{"id":"643c6583-87fe-4a19-8fc1-1d56f8bc07cd","arxiv_id":"2509.06302","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Membership in the closure of the entropic cone is undecidable, proved by recovering group structure from almost entropic partial Dowling geometries via a Desargues-type theorem.","lead":"This paper proves that no algorithm can decide whether a given integer vector lies in the closure of the entropic cone, the set of all joint-entropy vectors of random variables. It settles the last open case in a line of undecidability results, via a new Desargues-type theorem for almost entropic polymatroids.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"","rationale":"The reader's verdict is CONDITIONAL with moderate confidence, and the weakest assumption identified is exactly Theorem 15 (three-line intersection), with Lemma 16 as secondary. After re-examining the paper, I find no additional load-bearing concern that is not downstream of Theorem 15. The proof of Theorem 18 is a direct reduction to Theorem 15, and the group recovery (Thm 36) and lifting (Thm 37) depend on Theorem 18 and the copy lemma. The reduction framework from [KY22b] is explicitly cited and the missing direction is the new almost-entropic one; the paper supplies the new tool (Desargues) but that tool rests on the cited three-line theorem.\n\nI spot-checked the rank computations in the Desargues proof (Thm 18) and the group recovery and found no internal inconsistency. The final proof of Theorem 1 skips citing Theorem 34, but this is a minor expository omission that is easily fixed by inserting the product-closed extension step. The paper's own Remark 41 acknowledges the argument is an extension of [KY22b, Thm. 9.12], which is honest and does not undermine novelty.\n\nThe verdict should remain CONDITIONAL: the central claim is structurally sound if Theorem 15 and Lemma 16 hold as stated, but those are unproved in the manuscript and the cited sources must be checked. I therefore recommend no change to the reader's verdict, and the concrete test is to verify the cited propositions or independently prove Theorem 15.","tokens_in":23099,"tokens_out":11969,"duration_ms":127576,"concrete_test":"Directly verify Theorem 15 against its cited sources: open [BFP23, Prop. 3.16] and [MMRV02, Lemma 5] and check that the stated hypotheses cover an arbitrary almost entropic polymatroid (T,g) (not just entropic polymatroids, matroids, or PDGs) and that the conclusion provides an almost entropic extension of the whole (T,g), not merely an extension of the embedded matroid. If the statements do not match, attempt an independent proof: for a finite almost entropic (T,g), take approximating entropic h_n, use the entropic three-line construction on each h_n after applying the copy lemma to force the exact incidences on E while perturbing values on T by O(1/n), then pass to a limit of the resulting extensions and check that the limit is an almost entropic polymatroid extending (T,g). If the limit construction fails, try to build a concrete almost entropic (T,g) containing the three-line matroid","verdict_should_be":"UNCHANGED","load_bearing_attack":"","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the membership problem for the closure of the entropic cone is undecidable: there is no algorithm that, given n and an integer vector v in Z^{2^n}, decides whether v lies in \\overline{\\Gamma_n^*}. The strategy is to show that no algorithm can decide whether a finite matroid is almost entropic. The new technical engine is a Desargues-type theorem for almost entropic polymatroids (Theorem 18), built on a three-line intersection theorem (Theorem 15) and the copy lemma (Lemma 16). These tools are used to recover the underlying group from an almost entropic rank-4 partial Dowling geometry (Theorems 34 and 36) and to lift almost entropic rank-3 PDGs to rank 4 (Theorem 37). Combining these with the almost multilinear undecidability results of Kühne and Yashfe [KY22b] yields the main theorem.","tokens_in":22896,"tokens_out":22156,"duration_ms":221761,"significance":"The main result is significant: it settles the last open recognition problem for entropic limit cones and strengthens the undecidability program initiated in [KY22b]. The paper gives a genuinely new synthetic-geometric tool, a Desargues theorem valid in the almost entropic setting, and the reduction from group triviality to almost entropic representability is coherent. The author is explicit that the missing step in [KY22b] was precisely the almost entropic case (Remark 41), and the new Desargues machinery is a credible replacement for the approximate-linear-representation arguments used there. The paper does not provide machine-checked proofs or reproducible code; its strengths are conceptual and structural.","major_comments":[{"comment":"Theorem 15 is the foundation of the new Desargues theorem (Theorem 18) and hence of the group-recovery and lifting arguments. It is stated without proof and attributed to [MMRV02, Lemma 5] and [BFP23, Prop. 3.16]. Since the paper uses the result for arbitrary almost entropic polymatroids, not only for positive multiples of entropic