{"id":"820af92a-4e17-42b4-8769-b989143f8677","arxiv_id":"2501.01609","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new stable rank filtration on direct sum K-theory is defined via Gamma-sapces, with filtration quotients described as suspension spectra of decomposition posets.","lead":"This paper defines a new 'stable rank filtration' on algebraic K-theory using Gamma-spaces, and identifies the filtration quotients as homotopy coinvariants and suspension spectra of decomposition posets. It gives a fresh computational framework for K-theory spectra, generalizing Rognes's earlier rank filtration and recovering the Barratt-Priddy-Quillen theorem.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Connectivity hypotheses for Theorem C are false: the filtered pieces are configuration spaces, only (n−2)-connected, not (2n−1)-connected.","rationale":"The reader identified the connectivity hypothesis of Corollary 1.19 as the weakest assumption and flagged it as needing verification in applications. My analysis goes further: the hypothesis is not merely unverified but is actually false for the filtered pieces of S_A. Proposition 3.13 shows F_p S_A(L) ≅ L^p, where L^p is the space of injective functions from the finite set p to L, i.e., the ordered configuration space. When L = S^n, this is Conf_{|p|}(S^n). Standard results on configuration spaces give connectivity about n−2, not 2n−1. Concrete examples include Conf_2(S^2) ≅ RP^3 (π_1 = Z/2) and Conf_2(S^1) ≃ S^1 (π_1 = Z). Thus the central theorem, which identifies K(S_A) with the suspension spectrum of the decomposition poset, is not proven by the paper's stated methods. The applications in Section 4 depend directly on this theorem, so the paper cannot be accepted in its current form. The concern is not an external disagreement about consensus but an internal inconsistency: the stated hypotheses of the paper's own corollary are contradicted by the computed filtration pieces. I credit the paper for its clear framework and many correct intermediate results, but the main computational conclusion lacks a valid proof. A revised proof would need to establish Theorem C by a different route or substantially weaken the connectivity demands while preserving the stable equivalence.","tokens_in":12,"tokens_out":30511,"duration_ms":947462,"concrete_test":"Verify the connectivity claim directly: let A be a 2-dimensional vector space and let p be a decomposition of A into two 1-dimensional subspaces. By Proposition 3.13, F_p S_A(S^2) ≅ Conf_2(S^2) ≅ RP^3. Compute its fundamental group: π_1(RP^3) ≅ Z/2, so it is not even 1-connected, whereas Corollary 1.19 requires (2·2−1) = 3-connected. This falsifies the assertion that S_A satisfies the conditions of Corollary 1.20/1.19, and thereby invalidates the proof of Corollary 3.14.","verdict_should_be":"REJECT","load_bearing_attack":"Proposition 3.13 identifies the filtered piece F_p S_A(L) with L^p, the simplicial set of pointed maps from p to L that are either constant at the basepoint or injective. For L = S^n, L^p is the ordered configuration space Conf_{|p|}(S^n). For a nontrivial decomposition p (|p| ≥ 2), Conf_{|p|}(S^n) is at best (n−2)-connected: for instance, Conf_2(S^2) ≅ RP^3 has π_1 = Z/2 and Conf_2(S^1) ≃ S^1 has π_1 = Z. Corollary 1.19 requires the filtered piece at spectrum level n to be (2n−1)-connected; for n = 2 this means 3-connected, but RP^3 is not even 1-connected. Corollary 1.20 requires even more, (2m+1)-connected for every m-connected L. Hence S_A does not satisfy the hypotheses of Corollary 1.20/1.19 (or the equivariant Corollary 1.22), so the proof of Corollary 3.14 (Theorem C) does not go through. Since the spectral sequences in Section 4 rely on Theorem C, the main computational claims are unsupported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a stable rank filtration on the direct-sum K-theory of 'convenient addition categories', built from P-valuations on Γ-spaces. It proves that the associated graded pieces of the filtration are homotopy coinvariants of the K-theory of a subobject structure S_A (Theorem A / Theorem 3.6), and it identifies the K-theory spectrum of S_A with the suspension spectrum of