{"id":"4f7fa498-4d0a-4d63-b571-49807cdab03c","arxiv_id":"2507.10709","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A connected matroid is skew-representable if and only if it admits tensor products with the uniform matroid U_{2,3} at every order, giving verifiable certificates for non-representability and a new rank inequality.","lead":"Mathematicians show that whether a matroid can be built from vectors over a skew field can be decided by checking an infinite family of tensor product puzzles, and that failures have verifiable certificates. The same technique yields a new inequality that every folded skew-representable matroid obeys, the first claimed one not derivable from the common information property.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'first known' novelty claim for inequality (26) depends on an unsupported survey assertion that all previously known non-characteristic-specific linear rank inequalities follow from the common information property; if that assertion is false, the inequality is still new beyond CI but is not…","rationale":"The reader's conditional verdict targets the novelty framing, and I agree that the 'first known' label is not adequately supported. However, the reader bundles two assumptions, and only the second is a genuine gap. The first assumption — that every rank-3 matroid, in particular non-Desargues, satisfies the common information property — is effectively supplied by the paper itself: Theorem 3.4 proves fully modular extendability for rank-3 components via [1, Proposition 3.18], and fully modular extendability for integer-valued polymatroid functions implies the fractional common-information property. The genuinely unsupported step is the survey claim in Section 1.1 that all known non-characteristic-specific linear rank inequalities derive from the common information property. Without a proof or a precise citation isolating that theorem, the assertion 'first known inequality not following from common information' is not verified. This does not undermine the tensor-product characterization, the co-r.e. certificates, or the validity of inequality (26); those arguments appear coherent and the rank inequality is explicitly checked against the non-Desargues matroid. I also checked the proof of Theorem 5.5 and note a likely typo in the induction step ('N'' in T_{k-1}(N,U2,3)' should express that N' is (k-1)-tensor-compatible with U2,3), but the needed minor-closure property follows by induction from Lemma 4.1, so I do not regard it as a load-bearing flaw. The verdict should remain CONDITIONAL: the mathematical results are acceptable, but the novelty claim needs to be either proved, precisely cited, or softened.","tokens_in":51504,"tokens_out":31957,"duration_ms":367076,"concrete_test":"Enumerate the non-characteristic-specific linear rank inequalities catalogued in [13] and [1] and evaluate each on the rank function of the non-Desargues matroid (10 elements). Because non-Desargues is CI-compliant, any catalogued inequality that is also violated by it would itself go beyond the common information property, falsifying the 'first known' claim. If none of the catalogued inequalities is violated, the novelty claim survives; in that case the authors should still supply a precise derivation or citation establishing that those inequalities are consequences of the common information property.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The weakest load-bearing point is the historical claim attached to Theorem 7.9 / Corollary 7.13. The proof that inequality (26) does not follow from the common information property is sound provided the non-Desargues matroid satisfies the common information property; this is supported by Theorem 3.4 together with [1, Proposition 3.18] for rank-3 matroids, so I do not treat that as a gap. What is not established is the Section 1.1 assertion that 'all of the currently known linear rank inequalities that are not characteristic-specific can be derived from the so-called common information property [1, 13]'. No derivation or precise theorem statement is given, and this is exactly what is needed to justify 'the first known'. If some previously published non-characteristic-specific inequality is not a consequence of the common information property, then inequality (26) may still be an independent/non-CI inequality, but the abstract's headline claim collapses. The central characterization (Theorem 5.8, Corollary 5.9) is unaffected by this issue, since it does not rely on that literature assertion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a tensor-product framework for skew-representability. Theorem 5.8 and Corollary 5.9 characterize connected skew-representable matroids, and matroids