{"id":"1aa7a008-348a-450e-b94a-451135de836e","arxiv_id":"2509.08144","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove that pointed matroids over perfect idylls form a combinatorial proto-exact category with duality and exact direct sum, and that tropical toric reflexive sheaves (modular ones in particular) are proto-exact and proto-abelian, with Khan-Maclagan Harder-Narasimhan filtrations realized","lead":"This paper shows that matroids over perfect idylls form a proto-exact category and that tropical toric reflexive sheaves form a proto-exact, proto-abelian category. It then recasts the Harder-Narasimhan filtrations of Khan and Maclagan as categorical slope filtrations, a framework for studying stability in these combinatorial-geometric objects.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved pointed adaptation of Prop 3.3 underlies Lemma 3.16 and propagates to simple matroids and TRS; Section 4 asserts the simple version without proof.","rationale":"The paper's central claims are that F-Mat\\bullet is a combinatorial proto-exact category with duality and exact direct sum (Theorem A), and that the Harder-Narasimhan filtrations of [KM24b] coincide with categorical slope filtrations (Corollary 6.25). The proof of Theorem A rests on Proposition 3.3, the characterization of F-matroid morphisms by submonomial matrices, which enters in Lemma 3.16 (the fundamental biCartesian square) and hence in Propositions 3.17–3.19. The paper explicitly states in Remark 3.4 that the pointed adaptation of Proposition 3.3 is 'immediate and has been omitted'. This is a genuine gap: the pointed setting adds a distinguished loop, and the characterization (2) quantifies only over non-loop tuples, while morphisms may send non-loops to the basepoint. The proof of Lemma 3.16 relies on the equality β(B)=∗_R to cancel a summand; this depends on the pointed morphism condition in a way that is not visible in the non-pointed theorem. The same gap propagates to F-SMat\\bullet, where Section 4 asserts without proof that Lemma 3.16 holds with flats, and this assertion is used in Lemma 6.12 to prove proto-exactness of TRS\\Sigma_\\bullet (Theorem C). I also note a smaller issue: Section 6.1 defines µ(E)=rk(E)/deg(E), which is inverted relative to the slope used in Subsection 2.8 and in [KM24b]; as written, Proposition 6.24 and Corollary 6.25 do not follow, though this is likely a typo. Neither issue appears fatal if the omitted proofs can be supplied, but they are load-bearing, so a conditional verdict is appropriate. I agree with the reader's identification of the pointed Prop 3.3 as the weakest assumption, and I add that the simple-matroid and TRS layers depend on the same unproved adaptation.","tokens_in":36199,"tokens_out":20347,"duration_ms":205620,"concrete_test":"Independently prove the pointed version of Proposition 3.3: for pointed F-matroids M,N, show that a submonomial matrix f satisfies f·V_M ⊆ V_N iff condition (2) holds for all x∈\\tilde{E}_M^{m+1}, y∈\\tilde{E}_N^{n-1}. Specifically test the case where f maps a non-loop element to ∗_N, so f_{f(x_k),x_k}=0_F in the sum. If the equivalence fails, re-run the proof of Lemma 3.16 with that counterexample; if coCartesianity fails, Theorem 3.11 collapses. Separately, verify the Section 4 assertion by checking that the biCartesian completion of a diagram in F-SMat\\bullet with A,B flats has all objects simple; a simple-matroid counterexample (e.g., N=U_{2,4}, A a hyperplane, B a singleton) would falsify Theorem B.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proto-exactness of F-Mat\\bullet (Theorem 3.11) hinges on Lemma 3.16, whose coCartesian proof uses Proposition 3.3 to verify that the constructed matrix γ is a morphism. Proposition 3.3 is cited from [JLV24] in the non-pointed setting, and Remark 3.4 dismisses the pointed adaptation as 'immediate and has been omitted'. This is load-bearing because the pointed case introduces a distinguished loop: the characterization (2) quantifies only over tuples in \\tilde{E}_M and \\tilde{E}_N, but morphisms may send non-loop elements to the basepoint, and the proof of Lemma 3.16 uses β(B)=∗_R to kill a summand via β_{β(b_j),b_j}=0_F. That step relies on the pointed orthogonality condition and is not obviously contained in the non-pointed statement. Moreover, Section 4 (after Remark 4.4) asserts without proof that Lemma 3.16 'holds' in F-SMat\\bullet with A,B flats, and this simple-matroid version is then used in Lemma 6.12 to prove proto-exactness of TRS\\Sigma_\\bullet. Thus the unproved pointed/simple adaptation of Prop 3.3 is load-bearing for Theorems A, B, and C.