{"id":"1ad69893-cac9-4b72-9e19-bf225d62fdd4","arxiv_id":"2508.09915","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new A1-invariant motivic cohomology for all qcqs schemes is constructed from the slice filtration of KGL, with a spectral sequence to homotopy K-theory and etale/syntomic comparisons.","lead":"Voevodsky's program for building A1-invariant motivic cohomology from his slice filtration is realized for all quasi-compact quasi-separated schemes. The new theory ties homotopy K-theory, etale cohomology, and syntomic cohomology together, and proves several of his open slice conjectures.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Key hypothesis Val(X,p,j) (§8.1) is conjectural and is load-bearing for mixed-characteristic mod-p comparisons (Thms 1.10, 1.13); unconditional core of Thm 1.1 survives, so verdict stays CONDITIONAL.","rationale":"The reader identified Val(X,p,j) as the weakest assumption, and I agree. It is the only explicitly conjectural input that is load-bearing for a substantial advertised contribution: the full Beilinson–Lichtenbaum-style comparisons in mixed characteristic. The paper is careful to separate the unconditional core (Theorem 1.1 in its main existence and equal-characteristic properties) from the conditional mixed-characteristic statements (Theorem 1.10, 1.13). Tracing the structure, the unconditional core relies on Theorem 9.13 (P1-bundle formula for A1-localisation of cdh motivic cohomology), which is proved without Val, and on deep but established input (Levine, Voevodsky, Geisser–Levine, Bhatt–Mathew, etc.). Therefore the central existence claim is not threatened. However, the abstract's promise—'we compare to étale and syntomic cohomology in the style of the original conjectures of Beilinson and Lichtenbaum'—is only fully delivered conditional on a conjecture about p-adic cohomology of valuation rings. This is not an internal inconsistency, but it is a genuine gap in the advertised scope. A conditional verdict is appropriate; my read does not change the reader's verdict. The concrete test I propose aims to isolate whether the reduction to rank-1 valuation rings (which the cdh machinery of §2.2 permits) makes Val more tractable, and to check the first unknown weight j=2 in that setting.","tokens_in":64705,"tokens_out":13956,"duration_ms":157771,"concrete_test":"Verify whether the proof of Theorem 8.14 uses Val only for rank-1 henselian valuation rings, via the Milnor-excision reductions of Theorem 2.20/Proposition 2.22. If so, the load-bearing conjecture is the rank-1 case: prove that every rank-1 mixed-characteristic henselian valuation ring is F-smooth in the sense of Bhatt–Mathew (e.g., by showing its Nygaard-completed prismatic cohomology is concentrated in the expected degrees), which would imply Val and turn Theorems 1.10 and 1.13 unconditional. If one can exhibit a rank-1 V that is not F-smooth, compute the cofiber of F_p(2)_syn(V) → RΓ_ét(V[1/p], F_p(2)); a violation of the §8.1 degree bound would show the hypothesis fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing weakness is the 'key hypothesis' Val(X,p,j) introduced in §8.1. It requires, for every mixed-characteristic (0,p) henselian valuation ring V with Spec(V)→X, that F_p(j)_syn(V) is 'mostly given by' RΓ_ét(V[1/p], F_p(j)); the precise degree bound is in §8.1 and would follow from F-smoothness in the sense of Bhatt–Mathew [31]. The authors state this is expected but 'for the moment this remains conjectural' (Example 8.6). This hypothesis is used to prove Theorem 8.14 (=Thm 1.10, conditional A1-invariance of Z(j)_cdh) and Theorem 1.13 (optimal Beilinson–Lichtenbaum comparison in mixed characteristic). If Val fails, those theorems fail. The unconditional core of Theorem 1.1 (existence, AHSS, low weights, equal-characteristic p-syntomic comparison, étale comparison away from residue characteristics, and determination by smooth schemes via Thm 9.13) is not affected. However, the abstract advertises comparisons 'in the style of the original conjectures of Beilinson and Lichtenbaum' for arbitrary qcqs schemes; that promise is fully realized only conditionally on Val. Thus the paper is internally coherent and transparent, but the scope is conditional.