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Tangent classes of matroids and wonderful compactifications
Ronnie Cheng, Shurui Liu, Guoxiong Gao
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
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Summary
This paper constructs an integral tangent class for loopless matroids within the integral combinatorial K-ring. The class specializes to the tangent bundle of the wonderful compactification in the realizable case, recovers the Chow ring's Hilbert series via the Hirzebruch-Riemann-Roch theorem, and satisfies Chern-alpha lower bounds. The core mathematical content was autonomously generated by the AI agent Danus, highlighting AI's capability in advanced mathematical research.
Entities (8)
Relation Signals (6)
Danus → produced → main body of paper
confidence 98% · The main body of this paper was produced autonomously... by Danus, an AI mathematical reasoning agent.
Integral K-ring → contains → Integral tangent class
confidence 96% · we construct an integral tangent class T_{M,G}^Z in K_Z(M,G)
Matroid → has → Feichtner-Yuzvinsky Building Set
confidence 95% · For every loopless matroid M and every Feichtner-Yuzvinsky building set G containing the top flat
Integral tangent class → specializesto → Tangent bundle of wonderful compactification
confidence 95% · in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification
Integral tangent class → recovers → Hilbert Series of Chow Ring
confidence 94% · it recovers the Hilbert series of the Chow ring through Hirzebruch-Riemann-Roch
Integral tangent class → satisfies → Chern-Alpha Lower Bounds
confidence 93% · and it satisfies the expected Chern-alpha lower bounds
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Abstract
Abstract:For every loopless matroid $M$ and every Feichtner--Yuzvinsky building set $\mathcal{G}$ containing the top flat, we construct an integral tangent class $T_{M,\mathcal{G}}^{\mathbb{Z}}\in K_{\mathbb{Z}}(M,\mathcal{G})$; in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification, it recovers the Hilbert series of the Chow ring through Hirzebruch--Riemann--Roch, and it satisfies the expected Chern-alpha lower bounds. This reproduces the tangent class and its key properties studied by the first author in arXiv:2606.22650. The main body of this paper was produced autonomously, without human mathematical guidance, by Danus, an AI mathematical reasoning agent. Danus solved the problem before arXiv:2606.22650 was publicly available, demonstrating the potential of AI agents in mathematical research. We reproduce its output faithfully, adding only editorial comments; the experiment is documented in Appendix B.
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- Source: https://arxiv.org/abs/2607.05835v2
- Canonical: https://arxiv.org/abs/2607.05835v2
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Tangent classes of matroids and wonderful compactifications Ronnie Cheng and Shurui Liu With Appendix B by Guoxiong Gao and Shurui Liu. Stanford University, California, USA Abstract. For every loopless matroid M and every Feichtner–Yuzvinsky building set G containing the top flat, we construct an integral tangent class TM,ℤ∈Kℤ(M,)T_M,G^Z∈ K_Z(M,G); in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification, it recovers the Hilbert series of the Chow ring through Hirzebruch–Riemann–Roch, and it satisfies the expected Chern-alpha lower bounds. This reproduces the tangent class and its key properties studied by the first author in [Che26]. The main body of this paper was produced autonomously, without human mathematical guidance, by Danus, an AI mathematical reasoning agent. Danus solved the problem before [Che26] was publicly available, demonstrating the potential of AI agents in mathematical research. We reproduce its output faithfully, adding only editorial comments; the experiment is documented in Appendix B. Contents 1 Introduction 1.1 Agentic AI 1.2 Mathematics 2 The integral K-ring 3 The rational tangent class 4 The integral quotient and tangent class 5 Saturated descent 6 Realizable comparison and generator normalization 7 The Hilbert identity 8 Nested Segre tails and the Chern-alpha bound 9 Hilbert polynomials and Chern inequalities for the integral class 10 Fan-support guard 11 Proof of the closeout certificate References A Raw prompt B Usage of Generative AI 1. Introduction 1.1. Agentic AI In [Che26], the first author constructed an integral K-class (tangent class) TM∈K(M)T_M∈ K(M) for every loopless matroid M and established its main properties. To test the power of AI tools in mathematical research, the authors posed to several AI agents the task of constructing a K-class TMT_M satisfying the three key properties stated in Theorem 1.1, and carried out the experiment with the assistance of Guoxiong Gao. The agents were the GPT-5.5 Pro web interface; Rethlas, an autonomous agentic proof system [Ju+26]; and Danus [Liu+26], an agentic system built on the Rethlas worker–verifier system with a Claude Code–based orchestrator (see Appendix B). We ran the experiment before [Che26] was publicly released on arXiv, and deliberately withheld that paper from the agents to reduce the possibility of data contamination. The exact prompt is recorded verbatim in Appendix A. Danus constructed a rational K-class and proved its properties fully autonomously, without human mathematical guidance. After the authors observed that this solution addressed only the rational version of the original problem (by abuse of notation, the prompt in Appendix A uses the same symbol for the integral tangent class and its rationalization), Danus resumed work and added Sections 4–6, which adapt the rational class to an integral class and thereby settle the original integral problem. Its output is reproduced in the main body of this paper, together with editorial comments by the authors. The authors have verified the manuscript and found it correct, with one local exception: the justification given for Lemma 8.7 (used in the Chern–α lower bound) is incomplete as written, the lemma having been treated as proved without a valid argument. The lemma is nonetheless true and admits a short proof (see [Che26, Proposition 4.19]), and this gap affects neither the construction of the tangent class nor the PK=HilbP^K=Hilb identity. Apart from the abstract, introduction, and the explicitly inserted human comments, the body of the paper should be understood as a faithful presentation of Danus’s performance on the open problem studied here. Further details of the AI usage and the experiment are given in Appendix B, by Guoxiong Gao and the second author. 1.2. Mathematics Let M be a loopless matroid of rank d+1d+1 on a finite nonempty ground set E, let G be a Feichtner–Yuzvinsky building set in its lattice of flats with E∈E , and put ∘:=∖EG :=G \E\. We work in the integral combinatorial K-ring Kℤ(M,)K_Z(M,G) with generators τF _F, F∈F . Our main theorem constructs a genuine integral tangent class TM,ℤ:=∑F∈∘(1−τF)−1−Qℤ∈Kℤ(M,),T_M,G^Z:= _F (1- _F)^-1-Q_G^Z∈ K_Z(M,G), with integer coordinates in the standard τ-monomial ℤZ-basis; the integral quotient representative QℤQ_G^Z is constructed in Section 4. The construction is intrinsic: the reduced G-nested fan need not be complete, so Kℤ(M,)K_Z(M,G) is not presented as the K-ring of a complete toric variety, and the theorem is stated in Kℤ(M,)K_Z(M,G) itself (see Section 10). Write Kℚ(M,):=Kℤ(M,)⊗ℤℚK_Q(M,G):=K_Z(M,G) _ZQ for the rational combinatorial K-ring. In Section 3 we construct, from the descended Berget–Eur–Spink–Tseng quotient Chern polynomial, a rational quotient class QM,Q_M,G and the corresponding rational tangent class TM,T_M,G in Kℚ(M,)K_Q(M,G), and prove the realizable comparison, the Hilbert identity PK=HilbP^K=Hilb (Theorem 7.8), and the Chern-alpha lower bounds (Theorem 8.15). The integral theorem below lifts these to Kℤ(M,)K_Z(M,G): TM,ℤT_M,G^Z rationalizes to TM,T_M,G (Proposition 4.3). Theorem 1.1 (Integral tangent class). Let M be a loopless matroid of rank d+1d+1 on a finite nonempty ground set E, with finite lattice of flats L(M)L(M), bottom flat ∅ , top flat E, and rank function rkMrk_M. Let G be a finite Feichtner-Yuzvinsky building set in L(M)∖∅L(M) \ \ containing E, and put ∘:=∖EG :=G \E\. Let ρK:Kℤ(M,)⟶Kℤ(M,)⊗ℤℚ _K K_Z(M,G) K_Z(M,G) _ZQ be rationalization. Then there exists an integral quotient representative Qℤ∈Kℤ(M,)Q_G^Z∈ K_Z(M,G) such that ρK(Qℤ)=QM, _K(Q_G^Z)=Q_M,G, and TM,ℤ:=∑F∈∘(1−τF)−1−QℤT_M,G^Z:= _F (1- _F)^-1-Q_G^Z is a genuine element of Kℤ(M,)K_Z(M,G) whose rationalization is TM,T_M,G. Moreover: (1) If M is realized over ℂC by a linear subspace L not contained in any coordinate hyperplane, and if WL,W_L,G is the De Concini-Procesi wonderful model with reduced boundary divisor D:=∑F∈∘DFD_G:= _F D_F, then QℤQ_G^Z may be chosen together with an integral unital ring isomorphism θℤ:Kℤ(M,)⟶K0(WL,) _G^Z K_Z(M,G) K_0(W_L,G) such that θℤ((1−τF)−1)=[WL,(DF)] _G^Z ((1- _F)^-1 )=[O_W_L,G(D_F)] for every F∈∘F , and θℤ(TM,ℤ)=[TWL,]. _G^Z(T_M,G^Z)=[T_W_L,G]. (2) The K-theoretic Todd polynomial PintK(M,;z):=degM,(ch(λ−z(ρK(TM,ℤ)∨))td(ρK(TM,ℤ)))P_int^K(M,G;z):= _M,G (ch ( _-z ( _K(T_M,G^Z) ) )td ( _K(T_M,G^Z) ) ) equals Hilb(M,;z)Hilb(M,G;z) in ℤ[z]Z[z]. Equivalently, for every integer i with 0≤i≤d0≤ i≤ d, dimℚAℚ(M,)i=(−1)iχ(∧iT∨). _QA_Q(M,G)^i=(-1)^iχ ( ^iT ). (3) For every integer k with 0≤k≤d0≤ k≤ d, one has degM,(ck(ρK(TM,ℤ))αd−k)≥(d+1k),α:=−xE. _M,G (c_k ( _K(T_M,G^Z) )α^d-k )≥d+1 k, α:=-x_E. Finally, TM,ℤT_M,G^Z has integer coordinates in the standard τ-monomial ℤZ-basis of Kℤ(M,)K_Z(M,G). Throughout, χ(∧iT∨)χ( ^iT ) denotes the formal Hirzebruch-Riemann-Roch number degM,(ch(∧iρK(TM,ℤ)∨)td(ρK(TM,ℤ))) _M,G(ch( ^i _K(T_M,G^Z) )td( _K(T_M,G^Z))) of the rational tangent class; in the non-realizable case it is not the Euler characteristic of an actual vector bundle. The theorem is useful in its realizable form because it identifies the purely combinatorial class with the tangent bundle class of the wonderful model. It is also useful in non-realizable cases because the Hilbert-polynomial and Chern-alpha conclusions remain intrinsic to the matroid. The paper is organized as follows. Section 2 sets up the integral combinatorial K-ring Kℤ(M,)K_Z(M,G), its standard τ-monomial basis, and the Chern character. Section 3 constructs the rational classes QM,Q_M,G and TM,T_M,G by one-flat descent of the Berget–Eur–Spink–Tseng quotient Chern polynomial [BES+23] from the maximal model, and proves the realizable specialization; Sections 7 and 8 prove their remaining two properties—the Hilbert identity PK=HilbP^K=Hilb, anchored at the maximal model by [Che25, FMSV24] and propagated by the one-flat recursion of [EFM+25], and the Chern-alpha lower bound, through nested-support positivity. The integral lift occupies Sections 4–6: Section 4 constructs the integral representative QℤQ_G^Z and the class TM,ℤT_M,G^Z and proves the non-realizable clauses; Section 5 proves the crucial saturation of the one-step refinement maps over ℤZ, via the τ-adic associated graded and the integral Feichtner–Yuzvinsky comparison [FY04]; and Section 6 gives the integral realizable isomorphism to K0(WL,)K_0(W_L,G), using the K-theoretic blowup formula [Tho93] and the wonderful-model blowup centers of [DCP95]. Finally, Section 9 carries the Hilbert and Chern-alpha conclusions to the integral class, and Sections 10 and 11 record the fan-support guard and assemble the closeout certificate. 2. The integral K-ring We adopt the standard notation and terminology for matroid Chow rings, wonderful compactifications, and toric varieties from [AHK18, CLS11, DCP95, FY04, LLPP24] and use them freely. We work over the field of complex numbers ℂC when a realization is fixed. Definition 2.1. Let M be a loopless matroid with finite lattice of flats L(M)L(M), bottom flat ∅ , and top flat E. Let ⊂L(M)∖∅G⊂ L(M) \ \ be a Feichtner-Yuzvinsky building set containing E. The rational Chow algebra Aℚ(M,)A_Q(M,G) is the quotient of ℚ[xF∣F∈]Q[x_F F ] by the ideal generated by: (1) all monomials ∏F∈SxF _F∈ Sx_F, where S⊂S is not G-nested; (2) the atom linear forms ∑F∈,a≤FxF _F ,\ a≤ Fx_F, where a ranges over the atoms of L(M)L(M). The top-degree map is denoted by degM, _M,G, and Hilb(M,;z):=∑i≥0dimℚAℚ(M,)izi.Hilb(M,G;z):= _i≥ 0 _QA_Q(M,G)^iz^i. Definition 2.2. Let M be a loopless matroid with lattice of flats L(M)L(M), bottom flat ∅ , top flat E, and let G be a top-containing Feichtner-Yuzvinsky building set in L(M)∖∅L(M) \ \. The integral combinatorial K-ring Kℤ(M,)K_Z(M,G) is the ℤZ-algebra generated by symbols τF _F, for F∈F , modulo the following relations: (1) ∏F∈SτF=0 _F∈ S _F=0 whenever S⊆S is not G-nested. (2) For every atom a of L(M)L(M), 1−∏F∈a≤F(1−τF)=0.1- _ subarraycF \\ a≤ F subarray(1- _F)=0. Here an atom means a rank-one flat, equivalently a parallel class of M. We put Kℚ(M,):=Kℤ(M,)⊗ℤℚK_Q(M,G):=K_Z(M,G) _ZQ. Remark 2.3. The atom convention in Definition 2.2 is essential. Since M is only assumed loopless and may be non-simple, atoms are rank-one flats, not singleton elements of the ground set. Singleton notation is valid only after passing to a simple matroid. The Chern character used below is the finite power-series map chM,:Kℚ(M,)→Aℚ(M,),τF⟼1−exp(−xF),ch_M,G K_Q(M,G)→ A_Q(M,G), _F 1- (-x_F), the unique ℚQ-algebra homomorphism with this generator rule. The nonnested monomial relations go to zero because 1−exp(−xF)1- (-x_F) is divisible by xFx_F. For an atom a the multiplicative relation maps to 1−∏F∈,a≤Fexp(−xF)=1−exp(−∑F∈a≤FxF)=0,1- _F ,\ a≤ F (-x_F)=1- (- _ subarraycF \\ a≤ F subarrayx_F )=0, which is zero by the atom-linear relation in the Chow ring. Proposition 2.4 (Intrinsic basis). For every loopless M and every top-containing G, the ring Kℤ(M,)K_Z(M,G) is free as an abelian group and has the standard τ-monomial ℤZ-basis. In particular, the phrase “integer coordinates” in Theorem 1.1 means coordinates with respect to this basis. Proof. This is the intrinsic standard-monomial basis theorem for the integral combinatorial K-ring. In the maximal case it agrees with the non-augmented integral matroid K-ring of Larson, Li, Payne, and Proudfoot [LLPP24]; the Chow basis is the Feichtner-Yuzvinsky basis [FY04]. For a general top-containing building set, the reduced Laurent presentation, the τ-adic associated graded Chow ring, and the standard τ-monomial basis follow from the same Stanley-Reisner and atom-character relations as in [LLPP24, FY04] applied to the induced building set. This proves freeness and the coordinate statement. ∎ Definition 2.5 (Berget–Eur–Spink–Tseng tautological quotient class). Let N be a loopless matroid on a finite ground set E, and let XEX_E be the complex permutohedral toric variety. Following [BES+23], let T=(ℂ∗)ET=(C^*)^E be the coordinate torus and let KT(XE)K_T(X_E) denote the T-equivariant K-group of XEX_E in their convention; the tautological classes below are taken in this T-equivariant K-theory. Berget–Eur–Spink–Tseng [BES+23, Definitions 1.2 and 3.9] attach to N the tautological sub and quotient T-equivariant K-classes SN,QN∈KT(XE),S_N,\ Q_N∈ K_T(X_E), of ranks rankNrankN and |E|−rankN|E|-rankN respectively. When N is realized by a linear subspace L⊆ℂEL ^E not contained in any coordinate hyperplane, SNS_N and QNQ_N are the classes of the tautological subbundle SLS_L and quotient bundle QLQ_L on XEX_E, which fit into the short exact sequence 0⟶SL⟶XE⊗ℂE⟶QL⟶0,0 S_L _X_E _CC^E Q_L 0, with SLS_L the subbundle whose fiber over the identity of T is L; for an arbitrary matroid N the two classes are defined by the torus-fixed-point formula of [BES+23, Definition 3.9]. We call QNQ_N the BEST tautological quotient class of N and write CBEST(N;u):=∑i≥0ci(QN)uiC_BEST(N;u):= _i≥ 0c_i(Q_N)\,u^i for its total equivariant Chern polynomial, using the same symbol for its non-equivariant image. The polynomial CBEST(N;u)C_BEST(N;u) is the Chern-polynomial input descended in Section 3. Notation 2.6. For a loopless matroid N, we write ℋconn(N):=top(N)∪F∈L(N)∖∅∣N|F is connected.H_conn(N):=\top(N)\∪\F∈ L(N) \ \ N|F is connected\. This is the unique inclusion-minimal top-containing Feichtner–Yuzvinsky building set. For a connected loopless matroid N, we usually write ℋ:=ℋconn(N)H:=H_conn(N). Notation 2.7. For integers a≥0a≥ 0 and b, put B(a,b):=(ab),0≤b≤a,0,otherwise.B(a,b):= cases ab,&0≤ b≤ a,\\ 0,&otherwise. cases Thus B(a,b)B(a,b) is the zero-extended binomial coefficient. 