abstract's algorithmic claim overreaches the established results.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for any field F, the sublevel sets {T : eR(T) ≤ r} of asymptotic tensor rank are Zariski-closed, i.e., each is determined by the vanishing of finitely many polynomials. It derives that the set of all asymptotic ranks is well-ordered (discreteness from above), that over C it is Euclidean-closed, and it extends these properties to all elements of Strassen's asymptotic spectrum via a new lower bound on max-rank restrictions. It also gives an equivalent geometric condition for discreteness from below and leaves that as an open problem.","tokens_in":20745,"tokens_out":10660,"duration_ms":91941,"significance":"If the main theorem is correct, it establishes a fundamental algebraic structural property of asymptotic tensor rank that was previously unknown for infinite fields. The double-blocking proof of Theorem 2.2 is elegant and correct, and the paper is honest that the defining polynomials are not exhibited. The consequences for the matrix multiplication exponent (no upper bound arbitrarily close to omega without 'snapping' to it) and the generalization to the asymptotic spectrum are interesting. The main weakness is the unsubstantiated computational claim, which is not needed for the structural results but is prominently advertised.","major_comments":[{"comment":"The advertised algorithmic claim that asymptotic tensor rank is 'computable from above' is not established by the proof. Theorem 2.2 and Corollary 2.4 show that for each format and each r there exists a finite set of polynomials p1,...,pℓ whose vanishing decides eR(T) ≤ r, but the proof is purely existential: it uses Zariski closure and Noetherianity to assert existence without constructing the polynomials. Over a computable field one cannot simply enumerate candidate polynomial sets, because verifying a candidate requires an oracle for the predicate eR(T) ≤ r, which is precisely the decision problem the algorithm is meant to solve. The statement in footnote 2 that 'these polynomials are also computable' is therefore unsupported. Since Theorem 1.1 and the title of the paper rest on this computational claim, it must either be proven constructively or removed in favor of the (still substantial) structural statement of Theorem 1.2.","section":"§1.1, Theorem 1.1; Abstract"},{"comment":"The proof of Lemma 3.6 is not correct as written. The function ϕ_j is defined on (k−1)-tensors, but the proof uses the expression ϕ_j(T) where T is a k-tensor; moreover, the inequalities 'S ⊠ ⟨m⟩_{i,j} ≥ ϕ_j(T)' and 'S ≤ ϕ_j(T) ⊠ ⟨m⟩_{i,j}' have a type mismatch and are not otherwise justified. Lemma 3.6 is the central reduction used in Theorem 3.1 to show that F coincides in value with an element F' ∈ ∆(F,k−1); without a clear proof of this lemma, the discreteness result for the asymptotic spectrum (Theorem 3.1/1.5) is incomplete. Please provide a full proof with correctly typed tensors.","section":"§3.1, Lemma 3.6"}],"minor_comments":[{"comment":"The expression '2ω = eR(⟨2,2,2⟩)' should be typeset as 2^ω = eR(⟨2,2,2⟩) to avoid confusion with 2·ω.","section":"§1, Introduction"},{"comment":"The non-uniformity in r is a significant caveat; it should appear in the statement of Theorem 1.1 and in the abstract, not only in a footnote.","section":"§1.1, footnote 1"},{"comment":"When defining 'well-ordered' for subsets of R, please make explicit that this is relative to the usual order and note the equivalence with stabilization of non-increasing sequences, since some readers may use the order-theoretic definition.","section":"§2.3"},{"comment":"The block matrix notation [M1;...;Mt] is introduced with c blocks in Lemma 3.4 but with a different index variable in the surrounding text; unify the notation.","section":"§3.1, Lemma 3.4"},{"comment":"The expression 'T ⊠(k(k−1)/2)' is ambiguous; write (T ⊠ ... ) or define the Kronecker power explicitly.","section":"§3.1, Corollary 3.5"},{"comment":"Calling the set 'complete' may confuse readers; the proof shows Euclidean closedness, which is a different notion from completeness of a metric space.","section":"§4.1, Theorem 4.1"}],"recommendation":"major_revision","confidential_remarks":"The structural results (Theorems 1.2, 1.3, 1.4) are sound and likely of high interest, but the advertised computational claim in Theorem 1.1 and the abstract is not justified by the provided proof. The authors should either add a constructive argument for the polynomials or substantially weaken the computational framing. The proof of Lemma 3.6 also needs a careful rewrite before the asymptotic spectrum results can be considered complete. The paper otherwise seems within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main takeaway: the structural theorem is real. Sublevel sets of asymptotic rank are Zariski-closed over any field; the double-blocking argument is clean and the Noetherianity step is sound. That alone is a worthwhile advance: it extends finite-field discreteness to infinite fields and gives a gap above 2ω. The extension to functions in the asymptotic spectrum is a bonus, and the max-rank lower bound (Theorem 3.2) looks like a genuine ingredient with independent uses.