representable over skew fields of fixed prime characteristic, in terms of k-tensor-compatibility with a suitable test matroid for all positive integers k; Corollaries 5.12 and 5.15 turn this into co-recursively enumerable certification of non-representability. Section 6 constructs a freest tensor product of any rank-3 matroid with any uniform matroid. Section 7 uses tensor products to give a new proof of Ingleton's inequality, to derive characteristic-dependent inequalities from the Fano and non-Fano matroids, and to prove inequality (26), which is valid for folded skew-representable polymatroids and is violated by the non-Desargues matroid. The paper claims this is the first known characteristic-independent linear rank inequality not following from the common information property.","tokens_in":51581,"tokens_out":18519,"duration_ms":205951,"significance":"If the mathematical results stand, the tensor-product characterization is a notable structural contribution: it links representability to an iterated extension property and yields uniform certificates of non-representability. The freest tensor product theorem for rank-3 matroids is a strong constructive result with independent interest. The paper is largely self-contained, gives detailed proofs rather than fitted or numerically generated claims, and the new inequality (26) comes with an explicit violation witness. The 'first known' novelty assertion, however, is not established within the manuscript; the inequality itself and its derivation are still a substantial contribution once the historical claim is either verified or suitably weakened.","major_comments":[{"comment":"The headline claim that inequality (26) is 'the first known linear rank inequality for folded skew-representable matroids that does not follow from the common information property' rests on the unproved survey assertion in Section 1.1 that all currently known non-characteristic-specific linear rank inequalities can be derived from the common information property [1,13]. No precise theorem statement or derivation is given for this assertion, and it is exactly what is needed to justify the word 'first'. The validity of Theorem 7.9 and Corollary 7.13 does not depend on this historical assertion, but the advertised novelty of the paper does. Please either prove or carefully locate this assertion in the literature, or weaken the wording in the abstract, Section 1.2, and Section 7.3 to say that (26) is an inequality not following from the common information property rather than the first known such inequality.","section":"Section 1.1; Section 7.3; Abstract"},{"comment":"The statement that the non-Desargues matroid satisfies the common information property is asserted without proof or citation at the point where it is used. The conclusion that (26) goes beyond the common information property depends on this fact, since the violation witness must be a matroid that does satisfy that property. The needed chain can be supplied from the paper's own Theorem 3.4 together with [1, Proposition 3.18]: rank-3 matroids are fully modular extendable, and full modular extendability implies the fractional version, the common information property. This chain should be stated explicitly in Section 7.3.","section":"Section 7.3, paragraph after Theorem 7.9"}],"minor_comments":[{"comment":"The phrase 'The rest of of the paper' contains a duplicated word and should be corrected.","section":"Section 1.3"},{"comment":"Corollary 7.6(b) states that φ1 satisfies inequality (6), but the proof refers to Theorem 7.5(b), which establishes inequality (7); the displayed inequality reference appears to be a typo.","section":"Corollary 7.6(b)"},{"comment":"In the proof of Claim 7.12, the justification 'φ1({e1, e5, e5}) = φ1(S2)' is not a valid matroid statement; it should presumably refer to a triangle such as {e1, e4, e5} with rank equal to the full ground set rank.","section":"Proof of Claim 7.12"},{"comment":"In the proof of Claim 6.2(c), the text says 'Using the submodularity of rN' but the set in question lies in the tensor product ground set S × [n]; the rank function being used should be rP.","section":"Proof of Claim 6.2(c)"},{"comment":"In the second case of the basis-exchange proof, the sentence 'For i ∈ I, we have e ∈ A′_i' is not literally true when e is an added point of M• outside the original ground set S; the intended inequality should be justified by a spanning argument rather than by membership of e in A′_i.","section":"Proof of Claim 6.6"},{"comment":"Remark 4.16 asserts, without proof, that the non-Pappus matroid admits a tensor product with itself but that none of the resulting matroids is representable over the quaternions, and the text says the proof is omitted. Please either