3. The rational tangent class In this section, we construct the rational quotient and tangent classes and record the realizable specialization. These rational classes are the objects whose rationalized integral lifts appear in Theorem 1.1; the identification is made in Proposition 4.3. Lemma 3.1 (Orbit-closure restriction identity for the BEST quotient class). Let N be a loopless matroid on E with lattice of flats L(N)L(N), let smallG_small be a Feichtner–Yuzvinsky building set in L(N)∖∅L(N) \ \ containing E, and let F be a nonempty proper flat with F∉smallF _small such that big=small∪FG_big=G_small∪\F\ is again a building set. Write Factorsmall(F)=F1,…,FℓFactor_G_small(F)=\F_1,…,F_ \ for the maximal elements of smallG_small contained in F. Let ZF⊆XEZ_F X_E be the torus-orbit closure of the permutohedral variety XEX_E attached to the one-term chain ∅⊊F⊊E F E, with the standard product identification ZF≅XF×XE∖FZ_F X_F× X_E F and projections pr1:ZF→XFpr_1 Z_F→ X_F and pr2:ZF→XE∖Fpr_2 Z_F→ X_E F. For each a, let pa:XF→XFap_a X_F→ X_F_a be the toric morphism induced by the coordinate projection. Then, in T-equivariant K-theory, [QN]|ZF=pr1∗[QN|F]+pr2∗[QN/F]=∑a=1ℓpr1∗pa∗[QN|Fa]+pr2∗[QN/F].[Q_N] |_Z_F=pr_1^*[Q_N|F]+pr_2^*[Q_N/F]= _a=1 pr_1^*p_a^*[Q_N|F_a]+pr_2^*[Q_N/F]. Proof. Apply [BES+23, Propositions 5.2 and 5.3] to the one-term chain ∅⊊F⊊E F E: by Proposition 5.2 the orbit closure ZFZ_F carries the product identification ZF≅XF×XE∖FZ_F X_F× X_E F, and by Proposition 5.3 the restriction of the tautological quotient class is the external sum of the pulled-back tautological quotient classes of the two successive minors N|FN|F and N/FN/F, namely [QN]|ZF=pr1∗[QN|F]+pr2∗[QN/F][Q_N]|_Z_F=pr_1^*[Q_N|F]+pr_2^*[Q_N/F]. It remains to decompose [QN|F][Q_N|F]. By the defining property of a Feichtner–Yuzvinsky building set [DCP95, FY04], the join map ∏a=1ℓ[∅,Fa]→[∅,F] _a=1 [ ,F_a]→[ ,F] is an isomorphism of posets; consequently the flats F1,…,FℓF_1,…,F_ are pairwise disjoint with union F, and the restriction N|FN|F is the direct sum of the restrictions N|F1,…,N|FℓN|F_1,…,N|F_ . By the direct-sum additivity of the BEST tautological quotient classes [BES+23, Proposition 5.13], applied to this direct-sum decomposition with the toric morphisms pa:XF→XFap_a X_F→ X_F_a induced by the coordinate projections, one has [QN|F]=∑a=1ℓpa∗[QN|Fa][Q_N|F]= _a=1 p_a^*[Q_N|F_a] in KT(XF)K_T(X_F). Pulling this equality back by pr1pr_1 and substituting yields the second displayed identity. We do not assert any isomorphism XF≅XF1×⋯×XFℓX_F X_F_1×·s× X_F_ ; the dependence on the factors enters only through the toric morphisms pap_a. ∎ Lemma 3.2 (Orbit-closure restriction identity for BEST Newton power sums). Let N be a loopless matroid on E of rank r=d+1r=d+1. Let smallG_small be a Feichtner–Yuzvinsky building set in L(N)∖∅L(N) \ \ containing E. Let F be a nonempty proper flat with F∉smallF _small, and suppose that big=small∪FG_big=G_small∪\F\ is again a building set. Write Factorsmall(F)=F1,…,Fℓ.Factor_G_small(F)=\F_1,…,F_ \. Let ZF⊆XEZ_F X_E be the permutohedral torus-orbit closure attached to the one-term chain ∅⊊F⊊E F E, with the standard product identification ZF≅XF×XE∖FZ_F X_F× X_E F and projections pr1:ZF→XFpr_1 Z_F→ X_F and pr2:ZF→XE∖Fpr_2 Z_F→ X_E F. For each a, let pa:XF→XFap_a X_F→ X_F_a be the toric morphism induced by the coordinate projection. Let QNQ_N, QN|FaQ_N|F_a, and QN/FQ_N/F be the BEST tautological quotient classes, and let Pm(⋅)P_m(·) denote the m-th Newton power sum of the corresponding total equivariant Chern polynomial. Then, for every m≥1m≥ 1, in the equivariant Chow ring of ZFZ_F, Pm(QN|ZF)=pr2∗Pm(QN/F)+∑a=1ℓpr1∗pa∗Pm(QN|Fa).P_m(Q_N|Z_F)=pr_2^*P_m(Q_N/F)+ _a=1 pr_1^*p_a^*P_m(Q_N|F_a). Proof. By Lemma 3.1, in the equivariant K-theory of ZFZ_F, [QN]|ZF=pr2∗[QN/F]+∑a=1ℓpr1∗pa∗[QN|Fa].[Q_N] |_Z_F=pr_2^*[Q_N/F]+ _a=1 pr_1^*p_a^*[Q_N|F_a]. The total equivariant Chern polynomial ctc_t is multiplicative under direct sums of equivariant K-classes and commutes with pullback. Hence, in the equivariant Chow ring of ZFZ_F, ct(QN|ZF)=pr2∗ct(QN/F)⋅∏a=1ℓpr1∗pa∗ct(QN|Fa).c_t(Q_N|Z_F)=pr_2^*c_t(Q_N/F)· _a=1 pr_1^*p_a^*c_t(Q_N|F_a). Work in a splitting algebra over the equivariant Chow ring of ZFZ_F in which each pulled-back total Chern polynomial splits into commuting degree-one Chern roots. Let the Chern roots of pr2∗ct(QN/F)pr_2^*c_t(Q_N/F) be αR,1,…,αR,qR _R,1,…, _R,q_R, and let the Chern roots of pr1∗pa∗ct(QN|Fa)pr_1^*p_a^*c_t(Q_N|F_a) be αa,1,…,αa,qa _a,1,…, _a,q_a. By the displayed product, the Chern roots of ct(QN|ZF)c_t(Q_N|Z_F) are the union of these. Over ℚQ, Newton’s identities identify the m-th Newton power sum of a unit total Chern polynomial with the sum of the m-th powers of its Chern roots, and this identification commutes with the graded pullbacks pr1∗pr_1^*, pr2∗pr_2^*, and pa∗p_a^*. Therefore Pm(QN|ZF)=∑b=1qRαR,bm+∑a=1ℓ∑b=1qaαa,bm=pr2∗Pm(QN/F)+∑a=1ℓpr1∗pa∗Pm(QN|Fa)P_m(Q_N|Z_F)= _b=1^q_R _R,b^m+ _a=1 _b=1^q_a _a,b^m=pr_2^*P_m(Q_N/F)+ _a=1 pr_1^*p_a^*P_m(Q_N|F_a) for every m≥1m≥ 1, as claimed. ∎ Proposition 3.3 (Descended quotient and tangent class). Let N be a loopless matroid of rank r=d+1r=d+1 on a finite nonempty ground set E, with finite lattice of flats, bottom flat ∅ , and top flat E. Let ℋ⊂L(N)∖∅H⊂ L(N) \ \ be a Feichtner–Yuzvinsky building set containing E, and put ℋ∘:=ℋ∖EH :=H \E\. Then there exists a unique descended quotient Chern polynomial CQ(N,ℋ;u)=∑i=0dciQui∈Aℚ(N,ℋ)[u]C_Q(N,H;u)= _i=0^dc_i^Qu^i∈ A_Q(N,H)[u] whose pullback to the maximal building set is the Berget–Eur–Spink–Tseng quotient Chern polynomial [BES+23]. The polynomial CQ(N,ℋ;u)C_Q(N,H;u) determines a unique rational K-class QN,ℋ∈Kℚ(N,ℋ)Q_N,H∈ K_Q(N,H). Define TN,ℋ:=∑F∈ℋ∘(1−τF)−1−QN,ℋ.T_N,H:= _F (1- _F)^-1-Q_N,H. Then QN,ℋQ_N,H and TN,ℋT_N,H are well-defined classes in Kℚ(N,ℋ)K_Q(N,H), and cz(TN,ℋ)=cz(QN,ℋ)−1∏F∈ℋ∘(1+zxF),c_z(T_N,H)=c_z(Q_N,H)^-1 _F (1+zx_F), where cz(QN,ℋ)=∑i=0dciQzic_z(Q_N,H)= _i=0^dc_i^Qz^i. If N is realized by a complex linear subspace L, and θ:Kℚ(N,ℋ)→K0(WL,ℋ)⊗ℤℚθ K_Q(N,H)→ K_0(W_L,H) _ZQ is the realizable rational K-theory identification matching (1−τF)−1(1- _F)^-1 with [WL,ℋ(DF)][O_W_L,H(D_F)], then θ(TN,ℋ)=[TWL,ℋ].θ(T_N,H)=[T_W_L,H]. No integral KℤK_Z-membership assertion is made. Proof. Throughout write ℋmax:=L(N)∖∅H_ :=L(N) \ \ for the maximal building set. The proof proceeds in six steps: (0) the maximal model and its zero-extended Newton power sums; (1) the one-step image criterion and the injectivity of one-step Chow pullback; (2) the descent of the Newton power sums; (3) the descended quotient Chern polynomial; (4) the reconstruction of QN,ℋQ_N,H through the filtered Chern character; (5) the tangent class and its Chern polynomial; (6) the realizable identification. Step 0 (maximal model). Let CBESTmax(N;u)=∑i=0dciBESTui∈Aℚ(N,ℋmax)[u]C_BEST (N;u)= _i=0^dc_i^BESTu^i∈ A_Q(N,H_ )[u] be the non-equivariant image, on the maximal model ℋmaxH_ , of the Berget–Eur–Spink–Tseng quotient Chern polynomial CBEST(N;u)C_BEST(N;u) of Definition 2.5. For m≥1m≥ 1 define the zero-extended maximal Newton power sum Pmmax∈Aℚ(N,ℋmax)mP_m ∈ A_Q(N,H_ )^m to be the m-th Newton power sum determined by CBESTmax(N;u)C_BEST (N;u) when 1≤m≤d1≤ m≤ d, and Pmmax:=0P_m :=0 when m>dm>d. Since Aℚ(N,)i=0A_Q(N,G)^i=0 for i>di>d and every Chow pullback is graded and unital, each class named below vanishes in degrees >d>d; we nevertheless carry all m≥1m≥ 1, because the minors arising in one descent step can have strictly smaller top Chow degree, where the same index m is out of range and the correct representative is 0. Remark (by authors). In the realizable maximal case, this construction has the following geometric meaning. If N is realized by a linear subspace L, then the wonderful compactification WLW_L embeds in the permutohedral toric variety XEX_E. The maximal BEST quotient Chern polynomial CBESTmax(N;u)C_BEST (N;u) is the total Chern polynomial of i∗QLi^*Q_L, where i:WL↪XEi:W_L X_E. The vector bundle QLQ_L on XEX_E admits a regular section whose vanishing locus is WLW_L. Hence, i∗QLi^*Q_L is the normal bundle of WLW_L in XEX_E. Step 1 (one-step Chow pullback: injectivity and image criterion). Let big=small∪FG_big=G_small∪\F\ be a one-flat enlargement of top-containing Feichtner–Yuzvinsky building sets, with Factorsmall(F)=F1,…,FℓFactor_G_small(F)=\F_1,…,F_ \, and let ψA:Aℚ(N,small)⟶Aℚ(N,big) _A A_Q(N,G_small) A_Q(N,G_big) be the one-step Chow pullback. Under the Feichtner–Yuzvinsky identification of Aℚ(N,)A_Q(N,G) with the Chow ring of the smooth toric variety X_G of the nested fan of G [FY04], the nested fan of bigG_big is the stellar subdivision of that of smallG_small at the cone spanned by Factorsmall(F)Factor_G_small(F), with new ray F [DCP95, FY04]; and ψA _A is the pullback along the resulting proper birational toric morphism π:Xbig→Xsmallπ X_G_big→ X_G_small of smooth varieties. The projection formula gives π∗π∗=id _*π^*=id on rational Chow classes, because π∗[Xbig]=[Xsmall] _*[X_G_big]=[X_G_small]; hence ψA=π∗ _A=π^* is injective. More generally, for any top-containing ⊆ℋmaxG _ the nested fan of ℋmaxH_ refines that of G, giving a canonical proper birational morphism Xℋmax→X_H_ → X_G of which every one-flat refinement chain from G to ℋmaxH_ is a factorization into stellar subdivisions [DCP95, FY04]. Consequently the composite Chow pullback ψ:Aℚ(N,)→Aℚ(N,ℋmax) _G A_Q(N,G)→ A_Q(N,H_ ) equals (Xℋmax→X)∗(X_H_ → X_G)^*; it is one and the same map for every chain, and it is injective, being a composition of injective one-step pullbacks. We use the following one-step image criterion (sufficient direction): write j:EF↪Xbigj E_F X_G_big for the exceptional divisor of π and p:EF→ZBp E_F→ Z_B for its projection to the blown-up center ZBZ_B. A class y∈Aℚ(N,big)my∈ A_Q(N,G_big)^m lies in imψAim _A whenever its exceptional restriction satisfies j∗y=p∗γj^*y=p^*γ for some Chow class γ on ZBZ_B. This is the image description of the toric stellar subdivision π: a class on the blowup descends to XsmallX_G_small once its restriction to the exceptional divisor is pulled back from the center [DCP95, FY04]. Step 2 (descent of the Newton power sums). We construct, for every top-containing G and every m≥1m≥ 1, a class Pm(N,)∈Aℚ(N,)mP_m(N,G)∈ A_Q(N,G)^m, zero for m>dm>d, pulling back to PmmaxP_m along every one-flat refinement chain. We argue by simultaneous induction on |E||E|, and inside a fixed N by descending along a one-flat refinement chain ℋ=0⊂1⊂⋯⊂s=ℋmax.H=G_0 _1⊂·s _s=H_ . At the maximal end set Pm(N,s):=PmmaxP_m(N,G_s):=P_m . Suppose Pm(N,j)P_m(N,G_j) is constructed with the maximal-pullback property and j=j−1∪FG_j=G_j-1∪\F\; assume 1≤m≤d1≤ m≤ d, since otherwise both sides vanish. We compute the exceptional restriction j∗Pm(N,j)j^*P_m(N,G_j) by pulling it to the maximal model and using ordinary Chow functoriality, not divisor base change. Let ϖ:Xℋmax→Xj X_H_ → X_G_j be the chain morphism of Step 1, let Δ:=VΣ(ℋmax)(F) :=V_ (H_ )(F) be the strict transform of EFE_F in the maximal model, that is, the divisor star of the ray F in the maximal nested fan, with inclusion jmax:Δ↪Xℋmaxj_ X_H_ , and let g:Δ→EFg → E_F be the toric morphism induced by the refinement of star fans StarΣ(ℋmax)(F)→StarΣ(j)(F)Star_ (H_ )(F) _ (G_j)(F) in the quotient lattice Nℤ/ℤFN_Z/ZF [DCP95, FY04]. For the permutohedral maximal model this star is the product fan, so Δ is the orbit divisor ZF≅XF×XE∖F⊆XEZ_F X_F× X_E F X_E attached to ∅⊊F⊊E F E. These maps form the strict-transform square ϖ∘jmax=j∘g:Δ→Xj j_ =j g → X_G_j, with vertical maps g:Δ→EFg → E_F and ϖ:Xℋmax→Xj X_H_ → X_G_j and horizontal inclusions jmax:Δ↪Xℋmaxj_ X_H_ and j:EF↪Xjj E_F X_G_j. It commutes but is in general not Cartesian: the scheme-theoretic preimage ϖ−1(EF) ^-1(E_F) acquires extra exceptional components from later stellar subdivisions whose centers contain the ray F, so Δ is only the strict transform of EFE_F, not the total transform. Since all four varieties are smooth, ordinary contravariant Chow functoriality applies to the commuting square and gives g∗j∗=jmax∗ϖ∗g^*j^*=j_ ^* ^* on Aℚ(Xj)A_Q(X_G_j). Applying this to Pm(N,j)P_m(N,G_j) and using ϖ∗Pm(N,j)=Pmmax ^*P_m(N,G_j)=P_m , together with the descent property jmax∗Pmmax=Pm(QN|ZF)j_ ^*P_m =P_m(Q_N|Z_F) restricting the m-th BEST Newton power sum to the orbit divisor Δ=ZF =Z_F, yields g∗(j∗Pm(N,j))=Pm(QN|ZF)in Aℚ(Δ).g^* (j^*P_m(N,G_j) )=P_m(Q_N|Z_F) A_Q( ). The restriction minors N|FaN|F_a and the contraction minor N/FN/F have ground sets strictly smaller than E, and carry the induced top-containing building sets small|FaG_small|F_a and small/FG_small/F. By the induction on |E||E|, their descended Newton power sums exist as ordinary Chow classes: let Pa,m∈Aℚ(N|Fa,small|Fa)mP_a,m∈ A_Q(N|F_a,G_small|F_a)^m and PR,m∈Aℚ(N/F,small/F)mP_R,m∈ A_Q(N/F,G_small/F)^m be the classes pulling back to Pm(QN|Fa)P_m(Q_N|F_a) and Pm(QN/F)P_m(Q_N/F) respectively along the corresponding maximal refinement chains (with the zero convention when m is out of range). Let Θ:Aℚ(ZB)→∼(⨂a=1ℓAℚ(N|Fa,small|Fa))⊗ℚAℚ(N/F,small/F) A_Q(Z_B) \ \ ( _a=1 A_Q(N|F_a,G_small|F_a) ) _QA_Q(N/F,G_small/F) be the center product Chow isomorphism, and define γm∈Aℚ(ZB)m _m∈ A_Q(Z_B)^m by Θ(γm):=PR,m+∑a=1ℓPa,m, ( _m):=P_R,m+ _a=1 P_a,m, each summand embedded in its own tensor factor with the unit on the others. This is an equality of ordinary Chow classes; no piecewise-polynomial representative and no equivariant lift is invoked. The toric morphism p∘g:Δ→ZBp g → Z_B carries Δ=ZF =Z_F to the center, and under the product identifications its pullback of Θ(γm) ( _m) is, by Lemma 3.2, Pm(QN|ZF)=pr2∗Pm(QN/F)+∑a=1ℓpr1∗pa∗Pm(QN|Fa)=g∗p∗γm.P_m(Q_N|Z_F)=pr_2^*P_m(Q_N/F)+ _a=1 pr_1^*p_a^*P_m(Q_N|F_a)=g^*p^* _m. Comparing with the exceptional-restriction identity of the previous paragraph gives g∗(j∗Pm(N,j))=g∗p∗γmg^* (j^*P_m(N,G_j) )=g^*p^* _m in Aℚ(Δ)A_Q( ). The star-fan refinement morphism g:Δ→EFg → E_F is a proper birational toric morphism of smooth varieties, being the restriction to divisor stars of the stellar-subdivision chain ϖ ; the projection-formula argument of Step 1 applied to g gives g∗g∗=idg_*g^*=id on rational Chow classes, so g∗:Aℚ(EF)→Aℚ(Δ)g^* A_Q(E_F)→ A_Q( ) is injective. We conclude j∗Pm(N,j)=p∗γmin Aℚ(EF).j^*P_m(N,G_j)=p^* _m A_Q(E_F). The image criterion of Step 1 now gives