the nerve of the decomposition poset Dcp^∘_A (Theorem C / Corollary 3.14). This is then used to give new proofs and spectral sequences: the Barratt–Priddy–Quillen theorem, a rational spectral sequence for inner product spaces over ordered fields, a rational spectral sequence for free modules over fields or Dedekind domains, and a comparison with Rognes's common basis complex. The appendix by Kupers compares the homotopy type of decomposition posets with their ordered variants, following Mirzaii–van der Kallen.","tokens_in":23911,"tokens_out":14922,"duration_ms":140517,"significance":"If the main theorems hold, the paper gives a clean categorical framework that unifies and generalizes Rognes's stable rank filtration and common basis complex, and it supplies concrete spectral-sequence applications. The paper is well structured, contains substantial detailed proofs, and the appendix by Kupers is a useful contribution in its own right. The central computational engine, however, is Theorem C, and its proof currently relies on a connectivity hypothesis that is not satisfied; the issue is real but appears repairable by weakening the connectivity bounds, since only unbounded growth of the connectivity at spectrum level n is needed for a stable equivalence.","major_comments":[{"comment":"The proof of Corollary 3.14 asserts that S_A satisfies the hypotheses of Corollary 1.20, but this is false. By Proposition 3.13, the filtered piece F_p S_A(L) is isomorphic to L^p, the pointed simplicial set of maps p → L that are constant or injective. For L = S^n and a nontrivial decomposition p with |p| ≥ 2, this is the ordered configuration space Conf_{|p|}(S^n) (up to the basepoint). Already for |p| = 2, Conf_2(S^n) is homotopy equivalent to S^n, which has connectivity n−1, not the 2n−1 required by Corollary 1.20 when applied to L = S^n (which is (n−1)-connected and forces m = n−1). For n = 2 this requires 3-connectivity, while Conf_2(S^2) ≃ S^2 is only 1-connected. Thus the hypotheses of Corollary 1.20 fail for every n ≥ 1. The same failure affects Corollary 1.22 and therefore the equivariant statements used in Sections 4.2 and 4.3. The gap is repairable: for a stable equivalence it is enough that the level-n map induce isomorphisms on π_i for i < n−2 (equivalently, for i < m in the notation of Corollary 1.20), because for fixed i this holds for all sufficiently large n. I recommend weakening the connectivity hypotheses in Corollaries 1.19–1.22 accordingly and rechecking the proof of Corollary 3.14.","section":"Corollary 1.20, Proposition 3.13, Corollary 3.14"},{"comment":"The computations in Sections 4.2 and 4.3 depend on the equivariant equivalences K(S^n_k) ≃ Σ^∞ Σ^1 N Dcp^∘_{k^n} supplied by Corollary 3.14 via Corollary 1.22. Because the hypotheses of Corollary 1.22 are not satisfied, the displayed spectral sequences and the group identifications in Corollary 4.4 are currently unsupported. If the connectivity hypotheses are weakened as suggested in the previous comment, these applications should be verified levelwise; as written, the applications do not follow from the results proved in the paper.","section":"Sections 4.2–4.3, Corollary 4.4"}],"minor_comments":[{"comment":"The abstract and introduction state that the spectral sequences converge to the homology of algebraic K-theory, but the body of the paper only proves convergence to rationalized K-groups. Please add the rational qualifier throughout the introductory statements.","section":"Abstract and Section 4"},{"comment":"In the proof of Lemma 1.18, the word 'isomoprhism' should read 'isomorphism'.","section":"Lemma 1.18 proof"},{"comment":"The sentence about 'groups in green boxes being torsion' refers to colors that are absent in a monochrome printout; please use a color-independent designation or a displayed annotation.","section":"Section 4.3"},{"comment":"The phrase 'square-root closed ordered field' is not defined; please define it or give a reference, since it is a condition on the field in an example.","section":"Example 3.9"}],"recommendation":"major_revision","confidential_remarks":"The paper has a genuine gap in the proof of the main computational theorem, but it is not