include the proof or mark the assertion as a conjecture, since it is currently an unsupported claim in the paper.","section":"Remark 4.16"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core of the paper appears sound and the main obstacle is the unverified novelty claim. I would ask the authors to treat the 'first known' statement strictly: either provide a precise literature-based proof or remove the 'first' claim. The same care should be applied to the assertion about rank-3 matroids and the common information property. This is fixable within the scope of the manuscript, so I do not see a need for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central characterization is real and the paper deserves a serious referee. The tensor product characterization of skew-representability (Theorem 5.8 and Corollary 5.9) is new and proved from modular extension theory plus Veblen–Young, not from the target result. The co-recursively enumerable certificate consequence is a clean and valuable observation, especially combined with the known undecidability results: it shows there is no certificate for skew-representability. The freest tensor product theorem for rank-3 matroids with uniform matroids is technical but carefully argued. The new rank inequality for folded skew-representable matroids (Theorem 7.9, Corollary 7.13) is derived from the tensor product with M(K4) and is violated by the non-Desargues matroid, so the core inequality claim checks out.\n\nThe soft spot is the novelty framing. The abstract says \"first known linear rank inequality... that does not follow from the common information property.\" Section 7.3 phrases it as \"to the best of our knowledge.\" The proof that the inequality does not follow from the common information property rests on two literature assertions: that the non-Desargues matroid satisfies the common information property, and that all currently known non-characteristic-specific linear rank inequalities can be derived from the common information property. The first is supported by Theorem 3.4 together with [1]; the second is stated in Section 1.1 without a precise theorem or derivation. If that survey assertion is wrong, inequality (26) might still be new and independent, but the \"first known\" claim as stated would not be established. This is a historical/literature claim, not a mathematical gap in the central characterization. It should be fixed before publication, either by proving the derivation or by carefully rewriting the claim.\n\nMinor: the paper omits some proofs in remarks (e.g., Remark 4.16) and is very long, but those are not load-bearing. The authors are refreshingly explicit in Remark 7.7 that they do not know whether their characteristic-dependent inequalities are consequences of known ones.\n\nWho is this for? Matroid theorists, especially those working on representability, tensor products, and information inequalities. It is a substantial contribution with a new angle. It deserves peer review, and I would engage with it.","headline":"A substantial tensor-product characterization of skew-representability with a mostly sound rank-inequality application; the 'first known beyond common information' claim needs a more careful literature check.","tokens_in":52296,"tokens_out":2455,"would_cite":true,"duration_ms":26926,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B35","51A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a connected matroid is skew-representable if and only if it is $k$-tensor-compatible with the uniform matroid $U_{2,3}$ for every positive integer $k$, and derives from this a new rank inequality that the…","keywords":["matroids","skew-representability","tensor products","modular extensions","linear rank inequalities","common information property","non-Desargues matroid","co-recursively enumerable"],"falsifier":"The central characterization would collapse if a connected matroid $M$ failed to be skew-representable while $T_k(M,U_{2,3})$ was non-empty for every positive integer $k$; searching for such an $M$ among rank-3 matroids, where the first possible failure is $k=2$ by Theorem 6.1, is the direct falsification test.","tokens_in":51204,"feed_emoji":"🧮","tokens_out":15090,"duration_ms":158200,"temperature":0.7,"pith_summary":"The paper establishes a characterization of skew-representability in purely combinatorial tensor-product terms. A connected matroid is skew-representable, meaning representable by vectors over some division ring, if and only if it admits a $k$-fold tensor product with the three-element uniform matroid $U_{2,3}$ for every positive integer $k$. The same method characterizes representability over skew fields of a fixed prime characteristic, with $U_{2,3}$ replaced by a suitable test