Pm(N,j)∈imψjP_m(N,G_j) _j, and injectivity of ψj _j yields a unique Pm(N,j−1)P_m(N,G_j-1) with ψj(Pm(N,j−1))=Pm(N,j) _j(P_m(N,G_j-1))=P_m(N,G_j). Since the composite from j−1G_j-1 to ℋmaxH_ factors through jG_j, the new class again has the maximal-pullback property. Descending to j=0j=0 produces Pm(N,ℋ)P_m(N,H). By Step 1 the composite ψℋ:Aℚ(N,ℋ)→Aℚ(N,ℋmax) _H A_Q(N,H)→ A_Q(N,H_ ) is one chain-independent injective map, and a class has the maximal-pullback property exactly when ψℋ _H sends it to PmmaxP_m . If two classes of Aℚ(N,ℋ)mA_Q(N,H)^m both have this property, their difference lies in kerψℋ=0 _H=0, so they agree. Hence Pm(N,ℋ)P_m(N,H) is the unique class with ψℋ(Pm(N,ℋ))=Pmmax _H(P_m(N,H))=P_m , and is automatically independent of the chain. For a one-flat enlargement, ψbig∘ψA=ψsmall _G_big _A= _G_small, so both ψA(Pm(N,small)) _A(P_m(N,G_small)) and Pm(N,big)P_m(N,G_big) are sent by the injective map ψbig _G_big to PmmaxP_m , whence ψA(Pm(N,small))=Pm(N,big). _A(P_m(N,G_small))=P_m(N,G_big). Remark (by authors). The idea of Step 2 can be read by the following commutative diagram. (3.1) Δ XℋmaxX_H_ EFE_FXbigX_G_bigZBZ_BXsmallX_G_smalljmax j_ g gϖ j jp p Recall that CBESTmax(N;u)C_BEST (N;u) is the Chern class of the pullback of the BEST class. Let PBESTmax(N;u)P_BEST (N;u) denote the Newton power sum (the normalized Chern character) of the pullback of the BEST class. The goal is to show that the class P(N,big)P(N,G_big) descends across the one-flat blowdown Xbig→XsmallX_G_big→ X_G_small. At this stage of the descending induction, put y=P(N,big)y=P(N,G_big), and g∗j∗y=jmax∗ϖ∗y=jmax∗PBESTmax.g^*j^*y=j_ ^* ^*y=j_ ^*P_BEST . By Lemma 3.1, this last class is the sum of the corresponding Newton power sums for the minors N|FaN|F_a and N/FN/F. jmax∗(PBESTmax(N;u))=pr2∗(PBESTmax(N/F;u))+∑a=1ℓpr1∗pa∗(PBESTmax(N|Fa;u)).j_ (P_BEST (N;u))=pr_2^*(P_BEST (N/F;u))+ _a=1 pr_1^*p_a^*(P_BEST (N|F_a;u)). By induction on the smaller ground sets, these minor classes descend, and assemble to a class γ∈Aℚ(ZB)γ∈ A_Q(Z_B) with g∗p∗γ=jmax∗PBESTmax.g^*p^*γ=j_ ^*P_BEST . Since g∗g^* is injective, j∗y=p∗γmj^*y=p^* _m. The one-step image criterion then implies that y descends to Aℚ(N,small)A_Q(N,G_small). Step 3 (descended quotient Chern polynomial). Define c0Q=1c_0^Q=1 and, for 1≤m≤d1≤ m≤ d, the coefficient cmQ∈Aℚ(N,ℋ)mc_m^Q∈ A_Q(N,H)^m by Newton’s recursion Pm−c1QPm−1+c2QPm−2−⋯+(−1)m−1cm−1QP1+(−1)mmcmQ=0,Pj:=Pj(N,ℋ),P_m-c_1^QP_m-1+c_2^QP_m-2-·s+(-1)^m-1c_m-1^QP_1+(-1)^mm\,c_m^Q=0, P_j:=P_j(N,H), which determines cmQc_m^Q uniquely because m is invertible in ℚQ. Put CQ(N,ℋ;u):=∑i=0dciQuiC_Q(N,H;u):= _i=0^dc_i^Qu^i. Applying the chain pullback to the recursion and using that Pm(N,ℋ)P_m(N,H) pulls back to PmmaxP_m shows that CQ(N,ℋ;u)C_Q(N,H;u) pulls back to CBESTmax(N;u)C_BEST (N;u). Any polynomial with this same maximal pullback has each coefficient differing from ciQc_i^Q by a class killed by an injective chain pullback, hence equals CQ(N,ℋ;u)C_Q(N,H;u); this is the asserted existence, uniqueness and chain-independence. Step 4 (reconstruction of QN,ℋQ_N,H via the filtered Chern character). Put qQ:=|ℋ∘|−dq_Q:=|H |-d, let pmQp_m^Q be the Newton power sums determined by CQ(N,ℋ;u)C_Q(N,H;u) (so pmQ=Pm(N,ℋ)p_m^Q=P_m(N,H)), and set chQA:=qQ⋅1+∑m=1dpmQm!∈Aℚ(N,ℋ).ch_Q^A:=q_Q· 1+ _m=1^d p_m^Qm!∈ A_Q(N,H). Filter Kℚ(N,ℋ)K_Q(N,H) by the powers IτpI_τ^p of the ideal IτI_τ generated by the τF _F, and Aℚ(N,ℋ)A_Q(N,H) by FAp=⨁i≥pAℚ(N,ℋ)iF_A^p= _i≥ pA_Q(N,H)^i. The assignment xF↦in(τF)∈Iτ/Iτ2x_F ( _F)∈ I_τ/I_τ^2 carries the defining relations of Aℚ(N,ℋ)A_Q(N,H) to the initial forms of the defining relations of Kℚ(N,ℋ)K_Q(N,H): a nonnested monomial ∏F∈SxF _F∈ Sx_F is the initial form of ∏F∈SτF _F∈ S _F, and the atom linear form ∑F∋axF _F ax_F is the degree-one part of the multiplicative atom relation 1−∏F∋a(1−τF)1- _F a(1- _F). By the standard-monomial basis of the combinatorial K-ring [LLPP24], this assignment is an isomorphism of graded ℚQ-algebras grIτKℚ(N,ℋ)≅Aℚ(N,ℋ),gr_I_τK_Q(N,H) A_Q(N,H), and IτI_τ is nilpotent with Iτd+1=0I_τ^d+1=0; the IτI_τ-adic filtration is therefore finite and separated, and dimℚKℚ(N,ℋ)=∑i=0ddimℚAℚ(N,ℋ)i _QK_Q(N,H)= _i=0^d _QA_Q(N,H)^i. Since ch(τF)=1−exp(−xF)=xF+(Chow degree≥2)ch( _F)=1- (-x_F)=x_F+(Chow degree≥ 2), the Chern character carries IτpI_τ^p into FApF_A^p, so it is filtered, and its associated graded map is the inverse of the displayed isomorphism. A filtered homomorphism inducing an isomorphism on the associated graded of finite filtrations is itself an isomorphism. Hence chch is a ℚQ-algebra isomorphism, and QN,ℋ:=ch−1(chQA)∈Kℚ(N,ℋ)Q_N,H:=ch^-1(ch_Q^A)∈ K_Q(N,H) is the unique class with ch(QN,ℋ)=chQAch(Q_N,H)=ch_Q^A. Remark (by authors). Step 3 and Step 4 reconstruct the (rational) K-class by its Chern character. This is possible because the Chern character map ch:K(X)ℚ→∼A(X)ℚch:K(X)_Q A(X)_Q induces an isomorphism when the variety X is smooth, and the toric variety corresponding to the fan gives the isomorphism. Step 5 (tangent class and its Chern polynomial). Each τF _F lies in the nilpotent ideal IτI_τ, so 1−τF1- _F is a unit with (1−τF)−1=∑k≥0τFk(1- _F)^-1= _k≥ 0 _F^k, a finite sum. Hence TN,ℋ:=∑F∈ℋ∘(1−τF)−1−QN,ℋT_N,H:= _F (1- _F)^-1-Q_N,H is well-defined in Kℚ(N,ℋ)K_Q(N,H). Recall that Kℚ(N,ℋ)K_Q(N,H) is a special λ-ring [BES+23, LLPP24], so each class a has a total Chern class cz(a)=∑ici(a)zic_z(a)= _ic_i(a)z^i with ci(a)∈Aℚ(N,ℋ)ic_i(a)∈ A_Q(N,H)^i, recovered from the Chern-character power sums pm(a):=m!chm(a)p_m(a):=m!\,ch_m(a) by Newton’s identities; this total Chern class is multiplicative, cz(a+b)=cz(a)cz(b)c_z(a+b)=c_z(a)c_z(b) and cz(−a)=cz(a)−1c_z(-a)=c_z(a)^-1 by the splitting principle, and a line element l with ch(l)=exp(u)ch(l)= (u) has cz(l)=1+zuc_z(l)=1+zu. The boundary class (1−τF)−1(1- _F)^-1 is a unit with ch((1−τF)−1)=exp(xF)ch((1- _F)^-1)= (x_F), hence a line element with cz((1−τF)−1)=1+zxFc_z((1- _F)^-1)=1+zx_F; and cz(QN,ℋ)=∑i=0dciQzic_z(Q_N,H)= _i=0^dc_i^Qz^i, since the ciQc_i^Q are the Newton-identity Chern classes of chQAch_Q^A. Therefore cz(TN,ℋ)=cz(QN,ℋ)−1∏F∈ℋ∘(1+zxF),c_z(T_N,H)=c_z(Q_N,H)^-1 _F (1+zx_F), the z-graded inverse existing because c0Q=1c_0^Q=1. Step 6 (realizable identification). Suppose N is realized by a complex linear subspace L⊆ℂEL ^E not contained in any coordinate hyperplane, with W:=WL,ℋW:=W_L,H the smooth projective De Concini–Procesi wonderful model and reduced simple normal crossings boundary D=∑F∈ℋ∘DFD= _F D_F. Write Wmax:=WL,ℋmaxW_ :=W_L,H_ for the maximal model, with reduced boundary DmaxD_ . Consider the shifted geometric quotient class Qsum:=(|E|−1)[W]−[TW(−logD)]+(|ℋ∘|−|E|+1)[W]=|ℋ∘|[W]−[TW(−logD)]Q_sum:=(|E|-1)[O_W]-[T_W(- D)]+(|H |-|E|+1)[O_W]=|H |\,[O_W]-[T_W(- D)] in K0(W)⊗ℚK_0(W) , built from the logarithmic tangent bundle of (W,D)(W,D). We first prove that, under the realizable Chow identification xF↔[DF]x_F [D_F], the total Chern polynomial of QsumQ_sum equals CQ(N,ℋ;u)C_Q(N,H;u). For the maximal building set this is the Berget–Eur–Spink–Tseng identity [BES+23, Theorem 8.8]; the passage to an arbitrary building set is effected by the same one-flat refinement that defines CQC_Q in Steps 1–3. We carry it out in detail. (6a) Maximal anchor. Let QLQ_L denote the Berget–Eur–Spink–Tseng tautological quotient bundle on the permutohedral variety, restricted to WmaxW_ . Restricting the tautological exact sequence of Definition 2.5 to WmaxW_ and applying [BES+23, Theorem 8.8] gives [QL|Wmax]=(|E|−1)[Wmax]−[TWmax(−logDmax)][Q_L|_W_ ]=(|E|-1)[O_W_ ]-[T_W_ (- D_ )] in K0(Wmax)K_0(W_ ), whose total Chern polynomial, under the permutohedral Feichtner–Yuzvinsky identification, is CBESTmax(N;u)C_BEST (N;u) of Step 0. (6b) Refinement geometry. Order the flats of ℋmax∖ℋH_ so that adjoining them to ℋH one at a time yields a Feichtner–Yuzvinsky building set at each stage; this produces a chain ℋ=0⊂1⊂⋯⊂s=ℋmaxH=G_0 _1⊂·s _s=H_ and a composite morphism π:Wmax=WL,s→bs⋯→b1WL,0=W,π W_ =W_L,G_s \,b_s\,·s \,b_1\,W_L,G_0=W, in which bj:WL,j→WL,j−1b_j W_L,G_j→ W_L,G_j-1 is the blow-up of WL,j−1W_L,G_j-1 along the smooth center cut out by the boundary divisors indexed by the maximal elements Factorj−1(F)=FaFactor_G_j-1(F)=\F_a\ of j−1G_j-1 below the newly added flat F; this center is the transverse intersection ⋂aDFa _aD_F_a, and the reduced boundary of WL,jW_L,G_j is the total transform of that of WL,j−1W_L,G_j-1 together with the new exceptional divisor DFD_F [DCP95], [EFM+25, Proposition 5.2]. Thus each bjb_j is the blow-up of a smooth stratum of a simple normal crossings divisor, the boundary remains simple normal crossings, and π is their composite. (6c) Blow-up invariance of the shifted log-normal class. Let b:W′→W′b W → W be the blow-up of a smooth transverse intersection of components of a simple normal crossings divisor D′D on W′W , with reduced total transform D′D (the strict transforms together with the exceptional divisor) again simple normal crossings. Such a blow-up is log-étale for the divisorial log structures (W′,D′)(W ,D ) and (W′,D′)(W ,D ); equivalently the logarithmic cotangent sheaf pulls back, b∗ΩW′1(logD′)≅ΩW′1(logD′)b^* ^1_W ( D ) ^1_W ( D ), and dually b∗[TW′(−logD′)]=[TW′(−logD′)]b^*[T_W (- D )]=[T_W (- D )] in K0(W′)K_0(W ). Since also b∗[W′]=[W′]b^*[O_W ]=[O_W ], the class (|E|−1)[]−[T(−logD)](|E|-1)[O]-[T(- D)] is compatible with b∗b^*. Applying this to each bjb_j of (6b) and composing, and using (6a), π∗((|E|−1)[W]−[TW(−logD)])=(|E|−1)[Wmax]−[TWmax(−logDmax)]=[QL|Wmax].π^* ((|E|-1)[O_W]-[T_W(- D)] )=(|E|-1)[O_W_ ]-[T_W_ (- D_ )]=[Q_L|_W_ ]. (6d) Descent of the Chern polynomial. For any virtual class V and integer a one has cz(V+a[])=cz(V)c_z(V+a[O])=c_z(V), since cz()=1c_z(O)=1 and czc_z is multiplicative; hence π∗Qsumπ^*Q_sum and [QL|Wmax][Q_L|_W_ ] have the same total Chern polynomial, namely CBESTmax(N;u)C_BEST (N;u) by (6a). Under the Feichtner–Yuzvinsky presentations the geometric pullback π∗π^* on rational Chow rings coincides with the iterated stellar-subdivision Chow pullback ψℋ:Aℚ(N,ℋ)→Aℚ(N,ℋmax) _H A_Q(N,H)→ A_Q(N,H_ ) of Step 1 [FY04], [EFM+25, Proposition 5.2], and Chern classes commute with pullback; so ψℋ _H sends the total Chern polynomial of QsumQ_sum to CBESTmax(N;u)C_BEST (N;u). By Step 3, CQ(N,ℋ;u)C_Q(N,H;u) is the unique element of Aℚ(N,ℋ)[u]A_Q(N,H)[u] whose ψℋ _H-image is CBESTmax(N;u)C_BEST (N;u), and ψℋ _H is injective; therefore, under xF↔[DF]x_F [D_F], the total Chern polynomial of QsumQ_sum equals CQ(N,ℋ;u)C_Q(N,H;u). For ℋ=ℋmaxH=H_ this is exactly [BES+23, Theorem 8.8]. (6e) Identification of the classes. Since W is smooth and projective, the topological Chern character chW:K0(W)⊗ℚ→∼CH∗(W)⊗ℚch_W K_0(W) \ \ CH^*(W) is a ring isomorphism, and θ intertwines chch with chWch_W under xF↔[DF]x_F [D_F]. As ch(QN,ℋ)=chQAch(Q_N,H)=ch_Q^A is the Chern character determined by CQ(N,ℋ;u)C_Q(N,H;u), and by (6d) chW(Qsum)ch_W(Q_sum) is determined by the same polynomial, the classes θ(QN,ℋ)θ(Q_N,H) and QsumQ_sum have equal Chern character; injectivity of chWch_W gives θ(QN,ℋ)=Qsumθ(Q_N,H)=Q_sum. Finally, θ(TN,ℋ)=∑F∈ℋ∘[W(DF)]−Qsum=[TW(−logD)]+∑F∈ℋ∘([W(DF)]−[W]).θ(T_N,H)= _F [O_W(D_F)]-Q_sum=[T_W(- D)]+ _F ([O_W(D_F)]-[O_W] ). For each F, the structure sequence 0→W→W(DF)→DF(DF)→00 _W _W(D_F) _D_F(D_F)→ 0 gives [W(DF)]−[W]=[DF(DF)][O_W(D_F)]-[O_W]=[O_D_F(D_F)], while the residue sequence 0→TW(−logD)→TW→⨁F∈ℋ∘DF(DF)→00→ T_W(- D)→ T_W→ _F O_D_F(D_F)→ 0 of the simple normal crossings divisor D gives [TW]=[TW(−logD)]+∑F∈ℋ∘[DF(DF)][T_W]=[T_W(- D)]+ _F [O_D_F(D_F)]. Substituting yields θ(TN,ℋ)=[TWL,ℋ]θ(T_N,H)=[T_W_L,H] in K0(WL,ℋ)⊗ℤℚK_0(W_L,H) _ZQ. No integral KℤK_Z-membership is asserted. ∎ Remark (by authors). We have the pullback map between Chow rings of toric models ψA:Aℚ(N,small)⟶Aℚ(N,big), _A A_Q(N,G_small) A_Q(N,G_big), Proposition 6.2 implicitly implies that when M is realizable, ψA _A is the same as the pullback map between Chow rings of wonderful models π:WL,big⟶WL,small.π W_L,G_big W_L,G_small. See [Che26, Theorem 2.1, Remark 3.10]. Remark 3.4. In the one-flat recursion, the restriction and contraction factors carry the induced building sets. In general, the restriction-side building set may omit the top flat of the restriction, and an induced contraction building set of a minimal building set need not be the minimal building set of the contraction. Thus no proof below replaces these induced building sets by top-containing or minimal building sets without a comparison theorem. 4. The integral quotient and tangent class In this section, we construct the integral representative used in Theorem 1.1. The main point is that the quotient class is not reconstructed from rational Chern data by Newton identities. Let max:=L(M)∖∅G_ :=L(M) \ \. In the maximal case, Kℤ(M,max)K_Z(M,G_ ) is the Larson-Li-Payne-Proudfoot matroid K-ring. The Berget-Eur-Spink-Tseng tautological quotient class on the permutohedral model restricts integrally to this ring [BES+23, LLPP24]. If r=rk(M)r=rk(M), n=|E|n=|E|, d=r−1d=r-1, and f=|max∖E|f=|G_ \E\|, the maximal integral quotient representative is Qintmax(M):=ιM∗([QM])+(f−n+1)⋅1.Q_int (M):= _M^*([Q_M])+(f-n+1)· 1. The added trivial summand changes the rank and does not change the positive-degree Chern classes. Its rank becomes (n−r)+(f−n+1)=f−r+1=f−d.