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines the category F-Mat• of pointed matroids over a perfect idyll F, with morphisms given by submonomial matrices, and proves that it is a combinatorial proto-exact category with duality and exact direct sum (Theorem A, Theorems 3.11 and 3.22). It further shows that F-Mat• is proto-abelian (Corollary 3.12) and that the full subcategory F-SMat• of simple F-matroids is proto-abelian (Theorem B, Corollary 4.5). It then defines a category TRSΣ• of tropical toric reflexive sheaves, proves it is proto-exact and proto-abelian, and obtains the same for the full subcategory MTRSΣ• of modular sheaves (Theorem C, Theorem 6.16, Corollaries 6.17 and 6.18). Finally, it shows that the Harder–Narasimhan filtrations of Khan–Maclagan for modular tropical toric reflexive sheaves coincide with the categorical slope filtration of Li (Theorem D, Corollary 6.25). The paper also constructs Hall algebras and computes K0(F-Mat•) ≅ Z⊕Z.","tokens_in":36518,"tokens_out":7908,"duration_ms":83544,"significance":"If the main results hold, the paper provides a useful categorical framework for matroids over idylls and for tropical toric reflexive sheaves, unifying and extending earlier work of Eppolito–Jun–Szczesny and connecting it with the Harder–Narasimhan theory of Khan–Maclagan. The explicit use of submonomial matrices to describe morphisms is a strength, as is the careful formulation of admissible classes in terms of restrictions and contractions. The paper is honest about its deferred proofs: several load-bearing statements are explicitly left to the reader or cited from preprints. These gaps are likely fillable, but they currently prevent the paper from being fully self-contained at key technical points.","major_comments":[{"comment":"The pointed version of Proposition 3.3 is asserted to be immediate and is omitted (Remark 3.4). This adaptation is load-bearing: Lemma 3.16, which is the core of Theorem 3.11, uses Proposition 3.3 to verify that the constructed morphism γ is an F-matroid morphism, and its proof uses the pointed relation β(B)=∗_R to deduce that certain matrix coefficients vanish. The non-pointed statement of Proposition 3.3 quantifies only over tuples in \\tilde E_M and \\tilde E_N and does not directly address the case where a morphism sends a nonzero element to the basepoint. Please provide a complete statement and proof of the pointed adaptation, or a detailed explanation of why the non-pointed proof carries over verbatim.","section":"Remark 3.4; Lemma 3.16"},{"comment":"It is asserted without proof that Lemma 3.16 holds in F-SMat• with A and B flats, and that this implies Propositions 3.17 and 3.19 in the simple setting. This assertion is used to prove Corollary 4.5 (Theorem B) and is later invoked in Lemma 6.12 to establish proto-exactness of TRSΣ• (Theorem C). The sentence 'Lemma 3.16 is true on the nose...' is not a proof. Please give a proof of the flat version of Lemma 3.16 in F-SMat• or a precise reduction to the pointed version, and explain why the simple conditions preserve the biCartesian property.","section":"Section 4, after Remark 4.4"},{"comment":"The existence of cokernels is essential for the proto-abelian conclusions (Corollary 3.12 and Corollary 6.17). For Proposition 3.10, the proof that N/C is a cokernel is omitted ('completely analogous and so has been omitted'), and for Proposition 6.7 the cokernel part is similarly omitted. Since these are load-bearing for the main claims, the full cokernel proofs should be supplied, or at least the dual argument should be written out explicitly.","section":"Proposition 3.10; Proposition 6.7"},{"comment":"The strong slope inequality on MTRSΣ• is reduced to [KM24b, Lemma 7.24] in a single sentence. Since Theorem D (Corollary 6.25) depends on this, the reduction should be made fully explicit: state the relevant lemma, verify that its hypotheses (modularity of flats) are satisfied by the subobjects (E|F)/(F∧G) and (E|F∨G)/F, and show how the inequality proved there translates exactly to inequality (12) in the categorical framework. As written, the reader cannot check whether the cited lemma covers precisely this situation. Also, the slope function is defined on p. 37 as µ(E)=rk(E)/deg(E), which contradicts the definition µ=deg/rk used elsewhere; this should be corrected.","section":"Section 