matroid, and implies that non-skew-representability has finite certificates: deciding it is co-recursively enumerable. On the rank-inequality side, the paper derives a linear rank inequality for folded skew-representable matroids that it argues is the first not following from the common information property; the non-Desargues matroid violates the inequality. A construction shows every rank-3 matroid has a freest tensor product with every uniform matroid, so the obstruction to skew-representability of rank-3 matroids is invisible at the first tensor level.","feed_headline":"One uniform matroid, iterated, decides which matroids are skew-representable","feed_subtitle":"Repeated tensor products with U2,3 yield a first inequality beyond common information, violated by non-Desargues.","key_machinery":"The machinery is the matroid tensor product: given matroids $M_1=(S_1,r_1)$ and $M_2=(S_2,r_2)$, a tensor product is a matroid on $S_1\\times S_2$ with $r(X_1\\times X_2)=r_1(X_1)r_2(X_2)$ for all $X_i\\subseteq S_i$, and $k$-tensor compatibility means such a product can be iterated $k$ times. The paper couples tensor products with modular extensions, in which a pair of flats is forced to satisfy the modular rank equality. One tensor product with $U_{2,3}$ buys one modular extension step for any pair of flats, and iterating for all $k$ yields full modular extendability. Full modular extendability is then converted into skew-representability through projective spaces: modular matroids give generalized projective spaces, and the Veblen-Young theorem says projective spaces of dimension at least three are skew-representable. In the rank-inequality part, the same tensor product is analyzed by submodularity bounds on specially chosen cylinder sets, producing algebraic inequalities such as inequality (26).","core_discovery":"The paper's principal discovery is that skew-representability, a property normally defined by the existence of a linear representation over a division ring, can be characterized purely by the existence of iterated tensor products with the three-point uniform matroid $U_{2,3}$. For a connected matroid $M$, $M$ is skew-representable if and only if, for every positive integer $k$, there is a matroid on $M \\times U_{2,3}^k$ whose rank on rectangles $X \\times Y$ is $r_M(X)\\cdot r_{U_{2,3}}(Y)$. With a suitable test matroid $N$ built from a prescribed characteristic set $C$, the same statement characterizes direct sums of matroids representable over skew fields of characteristic in $C$. The proof shows that a connected matroid that is $k$-tensor-compatible with $U_{2,3}$ for every $k$ is fully modular extendable, and then uses the Veblen-Young theorem to conclude such a matroid is skew-representable; this yields co-recursively enumerable certificates of non-skew-representability. On the rank-inequality side, the framework proves a new linear rank inequality that all folded skew-representable polymatroids satisfy and the non-Desargues matroid violates, which the paper argues is the first such inequality not following from the common information property.","pith_inferences":["The matroid-level tensor characterization cannot directly extend to folded skew-representability: following the paper's Remark 5.11, a folded but not skew-representable matroid has a rank function that is $k$-tensor-compatible with $U_{2,3}$ for every $k$, so foldedness would have to be captured by polymatroid tensor compatibility rather than matroid tensor compatibility.","Inequality (26) is a candidate constraint for information-theoretic linear programs, since it holds for all folded skew-representable polymatroids but fails on the non-Desargues matroid; the paper does not run such computations, but this inequality could certify non-embeddability where extension-property constraints are inert.","The freest tensor product construction suggests a finite combinatorial invariant for rank-3 matroids: the smallest $k$ at which tensor compatibility with $U_{2,3}$ fails. Computing this value for other non-skew-representable sparse paving matroids would test how sharp the $k=2$ obstruction is."],"forward_implications":["Non-skew-representability of a connected matroid can be certified, in principle, by exhibiting some $k$ for which no $k$-fold tensor product with $U_{2,3}$ exists, making the decision problem co-recursively enumerable.","Since skew-representability of rank-3 matroids is undecidable while non-skew-representability is co-recursively enumerable, the problem of deciding skew-representability is not recursively enumerable either.","The non-Desargues matroid is not 2-tensor-compatible with $U_{2,3}$, so the infinite family of tensor tests is not overkill; the first level $k=1$ is never a blocker for