a fatal one: the connectivity bound in Corollaries 1.19–1.22 is stronger than necessary, and the argument can be repaired by requiring only that the connectivity at level n tends to infinity. I would encourage the editors to invite a revision rather than reject. The authors should also reconcile the abstract's unqualified convergence statement with the rationalized results actually proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper has a genuinely useful new framework—P-valuations on Γ-spaces and convenient addition categories—and Theorem 3.6, identifying filtration quotients as homotopy coinvariants of K(S_A), is a solid and attractive result. The writing is clear about the debts to Rognes and the GKRW circle, and the citation practice looks fine. But the main computational theorem, Theorem C, is not proved as written. The stress-test note is right. Proposition 3.13 identifies the filtered piece F_p S_A(L) with L^{|p|}, the pointed maps from |p| into L that are injective away from the basepoint. For L = S^n, that is the ordered configuration space Conf_{|p|}(S^n). For |p| ≥ 2, Conf_{|p|}(S^n) is at best (n−2)-connected—for example Conf_2(S^2) has π_1 = Z/2, and Conf_2(S^1) ≃ S^1. Corollary 1.19 needs the n-th filtered piece to be (2n−1)-connected, and Corollary 1.20 needs (2m+1)-connected for any m-connected L. The configuration spaces are nowhere near that connected. So the appeal to Corollaries 1.19/1.20 in Corollary 3.14 fails. That is load-bearing: the spectral sequences in Section 4 and the BPQ reproof for n>1 depend on Corollary 3.14. I do not see a quick patch in the text; the proof would need different connectivity bounds or a different identification of the filtered pieces. The theorem itself might be true, but it is unsupported here.\n\nSmaller issues: the abstract promises convergence to the homology of algebraic K-theory, while the body only has rationalized statements; and the spectrum-level half of Theorem 3.6 is passed over quickly. These are minor.\n\nWhat is good and likely durable: the P-valuation formalism, the convenient addition category setup, the decomposition poset analysis, and Kupers's appendix relating ordered and unordered decomposition posets. Those deserve to be in the literature.\n\nWho should read this: people working on rank filtrations in K-theory, and anyone using Rognes's common basis complex. This is not a desk-reject; it is a paper with a serious gap in its central application. I would send it to a knowledgeable referee, with explicit instructions to check the connectivity hypothesis and to ask whether Corollary 3.14 can be replaced by a statement that holds. If the authors can fix that, the paper would be a real contribution.","headline":"A promising filtration framework whose central computational theorem is not proved: the connectivity hypotheses in Corollary 1.19/1.20 fail for the configuration spaces that Proposition 3.13 produces.","tokens_in":24456,"tokens_out":15601,"would_cite":false,"duration_ms":138427,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19D10","19D23","55P42"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper builds a stable rank filtration on the K-theory spectrum of convenient addition categories, identifying filtration quotients as homotopy coinvariants and subobject K-theory as a suspension spectrum of decompositions.","keywords":["algebraic K-theory","stable rank filtration","Γ-spaces","convenient addition categories","decomposition posets","spectral sequences","common basis complex","Barratt–Priddy–Quillen theorem"],"falsifier":"For A the category of finite-dimensional F_2-vector spaces and object A = $F_2^{3}$, Proposition 4.5 says N Dcp^∘_A is a wedge of circles, so the central equivalence predicts π_2(K(S_A)) ≅ H_1(N Dcp^∘_A; Z), a free abelian group of rank equal to the number of circles. Compute that rank from the poset of direct-sum decompositions of $F_2^{3}$ and independently compute π_2 of the Γ-set S_A from its simplicial sets S_A(S^k) for k = 1,2,3; any disagreement between the two groups would refute the central identification.","tokens_in":23478,"feed_emoji":"🧩","tokens_out":20340,"duration_ms":183531,"temperature":0.7,"pith_summary":"The