6.1, Proposition 6.24"}],"minor_comments":[{"comment":"The displayed definition 'µ(E)=rk(E)/deg(E)' should read 'µ(E)=deg(E)/rk(E)' to match the rest of the paper and the standard convention.","section":"Section 6.1, p. 37"},{"comment":"In the final displayed equation, 'null(M)(M)[L]' appears to contain a typo; it should probably be 'null(M)[L]'.","section":"Proposition 5.4 proof"},{"comment":"The proof that PE4 and PE5 imply axiom (PA2) is a single sentence. A short explanation of how the biCartesian square axioms yield the required pullback/pushout properties would improve readability.","section":"Proposition 2.21"},{"comment":"The equality ∑ρ a_ρ = 0 for a principal divisor should reference explicitly the standard fact that the sum of coefficients of a principal divisor on a complete toric variety is zero; this is used critically in the degree computation.","section":"Lemma 6.23"},{"comment":"The argument using the 'algebra anti-automorphism' to identify the Hall algebra product with the dual of the matroid-minor Hopf algebra product is terse. A short explicit verification of the anti-automorphism property would be helpful.","section":"Section 5.1, Corollary 5.3"},{"comment":"The convention for restriction and contraction on pointed matroids is stated informally. It would be clearer to define the pointed restriction and contraction explicitly in terms of the distinguished loop.","section":"Section 2.2.3, Remark 2.5"}],"recommendation":"major_revision","confidential_remarks":"The paper's main results are plausible and the overall strategy is sound, but several load-bearing proofs are deferred, as the authors themselves note (Remark 3.4, Proposition 3.10, Proposition 6.7, and the assertion in Section 4). The most serious concern is the unproved pointed adaptation of Proposition 3.3, which underpins Lemma 3.16 and propagates to the simple matroid and tropical toric reflexive sheaf settings. These gaps appear fillable, so I recommend major revision rather than rejection. The dependence on the preprints [KM24b] and [Li23] should be made more precise, especially the translation of [KM24b, Lemma 7.24] into the categorical strong slope inequality. I would also encourage the authors to check the latest versions of these preprints for any changes to lemma statements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague — the short version: this is a real contribution that deserves referee time, but it should not pass as-is because a load-bearing proof is left out.\n\nWhat's genuinely new: the category TRS^Σ_• of tropical toric reflexive sheaves, with a morphism notion defined for the first time, and the proof that it and its modular subcategory are proto-exact and proto-abelian. The recasting of Khan–Maclagan's Harder–Narasimhan filtrations as Li's categorical slope filtration is a clean payoff, and the proto-exactness of F-Mat• over every perfect idyll genuinely extends EJS20b. The paper is clearly written and honest: it flags the places where proofs are omitted.\n\nThe soft spots are real. Remark 3.4 says the pointed version of Prop 3.3 is 'immediate and has been omitted.' That proposition is used in Lemma 3.16 to show the constructed γ is a morphism, and Lemma 3.16 is the engine for proto-exactness. The pointed case isn't literally the same: morphisms can send non-loops to the basepoint, and the non-pointed orthogonality condition doesn't obviously cover that. The stress-test is right that this is load-bearing. Section 4 then asserts the simple-matroid version of Lemma 3.16 without proof, and Section 6 uses it for TRS. So Theorems A, B, and C all depend on an unproved adaptation. Cokernel proofs in Props 3.10 and 6.7 are also omitted, though those look routine. Section 6.1 defines the slope as rk/deg, which is backwards relative to the deg/rk definition in Section 2.8—surely a typo, but it should be fixed. The stability part imports the strong slope inequality from [KM24b, Lemma 7.24], so Theorem D is only as solid as that lemma.