rank-3 matroids.","Every rank-3 matroid admits a freest tensor product with every uniform matroid, so obstructions to skew-representability of rank-3 matroids can only appear from $k=2$ onward.","Folded skew-representable polymatroid functions satisfy inequality (26), which the paper claims is the first known characteristic-independent linear rank inequality not following from the common information property."],"supporting_citations":[{"why":"Defines matroid tensor products and proves the basic properties used throughout, including the minor-closure and rectangle-rank lemmas.","marker":"[38]"},{"why":"Veblen-Young theorem: every projective space of dimension at least three is skew-representable, used to pass from full modular extendability to skew-representability.","marker":"[66]"},{"why":"Supplies the common-information and modular-extension statement that rank-3 matroids are fully modular extendable, used in Theorem 3.4 and as the baseline for the 'beyond common information' claim.","marker":"[1]"},{"why":"Gives characteristic-set realization by rank-3 matroids and the fact that $PG(2,p)$ is skew-representable exactly in characteristic $p$, enabling the characteristic-specific tensor characterization.","marker":"[34]"},{"why":"Cited as the source that all currently known characteristic-independent linear rank inequalities follow from the common information property, the contrast against which the new inequality is claimed.","marker":"[13]"},{"why":"Earlier proof that tensor compatibility with $U_{2,3}$ already yields Ingleton's inequality; the paper extends this tensor-product method to Fano, non-Fano, and non-Desargues-type inequalities.","marker":"[4]"}],"fun_headline_variants":["Iterated U2,3 tensors decide skew-representability","Skew-representability via tensor powers of U2,3","New rank inequality for folded skew-representable matroids","First rank inequality beyond common information","U2,3 tensor products characterize skew-representable matroids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim that inequality (26) is the first rank inequality beyond the common information property rests on two unproved literature assumptions: every rank-3 matroid, in particular the non-Desargues matroid, satisfies the common information property, and every previously known characteristic-independent linear rank inequality follows from it.","fun_headline_variants_meta":{"raw":{"variants":["Iterated U2,3 tensors decide skew-representability","Skew-representability via tensor powers of U2,3","New rank inequality for folded skew-representable matroids","First rank inequality beyond common information","U2,3 tensor products characterize skew-representable matroids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00034,"raw_usage":{"total_tokens":1951,"prompt_tokens":1098,"completion_tokens":853,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":714,"completion_tokens_details":{"reasoning_tokens":775}},"tokens_in":714,"tokens_out":853,"duration_ms":8749,"temperature":1.0,"reasoning_tokens":775,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:27:35.790209+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The central characterization would collapse if a connected matroid $M$ failed to be skew-representable while $T_k(M,U_{2,3})$ was non-empty for every positive integer $k$; searching for such an $M$ among rank-3 matroids, where the first possible failure is $k=2$ by Theorem 6.1, is the direct falsification test.","supporting_citations":[{"cited_title":"Las Vergnas","cited_arxiv_id":null,"evidence_quote":"Defines matroid tensor products and proves the basic properties used throughout, including the minor-closure and rectangle-rank lemmas."},{"cited_title":"Veblen and J","cited_arxiv_id":null,"evidence_quote":"Veblen-Young theorem: every projective space of dimension at least three is skew-representable, used to pass from full modular extendability to skew-representability."},{"cited_title":"Bamiloshin, A","cited_arxiv_id":null,"evidence_quote":"Supplies the common-information and modular-extension statement that rank-3 matroids are fully modular extendable, used in Theorem 3.4 and as the baseline for the 'beyond common information' claim."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives characteristic-set realization by rank-3 matroids and the fact that $PG(2,p)$ is skew-representable exactly in characteristic $p$, enabling the characteristic-specific tensor characterization."},{"cited_title":"B´ erczi, B","cited_arxiv_id":null,"evidence_quote":"Earlier proof that tensor compatibility with $U_{2,3}$ already yields Ingleton's inequality; the paper extends this tensor-product method to Fano, non-Fano, and non-Desargues-type inequalities."}],"review_version":1}