{"id":"e5ab161b-6d6c-4f26-9567-9f32e63d6584","arxiv_id":"2608.04877","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For bounded-below classical spectra with even MU homology, the effective slices of their C-motivic analogues are wedges of motivic Eilenberg-MacLane spectra indexed by the Adams-Novikov E2-page.","lead":"This paper computes the effective, connective, and very effective slices of certain C-motivic spectra by modeling them as filtered spectra, and obtains the slices of the motivic modular forms spectrum mmf as a new case. It also shows that the effective slice spectral sequence of these spectra is a re-indexed version of the classical Adams-Novikov spectral sequence.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of the main slice theorem depends on Proposition 7.8, whose proof uses Lemma 7.5 with an invalid justification; smashing with MZ need not preserve isomorphisms on a single stem, so the splitting of the slice from its homotopy groups is not established.","rationale":"The reader's weakest assumption concerned the connective and very effective filtrations not being identified with the full motivic filtrations. That is a legitimate caveat for secondary results, but it does not affect the effective slice theorem, which is the central claim and is covered by Heard's theorem as the paper states. The most load-bearing part of the effective slice proof is the splitting theorem (Proposition 7.8), and the proof of that theorem is not sound as written. In particular, Lemma 7.5 is essential to transfer the homotopy-group isomorphism to the MZ-free replacement, and its justification is incorrect: the Hurewicz map and the module multiplication do not automatically induce isomorphisms on a single stem, because smashing with MZ can introduce or annihilate classes. Lemma 7.3 similarly overlooks that the cofiber of a map of spheres carries adjacent-stem homotopy, so the constructed C_s is not a clean Eilenberg-MacLane-like object. These gaps mean Corollary 7.11 is not proven from the given arguments, even if the statement is likely true. I therefore recommend CONDITIONAL rather than UNCHANGED: the paper should be accepted only after a correct proof of Proposition 7.8 is provided, or an independent reference for the splitting of MZ-modules with free bigraded homotopy is cited. The verdict remains CONDITIONAL, matching the reader's overall recommendation, but for a different and more central reason.","tokens_in":35175,"tokens_out":34817,"duration_ms":298366,"concrete_test":"Take the simplest potential counterexample to Lemma 7.5: let C=S^{1,0}, Y=S^{1,0}∨MZ, and let α:C→Y be the inclusion of the first summand (an isomorphism on π_{1,0}). Compute the C-motivic, 2-complete homotopy group π_{1,0}(MZ∧MZ). If this group is nonzero, then id∧α:MZ∧S^{1,0}→MZ∧(S^{1,0}∨MZ) is not an isomorphism on π_{1,0}, disproving Lemma 7.5. If π_{1,0}(MZ∧MZ)=0, repeat with a stem where the sphere has torsion, such as C=S^{2,1} and Y=C∨MZ, checking π_{2,0}(MZ∧MZ).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Corollary 7.11 (Theorem 1.2) derives the effective slice of Γ⋆(X) from its homotopy groups by invoking Proposition 7.8, which claims that any MZ-module Y with π_{s,*}(Y) ≅ G_s[τ] is a wedge of M G_s. The proof of Proposition 7.8 constructs maps α_s:C_s→Y using Lemmas 7.3–7.5. Lemma 7.5 asserts that if α induces an isomorphism on π_{s,*}, then so does id∧α:MZ∧C_s→MZ∧Y; its proof claims the Hurewicz map C_s→MZ∧C_s and the module multiplication MZ∧Y→Y induce isomorphisms because a cellular approximation of MZ 'keeps π_{0,*} unchanged'. This does not imply isomorphisms on the relevant bigraded homotopy groups. For example, for C=S^{s,w}, the Hurewicz map S^{s,w}→MZ∧S^{s,w} is zero on stems t≠s and is not an isomorphism on π_{s,*} if π_{s,*}(S^{s,w}) contains torsion. Moreover, Lemma 7.3 constructs C_s as a cofiber of a wedge of spheres; such a compact cofiber has homotopy in adjacent stems (e.g. the cofiber of multiplication by 2 on S^{s,w} has nonzero π_{s+1,*}), so C_s is not concentrated in a single stem as assumed. Consequently the wedge in Proposition 7.8 may include spurious summands and the claimed equivalence is unproven. Since this splitting is the bridge from the homotopy computation of Proposition 7.2 to the closed form of the slice, the central claim of the paper is not yet established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the effective, connective, and very effective filtrations in the 2-complete, cellular C-motivic stable homotopy category, using the filtered-spectrum model of Gheorghe–Isaksen–Krause–Ricka and the motivic