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper constructs a theory of A1-invariant motivic cohomology for arbitrary qcqs schemes. The theory Z(j)_A is defined as slices of KGL (Definition 4.15) and is shown to provide an Atiyah-Hirzebruch filtration on KH, to be A1-invariant and a finitary cdh sheaf, to satisfy étale and syntomic comparisons in equal characteristic (Theorem 1.1(4)(5)), low-weight identifications, the projective bundle formula, and agreement with Bloch's cycle complex on smooth varieties. It also proves the zeroth-slice theorem s0(1_X) ≃ HZ^A_X and absolute base change (Theorem 1.6). A second theory Z(j)_cdh is introduced; its A1-invariance and the optimal Beilinson-Lichtenbaum comparison in mixed characteristic are proved only conditionally on the conjectural hypothesis Val(X,p,j) of Section 8.1.","tokens_in":64924,"tokens_out":8187,"duration_ms":95148,"significance":"If correct, this is a landmark contribution: it provides a well-behaved A1-invariant motivic cohomology for all qcqs schemes, with a spectral sequence to homotopy K-theory, and it resolves several of Voevodsky's slice conjectures in a non-vacuous form. The unconditional core is supported by a detailed 130+ page argument, and the paper is unusually transparent about what is conditional. No ad hoc free parameters enter; the theory is characterized by a universal property. The main caveat is that the mixed-characteristic comparisons advertised in the abstract are not unconditional: Theorem 1.13 depends on Val(X,p,j), which is explicitly conjectural. In addition, Remark 1.7 notes that the paper's definition makes Voevodsky's Conjectures 1 and 7 vacuous; the genuine content is Corollary 9.7. These caveats do not undermine the unconditional theorems, but they narrow the scope from what the abstract suggests.","major_comments":[{"comment":"The key hypothesis Val(X,p,j) is unproved and conjectural (Example 8.6). It is load-bearing for Theorem 1.10 (conditional A1-invariance of Z(j)_cdh modulo p) and for Theorem 1.13 (the optimal Beilinson-Lichtenbaum comparison in mixed characteristic). The abstract and Theorem 1.1(5) refer to the mixed-characteristic comparison without stating this assumption. Since Val is not established, the blanket claim of étale/syntomic comparisons 'in the style of the original conjectures' for arbitrary qcqs schemes is only conditional. Please qualify the abstract and the theorem statements, and make the conditional status of Theorem 1.13 visible in the introduction and in the cross-reference from Theorem 1.1(5).","section":"§8.1, Theorems 1.10/1.13"},{"comment":"The paper defines HZ^A_X := s0(KGL_X) and then notes in Remark 1.7 that Voevodsky's Conjectures 1 and 7 become vacuous under this definition. The paper should not be described as establishing those conjectures; the non-vacuous result is the equivalence s0(1_X) ≃ HZ^A_X of Corollary 9.7. This is a legitimate reformulation, but it is currently flagged only in a remark. The introduction and abstract should state clearly that the paper proves a substitute theorem, not a literal verification of the original conjectures.","section":"Remark 1.7 / Corollary 9.7"}],"minor_comments":[{"comment":"The mixed-characteristic syntomic comparison is stated unconditionally in Theorem 1.1(5) via the cross-reference 'see Theorem 1.13'. Since Theorem 1.13 is conditional on Val, the cross-reference should carry the hypothesis explicitly, and the abstract should say 'conditionally on the key hypothesis Val' for the mixed-characteristic comparisons.","section":"Abstract and Theorem 1.1(5)"},{"comment":"There is a typo: the display 'H^i_{A,cdh}(X,Z(j)) := H^i(Z(j)_A(X))' should read 'H^i(Z(j)_{A,cdh}(X))'.","section":"Definition 4.17"},{"comment":"Section 10 is described as a 'speed-run' and says 'we omit some details' with respect to the Hermitian K-theory analogues. If Theorem 10.22 is advertised as a main result, the omissions make verification difficult. Either include the omitted arguments or clearly label the section as a sketch.","section":"Section 10"},{"comment":"The observation that [156, Conjectures 1 and 7] are vacuous should also appear in the introduction's summary of Voevodsky's slice conjectures, not only in a remark after Theorem 1.6, to avoid overstating the historical claim.","section":"Remark 1.7"}],"recommendation":"major_revision","confidential_remarks":"This is likely to become an important paper. My recommendation of major revision is driven by scope and presentation, not by a detected mathematical error in the unconditional core. The key hypothesis Val(X,p,j) must be moved from an implicit assumption to