{"id":"680e1446-8f7d-445e-b58f-0d3627924a04","arxiv_id":"2502.06714","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A polymatroid with a tensor product by U2,3 is claimed to always admit common information extensions, but the proof has load-bearing errors.","lead":"The paper claims that any polymatroid admitting a tensor product with the tiny uniform matroid U2,3 also admits common information extensions for every pair of subsets, linking two necessary conditions for linear representability. The main proof, however, contains a wrong-direction inequality and an unjustified equality, so the result is not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.2 is not proved: its lower-bound argument uses the false equality g(A1_2A2_2)=g(A1_2A2_2A3_2), and Theorem 5.1 then applies Proposition 4.2's upper bound α where the lower bound β is needed. The central implication is unsupported.","rationale":"The paper's motivating construction is attractive and the linear-algebraic Proposition 4.1 is sound. However, the step from linear representations to arbitrary tensor products is exactly where the definition is too weak to justify the proof. The false equality in Prop 4.2 is not a typo in an inconsequential line; it is used to derive the lower bound, and Theorem 5.1's proof invokes the bounds in the wrong direction at the final inequality. The reader's verdict of REJECT with high correctness risk is therefore appropriate. I am not asserting the result is false, only that the manuscript does not demonstrate it; a repaired proof may exist. The proposed exhaustive check would settle whether Proposition 4.2 itself is salvageable.","tokens_in":5764,"tokens_out":9778,"duration_ms":79734,"concrete_test":"Run an exhaustive computational search over all polymatroids on n≤4 elements and all set functions g on E×{1,2,3} satisfying the rectangle condition g(X×Y)=f(X)u(Y), and test whether the inequalities β≤g(A1_1A2_2A3_3)≤α in Proposition 4.2 hold for every triple A1,A2,A3 and for every such g (or whether any g satisfying the rectangle condition exists for which (2) fails). If a counterexample is found, Theorem 5.1 is refuted; if none is found, the proposition may be true but its proof still needs a replacement for the false equality and a corrected use of the bounds in Theorem 5.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The tensor product definition only fixes g on rectangles X×Y. In Proposition 4.2, A1_2A2_2=(A1A2)×{2} and A1_2A2_2A3_2=(A1A2A3)×{2}, so the asserted equality would force f(A1A2)=f(A1A2A3), which is false in general (e.g. f=U3,3 with A1,A2,A3 distinct singletons gives 2≠3). The lower bound β≤g in (2) therefore does not follow from the written argument. This matters because Theorem 5.1 depends on Proposition 4.2 for the values g(X1Y2)=f(X)+f(Y), g(X1Y2X3)=f(X)+f(XY), and for the final check of f(z;z|A). Moreover, that final check writes g(X1Y2A3)-α(X,Y,A)≥0, but (2) gives g≤α, not g≥α; the correct requirement is the lower bound β≤g together with β≥f(XY)+f(A). As written, neither the bounds nor their application in Theorem 5.1 are established, so the proof of the main claim is incomplete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a new implication between two known necessary conditions for linear representability of polymatroids. The main theorem (Theorem 5.1) states that every polymatroid that admits a tensor product with the uniform matroid U_{2,3} is 1-CI, meaning that for every pair of subsets of the ground set there exists a common-information extension. The argument proceeds by deriving upper and lower bounds for the rank function of any such tensor product (Proposition 4.2), using them to reprove Ingleton's inequality, and then defining the common-information extension by the formula f(Az)=g(X_1Y_2A_3)-f(XY).","tokens_in":6110,"tokens_out":19156,"duration_ms":158888,"significance":"If the theorem were valid, it would establish a genuinely new connection between the tensor-product condition and common-information extensions, and it would give an explicit construction of CI extensions from a tensor product. The paper is concise, well motivated, and does not rely on numerical computation or parameter fitting. However, the proof as written contains several false or unjustified equalities in Proposition 4.2, and the final inequality in Theorem 5.1 uses the wrong