\n\nNone of this makes me think the main results are false. The strategy is sound and the omissions are flagged, not hidden. A referee should ask for the pointed Prop 3.3 to be written out, the simple version to be proven, and the cokernel proofs supplied. The paper is for people in matroid categories, proto-exact structures, or tropical toric geometry. I'd take it seriously, but not without revision.","headline":"Solid categorical treatment of matroids over idylls and tropical toric reflexive sheaves, but a load-bearing pointed version of Prop 3.3 is omitted and Section 6 defines the slope backwards.","tokens_in":37028,"tokens_out":3925,"would_cite":true,"duration_ms":40656,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18E10","05B35","14T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Matroids over any perfect idyll form a proto-exact category, and the same structure organizes tropical toric reflexive sheaves.","keywords":["proto-exact category","matroids over idylls","submonomial matrices","proto-abelian category","tropical toric reflexive sheaves","Harder–Narasimhan filtration","slope stability","Hall algebras"],"falsifier":"Test the omitted pointed case of Proposition 3.3 directly: find a perfect idyll F and a pointed F-matroid M together with a submonomial matrix f that maps vectors of M into vectors of another pointed F-matroid N but fails the explicit Grassmann–Plücker summation criterion (or conversely). Since Lemma 3.16 and Proposition 3.17 invoke Proposition 3.3 as their verification step, one such counterexample would collapse the proof that restriction–contraction squares are biCartesian, and with it Theorem 3.11.","tokens_in":36093,"feed_emoji":"🧮","tokens_out":8979,"duration_ms":91063,"temperature":0.7,"pith_summary":"This paper proves that the pointed matroids over any perfect idyll — one algebraic structure covering classical matroids, valuated matroids, oriented matroids, and linear subspaces over fields — form a proto-exact category, a non-additive analogue of an exact category. The admissible monomorphisms are matroid restrictions up to isomorphism, the admissible epimorphisms are matroid contractions up to isomorphism, and the whole structure rests on a matrix description of matroid morphisms by submonomial matrices. The paper then lifts this structure to the category of tropical toric reflexive sheaves, showing it too is proto-exact and proto-abelian. In the modular case, the Harder–Narasimhan filtration previously constructed by Khan and Maclagan is exactly the categorical slope filtration produced by Li's general theory. If correct, this means stability and filtration phenomena in tropical geometry are organized by the same proto-exact axioms that govern matroids.","feed_headline":"Proto-exact structure unifies matroids and tropical sheaves","feed_subtitle":"F-matroids satisfy the proto-exact axioms; tropical sheaves inherit them, and their HN filtrations become categorical.","key_machinery":"The load-bearing object is the submonomial matrix: a morphism f:M→N of F-matroids is exactly a matrix with at most one nonzero entry per row and column whose action sends vectors of M into vectors of N. From this description, restriction maps rA:M|A→M and contraction maps cA:M→M/A are morphisms, and the paper declares admissible monomorphisms to be restrictions up to isomorphism and admissible epimorphisms to be contractions up to isomorphism. The proto-exact axioms then reduce to a single geometric fact about these matrices: any square N|A→N→N/B with B⊆A is biCartesian, proved by deleting rows/columns and applying duality. For tropical toric reflexive sheaves the same squares are shown biCa","core_discovery":"The category F-Mat• of pointed F-matroids over a perfect idyll F, with morphisms given by submonomial matrices that send vectors to vectors, is a combinatorial proto-exact category with duality and exact direct sum: admissible monomorphisms are matroid restrictions up to isomorphism and admissible epimorphisms are matroid contractions. It is therefore proto-abelian in the sense of André. The same structure transfers to pointed tropical toric reflexive sheaves: TRSΣ• is proto-exact and proto-abelian, and within the modular subcategory MTRSΣ• the Khan–Maclagan slope function satisfies Li's strong slope inequality, making the Harder–Narasimhan filtration precisely the categorical slope filtrati","pith_inferences":["If the omitted pointed case of Proposition 3.3 holds, the same submonomial-matrix argument should give proto-exact structures for matroids over any band whose morphisms are tracked by such matrices, not only perfect idylls.","The paper's observation that non-modular sheaves can fail the desired slope inequality suggests a modified categorical slope (valued elsewhere than R) could restore unique HN filtrations for all TRSΣ•.","Knowing K0 is Z⊕Z for all idylls focuses attention on K1 and K2 as the true coefficient detectors; computing K1(Mat•) would show whether the sphere-stable