ones, the cited sources must be shown to cover exactly this statement. Please include a full proof, or a precise statement of the cited proposition together with a verification that it implies Theorem 15 as formulated here. This is load-bearing: if the cited proposition does not apply, the undecidability proof collapses.","section":"§3, Theorem 15"},{"comment":"The copy lemma for almost entropic polymatroids is stated but not proved. The references [DFZ06, Mat07a, DFZ11] are for the classical copy lemma, but the present lemma is used for almost entropic finite-type polymatroids, including an independence-over-base condition (property (3)) that is essential in the lifting theorem (Theorem 37). Please provide a proof or a precise citation to a result that establishes this exact almost entropic form. As with Theorem 15, this is a load-bearing point.","section":"§3.1, Lemma 16"},{"comment":"The sentence 'Now by theorem 36 we have an extension...' is incorrect as written: Theorem 36 assumes a PDG that is already closed under geometric products and does not construct an extension. The extension is presumably supplied by Theorem 34. This is likely a typo, but since the main proof depends on passing from a finite rank-4 PDG to a product-closed extension, the reference should be corrected and the use of Theorem 34 made explicit.","section":"§7, proof of Theorem 1"},{"comment":"The proof says 'Since the entire polymatroid has rank 4, this determines it completely...' The ground set E in Theorem 18 is not assumed to have rank 4; only the configuration C has rank 4. The intended meaning is that the six-element subconfiguration {a1,a2,b1,b2,x1,x2} has rank 4, as follows from earlier displayed equalities (e.g. f(a1,a2,b1,x1)=4). Please rephrase to avoid ambiguity, because the argument for identifying the rank function of the six-point matroid depends on this point.","section":"§3.2, proof of Theorem 18"}],"minor_comments":[{"comment":"The statement of Theorem 34 contains the sentence 'The details are routine but slightly longer, and the claim is not used in this paper, so it is omitted.' This is confusing because the proof given already constructs a countable extension by a union of a chain. If the countable case is not needed, the sentence should be removed or reformulated; if it is needed, the proof should be indicated.","section":"§5.1, Theorem 34"},{"comment":"The phrase 'Desargues’-type theorem' in the abstract and introduction mixes apostrophe styles; use a consistent possessive form, e.g. 'Desargues-type' or 'Desargues’s-type'.","section":"§3, opening"},{"comment":"The informal remark 'I was unable to find the finite type hypothesis here elsewhere in the literature' is not appropriate in a formal paper unless the author has made a genuine literature search; consider moving this to a footnote and citing the closest standard notion (finitary matroids of finite rank).","section":"§2.1, Remark 5"},{"comment":"The verification of condition (6) for the four remaining index triples is summarized by the four diagrams and a short paragraph. The diagrams are helpful, but the dependencies for the assumption checks are delicate; adding a table listing, for each of the four triples, the previously established relator used would make the proof easier to verify.","section":"§6, Theorem 37"},{"comment":"The proof of associativity chooses an element p with [p^{-1}]=[s]·[z]; this is justified by closure under geometric products, but the notation [s]·[z] is defined only after Theorem 32. The order of the argument is clear, but a one-sentence reminder of the two-step definition of the product on parallelism classes would improve readability.","section":"§5.2, Theorem 36"}],"recommendation":"major_revision","confidential_remarks":"The central reduction is coherent and the new Desargues theorem is a substantial contribution, but the two main technical inputs — Theorem 15 and Lemma 16 — are cited rather than proved or precisely located in the cited references. Given that the whole undecidability claim rests on these statements, I recommend that the editor ask the authors to provide either full proofs or precise quotations of the relevant propositions from [BFP23] and [DFZ06]/[Mat07a], before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: Yashfe proves that membership in the closed entropic cone Γ*_n is undecidable for integer vectors, closing the last open case in a sequence of undecidability results. The result is real and the proof is in good shape, but it rests on a couple of external results that a referee will need to check.\n\nWhat's new: Theorem 1 is genuinely new. Prior work (Li23, KY22b) handled exact conditional independence implication and almost multilinear representability; the almost entropic case was open. The paper's contribution is a Desargues-type theorem for almost entropic polymatroids (Thm 18), built from a three-line intersection theorem (Thm 15) and the copy lemma (Lemma 16). That theorem is a real new tool, and the reduction from the word problem is structurally sound. I spot-checked the rank computations in Thm 18, group recovery (Thm 36), and lifting (Thm 37); no internal error. The author is also honest: Remark 41 says it's an extension of [KY22b, Thm. 9.12] and the rest is identical.