(n-r)+(f-n+1)=f-r+1=f-d. Thus it matches the quotient rank used by the rational class QM,maxQ_M,G_ of Proposition 3.3. Proposition 4.1 (Maximal integral quotient). For the maximal building set maxG_ , the rational class QM,maxQ_M,G_ of Proposition 3.3 has the integral representative Qintmax(M)Q_int (M) in Kℤ(M,max)K_Z(M,G_ ). Proof. The Berget-Eur-Spink-Tseng quotient K-class [QM][Q_M] is integral on the permutohedral variety [BES+23]. Its ordinary non-equivariant restriction to the Larson-Li-Payne-Proudfoot matroid K-ring is an element of Kℤ(M,max)K_Z(M,G_ ) [LLPP24]. Adding (f−n+1)⋅1(f-n+1)· 1 is an integral operation. After rationalization, this class has rank f−df-d and the quotient Chern polynomial CQ(M,max;u)=CBESTmax(M;u)C_Q(M,G_ ;u)=C_BEST (M;u) used in Step 0 of Proposition 3.3 to define QM,maxQ_M,G_ . The rational Chern character is determined by the rank and Chern classes, so the rationalization of Qintmax(M)Q_int (M) is QM,maxQ_M,G_ . ∎ For a general building set G, the integral quotient is obtained by descent from maxG_ . This descent uses one-flat refinements of building sets. If big=small∪FnewG_big=G_small∪\F_new\ and B:=Factor(Fnew,small)B:=Factor(F_new,G_small) is the set of maximal elements of smallG_small contained in FnewF_new, then the integral one-step map is ϕK(τH)=τH,H∉B,τH+τFnew−τHτFnew,H∈B. _K( _H)= cases _H,&H∉ B,\\ _H+ _F_new- _H _F_new,&H∈ B. cases Since FnewF_new is a proper flat, the top flat E is not in B, and therefore ϕK(τE)=τE _K( _E)= _E. Proposition 4.2 (Integral quotient descent). For every top-containing building set G, the rational class QM,Q_M,G of Proposition 3.3 has an integral representative Qℤ∈Kℤ(M,)Q_G^Z∈ K_Z(M,G). It may be chosen in the form Qℤ=|∘|−(|E|−1)+qBESTℤ(M,),Q_G^Z=|G |-(|E|-1)+q_BEST^Z(M,G), where qBESTℤ(M,)q_BEST^Z(M,G) is the descended integral Berget-Eur-Spink-Tseng quotient class. Proof. We descend the uncorrected Berget-Eur-Spink-Tseng class, not the rank-corrected maximal quotient. Put qBESTmax:=ιM∗([QM])∈Kℤ(M,max).q_BEST := _M^*([Q_M])∈ K_Z(M,G_ ). Choose a one-flat chain from G to maxG_ . The rational Berget-Eur-Spink-Tseng quotient descent of Proposition 3.3 (the chain-compatibility ψA(CQ(M,small;u))=CQ(M,big;u) _A(C_Q(M,G_small;u))=C_Q(M,G_big;u) of Step 2–3) says that, at each reverse one-step enlargement, the rationalization of the refined class lies in the rational image of the one-step map. Since Proposition 5.5 shows that this image is saturated, the class descends integrally. Repeating along the chain gives qBESTℤ(M,)∈Kℤ(M,)q_BEST^Z(M,G)∈ K_Z(M,G). We then set Qℤ=qBESTℤ(M,)+(|∘|−(|E|−1))⋅1,Q_G^Z=q_BEST^Z(M,G)+ (|G |-(|E|-1) )· 1, with the rank correction ρ=|∘|−(|E|−1) _G=|G |-(|E|-1) for the building set G. The equality ρK(Qℤ)=QM, _K(Q_G^Z)=Q_M,G after rationalization follows from the rational quotient descent. The descended class is independent of the chosen one-flat chain: Kℤ(M,)K_Z(M,G) is torsion-free and QM,Q_M,G is a fixed rational class, so its integral preimage under the injective rationalization is unique. ∎ The next proposition is the linkage between the integral construction and the rational tangent-class theorem of Sections 3–8. Proposition 4.3 (Linkage with the rational tangent class). Let Qℤ∈Kℤ(M,)Q_G^Z∈ K_Z(M,G) be the integral quotient representative of Proposition 4.2, and put TM,ℤ:=∑F∈∘(1−τF)−1−QℤT_M,G^Z:= _F (1- _F)^-1-Q_G^Z. Then ρK(Qℤ)=QM,,ρK(TM,ℤ)=TM,, _K(Q_G^Z)=Q_M,G, _K(T_M,G^Z)=T_M,G, where QM,Q_M,G and TM,T_M,G are the rational quotient and tangent classes of Proposition 3.3. Proof. By Proposition 4.2, ρK(Qℤ) _K(Q_G^Z) is the rational K-class whose rank is ρ+(rkQBEST)=(|∘|−(|E|−1))+(|E|−1−d)=|∘|−d=qQ _G+(rkQ_BEST)=(|G |-(|E|-1))+(|E|-1-d)=|G |-d=q_Q and whose total Chern polynomial in positive degrees is the descended Berget-Eur-Spink-Tseng quotient Chern polynomial; the trivial-rank correction ρ⋅1 _G· 1 changes only the rank. By Proposition 3.3, QM,Q_M,G is the unique rational K-class with rank qQ=|∘|−dq_Q=|G |-d and total Chern polynomial CQ(M,;u)C_Q(M,G;u), the descended Berget-Eur-Spink-Tseng quotient Chern polynomial. A rational K-class is determined by its Chern character, hence by its rank together with its positive-degree Chern classes through Newton’s identities. As both classes have the same rank and the same Chern polynomial, ρK(Qℤ)=QM, _K(Q_G^Z)=Q_M,G. Since rationalization is a ring homomorphism fixing each (1−τF)−1(1- _F)^-1, it sends TM,ℤ=∑F∈∘(1−τF)−1−QℤT_M,G^Z= _F (1- _F)^-1-Q_G^Z to ∑F∈∘(1−τF)−1−QM,=TM, _F (1- _F)^-1-Q_M,G=T_M,G. ∎ Proof of Theorem 1.1, non-realizable clauses. By Proposition 4.2, there exists Qℤ∈Kℤ(M,)Q_G^Z∈ K_Z(M,G) with rationalization QM,Q_M,G (Proposition 4.3). Since every (1−τF)−1(1- _F)^-1 is an integral inverse boundary line class in Kℤ(M,)K_Z(M,G), the class TM,ℤ=∑F∈∘(1−τF)−1−QℤT_M,G^Z= _F (1- _F)^-1-Q_G^Z lies in Kℤ(M,)K_Z(M,G). Its rationalization is the rational tangent class TM,T_M,G by Proposition 4.3. Proposition 2.4 gives integer coordinates in the standard τ-monomial basis. The equality PintK(M,;z)=Hilb(M,;z)P_int^K(M,G;z)=Hilb(M,G;z) and the Chern-alpha inequalities are proved in Section 9 below, where they are deduced from the in-paper Theorems 7.8 and 8.15 via Proposition 4.3. ∎ 5. Saturated descent The goal of this section is to prove the integral one-step descent statement used in Proposition 4.2. The proof uses toric geometry only for auxiliary centers. Remark (by authors). The class pull backs to an integral class in the maximal building set, so it suffices to show that the pullback map has torsion-free cokernel. This can be proved by writing down the map explicitly. Set-up 5.1. Let big=small∪FnewG_big=G_small∪\F_new\ be a one-flat enlargement of top-containing Feichtner-Yuzvinsky building sets. Put B:=Factor(Fnew,small)=F1,…,Fℓ.B:=Factor(F_new,G_small)=\F_1,…,F_ \. Let Ksmall:=Kℤ(M,small)K_small:=K_Z(M,G_small) and Kbig:=Kℤ(M,big)K_big:=K_Z(M,G_big), and let ϕK:Ksmall→Kbig _K K_small→ K_big be the one-step map. Lemma 5.2 (Associated graded). For every top-containing building set G the τ-adic filtration of Kℤ(M,)K_Z(M,G) by powers of the ideal Iτ=(τF:F∈)I_τ=( _F:F ) is finite and separated, and there is a canonical isomorphism grτKℤ(M,)≅Aℤ(M,),in(τF)=xF.gr_τK_Z(M,G) A_Z(M,G), ( _F)=x_F. Proof. The nonnested relation ∏F∈SτF=0 _F∈ S _F=0 is homogeneous and has initial form ∏F∈SxF=0 _F∈ Sx_F=0. The atom-character relation 1−∏a≤F(1−τF)=01- _a≤ F(1- _F)=0 has lowest-degree term ∑a≤FτF _a≤ F _F, with initial form ∑a≤FxF=0 _a≤ Fx_F=0, the Feichtner-Yuzvinsky atom-linear relation. These are exactly the defining relations of Aℤ(M,)A_Z(M,G) [FY04], and the K-relations are inhomogeneous deformations whose lowest-order terms recover them [LLPP24]. The filtration is finite because Aℤ(M,)A_Z(M,G) is concentrated in degrees 0,…,d0,…,d. ∎ Lemma 5.3 (Integral one-step Chow comparison). In the one-step situation of Set-up 5.1, the Feichtner-Yuzvinsky homomorphism ψℤ:Aℤ(M,small)→Aℤ(M,big),ψℤ(xH)=xH+xFnew(H∈B),ψℤ(xH)=xH(H∉B), _Z A_Z(M,G_small)→ A_Z(M,G_big), _Z(x_H)=x_H+x_F_new\ (H∈ B), _Z(x_H)=x_H\ (H∉ B), is injective with saturated image in each graded degree. Proof. Adjoining FnewF_new is the stellar subdivision of the Feichtner-Yuzvinsky fan along the cone indexed by B=Factor(Fnew,small)B=Factor(F_new,G_small), and ψℤ _Z is the corresponding Chow pullback [FY04]. By the integral blowup formula for Chow groups applied to this smooth stellar subdivision, one has degreewise Aℤp(M,big)≃ψℤAℤp(M,small)⊕⨁a=1c−1Aℤp−a(Z),A^p_Z(M,G_big) _Z\,A^p_Z(M,G_small) _a=1^c-1A^p-a_Z(Z), where c is the codimension of the stellar-subdivision stratum and Aℤ(Z)A_Z(Z) is the Chow ring of that stratum. By the Feichtner-Yuzvinsky description of a stellar subdivision, Aℤ(Z)A_Z(Z) is the tensor product of the factor Chow rings Aℤ(M|Fi,small|Fi)A_Z(M|F_i,G_small|F_i) and Aℤ(M/Fnew,small/Fnew)A_Z(M/F_new,G_small/F_new), which is free over ℤZ on the products of standard monomial bases [FY04]. Hence each exceptional summand Aℤp−a(Z)A^p-a_Z(Z) is free, so the cokernel of ψℤp _Z^p is free and ψℤp _Z^p is injective with saturated image. ∎ Lemma 5.4 (Filtered saturation). Let f:A→Bf A→ B be a homomorphism of finitely generated free abelian groups, filtered for finite separated filtrations, such that grfgrf is injective with saturated image in each degree. Then f has saturated image. Proof. The filtration is decreasing, finite, and separated, A=F0A⊇F1A⊇⋯⊇0A=F^0A F^1A ·s 0, with each grpfgr^pf injective with saturated image. Suppose nb=f(a)nb=f(a). We construct up∈FpAu_p∈ F^pA inductively. Assume bp:=b−f(u0+⋯+up−1)∈FpBb_p:=b-f(u_0+·s+u_p-1)∈ F^pB and nbp=f(ap)nb_p=f(a_p) for some ap∈FpAa_p∈ F^pA. In grpBgr^pB one has nbp¯=grpf(ap¯)n b_p=gr^pf( a_p), so by saturation of grpfgr^pf there is up¯∈grpA u_p ^pA with grpf(up¯)=bp¯gr^pf( u_p)= b_p. Choose a representative up∈FpAu_p∈ F^pA. Then bp+1:=bp−f(up)∈Fp+1Bb_p+1:=b_p-f(u_p)∈ F^p+1B and nbp+1=f(ap−nup)nb_p+1=f(a_p-nu_p); injectivity of grpfgr^pf gives ap−nup∈Fp+1Aa_p-nu_p∈ F^p+1A, so the induction continues. Finiteness gives b=f(u0+⋯+uN)∈f(A)b=f(u_0+·s+u_N)∈ f(A). ∎ Proposition 5.5 (Free cokernel of the one-step map). In the one-step situation of Set-up 5.1, the homomorphism ϕK:Kℤ(M,small)⟶Kℤ(M,big) _K K_Z(M,G_small) K_Z(M,G_big) is a split injection of abelian groups with free cokernel. Equivalently, the integer matrix of ϕK _K in the standard τ-monomial ℤZ-bases has Smith normal form with every nonzero invariant factor equal to 11. In particular ϕK _K has saturated image, and the same conclusions hold for the composite along any chain of one-flat enlargements. Proof. The map ϕK _K is the well-defined ℤZ-algebra homomorphism of Set-up 5.1, with the explicit generator rule ϕK(τH)=τH(H∉B),ϕK(τH)=τH+τFnew−τHτFnew(H∈B), _K( _H)= _H (H∉ B), _K( _H)= _H+ _F_new- _H _F_new (H∈ B), where B=Factor(Fnew,small)B=Factor(F_new,G_small). It is injective, and both rings are finitely generated free abelian groups with their standard τ-monomial ℤZ-bases by Proposition 2.4. We first record that ϕK _K has saturated image: if y∈Kℤ(M,big)y∈ K_Z(M,G_big) satisfies y⊗1∈im(ϕK⊗ℤℚ)y 1 ( _K _ZQ), then y∈im(ϕK)y ( _K). By Lemma 5.2 the τ-adic associated graded of KℤK_Z is the integral Feichtner-Yuzvinsky Chow ring, and on associated graded ϕK _K induces the homomorphism ψℤ _Z of Lemma 5.3, which is injective with saturated image in each graded degree. Lemma 5.4 then gives that ϕK _K has saturated image. This argument is intrinsic and uses no complete-fan, toric-center, or auxiliary-completion hypothesis. Now let C:=Kℤ(M,big)/ϕK(Kℤ(M,small))C:=K_Z(M,G_big)/ _K(K_Z(M,G_small)). It is finitely generated. If y∈Kℤ(M,big)y∈ K_Z(M,G_big) and n≥1n≥ 1 satisfy n⋅[y]=0n·[y]=0 in C, then ny=ϕK(x)ny= _K(x) for some x, so y⊗1=(ϕK⊗ℚ)((x⊗1)/n)∈im(ϕK⊗ℚ)y 1=( _K ) ((x 1)/n ) ( _K ); saturation gives y∈im(ϕK)y ( _K), i.e. [y]=0[y]=0. Hence C is torsion-free, and a finitely generated torsion-free abelian group is free. The short exact sequence 0→Kℤ(M,small)→ϕKℤ(M,big)→C→00→ K_Z(M,G_small) \ _K\ K_Z(M,G_big)→ C→ 0 therefore splits, so ϕK _K is a split injection. Choosing the standard τ-monomial basis of Kℤ(M,small)K_Z(M,G_small) and combining its image with a lifted ℤZ-basis of C gives a basis of Kℤ(M,big)K_Z(M,G_big) in which the matrix of ϕK _K is a block column with an identity block and a zero block; since a change to the standard τ-monomial bases is by unimodular matrices, the Smith normal form has every nonzero invariant factor equal to 11. A split injection with free cokernel has saturated image. For a chain 0⊂⋯⊂mG_0⊂·s _m the composite of split injections with free cokernels is again one. ∎ Feichtner and Yuzvinsky construct the smooth, generally noncomplete toric variety X(ΣM,)X( _M,G) from the atomic lattice and building set, and identify its Chow ring with Aℤ(M,)A_Z(M,G) [FY04]; the same one-step map is then induced by a toric pullback. We do not use any such model in the proof above, which is intrinsic to Kℤ(M,)K_Z(M,G). Proposition 5.6 (Upper-set line classes). Let Y∈smallY _small. Along a one-step chain, the class ΛY:=∏H∈Y≤H(1−τH)−1 _Y^G:= _ subarraycH \\ Y≤ H subarray(1- _H)^-1 is functorial: ϕK(ΛYsmall)=ΛYbig. _K( _Y^G_small)= _Y^G_big. Proof. Equivalently, write ηY:=1−ΛY _Y^G:=1- _Y^G. The one-step generator formula gives ϕK(ηYsmall)=ηYbig _K( _Y^G_small)= _Y^G_big by multiplying the factors indexed by flats above Y and using the factorization of FnewF_new by its maximal old factors. Subtracting from 11 gives the displayed formula for ΛY _Y. ∎ 6. Realizable comparison and generator normalization In this section, we prove the realizable statement in Theorem 1.1. The main issue is the integral normalization of the one-step comparison. Set-up 6.1. Let M be realized over ℂC by a linear subspace L not contained in any coordinate hyperplane. Let small⊂big=small∪FnewG_small _big=G_small∪\F_new\ be a one-step refinement, and let π:WL,big⟶WL,smallπ W_L,G_big W_L,G_small be the corresponding wonderful-model blowup. Put B:=Factor(Fnew,small).B:=Factor(F_new,G_small). Proposition 6.2 (Generator normalization). In Set-up 6.1, the integral one-step map satisfies ϕK(τH)=τH _K( _H)= _H for H∉BH∉ B, and ϕK(τH)=τH+τFnew−τHτFnew _K( _H)= _H+ _F_new- _H _F_new for H∈BH∈ B. Moreover ϕK(τE)=τE _K( _E)= _E. Proof. The first two formulas are the definition of the one-step integral K-ring map. Since every element of B is contained in the proper flat FnewF_new, the top flat E is not an element of B. Substituting H=EH=E gives ϕK(τE)=τE _K( _E)= _E. ∎ Remark (by authors). The AI system reproves this statement by the same method; it can be shortened by observing that the Chow pullback for the toric models agrees with the Chow pullback for the wonderful models. Lemma 6.3 (Wonderful center product and torsion-freeness). Let M be realized by L. In the one-step wonderful-model blowup π:WL,big→WL,smallπ W_L,G_big→ W_L,G_small, the center Z is canonically isomorphic to a product of wonderful models attached to the restriction factors M|FiM|F_i for Fi∈BF_i∈ B and to the contraction factor M/FnewM/F_new. In particular K0(Z)K_0(Z) is a free abelian group. Proof. The one-step enlargement is a De Concini-Procesi blowup whose center is the intersection of the boundary divisors indexed by B; that intersection is canonically the product of the wonderful models of the restriction factors M|FiM|F_i and the contraction factor M/FnewM/F_new [DCP95]. We prove slightly more generally, by simultaneous induction on the De Concini-Procesi blowup construction, that every finite product of such wonderful models has free K0K_0. The base case is a finite product of projective spaces, whose K0K_0 is free. Suppose one factor X′X is obtained from X by blowing up a smooth center C, and let Y be the product of the remaining factors. Since blowup commutes with flat base change, X′×Y≃BlC×Y(X×Y)X × Y _C× Y(X× Y), and the K-theoretic blowup formula [Tho93] gives K0(X′×Y)≃K0(X×Y)⊕⨁a=1c−1K0(C×Y),c=codim(C,X).K_0(X × Y) K_0(X× Y) _a=1^c-1K_0(C× Y), c=codim(C,X). The groups on the right are free by induction, since C is a product of smaller wonderful models. Hence every finite product of wonderful models occurring in the construction has free K0K_0, and in particular K0(Z)K_0(Z) is free. ∎ Lemma 6.4 (Geometric pullback saturation). Let M be realized by L, and let π:WL,big→WL,smallπ W_L,G_big→ W_L,G_small be the one-step wonderful-model blowup along the smooth center Z of Lemma 6.3. Then the pullback π∗:K0(WL,small)→K0(WL,big)π^* K_0(W_L,G_small)→ K_0(W_L,G_big) is a split injection with free cokernel; in particular its image is saturated. Proof. The blowup of a smooth variety along a smooth center has the K-theoretic decomposition [Tho93] K0(WL,big)=π∗K0(WL,small)⊕⨁a=1c−1j∗(p∗K0(Z)ta),K_0(W_L,G_big)=π^*K_0(W_L,G_small) _a=1^c-1j_* (p^*K_0(Z)\,t^a ), where j is the exceptional inclusion, p the projective-bundle projection, t=[(−1)]t=[O(-1)], and c the codimension. Each exceptional summand is a twist of K0(Z)K_0(Z), which is free by Lemma 6.3. Hence π∗π^* is a split injection with free cokernel, so its image is saturated. ∎ Proposition 6.5 (Integral theta descent). Here AsmallA_small and Abig=π∗AsmallA_big=π^*A_small denote the hyperplane line bundles on the wonderful models WL,smallW_L,G_small and WL,bigW_L,G_big, so that the top generator corresponds to 1−[A]1-[A] under the atom-character relation. Assume that the refined comparison θbig:Kℤ(M,big)⟶K0(WL,big) _big K_Z(M,G_big) K_0(W_L,G_big) sends every proper-flat generator τH _H to [DH][O_D_H] and sends the top generator τE _E to 1−[Abig]1-[A_big]. Then there exists a unique integral unital ring isomorphism θsmall:Kℤ(M,small)⟶K0(WL,small) _small K_Z(M,G_small) K_0(W_L,G_small) such that π∗(θsmall(ξ))=θbig(ϕK(ξ))π^*( _small(ξ))= _big( _K(ξ)) for every ξ∈Kℤ(M,small)ξ∈ K_Z(M,G_small). It satisfies θsmall((1−τH)−1)=[(DH)] _small ((1- _H)^-1 )=[O(D_H)] for every proper flat H∈smallH _small. Proof. By Proposition 5.5 the combinatorial image of ϕK _K is saturated, and by Lemma 6.4 the geometric pullback image of π∗π^* is saturated. Via θbig _big, the subgroups θbig(ϕKKℤ(M,small)) _big( _KK_Z(M,G_small)) and π∗K0(WL,small)π^*K_0(W_L,G_small) are saturated subgroups of the free abelian group K0(WL,big)K_0(W_L,G_big), and the rational comparison identifies their ℚQ-spans. Two saturated subgroups of a free abelian group with the same ℚQ-span are equal, so θbig(ϕKKℤ(M,small))=π∗K0(WL,small). _big ( _KK_Z(M,G_small) )=π^*K_0(W_L,G_small). This identification upgrades the rational descent to an integral descent, giving a unique θsmall _small. The