analogue functor Γ⋆. The main construction associates to a bounded-below classical spectrum X the C-motivic analogue Γ⋆(X), gives filtered-spectrum models for its covers, and then computes the slices of these filtrations. The central result, Theorem 1.2 (Corollary 7.11), identifies the q-th effective slice of Γ⋆(X) with a wedge of C-motivic Eilenberg–MacLane spectra indexed by the classical Adams–Novikov E2-page of X, provided X has even MU-homology. This recovers earlier slice computations for S, MU, ku, ko, and MGL and yields a new computation for the motivic modular forms spectrum mmf. The paper also proves a comparison between the effective slice spectral sequence of Γ⋆(X) and the classical Adams–Novikov spectral sequence of X, stated as Theorem 1.3 (Theorem 8.1).","tokens_in":35520,"tokens_out":5175,"duration_ms":51039,"significance":"If the main results are correct, the paper gives a unified and computationally explicit framework for slice filtrations of C-motivic analogues, and the computation of the slices of mmf is a genuinely new contribution. The filtered-spectrum models for the effective, connective, and very effective covers are concrete and potentially useful for further calculations. The recovery of known results for MGL, the sphere, kq, and other spectra is a useful consistency check, and the comparison with the Adams–Novikov spectral sequence in Section 8 is conceptually attractive. The paper is not machine-checked, but many of the constructions are quite explicit, which is an asset.","major_comments":[{"comment":"The proof of Lemma 7.5 is not valid as written. The claim is that if α:C→Y induces an isomorphism on π_{s,∗}, then id∧α:MZ∧C→MZ∧Y also induces an isomorphism on π_{s,∗}. The argument uses a commutative diagram involving the Hurewicz map h:S→MZ and the module multiplication μ:MZ∧Y→Y, and asserts that all maps in the diagram induce isomorphisms on π_{s,∗}. This is not generally true for h∧id:S∧C→MZ∧C. For C=S^{s,w}, the map π_{s,∗}(S^{s,w})→π_{s,∗}(MZ∧S^{s,w}) is the Hurewicz map into an Eilenberg–MacLane spectrum, and it is not an isomorphism in positive stems where the sphere has torsion; the statement that cellular approximation of MZ 'keeps π_{0,∗} unchanged' does not imply isomorphisms on the relevant bigraded homotopy groups. Since Proposition 7.8 uses Lemma 7.5 to conclude that the maps Σ^{s,w}MG_s→Y induce isomorphisms on homotopy, the wedge decomposition in Corollary 7.11 is not established by the present proof.","section":"Section 7.1, Lemma 7.5"},{"comment":"Proposition 7.8 is load-bearing for Theorem 1.2, and its proof has a second gap beyond Lemma 7.5. Lemma 7.3 constructs C as the cofiber of a map between wedges of spheres. Such a compact cofiber generally has homotopy in adjacent stems, e.g. the cofiber of multiplication by 2 on S^{s,w} has nontrivial π_{s+1,∗}. Therefore an isomorphism on π_{s,∗}(C)→π_{s,∗}(Y) does not by itself control the behavior of the map α in neighboring stems, and the conclusion that the wedge map ⋁_s Σ^{s,w}MG_s→Y induces isomorphisms on all homotopy groups is not justified. A repair would need either a stronger splitting theorem for MZ-modules or a careful verification that the maps in the wedge decomposition have no spurious higher-stem contributions. As it stands, the central effective-slice formula rests on an unproven assertion.","section":"Section 7.1, Proposition 7.8"},{"comment":"The paper explicitly states in Section 1.5 that Heard's theorem identifies the cellular effective filtration with the full C-motivic effective filtration, but that no analogous statement has been proven for the connective and very effective filtrations. Nevertheless, Propositions 6.7 and 6.11 present the filtered spectra Γ≥q⋆(X) and Γ̃q⋆(X) as models for the q-th connective and q-th very effective covers, and Sections 7.2 and 7.3 use them to compute slices. As written, these computations are only established in the cellular subcategory, and unless a comparison theorem for the connective and very effective filtrations is supplied, the claims should be explicitly restricted to the cellular category. This is a substantial limitation of the paper's scope as stated.","section":"Sections 1.5, 6.2, and 6.3"}],"minor_comments":[{"comment":"In Lemma 5.13(1), the connective cover is written as f_q(Y), but the cover in question is Y_{\\ge q}; the notation is inconsistent with the surrounding text.","section":"Section 5.4, Lemma 5.13"},{"comment":"The notation τ^{<2q+2}_{\\ge 2q} is not defined in Section 2 or at its first use in Proposition 7.1; a brief definition would improve readability.","section":"Proposition 