direction of the proved bound. The central claim is therefore not established.","major_comments":[{"comment":"The proof of the lower bound in (2) uses the equality g(A1_2 A2_2)=g(A1_2 A2_2 A3_2). Under the tensor-product definition, g(A1_2 A2_2)=f(A1A2) and g(A1_2 A2_2 A3_2)=f(A1A2A3), because both are rectangle values with second factor {2}; these ranks are generally different. For example, with f=U_{3,3} and A1,A2,A3 distinct singletons, they are 2 and 3. Consequently the chain g(X)+g(Y) >= g(XY)+g(X∩Y) >= s3+s1+r1+r3 does not follow, and the claimed lower bound beta <= g(...) is unproved. This bound is load-bearing for Theorem 5.1.","section":"Section 4, Proposition 4.2 (lower bound)"},{"comment":"The proof asserts without justification that g(Z)=s1+s2, g(T)=s3+s, and g(ZT)=2s for Z=(A1A3)_1(A2A3)_2A3_3 and T=A1_1(A2A3)_2(A1A2A3)_3. These sets are not rectangles, so their g-values are not fixed by the tensor-product definition. For the canonical linear tensor product of f=U_{3,3} with U_{2,3} described in Proposition 4.1, with A1,A2,A3 distinct singletons, the set T has rank 6, whereas s3+s=5; thus the stated equality g(T)=s3+s is false for a genuine tensor product. Hence the upper bound alpha is also not established.","section":"Section 4, Proposition 4.2 (upper bound)"},{"comment":"The proof concludes f(z;z|A) >= g(X1Y2A3) - alpha(X,Y,A) >= 0. But Proposition 4.2 gives g(X1Y2A3) <= alpha(X,Y,A), so the expression g(X1Y2A3)-alpha(X,Y,A) is nonpositive in general, not nonnegative. To prove f(z;z|A)>=0 one needs g(X1Y2A3) >= f(XY)+f(A), which would follow from the lower bound beta <= g together with beta >= f(XY)+f(A); neither the lower bound nor the correct inequality is available in the written proof. Together with the use of Proposition 4.2 in the equalities for f(z) and f(Xz), this means the proof of the main theorem is incomplete.","section":"Section 5, Theorem 5.1 (final inequality)"}],"minor_comments":[{"comment":"There is a typo in 'commom information' in the definition of a common-information extension.","section":"Section 3"},{"comment":"The displayed definition 'f(Y|X)=f(Y;Y|Z)' appears to be a typo; it should be f(XY)-f(X).","section":"Section 2"},{"comment":"In the proof of Theorem 5.1, '(Ez,g)' should be '(Ez,f)' since the extension is on the ground set Ez with rank function f.","section":"Section 5"}],"recommendation":"reject","confidential_remarks":"The paper has not been carefully checked: the flaws in Proposition 4.2 are not mere presentation issues, and the final inequality in Theorem 5.1 uses the wrong direction. If the author can supply a correct proof that every tensor product with U_{2,3} satisfies the bounds in (2), the main theorem may become valid; in the current form, however, the contribution is not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Carles Padró's paper has a genuinely attractive idea: use a tensor product with U2,3 to construct a common information extension via f(Az)=g(X1Y2A3)-f(XY). Were that construction to work, it would answer the open question from [3] and give a more transparent proof of Ingleton's inequality. The paper is clearly written and the citations are appropriate.\n\nThe trouble is that the proof of the central Theorem 5.1 is not sound as written. Proposition 4.2, which supplies the bounds, contains an unjustified equality: the proof claims g(A1_2A2_2)=g(A1_2A2_2A3_2). But A1_2A2_2 is the rectangle (A1A2)×{2} and A1_2A2_2A3_2 is (A1A2A3)×{2}. The tensor product condition fixes these to f(A1A2) and f(A1A2A3), which are not equal in general. So the lower bound β≤g is not established. The upper bound α also looks suspect; the text derives it from a submodularity inequality but the equality g(ZT)=2s is asserted without justification and ZT is not a rectangle.\n\nThen in Theorem 5.1, the final verification writes f(z;z|A) ≥ g(X1Y2A3)−α ≥ 0, but Proposition 4.2 gives g ≤ α, not g ≥ α. So the direction is reversed. The construction may be repairable, but as it stands the main claim is not demonstrated.