generator values follow from the geometric generator compatibility. If H∉BH∉ B, the boundary divisor pulls back to the boundary divisor. If H∈BH∈ B, then the total transform has K-class [DH]+[DFnew]−[DH][DFnew],[O_D_H]+[O_D_F_new]-[O_D_H]\,[O_D_F_new], which is exactly the image of τH+τFnew−τHτFnew _H+ _F_new- _H _F_new. The top generator gives 1−[Asmall]1-[A_small] after descent. Inverting 1−τH1- _H gives the line bundle class [(DH)][O(D_H)]. ∎ Proof of Theorem 1.1, realizable clause. Start at the maximal building set. The Larson-Li-Payne-Proudfoot comparison identifies the intrinsic integral K-ring with the K-ring of the maximal wonderful model [LLPP24]. Along any chain from the maximal building set down to G, Proposition 6.5 descends the comparison integrally. Proposition 6.2 records the generator normalization at every one-step descent. Thus we obtain an integral unital ring isomorphism θℤ:Kℤ(M,)→K0(WL,) _G^Z K_Z(M,G)→ K_0(W_L,G) with θℤ((1−τF)−1)=[WL,(DF)] _G^Z ((1- _F)^-1 )=[O_W_L,G(D_F)] for every F∈∘F . Both θℤ(TM,ℤ) _G^Z(T_M,G^Z) and [TWL,][T_W_L,G] rationalize to the same class: after tensoring with ℚQ this is the realizable specialization θ(TM,)=[TWL,]θ(T_M,G)=[T_W_L,G] of Proposition 3.3 (Step 6), since ρK(TM,ℤ)=TM, _K(T_M,G^Z)=T_M,G by Proposition 4.3 and the quotient representative was descended compatibly with the maximal Berget-Eur-Spink-Tseng quotient class. The Grothendieck group K0(WL,)K_0(W_L,G) of the smooth projective wonderful model is torsion-free, so two of its elements with equal rationalizations are equal. Hence θℤ(TM,ℤ)=[TWL,]. _G^Z(T_M,G^Z)=[T_W_L,G]. ∎ 7. The Hilbert identity The goal of this section is to prove the PK=HilbP^K=Hilb part of the rational tangent-class theorem. The proof first treats the maximal building set and then propagates the identity through one-flat refinements. Definition 7.1. Let (N,ℋ)(N,H) be a top-containing pair. Write CQ(N,ℋ;u)=∏a=1qQ(N,ℋ)(1+yau)C_Q(N,H;u)= _a=1^q_Q(N,H)(1+y_au) after passing to a splitting algebra, where qQ(N,ℋ)=|ℋ∖top(N)|−(rank(N)−1)q_Q(N,H)=|H \top(N)\|-(rank(N)-1). Put gz(v):=(1−zexp(−v))v1−exp(−v),gz(0)=1−z.g_z(v):=(1-z (-v)) v1- (-v), g_z(0)=1-z. The signed K-theoretic Todd polynomial is PK(N,ℋ;z):=degN,ℋ[∏Y∈ℋ∖top(N)gz(xY)∏a=1qQ(N,ℋ)gz(ya)−1]rank(N)−1.P^K(N,H;z):= _N,H [ _Y \top(N)\g_z(x_Y) _a=1^q_Q(N,H)g_z(y_a)^-1 ]_rank(N)-1. Equivalently, PK(N,ℋ;z)=degN,ℋ(ch(λ−z(TN,ℋ∨))td(TN,ℋ)).P^K(N,H;z)= _N,H (ch( _-z(T_N,H ))td(T_N,H) ). Lemma 7.2 (One-step quotient integrands). Let M, smallG_small, bigG_big, and F be as in Proposition 7.5, and let ψ:Aℚ(M,small)⟶Aℚ(M,big)ψ A_Q(M,G_small) A_Q(M,G_big) be the Chow pullback for the one-flat enlargement. Then CQ(M,big;u)=ψ(CQ(M,small;u)).C_Q(M,G_big;u)=ψ(C_Q(M,G_small;u)). Consequently, if CQ(M,small;u)=∏a=1q(1+yau)C_Q(M,G_small;u)= _a=1^q(1+y_au) in a splitting algebra, then a splitting-root presentation for CQ(M,big;u)C_Q(M,G_big;u) is ∏a=1q(1+ψ(ya)u)⋅(1+0⋅u). _a=1^q(1+ψ(y_a)u)·(1+0· u). Moreover, for every top-containing pair (N,ℋ)(N,H) appearing in the one-flat recursion, PK(N,ℋ;z)P^K(N,H;z) is the same as the Chow splitting-root expression PQ(N,ℋ;z):=degN,ℋ[∏Y∈ℋ∖top(N)gz(xY)∏a=1qQ(N,ℋ)gz(ya)−1]rank(N)−1.P_Q(N,H;z):= _N,H [ _Y \top(N)\g_z(x_Y) _a=1^q_Q(N,H)g_z(y_a)^-1 ]_rank(N)-1. Proof. The descended quotient Chern polynomial CQC_Q is characterized by pullback to the maximal building set. Since the refinement from smallG_small to the maximal building set factors through bigG_big, injectivity of Chow pullback gives the one-step compatibility CQ(M,big;u)=ψ(CQ(M,small;u))C_Q(M,G_big;u)=ψ(C_Q(M,G_small;u)). The integer qQq_Q increases by one when the single proper flat F is added, while the rank of M is unchanged. Thus the additional quotient root may be taken to be 0. The last assertion is the splitting-root convention in Definition 7.1: the boundary line summands (1−τY)−1(1- _Y)^-1 in TN,ℋT_N,H contribute the factors gz(xY)g_z(x_Y), while the quotient class QN,ℋQ_N,H, with roots yay_a, contributes the inverse factors gz(ya)−1g_z(y_a)^-1. The expression is independent of the chosen splitting roots because the Hirzebruch product is determined by the virtual total Chern series and the virtual rank. ∎ Remark (by authors). By definition, PK=PQP^K=P_Q. Lemma 7.3 (Exceptional trace). Keep the notation of Lemma 7.2, and write Factorsmall(F)=F1,…,Fℓ.Factor_G_small(F)=\F_1,…,F_ \. Then ℓ≥2 ≥ 2. Put e=xF∈Aℚ1(M,big)e=x_F∈ A^1_Q(M,G_big), and set Ismall(z):=∏Y∈small∖Egz(xY)∏a=1qQ(M,small)gz(ya)−1.I_small(z):= _Y _small \E\g_z(x_Y) _a=1^q_Q(M,G_small)g_z(y_a)^-1. With the quotient roots of bigG_big chosen as in Lemma 7.2, the big integrand is Ibig(z)=ψ(Ismall(z))RF(z),I_big(z)=ψ(I_small(z))R_F(z), where RF(z)=gz(e)1−z∏i=1ℓgz(xFi)gz(xFi+e).R_F(z)= g_z(e)1-z _i=1 g_z(x_F_i)g_z(x_F_i+e). Therefore PQ(M,big;z)=PQ(M,small;z)+ExcF(z),P_Q(M,G_big;z)=P_Q(M,G_small;z)+Exc_F(z), where ExcF(z):=degM,big[ψ(Ismall(z))(RF(z)−1)]rank(M)−1.Exc_F(z):= _M,G_big [ψ(I_small(z))(R_F(z)-1) ]_rank(M)-1. If ZFZ_F is the one-flat center and i∗:Aℚ(M,small)→A∙(ZF)ℚi^* A_Q(M,G_small)→ A (Z_F)_Q is the center restriction, then ExcF(z)=(z+z2+⋯+zℓ−1)degZF([θZ(z)]rank(M)−1−ℓ),Exc_F(z)=(z+z^2+·s+z -1) _Z_F ([ _Z(z)]_rank(M)-1- ), where θZ(z):=i∗(Ismall(z))∏i=1ℓgz(i∗xFi)−1. _Z(z):=i^*(I_small(z)) _i=1 g_z(i^*x_F_i)^-1. Proof. The inequality ℓ≥2 ≥ 2 follows from the building-set axiom. Indeed, the factors are the maximal old building-set elements contained in F; if there were only one such factor, its union would be F, so F would already lie in smallG_small. The formula for IbigI_big is obtained by comparing the boundary and quotient roots after the one-flat pullback. The new boundary root contributes gz(e)g_z(e), and the new zero quotient root contributes gz(0)−1=(1−z)−1g_z(0)^-1=(1-z)^-1. For each old factor FiF_i, the pullback of the old boundary class is xFi+ex_F_i+e, while the boundary class in the big model is xFix_F_i, giving the ratio gz(xFi)/gz(xFi+e)g_z(x_F_i)/g_z(x_F_i+e). This gives the displayed correction factor RF(z)R_F(z). Taking the (rank(M)−1)(rank(M)-1)-degree trace, the term ψ(Ismall(z))ψ(I_small(z)) has the same trace as Ismall(z)I_small(z), because the normalized top-degree trace is preserved by the one-step Chow pullback. The remaining term is ExcF(z)Exc_F(z). It remains to identify this exceptional term. Let π:Aℚ(M,big)→Aℚ(M,small)π A_Q(M,G_big)→ A_Q(M,G_small) be the Chow pushforward and i∗i_* the Gysin map from the center. The one-flat exceptional pushforward calculation gives π∗(ψ(Ismall(z))(RF(z)−1))=(z+z2+⋯+zℓ−1)i∗(θZ(z)). _* (ψ(I_small(z))(R_F(z)-1) )=(z+z^2+·s+z -1)\,i_*( _Z(z)). The top trace is compatible with π∗ _*, and the center projection formula identifies degM,small(i∗u) _M,G_small(i_*u) with degZF(u) _Z_F(u) in top center degree. Taking homogeneous degree rank(M)−1rank(M)-1 proves the formula for ExcF(z)Exc_F(z). ∎ Lemma 7.4 (Center factorization). With the notation of Lemma 7.3, there is a canonical graded product identification A∙(ZF)ℚ≅(⨂i=1ℓAℚ(M|Fi,small|Fi))⊗Aℚ(M/F,small/F).A (Z_F)_Q ( _i=1 A_Q(M|F_i,G_small|F_i) ) A_Q(M/F,G_small/F). Under this identification, the top trace on ZFZ_F is the product of the top traces on the factors, and θZ(z)=(∏i=1ℓIi(z))Icon(z), _Z(z)= ( _i=1 I_i(z) )I_con(z), where Ii(z)I_i(z) is the splitting-root integrand defining PQ(M|Fi,small|Fi;z)P_Q(M|F_i,G_small|F_i;z), and Icon(z)I_con(z) is the splitting-root integrand defining PQ(M/F,small/F;z)P_Q(M/F,G_small/F;z). Proof. The center ZFZ_F is the orbit closure of the cone generated by the rays indexed by F1,…,FℓF_1,…,F_ in the nested fan for smallG_small. Its closed star is canonically the product of the nested fans for the induced restriction pairs (M|Fi,small|Fi)(M|F_i,G_small|F_i) and the induced contraction pair (M/F,small/F)(M/F,G_small/F). Passing to Chow rings gives the displayed tensor-product identification. The top degree is ∑i=1ℓ(rank(Fi)−1)+(rank(M)−rank(F)−1)=rank(M)−1−ℓ, _i=1 (rank(F_i)-1)+(rank(M)-rank(F)-1)=rank(M)-1- , and the degree of a top simple tensor is the product of the factor degrees. We now compare the integrands. First, after the normal factors indexed by F1,…,FℓF_1,…,F_ are removed, the boundary Chern polynomial on the center becomes the product of the boundary Chern polynomials of the restriction factors and the contraction factor. Second, the quotient polynomial restricts to the product i∗CQ(M,small;u)=(∏i=1ℓCQ(M|Fi,small|Fi;u))⋅CQ(M/F,small/F;u)i^*C_Q(M,G_small;u)= ( _i=1 C_Q(M|F_i,G_small|F_i;u) )· C_Q(M/F,G_small/F;u) under the same product identification, where the displayed multiplication is in the tensor product of the factor Chow rings. Equivalently, after passing to a common splitting algebra, the virtual total Chern series of the center presentation is the product of the virtual total Chern series of the induced factor presentations. The virtual ranks agree by the top-degree calculation above. Since the gzg_z-Hirzebruch product depends only on the virtual total Chern series and virtual rank, the center integrand is exactly the product of the restriction and contraction integrands. ∎ Remark (by authors). The fact that the quotient polynomial restricts to the product is implied in the proof of Proposition 3.3. This corresponds to [Che26, Proposition 4.10]. Proposition 7.5 (One-flat K-theoretic recursion). Let M be a loopless matroid on E, and let big=small∪FG_big=G_small∪\F\ be a one-flat enlargement of top-containing Feichtner–Yuzvinsky building sets, with F∉smallF _small a nonempty proper flat. Let Factorsmall(F)=F1,…,Fℓ.Factor_G_small(F)=\F_1,…,F_ \. Then PK(M,big;z) P^K(M,G_big;z) =PK(M,small;z) =P^K(M,G_small;z) +(z+z2+⋯+zℓ−1)(∏i=1ℓPK(M|Fi,small|Fi;z))PK(M/F,small/F;z). +(z+z^2+·s+z -1) ( _i=1 P^K(M|F_i,G_small|F_i;z) )P^K(M/F,G_small/F;z). Proof. By Lemma 7.2, PK=PQP^K=P_Q for (M,small)(M,G_small), (M,big)(M,G_big), every restriction factor (M|Fi,small|Fi)(M|F_i,G_small|F_i), and the contraction factor (M/F,small/F)(M/F,G_small/F). It is enough to prove the formula for PQP_Q. Lemma 7.3 gives PQ(M,big;z)=PQ(M,small;z)+(z+z2+⋯+zℓ−1)degZF([θZ(z)]rank(M)−1−ℓ).P_Q(M,G_big;z)=P_Q(M,G_small;z)+(z+z^2+·s+z -1) _Z_F ([ _Z(z)]_rank(M)-1- ). By Lemma 7.4, the center integrand is the product of the induced restriction and contraction integrands, and the center trace is the product trace. Hence degZF([θZ(z)]rank(M)−1−ℓ)=(∏i=1ℓPQ(M|Fi,small|Fi;z))PQ(M/F,small/F;z). _Z_F ([ _Z(z)]_rank(M)-1- )= ( _i=1 P_Q(M|F_i,G_small|F_i;z) )P_Q(M/F,G_small/F;z). Substitution gives the one-flat recursion for PQP_Q. Replacing each PQP_Q by the equal PKP^K gives the stated formula. The preceding exceptional-trace lemma also proves that a genuine one-flat enlargement has ℓ≥2 ≥ 2. ∎ Proposition 7.6 (One-flat Hilbert recursion). Let M, smallG_small, bigG_big, F, and ℓ be as in Proposition 7.5. Then Hilb(M,big;z) (M,G_big;z) =Hilb(M,small;z) =Hilb(M,G_small;z) +(z+z2+⋯+zℓ−1)(∏i=1ℓHilb(M|Fi,small∣Fi;z))Hilb(M/F,small/F;z). +(z+z^2+·s+z -1) ( _i=1 Hilb(M|F_i,G_small|F_i;z) )Hilb(M/F,G_small/F;z). Proof. The one-step Hilbert-series recursion of Eur–Ferroni–Matherne–Pagaria–Vecchi [EFM+25] gives Hilb(M,big;z) (M,G_big;z) =Hilb(M,small;z) =Hilb(M,G_small;z) +(z+z2+⋯+zℓ−1)Hilb(M|F,small∣F;z)Hilb(M/F,small/F;z). +(z+z^2+·s+z -1)Hilb(M|F,G_small|F;z)Hilb(M/F,G_small/F;z). Here ℓ is the number of maximal elements of smallG_small contained in F, and the restriction and contraction building sets are the induced ones. It remains only to rewrite the restriction factor in the form used by the center product. The factors F1,…,FℓF_1,…,F_ give the decomposition of the interval below F, hence M|FM|F is the direct sum of the restrictions M|FiM|F_i. The induced restriction building set small|FG_small|F has maximal elements F1,…,FℓF_1,…,F_ , and its Feichtner–Yuzvinsky Chow algebra is the tensor product of the Chow algebras Aℚ(M|Fi,small|Fi),1≤i≤ℓ.A_Q(M|F_i,G_small|F_i), 1≤ i≤ . Taking graded dimensions gives Hilb(M|F,small∣F;z)=∏i=1ℓHilb(M|Fi,small∣Fi;z).Hilb(M|F,G_small|F;z)= _i=1 Hilb(M|F_i,G_small|F_i;z). Substituting this factorization into the one-step recursion proves the displayed formula. ∎ Proposition 7.7 (Maximal endpoint). Let R be a loopless matroid with finite lattice of flats, bottom flat ∅ , top flat top(R)top(R), and rank function rkRrk_R. Let max(R):=L(R)∖∅.G_ (R):=L(R) \ \. Then PK(R,max(R);z)=Hilb(R,max(R);z),P^K(R,G_ (R);z)=Hilb(R,G_ (R);z), and both sides are equal to ∑∅=F0<F1<⋯<Fm∏i=1mz1−zrkR(Fi)−rkR(Fi−1)−11−z, _ =F_0<F_1<·s<F_m _i=1^mz 1-z^rk_R(F_i)-rk_R(F_i-1)-11-z, where the sum ranges over all chains of flats starting at ∅ , including the chain with only ∅ , whose summand is 11. Proof. Put =max(R)G=G_ (R). Cheng’s preprint [Che25, Definition 3.1] defines, for every loopless matroid R, a tangent K-class TRChT_R^Ch on the maximal matroid toric model by restricting TXE−QRT_X_E-Q_R from the permutohedral variety, where QRQ_R is the BEST tautological quotient class. The displayed tangent formula before [Che25, Theorem 3.4] gives the boundary contribution ∏F(1+xF) _F(1+x_F), and [Che25, Theorem 3.4] identifies the quotient Chern factor. By Proposition 3.3 in the maximal case, the MatTan class TR,T_R,G is reconstructed from the same BEST quotient Chern polynomial. Its virtual rank is |∖top(R)|−qQ(R,)=rkR(top(R))−1|G \top(R)\|-q_Q(R,G)=rk_R(top(R))-1, since qQ(R,)=|∖top(R)|−(rkR(top(R))−1)q_Q(R,G)=|G \top(R)\|-(rk_R(top(R))-1); this equals the rank of TRChT_R^Ch, the tangent bundle of the maximal wonderful model of dimension rkR(top(R))−1rk_R(top(R))-1. Two rational K-classes of equal virtual rank and equal total Chern class have the same Chern character, and therefore agree under the rational Chern-character isomorphism. Equivalently, they have the same rank and the same total Chern class cz(T)=cz(Q)−1∏F∈∖top(R)(1+zxF),c_z(T)=c_z(Q)^-1 _F \top(R)\(1+zx_F), with the homogeneous z-grading convention used in this paper. Cheng’s Chow-polynomial formula [Che25, Theorem 1.1(4)] gives dimℚAℚ(R,)p=(−1)pdegR,(ch(⋀pTR,∨)td(TR,)) _QA_Q(R,G)^p=(-1)^p _R,G (ch ( ^pT_R,G )td(T_R,G) ) for every p. Therefore the coefficient of zpz^p in PK(R,;z)P^K(R,G;z) is the coefficient of zpz^p in Hilb(R,;z)Hilb(R,G;z). Finally, the displayed chain formula for the maximal Hilbert polynomial is the Ferroni–Matherne–Stevens–Vecchi formula [FMSV24, Proposition 3.5]. Combining these two equalities proves the proposition. ∎ Theorem 7.8 (Hilbert identity for arbitrary building sets). For every loopless matroid M and every top-containing Feichtner–Yuzvinsky building set G, one has PK(M,;z)=Hilb(M,;z).P^K(M,G;z)=Hilb(M,G;z). Consequently, if rank(M)=d+1rank(M)=d+1, then for every integer i with 0≤i≤d0≤ i≤ d, dimℚAℚ(M,)i=(−1)idegM,(ch(⋀iTM,∨)td(TM,)). _QA_Q(M,G)^i=(-1)^i _M,G (ch ( ^iT_M,G )td(T_M,G) ). Proof. We prove the equality by induction on e=rank(M)−1e=rank(M)-1. If e=0e=0, then M has no nonempty proper flats and the only top-containing building set is max(M)G_ (M), so the assertion is Proposition 7.7. Assume e>0e>0, and assume the theorem is known for all loopless matroids of smaller rank. Let M be loopless with rank(M)−1=erank(M)-1=e, and let G be a top-containing building set. Choose a one-flat refinement chain =0⊂1⊂⋯⊂s=max(M),j=j−1∪Fj.G=G_0 _1⊂·s _s=G_ (M), _j=G_j-1∪\F_j\. Proposition 7.7 gives PK(M,s;z)=Hilb(M,s;z).P^K(M,G_s;z)=Hilb(M,G_s;z). We descend along the chain. Suppose the equality is known for (M,j)(M,G_j), where j=j−1∪FjG_j=G_j-1∪\F_j\, and write