7.1 and Section 7.1"},{"comment":"The diagram in the proof of Lemma 7.5 is not fully labeled; in particular the maps h∧id and μ are not marked, which makes the commutativity claim harder to check.","section":"Section 7.1, proof of Lemma 7.5"},{"comment":"The weight formula (s+f)/2 in Remark 8.2 silently assumes that s+f is even; since the even-MU-homology hypothesis makes odd antidiagonals vanish, this is consistent, but the assumption should be stated explicitly.","section":"Section 8, Remark 8.2"},{"comment":"The sentence 'tmf is bounded below since it is by definition a connective cover' is slightly imprecise: tmf is a connective ring spectrum by construction, but it is not literally defined as a connective cover of another fixed spectrum; the phrasing could be clarified.","section":"Section 1, Corollary 7.13"},{"comment":"The statement that the odd-primary analogues 'are likely to hold' is presented as a remark, but it would be helpful to specify which parts of the argument actually depend on 2-adic completion beyond the existence of τ.","section":"Section 1.5"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is potentially significant, but the current proof of the splitting behind Theorem 1.2 has a genuine gap in Lemma 7.5 and Proposition 7.8. I recommend major revision rather than rejection because the gap is local in nature: a successful proof will likely need a stronger splitting statement for MZ-modules, but the overall filtered-spectrum framework appears coherent. The cellular-versus-full-category issue for the connective and very effective filtrations should also be resolved or explicitly scoped."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things. First, the headline result is real: a uniform computation of effective slices of Γ⋆(X) for bounded below classical spectra with even MU homology, including the genuinely new case X = tmf (i.e., mmf). The filtered-spectrum model is the right framework, and the paper is clearly written. Second, the main proof currently has a gap that a referee will need to see closed. Proposition 7.8 — the splitting of an MZ-module with homotopy groups G_s[τ] into a wedge of Σ^{s,w}MG_s — depends on Lemma 7.5, and the justification for Lemma 7.5 is not sound. The claim that smashing with MZ preserves isomorphisms on a single stem is not established by the cellular approximation of MZ; keeping π_{0,*} unchanged says nothing about the relevant higher stems. The specific counterexample in the stress-test note using spheres does not quite land, because the fixed stem s for a sphere is torsion-free over Z[τ], but the general point stands: the proof as written is insufficient. The statement is probably true — a Künneth-type argument would likely give it — but the paper does not provide that argument.\n\nWhat the paper does well: the explicit filtered-spectrum models for the effective, connective, and very effective covers in Section 6 are clean and useful. The recovery of Voevodsky's slice conjectures for the sphere, MU, ku, and EM spectra in the 2-complete C-motivic setting is a nice sanity check. The comparison of the effective slice spectral sequence to the Adams-Novikov spectral sequence (Theorem 1.3) is clearly explained, and the décalage argument is credible, with the caveat that this comparison is somewhat built into the model.\n\nSoft spots, in proportion: the connective and very effective computations are honestly flagged as cellular statements; the author states in Section 1.5 that agreement with the full motivic filtrations for those two filtrations is unproven. That is a real limitation but it is explicit and does not affect the effective slice results. The larger issue is the missing proof of Proposition 7.8. The paper also has no machine-checked proofs, so the many spectral-sequence and compactness arguments will need expert checking.\n\nThis paper is for anyone working on motivic slices or C-motivic stable homotopy. The mmf slice computation is new and should be useful. I would send this to a serious referee: it deserves careful review, and the author should be asked to either prove Lemma 7.5 properly or replace that step with a cleaner argument.","headline":"Solid new slice computations for C-motivic analogues, but the proof of the key splitting theorem has a gap that needs to be closed before the central claim is fully established.","tokens_in":36116,"tokens_out":17351,"would_cite":true,"duration_ms":148513,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","55P42","55T99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a uniform antidiagonal formula for the effective