{"id":"3bd7372e-3817-4fc0-b5e4-b841f0a6651e","arxiv_id":"2606.22650","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A combinatorially defined tangent class T_{M,G} realizes the matroidal HRR package for every building set and produces Chern inequalities including a Miyaoka–Yau form.","lead":"The paper constructs a tangent class in the K-ring of any loopless matroid with a building set, extending the geometric tangent bundle of de Concini–Procesi wonderful models to non-realizable matroids. This class realizes a formal Hirzebruch–Riemann–Roch package and yields Chern-number inequalities, including a Miyaoka–Yau-type bound against the hyperplane class.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper's central claim (Theorem 4.4) is that the intrinsically defined tangent class realizes the formal HRR package for every building set. The only structural caveat the reader flags is that formal Hy is defined along a chain and independence is recovered from the realization rather than proved first. That is accurate as a description of the exposition, but it does not undermine the theorem: the induction that proves Hy=Φy(T) simultaneously shows every chain produces the same class, because the right-hand side is chain-independent by construction. The two supporting identities (relative Aluffi and star-normal) are established by direct K-theoretic calculation before the induction begins, so there is no hidden circularity. The subsequent numerical consequences (canonical class, Serre duality, Chern inequalities, Miyaoka–Yau form) rest only on the already-proved realization and on truncation lemmas that are standard. Consequently the reader's ACCEPT / high-confidence verdict stands; no adjustment is warranted.","tokens_in":22204,"tokens_out":622,"duration_ms":5669,"concrete_test":"Independently recompute Φy(TM,G) for the braid matroid K5 with the minimal building set (Example 3.4) by two different one-step chains from Gmin to Gmax; verify that both chains produce identical degree-d components of ch(λ y T∨) td(T) and that the resulting Chow polynomial matches the known Hilbert series of A(K5,Gmin). If they agree, the chain-independence claim is confirmed computationally.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption note (chain-independence of formal Hy recovered only after realization) is real but non-load-bearing for Theorem A. Hy is defined recursively along a chosen chain (4.4); the paper states a priori dependence is possible and that Theorem 4.4 implies independence. The proof of 4.4 proceeds by descending induction along that same chain, matching the relative Aluffi identity (Prop. 3.9) and the star-normal identity (Prop. 4.10) to the universal one-step Hirzebruch formula (Lemma 4.5). Because the right-hand side Φy(TM,G) is defined intrinsically (Def. 3.3) and is therefore chain-independent, the induction simultaneously proves both the equality and that every chain yields the same formal class. The circularity is only apparent; the argument is self-consistent once the two combinatorial identities are granted. No other soft spot in the central claim is visible: the identities are proved by direct expansion in the Feichtner–Yuzvinsky rings using only Stanley–Reisner relations and the definitions of cutting classes, and the Chern inequalities follow by truncation and induction without further assumptions.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs, for every loopless matroid M and every building set G containing the top flat E, a tangent class T_{M,G} in the Feichtner–Yuzvinsky K-ring K(M,G). The definition (3.5) is combinatorial and is motivated by Aluffi’s complete-intersection blow-up formula; when M is realizable it recovers the tangent bundle of the de Concini–Procesi wonderful model. The main result (Theorem 4.4) asserts that this class realizes the formal Hirzebruch–Riemann–Roch package previously defined by push-forward from the maximal building set: the Hirzebruch class of T_{M,G} equals the recursively defined H_y(M,G), its Todd class equals Td_{M,G}, and the degree of the Hirzebruch class at y=−t recovers the Chow polynomial of (M,G). The proof proceeds by verifying a relative Aluffi identity for one-step enlargements (Proposition 3.9) and a star-normal factorization on the blow-up center (Proposition 4.10), then matching both to a universal one-step formula for Hirzebruch classes (Lemma 4.5). Numerical consequences include Chern-number inequalities against the hyperplane class α and a Miyaoka–Yau-type inequality.","tokens_in":22426,"tokens_out":829,"duration_ms":6920,"significance":"The work cleanly extends