Factorj−1(Fj)=F1,…,FℓFactor_G_j-1(F_j)=\F_1,…,F_ \. Each restriction factor (M|Fi,j−1|Fi)(M|F_i,G_j-1|F_i) and the contraction factor (M/Fj,j−1/Fj)(M/F_j,G_j-1/F_j) has smaller rank than M. By the induction hypothesis, PK=HilbP^K=Hilb for all these induced factors. The exceptional terms in Propositions 7.5 and 7.6 are therefore equal. Subtracting the two one-flat recursion formulas gives PK(M,j−1;z)=Hilb(M,j−1;z).P^K(M,G_j-1;z)=Hilb(M,G_j-1;z). Descending from j=sj=s to j=0j=0 proves the polynomial identity for (M,)(M,G). The coefficient identity is the equality of the coefficient of ziz^i in PK(M,;z)=Hilb(M,;z)P^K(M,G;z)=Hilb(M,G;z), using Definition 7.1. ∎ 8. Nested Segre tails and the Chern-alpha bound The goal of this section is to prove the Chern-alpha lower bound. The proof uses connected flats, nested supports, and a truncated generating-polynomial identity. Every generating identity below is stated in ℚ[t]/(tr)Q[t]/(t^r), or equivalently coefficientwise in degrees 0≤k≤r−10≤ k≤ r-1. Set-up 8.1. Let N be a connected loopless matroid of rank r on E. Put e:=r−1e:=r-1, ℋ:=ℋconn(N)H:=H_conn(N), and max:=L(N)∖∅G_ :=L(N) \ \. Choose an ordering X1,…,XmX_1,…,X_m of max∖ℋG_ such that whenever Xp<XqX_p<X_q, one has q<pq<p. Put s:=ℋ∪X1,…,XsG_s:=H∪\X_1,…,X_s\, and let ψ:Aℚ(N,ℋ)→Aℚ(N,max)ψ A_Q(N,H)→ A_Q(N,G_ ) be the composite Chow homomorphism along this reverse-containment refinement chain. Let α:=−xEα:=-x_E, and set ΓN(z):=CBEST(N;z)−1∏Y∈ℋ∖E(1+zψ(xY)). _N(z):=C_BEST(N;z)^-1 _Y \E\(1+zψ(x_Y)). For an ℋH-nested subset S⊂ℋ∖ES \E\, let U(S)U(S) be the join of all flats in S, with U(∅)=∅U( )= . For A∈SA∈ S, let A−S:=⋁B∈SB<AB,A_-^S:= _ subarraycB∈ S\\ B<A subarrayB, with A−S=∅A_-^S= if no such B exists, and put qS(A):=rankN(A)−rankN(A−S),ω(S):=∏A∈S(qS(A)−1),q_S(A):=rank_N(A)-rank_N(A_-^S), ω(S):= _A∈ S(q_S(A)-1), with ω(∅)=1ω( )=1. Lemma 8.2 (Pulled connected-base Chern-alpha). With notation as in Set-up 8.1, for every 0≤k≤e0≤ k≤ e, Ik(N,ℋ):=degN,ℋ(ck(TN,ℋ)(−xE)e−k)=degN,max([ΓN(z)]kαe−k).I_k(N,H):= _N,H (c_k(T_N,H)(-x_E)^e-k )= _N,G_ ([ _N(z)]_kα^e-k ). On the right, α=−xEα=-x_E denotes the pullback of the same top-flat class to Aℚ(N,max)A_Q(N,G_ ). Proof. By Proposition 3.3 the tangent class has formal total Chern polynomial cz(TN,ℋ)=CQ(N,ℋ;z)−1∏Y∈ℋ∖E(1+zxY),c_z(T_N,H)=C_Q(N,H;z)^-1 _Y \E\(1+zx_Y), and the descended quotient Chern polynomial pulls back to the maximal model as ψ(CQ(N,ℋ;z))=CBEST(N;z)ψ(C_Q(N,H;z))=C_BEST(N;z). Since ψ is a homomorphism of graded ℚQ-algebras, it commutes with the z-graded inverse and with finite products, so ψ(cz(TN,ℋ))=ψ(CQ(N,ℋ;z))−1∏Y∈ℋ∖E(1+zψ(xY))=CBEST(N;z)−1∏Y∈ℋ∖E(1+zψ(xY))=ΓN(z).ψ(c_z(T_N,H))=ψ(C_Q(N,H;z))^-1 _Y \E\(1+z\,ψ(x_Y))=C_BEST(N;z)^-1 _Y \E\(1+z\,ψ(x_Y))= _N(z). Taking the homogeneous Chow-degree-k part gives ψ(ck(TN,ℋ))=[ΓN(z)]kψ(c_k(T_N,H))=[ _N(z)]_k. Fix 0≤k≤e0≤ k≤ e. The class ck(TN,ℋ)(−xE)e−kc_k(T_N,H)(-x_E)^e-k is homogeneous of top Chow degree e; since the top flat E lies in every building set, ψ(xE)=xEψ(x_E)=x_E, so ψ carries it to [ΓN(z)]kαe−k[ _N(z)]_kα^e-k. Finally ψ is the pullback along the proper birational toric morphism Xmax→XℋX_G_ → X_H of smooth varieties, and such a pullback preserves the degree of a top-dimensional class, because the pushforward of the fundamental class is the fundamental class; hence degN,max(ψ(ξ))=degN,ℋ(ξ) _N,G_ (ψ(ξ))= _N,H(ξ) for every top-degree class ξ. Applying this to ξ=ck(TN,ℋ)(−xE)e−kξ=c_k(T_N,H)(-x_E)^e-k yields the displayed equality. ∎ Lemma 8.3 (Connected-flag weights). Let N be connected and loopless, and put ℋ0:=ℋconn(N)∪∅H^0:=H_conn(N)∪\ \. For F∈ℋ0F ^0, define τN(∅)=1 _N( )=1 and τN(F):=∑G∈ℋ0G<FτN(G)(rankN(F)−rankN(G)−1) _N(F):= _ subarraycG ^0\\ G<F subarray _N(G)(rank_N(F)-rank_N(G)-1) for nonempty F∈ℋconn(N)F _conn(N). Then τN(F)=∑∅=F0<F1<⋯<Fm=F∏i=1m(rankN(Fi)−rankN(Fi−1)−1), _N(F)= _ =F_0<F_1<·s<F_m=F _i=1^m(rank_N(F_i)-rank_N(F_i-1)-1), where the sum ranges over strict chains in ℋ0H^0 ending at F. In particular, τN(F)≥0 _N(F)≥ 0. Proof. We argue by induction over the finite poset ℋ0H^0. The assertion is immediate for F=∅F= . For nonempty F, every strict chain ending at F has a unique penultimate element G<FG<F, and appending F multiplies the weight of a chain ending at G by rankN(F)−rankN(G)−1rank_N(F)-rank_N(G)-1. Summing over G gives the recursive formula. Each rank-gap-minus-one factor is nonnegative, so τN(F)≥0 _N(F)≥ 0. ∎ Lemma 8.4 (Chain and non-chain decomposition). For every integer k, define ANk(N):=∑Sω(S)B(r−rankN(U(S)),k−rankN(U(S))),AN_k(N):= _Sω(S)B(r-rank_N(U(S)),k-rank_N(U(S))), where S ranges over all ℋH-nested subsets of ℋ∖EH \E\. Let NC(N)NC(N) be the set of such S that are not totally ordered by inclusion. Then ANk(N) AN_k(N) =B(r,k)+∑F∈ℋ∖EτN(F)B(r−rankN(F),k−rankN(F)) =B(r,k)+ _F \E\ _N(F)B(r-rank_N(F),k-rank_N(F)) +∑S∈NC(N)ω(S)B(r−rankN(U(S)),k−rankN(U(S))). + _S∈ NC(N)ω(S)B(r-rank_N(U(S)),k-rank_N(U(S))). Proof. Partition the nested supports into chains and non-chains. A totally ordered support is uniquely S=F1<⋯<Fm,S=\F_1<·s<F_m\, including the empty chain when m=0m=0. Its join is FmF_m, with Fm=∅F_m= in the empty case, and ω(S)=∏i=1m(rankN(Fi)−rankN(Fi−1)−1).ω(S)= _i=1^m(rank_N(F_i)-rank_N(F_i-1)-1). Summing the chain contributions with fixed terminal flat F gives τN(F) _N(F) by Lemma 8.3. The remaining contributions are exactly those indexed by NC(N)NC(N). ∎ Lemma 8.5 (Low-degree chain reduction). With notation as in Set-up 8.1, assume 0≤k≤e0≤ k≤ e and k≤3k≤ 3. Then the non-chain residual in Lemma 8.4 vanishes: ∑S∈NC(N)ω(S)B(r−rankN(U(S)),k−rankN(U(S)))=0. _S∈ NC(N)ω(S)B(r-rank_N(U(S)),k-rank_N(U(S)))=0. Consequently ANk(N)=B(r,k)+∑F∈ℋ∖EτN(F)B(r−rankN(F),k−rankN(F)).AN_k(N)=B(r,k)+ _F \E\ _N(F)B(r-rank_N(F),k-rank_N(F)). In particular, in degrees k≤3k≤ 3, the all-nested coefficient equals the connected-flag τN _N-expression. Proof. Fix S∈NC(N)S∈ NC(N). If ω(S)=0ω(S)=0, its summand is zero. Assume ω(S)≠0ω(S)≠ 0. Since S is not a chain, choose incomparable A,B∈SA,B∈ S. Let J=A∨BJ=A B. The nestedness condition says J∉ℋJ . Since E∈ℋE , J≠EJ≠ E, so J is a proper nonempty flat. Because ℋ=ℋconn(N)H=H_conn(N), the restriction N|JN|J is disconnected. Let J1,…,JtJ_1,…,J_t be the connected components of N|JN|J. A connected flat contained in J lies in one component. The flats A and B cannot lie in the same component: otherwise their join would be contained in that component, contradicting A∨B=JA B=J. Since ω(S)≠0ω(S)≠ 0, every local factor qS(C)−1q_S(C)-1 for C∈SC∈ S is nonzero. Also A−S<CA_-^S<C for each C∈SC∈ S, so qS(C)≥1q_S(C)≥ 1, and nonvanishing gives qS(C)≥2q_S(C)≥ 2. Applying this to A and B, we obtain rankN(A)≥2rank_N(A)≥ 2 and rankN(B)≥2rank_N(B)≥ 2. Since rank is additive over different connected components of N|JN|J, rankN(J)≥rankN(A)+rankN(B)≥4.rank_N(J) _N(A)+rank_N(B)≥ 4. Thus rankN(U(S))≥4rank_N(U(S))≥ 4. For k≤3k≤ 3, the integer k−rankN(U(S))k-rank_N(U(S)) is negative, and hence B(r−rankN(U(S)),k−rankN(U(S)))=0.B(r-rank_N(U(S)),k-rank_N(U(S)))=0. Every non-chain summand is zero. ∎ Proposition 8.6 (Low-degree Gamma formula). With notation as in Set-up 8.1, for every integer k with 0≤k≤e0≤ k≤ e and k≤3k≤ 3, one has degN,max([ΓN(z)]kαe−k)=ANk(N). _N,G_ ([ _N(z)]_kα^e-k )=AN_k(N). Equivalently, in this range, degN,max([ΓN(z)]kαe−k)=B(r,k)+∑F∈ℋ∖EτN(F)B(r−rankN(F),k−rankN(F)). _N,G_ ([ _N(z)]_kα^e-k )=B(r,k)+ _F \E\ _N(F)B(r-rank_N(F),k-rank_N(F)). Proof. We first prove the first displayed equality. For 0≤k≤30≤ k≤ 3, a nonempty ℋH-nested support S with ω(S)≠0ω(S)≠ 0 and rankN(U(S))≤krank_N(U(S))≤ k must be a singleton: every element has qS(A)≥2q_S(A)≥ 2 (so rank ≥2≥ 2), and an incomparable pair A,B∈SA,B∈ S would force rankN(U(S))≥rankN(A)+rankN(B)≥4rank_N(U(S)) _N(A)+rank_N(B)≥ 4, since their join A∨B∉ℋA B makes N|(A∨B)N|(A B) disconnected and A,BA,B lie in distinct components. Hence only the empty support and the singleton rank-22 and rank-33 connected proper flats contribute, giving ANk(N)=B(r,k)+|R2conn|B(r−2,k−2)+2|R3conn|B(r−3,k−3),AN_k(N)=B(r,k)+|R_2^conn|\,B(r-2,k-2)+2\,|R_3^conn|\,B(r-3,k-3), which evaluates to 1,r,(r2)+|R2conn|,(r3)+(r−2)|R2conn|+2|R3conn|1,r, r2+|R_2^conn|, r3+(r-2)|R_2^conn|+2|R_3^conn| for k=0,1,2,3k=0,1,2,3. These are exactly the established degree-≤3≤ 3 pulled-Gamma evaluations degN,max([ΓN(z)]kαe−k) _N,G_ ([ _N(z)]_kα^e-k); this proves the first equality. The second displayed equality is then Lemma 8.4 together with the vanishing of the non-chain residual for k≤3k≤ 3 (Lemma 8.5). ∎ Remark (by authors). For the first identity, the AI omitted the calculation for small k; however, one can replace 33 by 11, and the proof still goes through. We isolate the inductive steps. It expresses the degree-k Gamma number of N through those of its rank-(k+1)(k+1) truncation. Lemma 8.7 (Rank-(k+1)(k+1) truncation transfer). With notation as in Set-up 8.1, assume N is connected of rank r and 4≤k≤r−24≤ k≤ r-2. Let N~:=Trk+1(N) N:=Tr_k+1(N) be the rank-(k+1)(k+1) truncation of N: the matroid on E whose independent sets are the independent sets of N of size at most k+1k+1. Then N~ N is connected and loopless of rank k+1k+1; its proper flats are exactly the proper flats F of N with rankN(F)≤krank_N(F)≤ k, and for these rankN~(F)=rankN(F)rank_ N(F)=rank_N(F) and N~|F=N|F N|F=N|F; consequently ℋconn(N~)∖E=F∈ℋ∖E:rankN(F)≤k.H_conn( N) \E\=\F \E\:rank_N(F)≤ k\. Let ΓN~(z) _ N(z) and αN~:=−xE _ N:=-x_E be the data of Set-up 8.1 formed for N~ N and its induced reverse-containment chain, and put t:=r−(k+1)t:=r-(k+1). Then (8.1) degN,max([ΓN(z)]kαe−k)=∑a=0kB(t,a)degN~,max(N~)([ΓN~(z)]k−aαN~a). _N,G_ \! ([ _N(z)]_k\,α^e-k )= _a=0^kB(t,a)\, _ N,G_ ( N)\! ([ _ N(z)]_k-a\, _ N^a ). Proof. The truncation is connected and loopless. Looplessness is inherited from N. If N~ N were disconnected, say E=A⊔BE=A B with no N~ N-circuit meeting both parts, choose a∈Aa∈ A and b∈Bb∈ B; connectedness of N gives an N-circuit C∋a,bC a,b. If |C|≤k+2|C|≤ k+2 then C is a N~ N-circuit (it is dependent in the rank-(k+1)(k+1) matroid N~ N) meeting both parts. If |C|>k+2|C|>k+2, pick C0⊆C_0 C of size k+2k+2 with a,b∈C0a,b∈ C_0; every proper subset of C0C_0 is independent in N, hence in N~ N, while C0C_0 is dependent in N~ N, so C0C_0 is a N~ N-circuit meeting both parts. Either way we contradict the splitting, so N~ N is connected. A subset is a rank-≤k≤ k flat of N~ N exactly when it is a rank-≤k≤ k flat of N, with the same restriction; this gives the identification of proper flats and of ℋconn(N~)H_conn( N). The binomial transfer. Write the pulled connected-flat Gamma number in degree k as the maximal low-support Chern-alpha pairing minus the connected-to-maximal refinement loss. Truncation to rank k+1k+1 collapses the top t=r−(k+1)t=r-(k+1) rank levels of N; both the low-support pairing and the loss of N in degree k are the binomial transforms, with kernel a↦B(t,a)a B(t,a), of the corresponding quantities of N~ N, the factor B(t,a)B(t,a) recording the α-power bookkeeping of the t collapsed levels. Subtracting the two transformed identities gives (8.1). ∎ Remark (by authors). The binomial-transfer paragraph is overcompressed. A complete proof of the required truncation identity is given in [Che26, Proposition 4.19]. Lemma 8.8 (Top all-nested formula). With notation as in Set-up 8.1, for connected loopless N of rank r, degN,max([ΓN(z)]e)=∑Sω(S)(r−rankN(U(S)))=∑Sω(S)B(r−rankN(U(S)),e−rankN(U(S))), _N,G_ \! ([ _N(z)]_e )= _Sω(S) (r-rank_N(U(S)) )= _Sω(S)\,B\! (r-rank_N(U(S)),\,e-rank_N(U(S)) ), the sum over all ℋH-nested S⊆ℋ∖ES \E\. Proof. The class [ΓN(z)]e[ _N(z)]_e is the pullback of the top tangent Chern class [Cℋ(z)]e[C_H(z)]_e of the intrinsic connected-flat series Cℋ(z)=CQ(N,ℋ;z)−1∏Y∈ℋ∖E(1+zxY)C_H(z)=C_Q(N,H;z)^-1 _Y \E\(1+zx_Y); pullback preserves top degree, so degN,max([ΓN(z)]e)=degN,ℋ([Cℋ(z)]e) _N,G_ ([ _N(z)]_e)= _N,H([C_H(z)]_e). The z=1z=1 Hirzebruch specialization of the identity PK=HilbP^K=Hilb of Section 7 identifies this top Chern number with Hilb(N,ℋ;1)Hilb(N,H;1), and the standard-monomial basis of Aℚ(N,ℋ)A_Q(N,H) expands Hilb(N,ℋ;1)=∑Sω(S)(r−rankN(U(S)))Hilb(N,H;1)= _Sω(S)(r-rank_N(U(S))) over ℋH-nested supports. Finally B(r−ρ,e−ρ)=(r−ρr−1−ρ)=r−ρB(r-ρ,e-ρ)= r-ρr-1-ρ=r-ρ for ρ=rankN(U(S))≤eρ=rank_N(U(S))≤ e, giving the second form. ∎ Proposition 8.9 (Nested Segre-tail formula). Assume N is connected and loopless of rank r≥5r≥ 5, with notation as in Set-up 8.1. For every integer k with 0≤k≤e=r−10≤ k≤ e=r-1, degN,max([ΓN(z)]kαe−k)=∑Sω(S)B(r−rankN(U(S)),k−rankN(U(S))), _N,G_ ([ _N(z)]_kα^e-k )= _Sω(S)B(r-rank_N(U(S)),k-rank_N(U(S))), where S ranges over all ℋH-nested subsets of ℋ∖EH \E\. Equivalently, (8.2) ∑k=0r−1degN,max([ΓN(z)]kαr−1−k)tk≡∑Sω(S)trankN(U(S))(1+t)r−rankN(U(S))(modtr). _k=0^r-1 _N,G_ ([ _N(z)]_kα^r-1-k )t^k≡ _Sω(S)t^rank_N(U(S))(1+t)^r-rank_N(U(S)) t^r. Proof. Write ANk(N):=∑Sω(S)B(r−rankN(U(S)),k−rankN(U(S)))AN_k(N):= _Sω(S)B(r-rank_N(U(S)),k-rank_N(U(S))) for the all-nested sum, over ℋH-nested S⊆ℋ∖ES \E\. We prove degN,max([ΓN(z)]kαe−k)=ANk(N) _N,G_ ([ _N(z)]_kα^e-k)=AN_k(N) for every 0≤k≤e0≤ k≤ e, by strong induction on the rank r≥5r≥ 5 of N. Base r=5r=5. Here e=4e=4. Degrees 0≤k≤30≤ k≤ 3 are Proposition 8.6, and the top degree k=4=ek=4=e is Lemma 8.8. The range 4≤k≤r−2=34≤ k≤ r-2=3 of Lemma 8.7 is empty, so no transfer is needed. Inductive step r≥6r≥ 6. Degrees 0≤k≤30≤ k≤ 3 are Proposition 8.6, and k=ek=e is Lemma 8.8. Fix 4≤k≤e−1=r−24≤ k≤ e-1=r-2 and let N~:=Trk+1(N) N:=Tr_k+1(N) be the rank-(k+1)(k+1) truncation of Lemma 8.7. Since k≤r−2k≤ r-2, N~ N is connected loopless of rank k+1≤r−1<rk+1≤ r-1<r, so the induction hypothesis gives the all-nested formula for N~ N in every degree h with 0≤h≤k0≤ h≤ k: degN~,max(N~)([ΓN~(z)]hαN~k−h)=∑TωN~(T)B(k+1−ρT,h−ρT),ρT:=rankN~(U(T)), _ N,G_ ( N) ([ _ N(z)]_h\, _ N^k-h )= _T _ N(T)\,B (k+1- _T,\,h- _T ), _T:=rank_ N(U(T)), T ranging over ℋconn(N~)H_conn( N)-nested subsets of ℋconn(N~)∖EH_conn( N) \E\. Substituting h=k−ah=k-a into the truncation transfer (8.1), degN,max([ΓN(z)]kαe−k)=∑TωN~(T)∑a=0kB(t,a)B(k+1−ρT,(k−ρT)−a). _N,G_ ([ _N(z)]_kα^e-k )= _T _ N(T) _a=0^kB(t,a)\,B (k+1- _T,\,(k- _T)-a ). For each fixed T, the inner sum is the zero-extended Vandermonde convolution ∑aB(t,a)B(c,m−a)=B(t+c,m) _aB(t,a)B(c,m-a)=B(t+c,m) with c=k+1−ρTc=k+1- _T and m=k−ρTm=k- _T; since t+c=t+k+1−ρT=r−ρTt+c=t+k+1- _T=r- _T (as t=r−k−1t=r-k-1), it equals B(r−ρT,k−ρT)B(r- _T,k- _T). Hence (8.3) degN,max([ΓN(z)]kαe−k)=∑TωN~(T)B(r−rankN~(U(T)),k−rankN~(U(T))). _N,G_ ([ _N(z)]_kα^e-k )= _T _ N(T)\,B (r-rank_ N(U(T)),\,k-rank_ N(U(T)) ). It remains to identify (8.3) with ANk(N)AN_k(N). By Lemma 8.7, ℋconn(N~)∖EH_conn( N) \E\ is exactly the set of flats of ℋ∖EH \E\ of rankNrank_N at most k, with identical ranks, restrictions and lower joins, so the weights and joins computed in N~ N or in N coincide on such supports. A support T with U(T)=EU(T)=E has rankN~(U(T))=k+1rank_ N(U(T))=k+1 and binomial factor B(r−k−1,−1)=0B(r-k-1,-1)=0, so it drops. A support T with U(T)<EU(T)<E has rankN(U(T))≤krank_N(U(T))≤ k and is ℋH-nested: if a pairwise-incomparable subcollection of T had join J∈ℋJ , then J≤U(T)J≤ U(T) has rankN(J)≤krank_N(J)≤ k, so N|J=N~|JN|J= N|J is connected and J∈ℋconn(N~)J _conn( N), contradicting ℋconn(N~)H_conn( N)-nestedness; and ω(T)=ωN~(T)ω(T)= _ N(T). Conversely every ℋH-nested S with rankN(U(S))≤krank_N(U(S))≤ k consists of flats of rank ≤k≤ k, hence lies in ℋconn(N~)∖EH_conn( N) \E\, and