a clearly displayed condition in the abstract and theorem statements; otherwise the paper overclaims the mixed-characteristic results. I would not reject the paper on this basis, but I would not accept it in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Tom,\n\nHere's my reading. The paper is the real thing: a 130-page development of A1-invariant motivic cohomology for all qcqs schemes, built from the slice filtration on KGL, with an Atiyah–Hirzebruch spectral sequence abutting to KH, cdh descent, and comparisons to étale and syntomic cohomology. If the main results hold up, the authors have realized Voevodsky's slice program in essentially the expected generality. The strongest unconditional results are Theorem 1.1 (the nine properties) and Theorem 1.6/Cor 9.7, identifying s0(1_X) with HZ^A_X; the latter settles the non-vacuous slice conjectures and is proven by reduction to Levine's theorem over fields plus cdh-descent machinery.\n\nWhat I appreciate: the paper is exceptionally honest about its own scope. Remark 1.7 says flat out that under their definition, Voevodsky's Conjectures 1 and 7 are vacuous; they don't oversell what follows formally. The key hypothesis Val(X,p,j) is introduced in §8.1 with a clear statement that it is expected but not proved, and the conditional theorems (1.10, 1.13) are labeled as such. That separation is exactly how a hard paper should be written.\n\nThe soft spot is precisely Val. It is load-bearing for the mixed-characteristic mod-p A1-invariance of cdh-motivic cohomology and for the optimal Beilinson–Lichtenbaum comparison in mixed characteristic. The concrete gap is a regularity statement about syntomic cohomology of henselian valuation rings, expected to follow from F-smoothness in the sense of Bhatt–Mathew. If Val fails, those theorems fail, though the unconditional core of Theorem 1.1 survives. So the abstract is slightly stronger than the body: the 'style of the original conjectures' comparison for arbitrary qcqs schemes is realized unconditionally only away from mixed characteristic. Minor issues: Section 10 is a compressed treatment of Hermitian K-theory and omits some details, and there are a few points where I'd want more proof (notably the proof of Theorem 9.13, which they admit is indirect).\n\nMy verdict: this deserves a serious referee. It is a landmark-scale piece of work, and the conditional material is handled responsibly. I'd rather referee this than ten incremental papers. If you're working on motivic homotopy theory, K-theory, or motives over general bases, it's worth a careful read.","headline":"Serious realization of Voevodsky's slice program for qcqs schemes, with an honest conditional hypothesis (Val) for mixed-characteristic mod-p comparisons.","tokens_in":65659,"tokens_out":2359,"would_cite":true,"duration_ms":27879,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","14F20","19D55","19E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A well-behaved $\\mathbb{A}^1$-invariant motivic cohomology exists for every quasicompact quasiseparated scheme, built from the slice filtration on homotopy $K$-theory.","keywords":["A1-invariant motivic cohomology","slice filtration","homotopy K-theory","cdh topology","Beilinson-Lichtenbaum","syntomic cohomology","qcqs schemes","motivic spectra"],"falsifier":"Find a henselian valuation ring $V$ of mixed characteristic $(0,p)$ and a weight $j$ for which the cofiber of $F_p(j)_{\\mathrm{syn}}(V) \\to \\tau_{\\le j} R\\Gamma_{\\mathrm{ét}}(V[1/p], \\mu_p^{\\otimes j})$ has nonzero cohomology in degree below $j-1$; equivalently, exhibit a qcqs scheme of finite valuative dimension where the map $Z(j)_{\\mathrm{cdh}}(X) \\to Z(j)_{\\mathrm{cdh}}(\\mathbb{A}^1_X)$ is not an equivalence modulo $p$. Such a discovery would refute the key hypothesis and with it the conditional Theorems 1.10 and 1.13, while leaving the unconditional core of Theorem 1.1 intact.","tokens_in":64397,"feed_emoji":"","tokens_out":12390,"duration_ms":125672,"temperature":0.7,"pith_summary":"The paper claims that every quasicompact quasiseparated scheme — in particular, every scheme of finite type over a field or Dedekind domain — carries a well-behaved $\\mathbb{A}^1$-invariant motivic cohomology. The theory is constructed from the slice filtration on the motivic spectrum representing homotopy $K$-theory, so that the graded pieces of a natural filtration on $KH$ are exactly the new cohomology groups. The