the author’s earlier tangent-class construction from the maximal building set to arbitrary building sets, thereby supplying a single geometric object that simultaneously produces the Todd class, the Hirzebruch class, and the Chow polynomial for every (M,G). The argument is entirely formal in the Feichtner–Yuzvinsky rings and therefore applies to non-representable matroids; the two combinatorial identities (relative Aluffi and star-normal) are proved by direct expansion using only Stanley–Reisner relations and the definitions of cutting classes. The resulting Chern inequalities and the Miyaoka–Yau comparison with projective space give concrete numerical content. The construction is intrinsic, parameter-free, and recovers the classical geometry whenever a realization exists.","major_comments":[],"minor_comments":[{"comment":"Section 4.3, after (4.4): the text notes that the recursive definition of H_y may a priori depend on the chosen chain from G to G_max, and that independence follows from Theorem 4.4. A short forward pointer or a one-sentence remark that the induction simultaneously proves independence would make the logical order clearer for the reader.","section":null},{"comment":"Definition 3.3 and equation (3.6): the total Chern class is written as a product involving rational functions of divisor classes. It would help to record explicitly that these expressions are well-defined in the Chow ring (i.e., that the denominators invert after the Stanley–Reisner relations are imposed).","section":null},{"comment":"Lemma 4.16: the self-intersection formula ∫ x_F^s α^{d−s} = (−1)^{s−1} is used repeatedly; a brief cross-reference to the corresponding geometric statement for realizable matroids (or a pointer to the toric orbit-closure computation) would improve readability.","section":null},{"comment":"References: the arXiv identifiers of the author’s related preprints [Che26] and [CL26] appear with future dates; once those papers are published or stabilized, the citations should be updated.","section":null}],"recommendation":"accept","confidential_remarks":"The paper is a natural and technically solid sequel to the author’s earlier work on the maximal building set. The central identities are carefully checked and the logical structure is self-consistent. I see no reason to delay acceptance; the minor remarks are purely presentational."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper does exactly what the title claims: it builds an intrinsic tangent class T_{M,G} in K(M,G) for every building set, not just the maximal one, and shows it realizes the formal Todd and Hirzebruch classes that recover the Chow polynomial. The construction is the Aluffi complete-intersection correction written combinatorially (Def. 3.3), and the two load-bearing identities—relative Aluffi under one-step enlargement (Prop. 3.9) and star-normal factorization on the center (Prop. 4.10)—are proved by direct expansion in the Feichtner–Yuzvinsky rings using only Stanley–Reisner and the cutting-class definitions. Once those hold, the universal blow-up formula for Φ_y matches the recursive definition of H_y, so Theorem A is formal and does not need realizability.\n\nWhat is new is the general-G class itself, the two identities, and the resulting Chern-number inequalities against α (Thm 4.20), including the Miyaoka–Yau form for k=2. The maximal case was already in the author’s earlier note; this is the natural and non-trivial extension. The formal package is defined by push-forward from G_max, so there is a mild dependence on prior work, but the new content is self-contained once that is granted. The chain-independence caveat the reader flagged is real but non-load-bearing: H_y is defined along a chosen chain, yet the induction simultaneously shows that Φ_y(T_{M,G}) (which is intrinsic) equals every such recursive class, so independence follows rather than being assumed.\n\nSoft spots are minor. The inequalities are combinatorial lower bounds that recover the projective-space numbers; they are clean but not deep. Citation pattern is appropriate and the math is careful. This is for people already working on matroid Chow rings, wonderful models, or tautological classes; they will use the formulas. It deserves a serious referee and I would cite the inequalities and the general tangent class. Send it out.","headline":"Solid extension of the maximal-building-set tangent class to arbitrary G, with a clean formal realization of the HRR package and usable Chern inequalities.","tokens_in":23070,"tokens_out":526,"would_cite":true,"duration_ms":5869,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C17","14M25","05B35","19E20"],"pacs":[],"model":"grok-4.5","headline":"A