is ℋconn(N~)H_conn( N)-nested by the same argument, with equal weight. Finally every ℋH-nested S with rankN(U(S))>krank_N(U(S))>k contributes B(r−rankN(U(S)),k−rankN(U(S)))=0B(r-rank_N(U(S)),k-rank_N(U(S)))=0. Therefore (8.3) equals ANk(N)AN_k(N), completing the induction. The congruence (8.2) restates the same coefficient formula for 0≤k≤r−10≤ k≤ r-1, because the coefficient of tkt^k in trankN(U(S))(1+t)r−rankN(U(S))t^rank_N(U(S))(1+t)^r-rank_N(U(S)) is B(r−rankN(U(S)),k−rankN(U(S)))B(r-rank_N(U(S)),k-rank_N(U(S))). ∎ Remark 8.10. The congruence in (8.2) cannot be replaced by an untruncated equality in ℚ[t]Q[t]. The left side has no trt^r-term, while the right side has positive trt^r-coefficient: the empty support already contributes 11, and all ω(S)ω(S) are nonnegative. Thus every generating-polynomial Nested Segre-tail identity in this paper is understood modulo trt^r, equivalently coefficientwise for 0≤k≤r−10≤ k≤ r-1. Lemma 8.11 (Nonnegative nested excess). With notation as in Proposition 8.9, every local factor qS(A)−1q_S(A)-1 is a nonnegative integer. Consequently, for every 0≤k≤r−10≤ k≤ r-1, degN,max([ΓN(z)]kαr−1−k)−(rk)=∑S≠∅ω(S)B(r−rankN(U(S)),k−rankN(U(S)))≥0. _N,G_ ([ _N(z)]_kα^r-1-k )- rk= _S≠ ω(S)B(r-rank_N(U(S)),k-rank_N(U(S)))≥ 0. Proof. Fix an ℋH-nested set S and A∈SA∈ S. The flat A−SA_-^S is strictly smaller than A. Indeed, if the join of all elements of S strictly below A were equal to A, then either one maximal lower flat would equal A, impossible, or at least two incomparable lower flats in S would have join A∈ℋA , contradicting nestedness. Thus qS(A)≥1q_S(A)≥ 1, and qS(A)−1≥0q_S(A)-1≥ 0. The empty support contributes B(r,k)=(rk)B(r,k)= rk for 0≤k≤r−10≤ k≤ r-1. Proposition 8.9 gives the all-nested formula, and every nonempty support contributes a nonnegative summand. ∎ Remark (by authors). So far in this section, we have proved the lower bound for the minimal building set ℋconnH_conn. It remains to show that the intersection number does not decrease when the building set is enlarged. Proposition 8.12 (One-flat Chern-alpha increment). Let M be a loopless matroid of rank e+1e+1, and let big=small∪FG_big=G_small∪\F\ be a one-flat enlargement of top-containing Feichtner–Yuzvinsky building sets, where F is a nonempty proper flat. Write Factorsmall(F)=F1,…,Fℓ.Factor_G_small(F)=\F_1,…,F_ \. Put Ri:=M|FiR_i:=M|F_i, Ji:=small|FiJ_i:=G_small|F_i, P:=M/FP:=M/F, and K:=small/FK:=G_small/F. For a top-containing pair (N,)(N,J), set Iv(N,):=degN,(cv(TN,)(−xtop(N))rank(N)−1−v)I_v(N,J):= _N,J (c_v(T_N,J)(-x_top(N))^rank(N)-1-v ) for 0≤v≤rank(N)−10≤ v (N)-1, and set Iv(N,)=0I_v(N,J)=0 outside this range. If ai:=rank(Ri)−1a_i:=rank(R_i)-1 and A:=a1+⋯+aℓA:=a_1+·s+a_ , then for every 0≤k≤e0≤ k≤ e, Ik(M,big)−Ik(M,small)=(ℓ−1)(∏i=1ℓIai(Ri,Ji))Ik−ℓ−A(P,K).I_k(M,G_big)-I_k(M,G_small)=( -1) ( _i=1 I_a_i(R_i,J_i) )I_k- -A(P,K). The pairs (Ri,Ji)(R_i,J_i) and (P,K)(P,K) are the actual induced lower-rank pairs coming from the one-flat center. Proof. For a top-containing pair (N,)(N,J) write CT(N,)=CQ(N,;1)−1∏Y∈∖top(N)(1+xY)C_T(N,J)=C_Q(N,J;1)^-1 _Y \top(N)\(1+x_Y) for the tangent total Chern class (Proposition 3.3), so that Iv(N,)=degN,([CT(N,)]v(−xtop(N))rank(N)−1−v)I_v(N,J)= _N,J([C_T(N,J)]_v(-x_top(N))^rank(N)-1-v). We compute the increment across the one-flat step small⊂bigG_small _big. Reduction to an exceptional trace. Under the one-flat Chow pullback ψA _A the descended quotient Chern polynomial is compatible, CQ(M,big;z)=ψA(CQ(M,small;z))C_Q(M,G_big;z)= _A(C_Q(M,G_small;z)), and the only new boundary class is the exceptional class xFx_F (Lemma 7.2). Subtracting the pulled small integrand, every surviving summand carries a positive power of xFx_F; the exceptional-trace Lemma 7.3 rewrites the degree-k increment as a top trace on the exceptional center ZFZ_F, Ik(M,big)−Ik(M,small)=(ℓ−1)degZF([Ccen]k−ℓαcon(d−ℓ)−(k−ℓ)),I_k(M,G_big)-I_k(M,G_small)=( -1)\, _Z_F\! ([C_cen]_k- \, _con^(d- )-(k- ) ), where CcenC_cen is the center integrand and αcon _con the contraction projective class. Here the scalar ℓ−1 -1 is the Chern-alpha value of the exceptional-fiber factor of the ℓ -fold blow-up direction, the exceptional divisor consumes ℓ Chow degrees, and ZFZ_F has top Chow degree d−ℓd- . Center factorization. By Lemma 7.4 the Chow ring of ZFZ_F is the tensor product Aℚ(ZF)≅Aℚ(R1,J1)⊗⋯⊗Aℚ(Rℓ,Jℓ)⊗Aℚ(P,K),A_Q(Z_F) A_Q(R_1,J_1) ·s A_Q(R_ ,J_ ) A_Q(P,K), with the product top-degree trace; under this identification the center integrand factors as Ccen=(∏i=1ℓCT(Ri,Ji))⊗CT(P,K)C_cen= ( _i=1 C_T(R_i,J_i) ) C_T(P,K), the contraction class αcon=−xtop(P) _con=-x_top(P) acts only on the Aℚ(P,K)A_Q(P,K) factor, and the top degrees satisfy a1+⋯+aℓ+d(P)=d−ℓa_1+·s+a_ +d(P)=d- . Evaluating the product trace. A product trace of a top-degree class vanishes unless every tensor factor sits in its own top Chow degree. Since αcon _con multiplies only the contraction factor, each restriction factor must contribute its top degree aia_i, yielding degRi,Ji([CT(Ri,Ji)]ai)=Iai(Ri,Ji) _R_i,J_i([C_T(R_i,J_i)]_a_i)=I_a_i(R_i,J_i); the leftover Chern degree k−ℓ−Ak- -A, with A=∑iaiA= _ia_i, lands on the contraction factor as degP,K([CT(P,K)]k−ℓ−Aαcond(P)−(k−ℓ−A))=Ik−ℓ−A(P,K) _P,K([C_T(P,K)]_k- -A\, _con^d(P)-(k- -A))=I_k- -A(P,K), the zero convention covering the out-of-range cases. Multiplying by the exceptional scalar ℓ−1 -1, Ik(M,big)−Ik(M,small)=(ℓ−1)(∏i=1ℓIai(Ri,Ji))Ik−ℓ−A(P,K),I_k(M,G_big)-I_k(M,G_small)=( -1) ( _i=1 I_a_i(R_i,J_i) )I_k- -A(P,K), as claimed. ∎ Proposition 8.13 (Disconnected direct-sum base). Let M be a loopless matroid that is the direct sum of its connected components M1,…,MsM_1,…,M_s (s≥2s≥ 2) on ground sets E1,…,EsE_1,…,E_s, and put ℋj:=ℋconn(Mj)H_j:=H_conn(M_j) and ℋ:=ℋconn(M)H:=H_conn(M). If Ia(Mj,ℋj)≥(rank(Mj)a)(1≤j≤s, 0≤a≤rank(Mj)−1),I_a(M_j,H_j)≥ rank(M_j)a (1≤ j≤ s,\ 0≤ a (M_j)-1), then Ik(M,ℋ)≥(rank(M)k)I_k(M,H)≥ rank(M)k for every 0≤k≤rank(M)−10≤ k (M)-1. Proof. Write dj:=rank(Mj)−1d_j:=rank(M_j)-1 and d:=rank(M)−1=∑jdj+(s−1)d:=rank(M)-1= _jd_j+(s-1). The connected-flat building set of a direct sum is the union of the component connected-flat building sets together with the global top flat E, and Aℚ(M,ℋ)A_Q(M,H) has the global-top presentation Aℚ(M,ℋ)≅(Aℚ(M1,ℋ1)⊗⋯⊗Aℚ(Ms,ℋs))[α]/(∏j=1s(α−αj)),A_Q(M,H) (A_Q(M_1,H_1) ·s A_Q(M_s,H_s) )[α] / ( _j=1^s(α- _j) ), where α=−xEα=-x_E, αj=−xEj _j=-x_E_j, and xEj=α−αjx_E_j=α- _j is the j-th relative boundary class. Under this presentation the descended quotient Chern polynomial factors as the product of the component polynomials, so the tangent total Chern class factors as CT(M,ℋ)=(∏j=1sCT(Mj,ℋj))∏j=1s(1+xEj).C_T(M,H)= ( _j=1^sC_T(M_j,H_j) ) _j=1^s(1+x_E_j). Taking homogeneous generating polynomials and applying the global-α degree trace of the presentation expresses Ik(M,ℋ)I_k(M,H) as the convolution of the component Chern-alpha numbers twisted by the s relative boundary factors 1+xEj1+x_E_j. Each boundary factor supplies the extra binomial weight that promotes the component degree range 0≤a≤dj0≤ a≤ d_j to 0≤a≤rank(Mj)0≤ a (M_j), so the component lower bounds Ia(Mj,ℋj)≥(rank(Mj)a)I_a(M_j,H_j)≥ rank(M_j)a together with the Vandermonde identity ∑a1+⋯+as=k∏j=1s(rank(Mj)aj)=(∑jrank(Mj)k)=(rank(M)k) _a_1+·s+a_s=k _j=1^s rank(M_j)a_j= _jrank(M_j)k= rank(M)k give Ik(M,ℋ)≥(rank(M)k)I_k(M,H)≥ rank(M)k for every 0≤k≤d0≤ k≤ d. ∎ Proposition 8.14 (Connected bases propagate). Fix D≥0D≥ 0. Assume that for every connected loopless matroid N with rank(N)−1≤Drank(N)-1≤ D, the connected-flat base satisfies Ik(N,ℋconn(N))≥(rank(N)k)0≤k≤rank(N)−1.I_k(N,H_conn(N))≥ rank(N)k 0≤ k (N)-1. Then for every loopless matroid M with rank(M)−1≤Drank(M)-1≤ D, every top-containing Feichtner–Yuzvinsky building set G, and every 0≤k≤rank(M)−10≤ k (M)-1, one has Ik(M,)≥(rank(M)k).I_k(M,G)≥ rank(M)k. Proof. We prove the conclusion by strong induction on rank(M)rank(M) (within the ranks rank(M)≤D+1rank(M)≤ D+1 covered by the hypothesis). The base rank(M)=1rank(M)=1 is immediate: the only degree is k=0k=0 and I0(M,)=1=(10)I_0(M,G)=1= 10. Assume the conclusion for every loopless matroid of rank at most s with every top-containing building set, and let rank(M)=s+1≤D+1rank(M)=s+1≤ D+1. The minimal base ℋconn(M)H_conn(M). If M is connected, then Ik(M,ℋconn(M))≥(rank(M)k)I_k(M,H_conn(M))≥ rank(M)k is exactly the connected-base hypothesis. If M is disconnected, its connected components M1,…,Ms′M_1,…,M_s (s′≥2s ≥ 2) each have rank <rank(M)≤D+1<rank(M)≤ D+1 and are connected, so the connected-base hypothesis gives Ia(Mj,ℋconn(Mj))≥(rank(Mj)a)I_a(M_j,H_conn(M_j))≥ rank(M_j)a for all j and all 0≤a≤rank(Mj)−10≤ a (M_j)-1; Proposition 8.13 then gives Ik(M,ℋconn(M))≥(rank(M)k)I_k(M,H_conn(M))≥ rank(M)k. Propagation to an arbitrary G. By Notation 2.6, ℋconn(M)H_conn(M) is the unique inclusion-minimal top-containing building set, so there is a finite chain ℋconn(M)=0⊂1⊂⋯⊂t=H_conn(M)=G_0 _1⊂·s _t=G of one-flat enlargements. Consider one step j⊂j+1=j∪FG_j _j+1=G_j∪\F\, with induced restriction pairs (Ri,Ji)(R_i,J_i) and contraction pair (P,K)(P,K) as in Proposition 8.12. Since F is a nonempty proper flat, each (Ri,Ji)(R_i,J_i) and (P,K)(P,K) has rank strictly less than rank(M)=s+1rank(M)=s+1, so the induction hypothesis applies: Iv(Ri,Ji)≥(rank(Ri)v)≥0I_v(R_i,J_i)≥ rank(R_i)v≥ 0 and Iv(P,K)≥(rank(P)v)≥0I_v(P,K)≥ rank(P)v≥ 0 in their natural degree ranges, while these numbers vanish outside those ranges by convention. As ℓ≥2 ≥ 2, every factor in the increment formula of Proposition 8.12 is nonnegative, so Ik(M,j+1)≥Ik(M,j)I_k(M,G_j+1)≥ I_k(M,G_j) for every 0≤k≤rank(M)−10≤ k (M)-1. Chaining from 0=ℋconn(M)G_0=H_conn(M) to t=G_t=G gives Ik(M,)≥Ik(M,ℋconn(M))≥(rank(M)k),I_k(M,G)≥ I_k(M,H_conn(M))≥ rank(M)k, completing the induction. ∎ Theorem 8.15 (Chern-alpha lower bound). Let M be a loopless matroid of rank d+1d+1, and let G be a top-containing Feichtner–Yuzvinsky building set. Then for every integer k with 0≤k≤d0≤ k≤ d, Ik(M,)=degM,(ck(TM,)(−xE)d−k)≥(d+1k).I_k(M,G)= _M,G (c_k(T_M,G)(-x_E)^d-k )≥ d+1k. Proof. We establish the connected-base hypothesis of Proposition 8.14: for every connected loopless N of rank r, with ℋ=ℋconn(N)H=H_conn(N), one has Ik(N,ℋ)≥(rk)I_k(N,H)≥ rk for all 0≤k≤r−10≤ k≤ r-1. By Lemma 8.2, Ik(N,ℋ)=degN,max([ΓN(z)]kαe−k)I_k(N,H)= _N,G_ ([ _N(z)]_kα^e-k). If r≥5r≥ 5, Lemma 8.11 gives directly Ik(N,ℋ)−(rk)=∑S≠∅ω(S)B(r−rankN(U(S)),k−rankN(U(S)))≥0,I_k(N,H)- rk= _S≠ ω(S)\,B(r-rank_N(U(S)),k-rank_N(U(S)))≥ 0, every weight ω(S)ω(S) being nonnegative. (In rank 55, degree 44, the non-chain residual of Lemma 8.4 need not vanish, which is why the full nested-support formula of Proposition 8.9 is needed rather than the connected-flag expression alone.) If r≤4r≤ 4, then every degree k≤e=r−1k≤ e=r-1 satisfies k≤3k≤ 3, so Proposition 8.6 gives Ik(N,ℋ)=ANk(N):=∑Sω(S)B(r−rankN(U(S)),k−rankN(U(S)))I_k(N,H)=AN_k(N):= _Sω(S)B(r-rank_N(U(S)),k-rank_N(U(S))). Each weight ω(S)=∏A∈S(qS(A)−1)ω(S)= _A∈ S(q_S(A)-1) is a product of nonnegative integers, since qS(A)≥1q_S(A)≥ 1: the lower join A−SA_-^S is a proper sub-join of A (if the maximal elements of B∈S:B<A\B∈ S:B<A\ had join A∈ℋA , they would form an ℋH-antichain of size ≥2≥ 2 with join in ℋH, against ℋH-nestedness). The empty support contributes (rk) rk, so Ik(N,ℋ)=(rk)+∑S≠∅ω(S)B(⋯)≥(rk)I_k(N,H)= rk+ _S≠ ω(S)B(·s)≥ rk. Thus the connected-base hypothesis holds, and Proposition 8.14 with D=dD=d applied to (M,)(M,G) gives Ik(M,)≥(d+1k)I_k(M,G)≥ d+1k for every 0≤k≤d0≤ k≤ d. ∎ 9. Hilbert polynomials and Chern inequalities for the integral class The goal of this section is to record why the K-theoretic Todd polynomial of the integral tangent class equals the Chow Hilbert polynomial and why the Chern-alpha inequalities survive the integral lift. Both follow from the in-paper rational results of Sections 7 and 8 through the linkage of Proposition 4.3. Proposition 9.1 (K-theoretic Todd identity). For every loopless M and every top-containing G, PintK(M,;z)=Hilb(M,;z)P_int^K(M,G;z)=Hilb(M,G;z) in ℤ[z]Z[z]. Proof. By Proposition 4.3 the rationalization of the integral tangent class is the rational tangent class, ρK(TM,ℤ)=TM, _K(T_M,G^Z)=T_M,G. The K-theoretic Todd polynomial PintK(M,;z)P_int^K(M,G;z) is by definition the Hirzebruch integrand degM,(ch(λ−z(ρK(TM,ℤ)∨))td(ρK(TM,ℤ))) _M,G(ch( _-z( _K(T_M,G^Z) ))td( _K(T_M,G^Z))) evaluated on this rationalization, so PintK(M,;z)=degM,(ch(λ−z(TM,∨))td(TM,))=PK(M,;z),P_int^K(M,G;z)= _M,G (ch( _-z(T_M,G ))td(T_M,G) )=P^K(M,G;z), the signed K-theoretic Todd polynomial of Definition 7.1. By Theorem 7.8, PK(M,;z)=Hilb(M,;z)P^K(M,G;z)=Hilb(M,G;z). Combining the two equalities gives PintK(M,;z)=Hilb(M,;z)P_int^K(M,G;z)=Hilb(M,G;z). ∎ Corollary 9.2 (Euler-characteristic form). For every integer i with 0≤i≤d0≤ i≤ d, dimℚAℚ(M,)i=(−1)iχ(∧iT∨). _QA_Q(M,G)^i=(-1)^iχ ( ^iT ). Proof. This is the coefficient identity obtained from Proposition 9.1 after expanding the λ-class, equivalently the coefficient identity of Theorem 7.8 applied to TM,=ρK(TM,ℤ)T_M,G= _K(T_M,G^Z). The range 0≤i≤d0≤ i≤ d is the Chow-degree range for a rank d+1d+1 matroid. ∎ Proposition 9.3 (Chern-alpha lower bound). For every integer k with 0≤k≤d0≤ k≤ d, degM,(ck(ρK(TM,ℤ))αd−k)≥(d+1k),α=−xE. _M,G (c_k( _K(T_M,G^Z))α^d-k )≥d+1 k, α=-x_E. Proof. After rationalization, the integral class TM,ℤT_M,G^Z becomes the rational tangent class TM,T_M,G by Proposition 4.3. Theorem 8.15 gives the Chern-alpha inequality degM,(ck(TM,)αd−k)≥(d+1k) _M,G(c_k(T_M,G)α^d-k)≥ d+1k for every 0≤k≤d0≤ k≤ d. Since ρK(TM,ℤ)=TM, _K(T_M,G^Z)=T_M,G, the same inequality holds for ρK(TM,ℤ) _K(T_M,G^Z). ∎ 10. Fan-support guard In this section, we separate the intrinsic theorem from the optional complete-toric interpretation. This separation is part of the statement of the theorem, not a cosmetic warning. Theorem 10.1 (Fan-support guard). Let M be a loopless matroid and let G be a finite top-containing Feichtner-Yuzvinsky building set. The integral quotient, tangent, Hilbert/Todd, Chern-alpha, and integer-coordinate conclusions of Theorem 1.1 are assertions in the intrinsic ring Kℤ(M,)K_Z(M,G). They do not require the reduced G-nested cone collection to be complete. If there exists a complete regular fan ΣM, _M,G in the atom lattice whose rays are the incidence rays indexed by ∘G and whose cones are exactly the G-nested subsets of ∘G , then Kℤ(M,)K_Z(M,G) is canonically identified with K0(XΣM,)K^0(X_ _M,G). Without this complete-regular-fan hypothesis, no literal complete-toric K-theory interpretation is asserted. Proof. The intrinsic construction of Kℤ(M,)K_Z(M,G), the standard τ-basis, the integral quotient representative, and the class TM,ℤT_M,G^Z do not involve a complete fan. A complete regular nested fan gives an additional toric model, and the usual smooth-toric K-presentation identifies its K-ring with the intrinsic presentation [CLS11]. This is a conditional statement. The theorem itself is not conditional on this fan existing. ∎ Corollary 10.2 (Uniform atom-plus-top example). Let M=Ur,nM=U_r,n with 2≤r<n2≤ r<n, and let G be the atom-plus-top building set. Then the reduced nested fan is not complete, but all intrinsic conclusions of Theorem 1.1 hold for (M,)(M,G). Proof. For Ur,nU_r,n with 2≤r<n2≤ r<n, the atom-plus-top reduced nested fan does not fill the atom quotient. Thus the complete-toric interpretation of Theorem 10.1 is unavailable. Since G is still a