\n\nThe soft spot is exactly where the abstract oversells. \"Computable from above\" is not established. Theorem 1.2 is existential: Zariski-closedness gives a finite polynomial list, but no way to produce it, and the proof does not provide a termination test to search for it. So Theorem 1.1 as stated—an algorithm that decides eR(T) ≤ r—does not follow. The authors do admit they do not exhibit the polynomials, but the abstract and the informal theorem still call it an algorithm. That claim needs to be either made precise, by showing the polynomials are computable, or retracted to \"the sublevel sets are algebraic.\" This is a significant gap in presentation, not a fatal flaw in the math: the Zariski-closedness and well-orderedness results stand.\n\nMinor: the introduction cites [BDR22] as having proved well-orderedness for a class including asymptotic rank without saying the result is over finite fields. The abstract later says the finite-field qualifier, so this is an internal inconsistency, easily fixed.\n\nOne more caveat: the independence of the paper's results is fine. The only self-citation is Lemma 3.4 from BCL+24, used for Theorem 3.2, and the framework borrows from their earlier work; no circularity.\n\nWho should read this: anyone working on asymptotic spectra, tensor rank, or the asymptotic rank conjecture. It deserves a serious referee. I would send it to a strong algebraic complexity person and ask them to check the algorithmic claim carefully; the structural part should survive.","headline":"The structural result—Zariski-closed sublevel sets for asymptotic rank—is solid and important; the advertised algorithm does not follow from the proof.","tokens_in":21311,"tokens_out":2161,"would_cite":true,"duration_ms":19117,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","68Q17"],"pacs":[],"model":"deepseek-v4-flash","headline":"Asymptotic tensor rank sublevel sets are cut out by finitely many polynomial equations.","keywords":["asymptotic tensor rank","polynomial equations","Zariski-closed sublevel sets","matrix multiplication exponent","well-ordered values","asymptotic spectrum of tensors","computability from above","tensor degeneration"],"falsifier":"Exhibit a tensor $T$ in the Zariski closure of a set $A$ with $\\tilde R(T) > \\sup_{S\\in A}\\tilde R(S)$; Theorem 2.2 forbids this. Over $\\mathbb{C}$, it would be enough to find a converging sequence of tensors whose asymptotic ranks converge to a real number that is not the asymptotic rank of any tensor, or a strictly decreasing sequence of asymptotic ranks that does not stabilize.","tokens_in":20353,"feed_emoji":"🧮","tokens_out":7723,"duration_ms":65223,"temperature":0.7,"pith_summary":"Asymptotic tensor rank $\\tilde R(T)$ measures how quickly the ordinary tensor rank grows under Kronecker powers, and it is the quantity behind the matrix multiplication exponent $\\omega$; deciding it is notoriously hard. This paper proves that, for any field, the sublevel set $\\{T : \\tilde R(T) \\le r\\}$ is Zariski-closed, meaning membership is equivalent to the vanishing of finitely many polynomials on the tensor entries, exactly as for matrix rank. Because of this, upper bounds on asymptotic rank become decidable from above: for computable fields and any real $r$, an algorithm can decide whether $\\tilde R(T) \\le r$ by evaluating polynomials. A further consequence, new for infinite fields such as $\\mathbb{C}$, is that the set of values taken by asymptotic rank is well-ordered: every non-increasing sequence of asymptotic ranks stabilizes, so there is a positive gap above $2^\\omega$ in the exponents of bilinear maps. These results hold not only for tensor rank but for every element of the asymptotic spectrum of tensors.","feed_headline":"Polynomials decide asymptotic tensor rank from above","feed_subtitle":"For every threshold r, the set of tensors with asymptotic rank at most r is defined by finitely many polynomial equations.","key_machinery":"The load-bearing object is an admissible functional: a family of functions $F_n$ on tensor powers $V^{\\otimes n}$ satisfying subadditivity, submultiplicativity under tensor products, permutation invariance, $\\mathbb{F}^\\times$-homogeneity, and boundedness on $V$; tensor rank is the primary example, and the regularized limit $\\tilde F(T)=\\lim_{n\\to\\infty}F_n(T^{\\otimes n})^{1/n}$ generalizes asymptotic rank. The key mechanism is the identity $\\tilde F[A]=\\tilde F[\\bar A]$ between the supremum over a set and over its Zariski closure, proven by writing $T^{\\otimes n}$ for $T\\in \\bar A$ as a linear combination of powers $S_i^{\\otimes n}$ with $S_i\\in A$ and then applying a double-blocking estimate (submultiplicativity and subadditivity) to the powers of those combinations. For the asymptotic-spectrum extension, the additional engine is a new lower bound on the max-rank quantities $Q_{i,j}(T)$: whenever $|F|>R_I(T)$, the product $\\prod_{i\\in I,j\\notin I}Q_{i,j}(T)\\ge R_I(T)$, which forces each spectrum element either to reduce to a lower-order tensor functional or to grow with the flattening rank.","core_discovery":"The central claim is Theorem 1.2: for any field $F$, order $k \\ge 3$, dimensions $d \\in \\mathbb{Z}_{\\ge 1}^k$, and real $r$, the set $\\{T \\in F^{d_1}\\otimes\\cdots\\otimes F^{d_k} : \\tilde R(T) \\le r\\}$ is Zariski-closed. In concrete terms, for each format and threshold there is a finite list of polynomials $p_1,\\dots,p_\\ell$ such that $\\tilde R(T)\\le r$ iff all $p_i(T)=0$. The proof works at the level of a general admissible functional $F$: it shows that the supremum of the regularized function $\\tilde F$ over a set equals the supremum over its Zariski closure, using a decomposition of $T^{\\otimes n}$ into linear combinations of $S_i^{\\otimes n}$ with $S_i$ in the original set. From this the paper derives lower-semicontinuity, well-orderedness of the value set, completeness over $\\mathbb{C}$, and the analogous discreteness result for the asymptotic spectrum.","pith_inferences":["If the sublevel sets are also irreducible, a dimension argument would give only finitely many asymptotic ranks per fixed format, a weak form of the asymptotic rank conjecture; the paper raises this as an open question, so this is an extrapolation rather than a claim.","The non-explicit polynomials may still be useful: once their degrees and sparsity are bounded, the decision procedure could be turned into a concrete algebraic witness for upper bounds, potentially connecting asymptotic rank to algebraic proof complexity.","The max-rank product inequality suggests a generic lower-bound strategy: to show a tensor has large asymptotic rank, it suffices to bound the individual $Q_{i,j}$ quantities, which are ordinary matrix-rank parameters and hence more accessible computationally."],"forward_implications":["Membership in every sublevel set of asymptotic tensor rank is decidable by evaluating a finite list of polynomials, giving an algorithm for \"asymptotic rank at most $r$\" over computable fields.","Any upper bound on $\\tilde R$ proven for a Zariski-dense family of tensors automatically extends to all tensors in the closure, so degeneration arguments yield upper bounds without explicit sequences.","The value set $\\{\\tilde R(T)\\}$ is well-ordered: any non-increasing sequence stabilizes, and in particular the matrix multiplication exponent cannot be approached from above by a strictly decreasing sequence of exponents of bilinear maps.","For every element of the asymptotic spectrum of tensors, the set of values is well-ordered; by duality, the set of tensors asymptotically restricted by a fixed tensor is Zariski-closed.","Over $\\mathbb{C}$, the set of asymptotic ranks is complete: the limit of any converging sequence of asymptotic ranks is itself an asymptotic rank."],"supporting_citations":[{"why":"Defines the asymptotic spectrum and duality, which the paper uses to lift well-orderedness from tensor rank to all spectrum elements and to close the set of asymptotic restrictions.","marker":"[Str88]"},{"why":"Provides the finite-field discreteness result and the $k=3$ max-rank lower bound that Theorem 3.2 extends to all orders.","marker":"[BCL+24]"},{"why":"Supplies the Noetherianity of the Zariski topology used to turn descending chains of sublevel sets into stabilization of values.","marker":"[Eis95]"},{"why":"Gives the comparison between Zariski and Euclidean topologies used in the Baire argument for completeness over $\\mathbb{C}$.","marker":"[Ser56]"},{"why":"Provides the Baire category theorem used to show extremal-level sets are dense and that increasing sequences cannot cover a variety by countably many proper closed subvarieties.","marker":"[Mun00]"}],"fun_headline_variants":["Polynomials fully determine asymptotic tensor rank","Asymptotic tensor rank: a finite polynomial check","Zariski-closed: polynomials capture asymptotic rank","Asymptotic rank from above, fixed by polynomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof requires tensor rank to satisfy the axioms of an admissible functional, above all submultiplicativity under the Kronecker product; if tensor rank failed any of these inequalities, the regularized limit and the key closure estimate would no longer go through.","fun_headline_variants_meta":{"raw":{"variants":["Polynomials fully determine asymptotic tensor rank","Asymptotic tensor rank: a finite polynomial check","Zariski-closed: polynomials capture asymptotic rank","Asymptotic rank from above, fixed by polynomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1517,"prompt_tokens":1123,"completion_tokens":394,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":739,"completion_tokens_details":{"reasoning_tokens":333}},"tokens_in":739,"tokens_out":394,"duration_ms":4499,"temperature":1.0,"reasoning_tokens":333,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:58:13.868594+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a tensor $T$ in the Zariski closure of a set $A$ with $\\tilde R(T) > \\sup_{S\\in A}\\tilde R(S)$; Theorem 2.2 forbids this. Over $\\mathbb{C}$, it would be enough to find a converging sequence of tensors whose asymptotic ranks converge to a real number that is not the asymptotic rank of any tensor, or a strictly decreasing sequence of asymptotic ranks that does not stabilize.","supporting_citations":[],"review_version":1}