paper's goal is a stable rank filtration on algebraic K-theory for a general class of symmetric monoidal categories, not just rings. It introduces convenient addition categories—categories in which direct sum behaves like a disjoint union—and shows that any rank-like valuation on such a category filters the K-theory spectrum at the spectrum level. The central result identifies the filtration quotients with homotopy coinvariants of the K-theory of subobject structures, and then identifies that subobject K-theory with the suspension spectrum of the nerve of the poset of nontrivial decompositions. From this it derives new spectral sequences converging to the homology of algebraic K-theory, recovers the Barratt–Priddy–Quillen theorem, and re-expresses Rognes's common basis complex.","feed_headline":"Stable rank filtration computes K-theory from decomposition posets","feed_subtitle":"Filtration quotients are homotopy coinvariants, yielding new spectral sequences for rational algebraic K-groups.","key_machinery":"The load-bearing mechanism is the P-valuation: a functor from the category of elements of a Set-valued functor (or, levelwise, of a symmetric spectrum or Γ-set) to a poset P, which records the 'rank' of each element and induces a filtration F_p. The key identity is Theorem C, K(S_A) ≃ Σ^∞ $Σ^{1}$ N(Dcp^∘_A), where Dcp^∘_A is the poset of nontrivial unordered decompositions of A into non-initial subobjects, ordered by refinement. It is proven by Proposition 3.13, which shows the filtered pieces of the Γ-set S_A are exactly smash powers $L^{{∧|p|}}$, and by Corollary 1.19, which assembles these highly connected pieces into a stable equivalence via a levelwise Mayer–Vietoris argument. The axioms of a convenient addition category—unit is initial, all morphisms monic, maps out of a sum determined by components, and the pullback of the two inclusions into a sum is the initial object—are precisely what makes the decomposition poset well-behaved and the smash-power identification hold.","core_discovery":"For a convenient addition category A with a rank-like valuation, the K-theory spectrum K(A) carries a filtration indexed by the rank poset whose associated graded pieces are computed in two steps (Theorems 3.6 and 3.14). The first step is an Aut(A)-equivariant equivalence between the p-th filtration quotient and the homotopy coinvariants K(S_A)_{hAut(A)}, where S_A is the Γ-set built from the subobject structure of an object A of rank p. The second step is a general connectivity theorem (Corollary 1.19) applied to a decomposition-valued filtration on S_A: because the filtered pieces are highly connected smash powers, K(S_A) is stably equivalent to Σ^∞ $Σ^{1}$ N(Dcp^∘_A), the suspension spectrum of the nerve of the poset of nontrivial decompositions of A. In the applications, these decomposition posets are wedges of spheres (using Kupers's comparison with ordered decompositions and classical Tits-building results), which makes the spectral sequences collapse enough to compute rational K-groups from group homology.","pith_inferences":["The suspension-spectrum form of K(S_A) suggests the whole rank filtration is cellular, so one could attempt to compute Steenrod operations or higher differentials in algebraic K-theory directly from the combinatorics of decomposition posets; the paper does not pursue this.","If the wedge-of-spheres conjecture for the minimal-spanning-poset space holds, the new spectral sequences would collapse at E2, yielding rational K-groups as a direct sum of group homologies of GL_n(k) (or O_n(k)) with Steinberg-like coefficients.","The recognition principle behind Theorem C—highly connected poset filtrations force a spectrum to be a suspension spectrum of a nerve—is stated for arbitrary spectra and could be applied to other filtered spectra in algebraic topology, such as Waldhausen's A-theory or L-theory, whenever a rank-like invariant exists.","The Aut(A)-equivariance in Theorem C provides extra structure not needed for the homology computations; tracking this action might yield homological stability theorems for automorphism groups of objects in convenient addition categories, generalizing