\n\nDespite these flaws, the paper deserves a serious referee. The question is important, the construction is natural, and the errors look fixable rather than fatal to the idea. A referee should focus on whether the CI extension defined in (5) can be proved polymatroid by a different argument, perhaps using the lower bound β instead of α. If the author can repair the proof, this would be a nice contribution. For now, I would not cite the main theorem.","headline":"A promising construction for CI extensions from tensor products, but the proof of the main theorem has a false equality and a reversed inequality.","tokens_in":6612,"tokens_out":3140,"would_cite":false,"duration_ms":24958,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A polymatroid that admits a tensor product with the uniform matroid U2,3 must have common information extensions for every pair of subsets.","keywords":["polymatroid","common information extension","tensor product of polymatroids","uniform matroid U2,3","Ingleton's inequality","linear representability","submodularity","matroid"],"falsifier":"To test Theorem 5.1, take a polymatroid that admits a tensor product with $U_{2,3}$ and check whether the proposed function $f(Az)=g(X_1Y_2A_3)-f(XY)$ is submodular for all choices of $X,Y$; a single violation of $f(x;z|A)\\ge 0$ in a concrete example would disprove the theorem. More narrowly, evaluating Proposition 4.2's bounds on a linear polymatroid with three independent elements shows the proof's equality $g((A_1A_2)_2)=g((A_1A_2A_3)_2)$ fails, so the bounds themselves are the critical step to verify.","tokens_in":5582,"feed_emoji":"🔗","tokens_out":6292,"duration_ms":51265,"temperature":0.7,"pith_summary":"This paper establishes a new connection between two necessary conditions a polymatroid must satisfy to be linearly representable. It proves that whenever a polymatroid has a tensor product with the uniform matroid U2,3, that polymatroid is 1-CI: for every pair of subsets, an extension of the polymatroid exists in which a new element has the mutual information value of the pair and is conditionally independent of it. The proof gives an explicit formula for such an extension directly from the tensor product's rank function. As a corollary, the paper obtains a simpler proof that tensor products with U2,3 imply Ingleton's inequality.","feed_headline":"Tensor products with U2,3 guarantee common information","feed_subtitle":"For any pair of subsets, a tensor product with U2,3 builds an extension where a new element carries their common information.","key_machinery":"The load-bearing object is the tensor product itself: a polymatroid $(E_1\\times E_2,g)$ of two polymatroids with rank functions $f_1,f_2$ satisfies $g(X\\times Y)=f_1(X)f_2(Y)$ for all subsets $X\\subseteq E_1$, $Y\\subseteq E_2$. Here $U_{2,3}$ is the uniform matroid of rank two on three elements. The proof also uses bounds $\\beta\\le g(A_1^1A_2^2A_3^3)\\le \\alpha$ on the rank of three-layer sets, derived from submodularity in Proposition 4.2. The common information extension is built by subtracting $f(XY)$ from the tensor rank value $g(X_1Y_2A_3)$, and submodularity of $g$ is shown to carry over to the extension.","core_discovery":"On the paper's own terms, the central claim is Theorem 5.1: every polymatroid $(E,f)$ that admits a tensor product $(E\\times\\{1,2,3\\},g)$ with $U_{2,3}$ is 1-CI. For any pair $X,Y\\subseteq E$, the extension defined by $f(Az)=g(X_1Y_2A_3)-f(XY)$ for all $A\\subseteq E$ is a polymatroid in which $z$ is a common information for $(X,Y)$. The theorem is reached by observing that in the linear case the tensor product's rank on $X_1Y_2A_3$ is $f(X)+f(Y)+f(A)-\\dim(U_X\\cap U_Y\\cap U_A)$, while a common information extension adds a subspace $V_z=U_X\\cap U_Y$, so the two constructions coincide; the paper then argues the same identity holds for any tensor product with $U_{2,3}$ via submodularity bounds.","pith_inferences":["The explicit formula suggests a recipe for iterating common information extensions: apply the same construction to the extended polymatroid, potentially producing new linear rank inequalities beyond Ingleton's, provided each iteration admits a suitable tensor product.","The theorem leaves open the converse: a 1-CI polymatroid might fail to admit a tensor product with $U_{2,3}$, which would show the two necessary conditions are not equivalent; the Vámos matroid is one natural place to look, since it has no such tensor product.","Because every linearly representable polymatroid has both properties, the construction gives a representation-free way to build common information extensions for linear polymatroids directly from tensor