\n\nSoft spots: The foundation of the new geometry is Thm 15, cited to [MMRV02] and [BFP23] rather than proved. The copy lemma is stated without proof. These are load-bearing: if Thm 15 fails for arbitrary almost entropic polymatroids, the group recovery collapses. I would want a referee to verify the citations actually cover the exact statement. Also, the proof of Thm 1 says 'by theorem 36 we have an extension' but it needs Theorem 34 (product-closed extension); Theorem 36 assumes it. That's a minor citation slip, not a mathematical gap. The heavy self-citation to unpublished [KY22b] is a practical barrier for verification, but not circular—the central claim was explicitly out of reach there.\n\nWho it's for: anyone working on entropy cones, network coding, or matroid representability. It deserves a serious referee. I'd send it to peer review.\n\nRecommendation: accept for review; the referee should double-check Thm 15 and Lemma 16.","headline":"Yashfe closes the last open case in entropy-cone undecidability with a sound-looking Desargues-type theorem, but the proof leans on two external results a referee must verify.","tokens_in":23558,"tokens_out":2445,"would_cite":true,"duration_ms":26269,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B35","03D35","94A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that no algorithm can decide whether a given integer vector is a pointwise limit of joint entropy functions: membership in the closure of the entropic cone is undecidable.","keywords":["undecidability","entropic cone","entropy functions","almost entropic polymatroids","matroid recognition","Desargues theorem","partial Dowling geometries","word problem"],"falsifier":"The most direct check is on Theorem 15: take the six-point rank-4 configuration (three pairs of points, each pair spanning a 'line', with the three lines pairwise coplanar), realize it by actual random variables on a finite probability space, and verify that a seventh point lying on all three lines can always be adjoined in an almost entropic extension. A single almost entropic embedding of the six-point matroid with no such extension would refute the base theorem and collapse the argument. On the group side, the equivalent test: for a presentation in which a generator x is explicitly trivial,","tokens_in":1957,"feed_emoji":"🚫","tokens_out":1858,"duration_ms":387169,"temperature":0.7,"pith_summary":"The paper proves that the recognition problem for the closed entropic cone is undecidable: there is no algorithm that, given a number n and an integer table of proposed entropies, decides whether that table is a pointwise limit of the joint entropy functions of n random variables on a finite probability space. In other words, no computer can certify that a proposed vector can be approximated arbitrarily well by genuine entropy vectors. The proof establishes the same undecidability for the more restricted question of whether a finite matroid is almost entropic, i.e. is itself a limit of entropy functions. The mechanism is group-theoretic: certain matroids built from a group presentation encode the presentation's relations in their rank function, and the paper shows that an encoded generator is nontrivial in the group exactly when one of the constructed matroids is almost entropic. Since deciding nontriviality of a generator is undecidable, so is the matroid question, which closes the last open case in a sequence of undecidability results for entropy-like cones and, via a known reduction, makes approximate conditional independence implication undecidable as well.","feed_headline":"No algorithm can tell if a vector is a limit of entropies","feed_subtitle":"Deciding whether entropy values can be approximated is undecidable — matroid geometry hides group word problems.","key_machinery":"Partial Dowling geometries (PDGs): matroids built from a group presentation whose rank function records the relations — a generator triple is a relator exactly when its three points lie on a rank-2 line. The load-bearing mechanism is a Desargues'-type theorem for almost entropic polymatroids (Theorem 18): it guarantees that the intersection points demanded by a projective configuration exist in some almost entropic extension. The theorem rests on two imported tools: the three-line intersection theorem (Theorem 15) and the copy lemma (Lemma 16). From these configurations a geometric product of generators is defined, unique up to parallelism and associative (Theorems 32, 36), recovering a grou","core_discovery":"Membership in the closed entropic cone is undecidable: no procedure decides whether an integer vector is a limit of entropy functions; equivalently, whether a finite matroid is almost entropic. The proof reduces the group word problem to matroids: Dowling-type geometries built from a presentation yield a computable family F of rank-3 matroids in which x is nontrivial iff some member is almost entropic. One direction was known for almost multilinear matroids; the converse uses a Desargues'-type theorem for almost entropic polymatroids