paper proves this theory has the properties that make motivic cohomology useful: it agrees with the classical cycle complex on smooth varieties, with étale cohomology away from residue characteristics, and with logarithmic de Rham–Witt cohomology in characteristic $p$, and it satisfies cdh descent. One conditional ingredient, a regularity hypothesis about henselian valuation rings of mixed characteristic, is needed only for the optimal mod-$p$ comparisons and for $\\mathbb{A}^1$-invariance of the auxiliary cdh-motivic cohomology. If the construction stands, it settles the slice-filtration conjectures identifying the zeroth slice of the motivic sphere with motivic cohomology, and provides a candidate derived category of $\\mathbb{A}^1$-invariant motives over arbitrary qcqs bases.","feed_headline":"A^1-motivic cohomology exists for every qcqs scheme","feed_subtitle":"The graded pieces of a filtration on homotopy K-theory satisfy the expected Beilinson–Lichtenbaum comparisons.","key_machinery":"The load-bearing object is the slice filtration on the stable motivic homotopy category $SH(X)$: the descending tower of full subcategories generated by motives of smooth $X$-schemes with $j$ Tate twists, whose associated graded pieces $s_j E$ are the slices of a motivic spectrum $E$. For $E = KGL$, the motivic spectrum representing homotopy $K$-theory, the zeroth slice $s_0 KGL$ is taken as the definition of the motivic ring spectrum $HZ^A_X$, and the higher slices give the cohomology theories $Z(j)_{\\mathbb{A}}$. Two auxiliary constructions carry the technical weight: cdh-motivic cohomology, obtained by cdh sheafifying the left Kan extension of motivic cohomology from smooth $\\mathbb{Z}$-s","core_discovery":"The paper defines, for each qcqs scheme $X$, the $\\mathbb{A}^1$-motivic cohomology groups as shifts of the slices of the motivic spectrum $KGL$: $Z(j)_{\\mathbb{A}}(X) = \\mathrm{map}_{SH(X)}(1_X, s_j KGL_X)[-2j]$. Its main theorem is that these assemble into a multiplicative family of presheaves of complexes with an Atiyah–Hirzebruch spectral sequence abutting to $KH$, with $\\mathbb{A}^1$-invariance, finitary cdh descent, a projective bundle formula, and the expected comparisons to étale cohomology, syntomic cohomology, and the cycle complex. The paper further proves that the unit map $1_X \\to KGL_X$ induces an equivalence $s_0(1_X) \\simeq HZ^A_X$ in $SH(X)$, where $HZ^A_X := s_0 KGL_X$; this","pith_inferences":["If the key hypothesis Val$(X,p,j)$ is ever disproved, the failure would be localized to the comparison between mod-$p$ syntomic cohomology and étale cohomology of valuation rings; the $\\mathbb{A}^1$-motivic cohomology itself, defined from slices, would remain a viable theory, though it would no longer be known to agree with the cdh-sheafified version in mixed characteristic.","The cdh-descent and Milnor-excision properties suggest an effective computational strategy for $KH$ on singular schemes: evaluate the filtration on henselian valuation rings, where the new cohomology reduces to étale and Milnor $K$-theory data; verifying the key hypothesis in low weights would make the mixed-characteristic computations unconditional.","The Hermitian variant in Section 10 indicates a parallel story for Grothendieck–Witt groups and the motivic spectrum $KO$, with an 8-periodic refinement of periodicity; if the same machinery works there, it would produce an Atiyah–Hirzebruch spectral sequence from Milnor–Witt motivic cohomology to Hermitian $K$-theory.","The paper leaves open integral versions of low-weight vanishing results it proves rationally; the slice definition gives a concrete candidate theory on which those integral statements could be tested."],"forward_implications":["An Atiyah–Hirzebruch spectral sequence with $E_2^{i,j} = H^{i-j}_{\\mathbb{A}}(X, \\mathbb{Z}(-j))$ abuts to $KH^{-i-j}(X)$ for every qcqs scheme, and the filtration is complete when the scheme has finite valuative dimension.","On smooth varieties over fields, the new groups agree with higher Chow groups, so the construction is a common generalization of the classical motivic cohomology theories.","Away from residue characteristics, $Z(j)_{\\mathbb{A}}/\\ell \\simeq L_{\\mathrm{cdh}} \\tau_{\\le j} R\\Gamma_{\\mathrm{ét}}(-,\\mu_\\ell^{\\otimes j})$; in characteristic $p$, $Z(j)_{\\mathbb{A}}/p^r \\simeq R\\Gamma_{\\mathrm{cdh}}(X, W_r\\Omega^j_{\\log})[-j]$. The expected comparison isomorphisms