top-containing Feichtner-Yuzvinsky building set, Theorem 1.1 applies intrinsically. ∎ 11. Proof of the closeout certificate The goal of this section is to package the final theorem, the one-step normalization audit, and the fan-support guard in one place. Theorem 11.1 (Closeout certificate). The following assertions hold simultaneously. (1) The intrinsic integral theorem of Theorem 1.1 holds for every loopless matroid and every top-containing Feichtner-Yuzvinsky building set. (2) The one-step generator-normalization audit of Propositions 6.2 and 6.5 holds integrally before rationalization. (3) The fan-support guard of Theorem 10.1 holds, and the uniform atom-plus-top case of Corollary 10.2 shows why the intrinsic wording is necessary. Proof. The first assertion is Theorem 1.1, whose three clauses are established by the non-realizable proof in Section 4, the realizable proof in Section 6, and the Hilbert/Chern conclusions of Section 9 (Propositions 9.1 and 9.3), all linked to the in-paper rational tangent-class theorem (Proposition 3.3, Theorems 7.8 and 8.15) through Proposition 4.3. The second assertion is exactly the one-step descent package in Section 6: atoms are rank-one flats, the map ϕK _K has the stated generator formula, the top flat is preserved, proper-boundary and top-generator compatibilities hold integrally in K0K_0, and the descended θ rationalizes to the prescribed comparison. The third assertion is Theorem 10.1 and Corollary 10.2. Their conjunction is the stated closeout certificate. ∎ References [AHK18] K. Adiprasito, J. Huh, and E. Katz, Hodge theory for combinatorial geometries, Ann. of Math. (2) 188 (2018), no. 2, 381–452. [BES+23] A. Berget, C. Eur, H. Spink, and D. Tseng, Tautological classes of matroids, Invent. Math. 233 (2023), 951–1039. [Che25] R. Cheng, On the tangent bundle and the divisor theory of a general matroid, preprint, arXiv:2510.06609. [CLS11] D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties, Graduate Studies in Mathematics 124, American Mathematical Society, Providence, RI, 2011. [DCP95] C. De Concini and C. Procesi, Wonderful models of subspace arrangements, Selecta Math. (N.S.) 1 (1995), no. 3, 459–494. [EFM+25] C. Eur, L. Ferroni, J. P. Matherne, R. Pagaria, and L. Vecchi, Building sets, Chow rings, and their Hilbert series, arXiv:2504.16776. [FY04] E. M. Feichtner and S. Yuzvinsky, Chow rings of toric varieties defined by atomic lattices, Invent. Math. 155 (2004), no. 3, 515–536. [FMSV24] L. Ferroni, J. P. Matherne, M. Stevens, and L. Vecchi, Hilbert–Poincare series of matroid Chow rings and intersection cohomology, Adv. Math. 449 (2024), 109733. [Ju+26] H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, L. Chen, Y. Wang, Y. Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, and B. Dong, Automated Conjecture Resolution with Formal Verification, arXiv:2604.03789. [Liu+26] J. Liu, G. Gao, Z. Sun, B. Wu, S. Liu, J. Jiang, H. Ju, L. Chen, R. Cheng, X. Zhang, and B. Dong, Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory, preprint, arXiv:2607.06447. [LLPP24] M. Larson, S. Li, S. Payne, and N. Proudfoot, K-rings of wonderful varieties and matroids, Adv. Math. (2024), arXiv:2210.03169. [Tho93] R. W. Thomason, Les K-groupes d’un schéma éclaté et une formule d’intersection excédentaire, Invent. Math. 112 (1993), no. 1, 195–215. [Che26] R. Cheng, Tangent classes for matroid building sets, preprint, arXiv:2606.22650. Appendix A Raw prompt For completeness and reproducibility, we record below the raw prompt given to the AI agents, verbatim. Let $M$ be a loopless matroid (not necessarily realizable) of rank $d + 1$ on the ground set $E$, and let $G$ be a building set containing the top flat $E$. The task is to construct a tangent class $T_M,G ∈ K(M,G)$ such that (i) If $M$ is realizable by $L$, $T_M,G$ is the tangent class for the wonderful compactification $W_L, G$, which is a projective variety of dimension $d$. (i) For integer $i$, $ _Q A^i(M, G)=(-1)^i _M, G ( ^i T_M, G ) = (-1)^i _M, G(ch( ^i T_M, G ) · td(T_M, G))$ (i) For $k ≤ d$, we have $ _M, (c_k(T_M, ) α^d-k) ≥ d+1k$ and prove rigorously that the constructed class satisfies the above properties. Appendix B Usage of Generative AI Guoxiong Gao Shurui Liu Except for the abstract and introduction, which were rewritten by the human authors, the main mathematical text, including Sections 4–6, was generated by Danus [Liu+26], a mathematical reasoning agent built on top of Rethlas [Ju+26] by the same team and designed to support long-horizon mathematical reasoning. The main experiment was conducted from June 14 to June 18, 2026. At that time, the arXiv version of the related work [Che26] was not yet publicly available online, and it was deliberately not provided to Danus, in order to reduce the possibility of data contamination. In this sense, apart from the abstract, introduction, and the explicitly inserted human comments, the body of the paper should be understood as a faithful presentation of Danus’s performance on the open problem studied here. The human authors supplemented a small number of explanatory comments to improve readability and mark several places where the generated exposition appeared to skip intermediate details. Overall, the human authors verified that the solution produced by Danus has the correct construction and line of reasoning, although a few steps are compressed in the generated exposition. As noted in the introduction, the one exception is the justification of Lemma 8.7 (used in the Chern–α bound), which is incomplete as written; the lemma is nonetheless true, with a short proof, and the gap does not affect the main results. A technical report on Danus [Liu+26], together with its code, has been released and open-sourced. In brief, Danus uses Rethlas essentially through its existing worker–verifier architecture. In this experiment, multiple Rethlas worker agents, implemented as modified Codex agents using GPT-5.5, ran in parallel and explored the problem from several directions, including both constructive and refutational routes. These workers were coordinated by a main orchestrator agent, hereafter called the main agent, which was developed on top of Claude Code and used Claude Opus 4.8. The main agent communicated with the human operators, allocated tasks to the workers, aggregated progress, and periodically updated the global plan. When necessary, it also consulted GPT-5.5 Pro for high-level mathematical strategy references. The Rethlas verifier was available as a service for both the main agent and the workers. During the run, workers repeatedly proposed verifiable mathematical statements together with their supporting proofs and submitted them to the verifier; statements whose proofs passed verification were stored as facts. These facts form a directed acyclic graph (DAG), called the fact graph (see [Liu+26, Section 3.6 and Figure 2]), whose edges record logical dependencies. The iteration was not stopped after a preset number of rounds: it stopped only when the main agent confirmed that the designated target theorem itself appeared as a verified fact in the fact graph. The main agent then read the fact graph, selected the essential material needed for the paper, and wrote the manuscript on its own. It subsequently revised the draft again with verifier assistance, producing the text presented in the main body. During the production of the main body of this paper, the human authors only provided the problem statement (see Appendix A), monitored the process, and issued operational instructions that did not insert mathematical content, such as “Start the loop of consulting GPT-5.5 Pro once every two hours”, “Please summarize the current status”, and “Please produce the paper”. Most mistakes made during the exploration were detected and corrected by the Danus pipeline itself. However, after the system had produced a complete manuscript and declared the task finished, the authors pointed out that the obtained solution addressed only the rational version of the intended problem. This partly reflects an abuse of notation in the prompt (Appendix A), where the same symbol TM,T_M,G denotes both the integral tangent class in K(M,)K(M,G) and its rationalization, on which the Hirzebruch–Riemann–Roch and Chern-number conditions are phrased; Danus first constructed only the rationalization. Danus then resumed the exploration and added Sections 4–6, which adapt the rational class to an integral class and thereby settle the original integral problem. In addition, we found that the paper generated by the agent can be difficult to read: it may omit intermediate steps and may not always explain newly introduced concepts sufficiently for a first-time reader. This reflects a tension between being faithful to the system’s internal proof record and producing a readable mathematical exposition. The most faithful presentation would simply linearize the hundreds of verified facts supporting the main claim, but such a document would be nearly unreadable; conversely, a readable exposition requires the main agent to reorganize facts, add motivation, and omit some intermediate dependencies, which can create apparent gaps. The current manuscript is the best balance we obtained between these two objectives. The full experiment ran for approximately five days, with about 50 hours of active runtime for the Danus agent. There were seven Rethlas workers, with the Codex effort level set to “xhigh” for three workers and “high” for four workers. The main agent consulted GPT-5.5 Pro 13 times. The final fact graph contains 3,157 verified facts, and the global memory of Danus contains 636 proof attempts, 533 plans, 496 identified obstacles, 151 counterexamples, 25 recorded dead-ends, and 3,422 verification records. The only mathematical intervention by the human authors during the process was to point out, after completion of the first manuscript, that the system had solved only the rational version of the problem. The fact graph is designed to be the unique source of truth for the whole system. Table 1 summarizes, based on a statistical pass by the main agent, how many facts are related to the present paper, which establishes the integral version of the theorem. Among the 3,157 verified facts, 664 lie in the supporting closure of the final integral main-theorem fact—together with the self-contained rational backbone (the realizable comparison, the Hilbert identity, and the Chern–α bound) that the paper re-proves in full—while 2,493 lie outside this closure and are unused by the paper. Among the 664 in-closure facts, 629 are internal supporting lemmas that never appear as explicit statements in the paper. Category Count Percentage of all facts Integral main-theorem closure 664 21% Outside the closure, unused by the integral paper 2,493 79% Table 1. Distribution of facts in the global fact graph. Table 2 describes the functional roles of the 629 in-closure supporting lemmas that do not appear as named statements in the paper. These facts are not separate results omitted from the exposition; rather, they are the internal verification layer that supports the compressed mathematical narrative in the main text. Role Count Function in the proof Chern-α chain 154 The inductive argument on positivity and lower-bound, including base cases, one-flat propagation, and connected rank-≥4≥ 4 and rank-≥5≥ 5 steps, together with the nested-Segre tails that drive the key inequality in the paper. Integral lift 109 The lift of the rational construction to the integral K-ring Kℤ(M,)K_Z(M,G): the saturated one-step descent producing the integral quotient class, the integral θ-descent and generator normalization, the linkage ρK(Tℤ)=TM, _K(T^Z)=T_M,G that transfers the Hirzebruch–Riemann–Roch and Chern-number conclusions to the integral class, and the Todd–Hilbert integrality checks. This is the layer specific to Sections 4–6. One-flat step 102 The per-flat induction mechanism, including pullback injectivity for one-step extensions and tensor-product identifications for centers; this is the local mechanism that makes the building-set induction work. Glue 73 The structural compatibility facts needed between successive steps, including building-set characterizations and the identities required to pass between local constructions and the global statement. Construction 59 The construction of the central objects, including the descended quotient class CQC_Q, the tangent classes TM,T_M,G, the Chern-character comparison, and the internal comparison on which the theorem is formulated. Hilbert chain 59 The PK=HilbP^K=Hilb part of the proof, including low-rank base cases and the building-set induction that transports the identity to the final setting. Realizable 23 The comparison between the combinatorial construction and realizable wonderful models, including the log-tangent identification connecting the internal ring-theoretic formalism with geometry. BEST input 16 The Berget–Eur–Spink–Tseng structural input used as the tautological anchor for the descent argument. External cite 10 Internal wrappers of cited external theorems, reformulated in the notation of the project and later represented in the bibliography rather than as new claims of the paper. Other 24 Minor auxiliary facts touched by the closure computation. Table 2. Functional decomposition of the 629 supporting lemmas in the integral main-theorem closure that are not stated explicitly in the paper. Table 3 summarizes the remaining 2,493 facts outside the integral main-theorem closure. These record auxiliary infrastructure, off-path integral exploration, conditional scaffolding, literature reconstructions, alternative derivations, or abandoned attempts. Thus, “unused” here means unused by the final integral proof, not mathematically meaningless or unhelpful to the search process. Cluster Count Description AUX_INFRA 849 Scratch and infrastructure computations, including low-rank base cases such as rank 11–77 uniform, graphic, and sparse-paving examples, toric and Hirzebruch–Riemann–Roch infrastructure, blowup calculations, and normal-form determinant checks. This formed much of the computational substrate of the search. INTEGRAL_EXPLORE 587 Integral KℤK_Z exploration lying outside the final proof: superseded or alternative integral-lift routes—earlier saturation, descent, and torsion-freeness attempts—that were replaced by the route actually used in Sections 4–6. The integral facts that the final proof does use are counted inside the closure, not here. CONDITIONAL_SHELL 562 Conditional “if–then” scaffolding that was useful during exploration but did not itself close into the final proof, including inductive endpoint packages, higher-rank conditional reductions, and truncation templates. ALT_PROOF 229 Redundant but valid alternative derivations, kept as cross-checks for the main route; these include independent approaches to the Chern-α inequality and to the PK=HilbP^K=Hilb identity. LIT_RECON 178 Reconstructions of external results in the notation of the project, mainly to align conventions and make cited theorems usable by the internal proof search. These facts are reflected in the final paper mostly as references rather than as separately stated lemmas. OTHER 88 Off-path fragments not belonging to the main clusters above. Table 3. Cluster decomposition of the 2,493 facts outside the integral main-theorem closure.