single tangent class for any matroid building set realizes the formal Hirzebruch–Riemann–Roch package and recovers the Chow polynomial as a Hirzebruch genus.","keywords":["matroid","building set","tangent class","Hirzebruch–Riemann–Roch","Chow polynomial","wonderful model","Chern numbers","Miyaoka–Yau"],"falsifier":"Compute both sides of the claimed equality Hy(M,G)=Φ_y(T_{M,G}) for a small non-realizable matroid (for example the Fano matroid with a non-maximal building set) by expanding the Feichtner–Yuzvinsky presentations and check whether the coefficients of the two polynomials in y agree.","tokens_in":23053,"feed_emoji":"📐","tokens_out":683,"duration_ms":6468,"temperature":0.7,"pith_summary":"The paper constructs a tangent class in the K-ring of any loopless matroid equipped with a building set. This class extends the ordinary tangent bundle of the de Concini–Procesi wonderful model from the realizable case to arbitrary matroids. Its Hirzebruch class is shown to equal the formally defined Hirzebruch class of the pair, so the Todd class is the Todd of this tangent class and the degree of the specialized Hirzebruch class recovers the Chow polynomial. The construction is combinatorial, using Aluffi’s blow-up correction term interpreted entirely inside Feichtner–Yuzvinsky rings, and therefore does not require a linear realization. As a numerical consequence the same class yields Chern-number inequalities against the hyperplane class, including a Miyaoka–Yau-type inequality. A sympathetic reader cares because algebraic-geometry constraints on Chern numbers and duality become available for every matroid, not merely the representable ones.","feed_headline":"Tangent class recovers Chow polynomial for every matroid","feed_subtitle":"One K-class extends wonderful-model geometry to non-representable matroids and yields Chern inequalities","key_machinery":"The tangent class T_{M,G} itself—an explicit K-class built from Euler, boundary-normal and cutting-class correction terms that obeys the relative Aluffi recursion under every one-step building-set enlargement and the star-normal identity on blow-up centers. These two identities make the universal blow-up formula for Hirzebruch classes coincide with the recursive definition of Hy.","core_discovery":"For every loopless matroid M and every building set G containing the top flat, the intrinsically defined tangent class T_{M,G} satisfies Hy(M,G)=ch(λ_y T^∨_{M,G}) td(T_{M,G}). In particular the formal Todd class equals td(T_{M,G}) and the Chow polynomial of (M,G) is recovered as the Hirzebruch genus of T_{M,G}.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Tangent class recovers Chow polynomial for every matroid","Intrinsic matroid tangent class yields Hirzebruch-Riemann-Roch","Building-set tangent class computes Chow polynomial of (M,G)","K-class extends wonderful models and gives matroid Chern inequalities","Matroidal tangent class equals formal Todd and recovers Chow genus"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The recursive definition of the formal Hirzebruch class along a chain of one-step enlargements from G to the maximal building set is independent of the chosen chain; the paper treats this independence as a consequence of the realization theorem rather than proving it first.","fun_headline_variants_meta":{"raw":{"variants":["Tangent class recovers Chow polynomial for every matroid","Intrinsic matroid tangent class yields Hirzebruch-Riemann-Roch","Building-set tangent class computes Chow polynomial of (M,G)","K-class extends wonderful models and gives matroid Chern inequalities","Matroidal tangent class equals formal Todd and recovers Chow genus"]},"model":"grok-4.5","effort":"low","cost_usd":0.005708,"raw_usage":{"total_tokens":1532,"prompt_tokens":775,"num_sources_used":0,"completion_tokens":89,"cost_in_usd_ticks":57080000,"prompt_tokens_details":{"text_tokens":775,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":668,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":775,"tokens_out":89,"duration_ms":5683,"temperature":1.0,"reasoning_tokens":668,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T12:51:47.985562+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute both sides of the claimed equality Hy(M,G)=Φ_y(T_{M,G}) for a small non-realizable matroid (for example the Fano matroid with a non-maximal building set) by expanding the Feichtner–Yuzvinsky presentations and check whether the coefficients of the two polynomials in y agree.","supporting_citations":[],"review_version":2}