therefore hold in equal characteristic unconditionally.","The category of $HZ^A_X$-modules inherits a six-functor formalism from $SH(X)$, giving a candidate for the derived category of $\\mathbb{A}^1$-invariant motives over any qcqs base.","In mixed characteristic, the mod-$p$ comparisons hold conditionally on the key hypothesis about henselian valuation rings; rationally, all comparisons are unconditional."],"supporting_citations":[{"why":"Sets out the slice-filtration programme whose conjectures the paper realizes, including the identification of the zeroth slice of the sphere with motivic cohomology.","marker":"[156, 157]"},{"why":"Supplies the classical cycle complex to which the new theory is compared on smooth varieties.","marker":"[37]"},{"why":"Provides the theorem over fields identifying the zeroth slice of the motivic sphere with that of KGL, the key input carried from fields to all qcqs schemes.","marker":"[110]"},{"why":"Establishes cdh descent for homotopy K-theory, which underlies the sheafiness and blow-up descent properties of the filtration.","marker":"[52]"},{"why":"Supplies Adams operations on KGL, used to split the filtration rationally and to identify rational motivic cohomology with Adams eigenspaces of K-theory.","marker":"[134]"},{"why":"Extends syntomic cohomology to all schemes, providing the gluing of p-adic étale and p-complete contributions used in the Beilinson–Lichtenbaum comparisons.","marker":"[29]"},{"why":"Introduces the F-smooth regularity condition expected to imply the key hypothesis about valuation rings that controls the mixed-characteristic mod-p theorems.","marker":"[31]"},{"why":"Provides the framed effective spectrum V used to control the effective cover of KGL and to transfer statements about the motivic sphere.","marker":"[93]"}],"fun_headline_variants":["A^1 motivic cohomology now works on all qcqs schemes","Slices of KGL yield A^1 cohomology for every qcqs scheme","A^1-invariant motivic cohomology: from smooth to qcqs","Homotopy K-theory spectral sequence meets A^1 motives","Voevodsky's slice filtration realized for all qcqs schemes"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The load-bearing premise is that for every henselian valuation ring of mixed characteristic $(0,p)$ mapping to the scheme, mod-$p$ syntomic cohomology is essentially determined by the étale cohomology of the $p$-inverted locus — a regularity condition that would follow from F-smoothness and is expected but not yet proved.","fun_headline_variants_meta":{"raw":{"variants":["A^1 motivic cohomology now works on all qcqs schemes","Slices of KGL yield A^1 cohomology for every qcqs scheme","A^1-invariant motivic cohomology: from smooth to qcqs","Homotopy K-theory spectral sequence meets A^1 motives","Voevodsky's slice filtration realized for all qcqs schemes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000287,"raw_usage":{"total_tokens":1642,"prompt_tokens":985,"completion_tokens":657,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":729,"completion_tokens_details":{"reasoning_tokens":553}},"tokens_in":729,"tokens_out":657,"duration_ms":6485,"temperature":1.0,"reasoning_tokens":553,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:44:23.067958+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a henselian valuation ring $V$ of mixed characteristic $(0,p)$ and a weight $j$ for which the cofiber of $F_p(j)_{\\mathrm{syn}}(V) \\to \\tau_{\\le j} R\\Gamma_{\\mathrm{ét}}(V[1/p], \\mu_p^{\\otimes j})$ has nonzero cohomology in degree below $j-1$; equivalently, exhibit a qcqs scheme of finite valuative dimension where the map $Z(j)_{\\mathrm{cdh}}(X) \\to Z(j)_{\\mathrm{cdh}}(\\mathbb{A}^1_X)$ is not an equivalence modulo $p$. Such a discovery would refute the key hypothesis and with it the conditional Theorems 1.10 and 1.13, while leaving the unconditional core of Theorem 1.1 intact.","supporting_citations":[{"cited_title":"The homotopy coniveau tower.J","cited_arxiv_id":null,"evidence_quote":"Provides the theorem over fields identifying the zeroth slice of the motivic sphere with that of KGL, the key input carried from fields to all qcqs schemes."},{"cited_title":"Algebraic K-theory, A1-homotopy and Riemann-Roch theorems","cited_arxiv_id":null,"evidence_quote":"Supplies Adams operations on KGL, used to split the filtration rationally and to identify rational motivic cohomology with Adams eigenspaces of K-theory."}],"review_version":1}