inclusion from pointed sets is proper.","Because proto-exact categories carry 2-Segal structures, Theorem A positions matroids over idylls inside Hall-algebra and K-theory machinery, connecting directly to the moduli space of matroids the paper aims to clarify."],"forward_implications":["Every perfect idyll F produces a finitary proto-exact category F-Mat• whose Hall algebra is the graded Hopf dual of the F-matroid-minor Hopf algebra when F is finite.","Since K0(F-Mat•) ≅ Z⊕Z for every perfect idyll, any coefficient-sensitive information must live in higher K-theory — a concrete place to look next.","The tropical toric reflexive sheaf category TRSΣ• and its modular subcategory inherit kernels, cokernels, and well-behaved sums and intersections of strict subobjects at flats — a package unavailable before.","The modular Harder–Narasimhan filtration of Khan–Maclagan is unique and agrees with the categorical slope filtration; the obstruction to uniqueness for non-modular sheaves is precisely the failure of the strong slope inequality."],"supporting_citations":[{"why":"Defines submonomial matrices and proves Theorem 2.8, the test that characterizes F-matroid morphisms; the paper's whole category F-Mat• is built on this.","marker":"[JLV24]"},{"why":"Introduces matroids over idylls, Grassmann–Plücker functions, duality, and restriction/contraction; supplies the objects and minor operations Theorem A governs.","marker":"[BB19]"},{"why":"Supplies the definition of proto-exact categories and the biCartesian-square axioms that Theorem 3.11 verifies for F-Mat•.","marker":"[DK19]"},{"why":"Establishes the pointed classical matroid case that this paper generalizes; its Lemma 4.9 is the template for the biCartesian restriction–contraction squares in Lemma 3.16.","marker":"[EJS20b]"},{"why":"Introduces tropical toric reflexive sheaves, degree/slope, modular flats, and the Harder–Narasimhan filtration; Section 6 re-derives this filtration categorically.","marker":"[KM24b]"},{"why":"Defines proto-abelian categories and slope filtrations; the paper uses Proposition 2.21 to promote proto-exactness to proto-abelianness.","marker":"[And09]"},{"why":"Proves existence and uniqueness of categorical slope filtrations from the strong slope inequality (Theorem 2.24), the target notion of Corollary 6.25.","marker":"[Li23]"},{"why":"Develops vectors of matroids over tracts and records that contraction maps are matroid morphisms; this supplies the contraction morphisms used in Definition 3.1 and Proposition 3.9.","marker":"[And19]"}],"fun_headline_variants":["Proto-exact categories unite F-matroids and tropical sheaves","F-matroids and tropical sheaves share proto-exact structure","From matroid restrictions to sheaf contractions: proto-exact","Proto-abelian theory links matroids and tropical sheaves"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The paper invokes Proposition 3.3 — the equivalence between submonomial matrices defining matroid morphisms and a Grassmann–Plücker summation test — in the pointed setting, saying the adaptation is 'immediate and has been omitted' (Remark 3.4); if that pointed adaptation fails, the biCartesian-square proofs establishing proto-exactness of F-Mat• break, and with them the tropical sheaf applications.","fun_headline_variants_meta":{"raw":{"variants":["Proto-exact categories unite F-matroids and tropical sheaves","F-matroids and tropical sheaves share proto-exact structure","From matroid restrictions to sheaf contractions: proto-exact","Proto-abelian theory links matroids and tropical sheaves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1332,"prompt_tokens":703,"completion_tokens":629,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":556}},"tokens_in":447,"tokens_out":629,"duration_ms":7680,"temperature":1.0,"reasoning_tokens":556,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T21:10:23.786497+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the omitted pointed case of Proposition 3.3 directly: find a perfect idyll F and a pointed F-matroid M together with a submonomial matrix f that maps vectors of M into vectors of another pointed F-matroid N but fails the explicit Grassmann–Plücker summation criterion (or conversely). Since Lemma 3.16 and Proposition 3.17 invoke Proposition 3.3 as their verification step, one such counterexample would collapse the proof that restriction–contraction squares are biCartesian, and with it Theorem 3.11.","supporting_citations":[],"review_version":1}