products, which may be useful when explicit rank formulas are needed."],"forward_implications":["Every polymatroid admitting a tensor product with $U_{2,3}$ automatically satisfies the 1-CI necessary condition for linear representability, not just Ingleton's inequality.","There is an explicit rank formula for a common information extension in terms of any tensor product: $f(Az)=g(X_1Y_2A_3)-f(XY)$.","The result ties together two previously separate necessary conditions, showing that tensor-product existence is at least as restrictive as common information at the first iteration.","The new proof of Ingleton's inequality is shorter and more direct than the earlier proof via submodular coupling."],"supporting_citations":[{"why":"Defines k-CI polymatroids, the property Theorem 5.1 targets.","marker":"[2]"},{"why":"Introduced tensor products via submodular coupling and proved tensor products with U2,3 imply Ingleton's inequality, the point of departure.","marker":"[3]"},{"why":"Introduced common information extensions and proved every linear polymatroid has them.","marker":"[8]"},{"why":"The original Ingleton inequality that motivates these necessary conditions.","marker":"[9]"},{"why":"One of the two sources that introduced tensor products of polymatroids and matroids.","marker":"[11]"},{"why":"Provides the characterization of polymatroids used to verify submodularity of the constructed extension.","marker":"[13]"}],"fun_headline_variants":["Tensor product with U2,3 forces common information","U2,3 tensor product implies common information for all pairs","Polymatroid tensor products with U2,3 guarantee CI","Tensor product with U2,3 builds common information extensions","Common information from U2,3 tensor product"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument for the key rank bounds assumes that replacing the set $(A_1A_2)$ by the larger set $(A_1A_2A_3)$ in the second factor of the tensor product leaves the rank unchanged; the tensor product definition gives the two values as $f(A_1A_2)$ and $f(A_1A_2A_3)$, which need not be equal.","fun_headline_variants_meta":{"raw":{"variants":["Tensor product with U2,3 forces common information","U2,3 tensor product implies common information for all pairs","Polymatroid tensor products with U2,3 guarantee CI","Tensor product with U2,3 builds common information extensions","Common information from U2,3 tensor product"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000216,"raw_usage":{"total_tokens":1353,"prompt_tokens":788,"completion_tokens":565,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":404,"completion_tokens_details":{"reasoning_tokens":484}},"tokens_in":404,"tokens_out":565,"duration_ms":4923,"temperature":1.0,"reasoning_tokens":484,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T14:34:49.853921+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test Theorem 5.1, take a polymatroid that admits a tensor product with $U_{2,3}$ and check whether the proposed function $f(Az)=g(X_1Y_2A_3)-f(XY)$ is submodular for all choices of $X,Y$; a single violation of $f(x;z|A)\\ge 0$ in a concrete example would disprove the theorem. More narrowly, evaluating Proposition 4.2's bounds on a linear polymatroid with three independent elements shows the proof's equality $g((A_1A_2)_2)=g((A_1A_2A_3)_2)$ fails, so the bounds themselves are the critical step to verify.","supporting_citations":[{"cited_title":"Matroid products via submodular coupling","cited_arxiv_id":"2411.02197","evidence_quote":"Introduced tensor products via submodular coupling and proved tensor products with U2,3 imply Ingleton's inequality, the point of departure."},{"cited_title":"Journal of Computer and Systems Sciences 60, 442–464 (2000)","cited_arxiv_id":null,"evidence_quote":"Introduced common information extensions and proved every linear polymatroid has them."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The original Ingleton inequality that motivates these necessary conditions."},{"cited_title":"InCombinatorial Surveys (Proc","cited_arxiv_id":null,"evidence_quote":"One of the two sources that introduced tensor products of polymatroids and matroids."},{"cited_title":"Polyhedra and Efficiency","cited_arxiv_id":null,"evidence_quote":"Provides the characterization of polymatroids used to verify submodularity of the constructed extension."}],"review_version":1}