stability for GL_n and symmetric groups."],"forward_implications":["For finite sets, the filtration quotients above rank one vanish, giving a new proof of the Barratt–Priddy–Quillen theorem: K(FinSet_*) is the sphere spectrum.","For a field or Dedekind domain k, there is a spectral sequence with E1-page H_s(GL_n(k); Dec_n) converging to the rationalized K-groups of k, where Dec_n is the rational homology of the decomposition poset of k^n.","For inner product spaces over an ordered field, an analogous spectral sequence with orthogonal groups O_n(k) converges to the rational K-theory of that category.","Rognes's common basis complex C_n is shown to be stably equivalent to the suspension of the poset of nontrivial minimal spanning posets of R^n, and the paper conjectures this is an equivariant equivalence and that the poset is a wedge of (2n−3)-spheres.","The valuation formalism extends to Waldhausen categories, giving an alternate spectrum-level model of Rognes's rank filtration."],"supporting_citations":[{"why":"supplies the original spectrum-level rank filtration, the common basis complex, and the poset-filtration technique the paper generalizes and recovers.","marker":"[Rog92]"},{"why":"introduces the unstable rank filtration and the Tits-building wedge-of-spheres computation used in Section 4.","marker":"[Qui73]"},{"why":"introduces the Γ-space model of K-theory whose simplicial structure the new filtration is built from.","marker":"[Seg74]"},{"why":"provides the S_•-construction and the homotopy fiber sequence that the U_•-construction and Proposition 2.9 follow.","marker":"[Wal85]"},{"why":"supplies the E_k-algebra splitting complexes and the E_1/E_8 comparison that Kupers's appendix relates to decomposition posets.","marker":"[GKRW18]"},{"why":"gives the connectivity lemma (Theorem 3.8) used in Kupers's proof comparing ordered and unordered decomposition posets.","marker":"[MvdK02]"},{"why":"states the Barratt–Priddy–Quillen theorem that the paper recovers as a corollary of its filtration.","marker":"[BP72]"},{"why":"provides the vanishing result for q < p used to simplify the spectral sequence for infinite fields in Section 4.3.","marker":"[GKRW20]"}],"fun_headline_variants":["Gamma-space stable rank filtration gives homotopy coinvariants","Alternate stable rank filtration via decomposition posets","Stable rank filtration yields spectral sequences for K-theory","Decomposition posets compute K-theory via stable rank filtration"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument that a filtered spectrum is a suspension spectrum of a nerve depends on the connectivity of the filtered pieces: at level n, every piece in the distinguished cosieve must be (2n−1)-connected, since only then does the levelwise approximation become a true stable equivalence; without that, the identification of K(S_A) and the spectral sequences built on it collapse.","fun_headline_variants_meta":{"raw":{"variants":["Gamma-space stable rank filtration gives homotopy coinvariants","Alternate stable rank filtration via decomposition posets","Stable rank filtration yields spectral sequences for K-theory","Decomposition posets compute K-theory via stable rank filtration"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000841,"raw_usage":{"total_tokens":3652,"prompt_tokens":920,"completion_tokens":2732,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":2665}},"tokens_in":536,"tokens_out":2732,"duration_ms":21737,"temperature":1.0,"reasoning_tokens":2665,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:23:56.652405+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For A the category of finite-dimensional F_2-vector spaces and object A = $F_2^{3}$, Proposition 4.5 says N Dcp^∘_A is a wedge of circles, so the central equivalence predicts π_2(K(S_A)) ≅ H_1(N Dcp^∘_A; Z), a free abelian group of rank equal to the number of circles. Compute that rank from the poset of direct-sum decompositions of $F_2^{3}$ and independently compute π_2 of the Γ-set S_A from its simplicial sets S_A(S^k) for k = 1,2,3; any disagreement between the two groups would refute the central identification.","supporting_citations":[],"review_version":1}