to define a geometric product on generators, prove it associative, and recover a quotient of the presented group from a rank-4 geometry.","pith_inferences":["A consequence the paper leaves implicit: the undecidable instances are matroids of bounded rank (3 and 4) on growing ground sets, so the hardness is carried by the incidence structure rather than by high rank; restricting attention to small-rank entropy regions would not bypass the problem.","Connection to a neighbouring problem: the paper notes that algebraic matroids form a proper subclass of almost entropic ones in which even the simpler two-line intersection theorem fails; whether the three-line/Desargues incidence still holds inside algebraic matroids is a natural open probe, and if it does, algebraic-matroid recognition would inherit undecidability.","The construction never bounds the size of the almost entropic extension that manufactures each geometric product, and the recovered group is typically infinite; read quantitatively, the minimal size of an approximating entropic structure would give each group presentation a concrete hardness measure — a direction the paper leaves untouched."],"forward_implications":["No algorithm decides membership in the closed entropic cone: given n and an integer vector in Z^{2^n}, the question 'is this vector a limit of joint entropy functions?' is undecidable (Theorem 1).","No algorithm decides whether a finite matroid is almost entropic; this is the statement the proof actually establishes, and it is equivalent to Theorem 1.","It is undecidable whether a given integer-valued set function h on P({1,...,n}) is epsilon-approximable by joint entropies of n random variables for every epsilon > 0 — that is, whether h is a pointwise limit of entropy functions.","Approximate conditional independence implication is undecidable: via a reduction the paper cites from earlier work, Theorem 1 makes the approximate version of the conditional-independence implication problem unsolvable.","Any class of representations that admits both a copy lemma and a three-line intersection theorem inherits the undecidability, since these are the only properties of the almost entropic setting the proof uses."],"supporting_citations":[{"why":"Supplies the undecidability results this paper extends: the computable family of PDGs, the nontrivial-to-representability implications, and the reduction to conditional independence.","marker":"[KY22b]"},{"why":"Cited as the source of the three-line intersection theorem for almost entropic polymatroids, the base of the new Desargues theorem.","marker":"[MMRV02, Lemma 5]"},{"why":"Provides a published proof of the almost entropic extension property used for Theorem 15.","marker":"[BFP23, Prop. 3.16]"},{"why":"Classical copy lemma, adapted as Lemma 16 to lift rank-3 configurations into higher rank.","marker":"[DFZ06, Mat07a, DFZ11]"},{"why":"Introduced partial Dowling geometries and their construction from group presentations, the encoding objects of the reduction.","marker":"[KY22a]"}],"fun_headline_variants":["No algorithm decides if a vector is an entropy limit","Entropic cone membership is undecidable","Undecidable: recognizing limits of entropy functions","Desargues' theorem exposes undecidable entropy limits","Matroid geometry proves entropy limits undecidable"],"cache_read_input_tokens":25472,"weakest_assumption_plain":"Everything rests on the three-line intersection theorem for almost entropic polymatroids (Theorem 15, Section 3), which the paper states without a full proof and attributes to earlier work: if that theorem does not actually hold for every polymatroid that is a limit of entropy functions, the new Desargues theorem, the group recovery, and the undecidability result all collapse; the copy lemma (Lemma 16, Section 3.1) is a second unproved input.","fun_headline_variants_meta":{"raw":{"variants":["No algorithm decides if a vector is an entropy limit","Entropic cone membership is undecidable","Undecidable: recognizing limits of entropy functions","Desargues' theorem exposes undecidable entropy limits","Matroid geometry proves entropy limits undecidable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1264,"prompt_tokens":722,"completion_tokens":542,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":469}},"tokens_in":466,"tokens_out":542,"duration_ms":7010,"temperature":1.0,"reasoning_tokens":469,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T23:54:57.755269+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The most direct check is on Theorem 15: take the six-point rank-4 configuration (three pairs of points, each pair spanning a 'line', with the three lines pairwise coplanar), realize it by actual random variables on a finite probability space, and verify that a seventh point lying on all three lines can always be adjoined in an almost entropic extension. A single almost entropic embedding of the six-point matroid with no such extension would refute the base theorem and collapse the argument. On the group side, the equivalent test: for a presentation in which a generator x is explicitly trivial,","supporting_citations":[],"review_version":1}