slices of C-motivic analogues of classical spectra, and applies it to compute the effective slices of the motivic modular forms spectrum mmf from the Adams-Novikov E2-page…","keywords":["C-motivic homotopy theory","slice filtrations","filtered spectra","Adams-Novikov spectral sequence","motivic modular forms","effective filtration","connective filtration","very effective filtration"],"falsifier":"Take $X=\\mathrm{tmf}$ and let $q=1$. The theorem predicts that $s_1(\\mathrm{mmf})$ is the wedge of $\\Sigma^{s,1}M E^{s,f}_2$ over all $s+f=2$, so its homotopy groups should be free $\\mathbb{Z}[\\tau]$-modules on exactly the classes of the $s+f=2$ antidiagonal of the Adams-Novikov $E_2$-page of $\\mathrm{tmf}$. A reader could compute $\\pi_{*,*}(s_1(\\mathrm{mmf}))$ from the filtered-spectrum description of the slice in Proposition 7.1 and the known $\\mathrm{tmf}$ Adams-Novikov chart; any missing class, extra $\\tau$-torsion, or non-split extension would falsify the wedge decomposition.","tokens_in":34925,"feed_emoji":"🧮","tokens_out":13126,"duration_ms":97880,"temperature":0.7,"pith_summary":"The paper tries to establish that the slice filtrations of $\\mathbb{C}$-motivic spectra of the form $\\Gamma_\\star(X)$ can be computed entirely from classical data attached to $X$, using the filtered-spectrum model for the $\\mathbb{C}$-motivic category. The main theorem gives a closed formula for the $q$-th effective slice of $\\Gamma_\\star(X)$ as a wedge of $\\mathbb{C}$-motivic Eilenberg-MacLane spectra indexed by the antidiagonal $s+f=2q$ of the classical Adams-Novikov $E_2$-page, whenever $X$ is bounded below and has even $MU$-homology. This matters because it simultaneously reproves several slice conjectures and computes the previously unknown effective slices of the motivic modular forms spectrum $\\mathrm{mmf}$. The paper also proves that the effective slice spectral sequence of $\\Gamma_\\star(X)$ is just a re-indexing of the classical Adams-Novikov spectral sequence of $X$.","feed_headline":"Slice formula computes C-motivic spectra from Adams-Novikov pages","feed_subtitle":"For tmf the formula yields the slices of motivic modular forms, and it reproves earlier slice conjectures.","key_machinery":"The carrying object is the $\\mathbb{C}$-motivic analogue functor $\\Gamma_\\star\\colon \\mathrm{SH}^{\\mathrm{cl}}\\to \\mathrm{SH}^{\\mathrm{fil}}$, defined by $\\Gamma_w(X)=\\mathrm{Tot}(\\tau_{\\ge 2w}(X\\wedge MU^{\\wedge \\bullet+1}))$, together with the equivalence $\\mathrm{SH}_{\\mathbb{C}}\\simeq \\mathrm{Mod}_{\\Gamma_\\star(S)}$ between the $2$-complete cellular $\\mathbb{C}$-motivic category and modules over the filtered spectrum $\\Gamma_\\star(S)$. The paper modifies this construction in three ways—freezing weights below $q$ for the effective cover, applying classical connective covers $\\tau_{\\ge w+q}$ for the connective cover, and both for the very effective cover—to produce filtered-spectrum models for the covers. The homotopy groups of these models are computed by Bousfield-Kan spectral sequences that are truncations of the Adams-Novikov spectral sequence of $X$ below the antidiagonal $s+f=2w$. A $\\mathbb{C}$-motivic analogue of Adams' theorem on modules over $M\\mathbb{Z}$ (Proposition 7.8) then turns homotopy groups of the form $G[\\tau]$ into wedges of Eilenberg-MacLane spectra.","core_discovery":"The central claim is Corollary 7.11: for every bounded below classical spectrum $X$ with even $MU$-homology, the $q$-th effective slice of the $\\mathbb{C}$-motivic analogue $\\Gamma_\\star(X)$ is equivalent to\n$$s_q(\\Gamma_\\star(X)) \\simeq \\bigvee_{s+f=2q} \\$Sigma^{{s,q}}$ M $E^{{s,f}}$_2,$$\nwhere $E^{s,f}_2$ is the $E_2$-page of the classical Adams-Novikov spectral sequence of $X$ in stem $s$ and filtration $f$, and $M$ is the functor taking a group to the associated $\\mathbb{C}$-motivic Eilenberg-MacLane spectrum. The argument models each effective cover of $\\Gamma_\\star(X)$ by an explicit filtered spectrum, checks the cover is correct through homotopy-group characterizations, and then uses a $\\mathbb{C}$-motivic analogue of Adams' splitting theorem for modules over $M\\mathbb{Z}$ to assemble the slice from its homotopy groups. Applied to $X=\\mathrm{tmf}$, this says the effective slices of $\\mathrm{mmf}=\\Gamma_\\star(\\mathrm{tmf})$ are wedges of Eilenberg-MacLane spectra indexed by the Adams-Novikov $E_2$-page of $\\mathrm{tmf}$; applied to $S$, $MU$, $ku$, $ko$, and Eilenberg-MacLane spectra it recovers the earlier slice conjectures in the $2$-complete cellular setting.","pith_inferences":["Our inference: if odd-primary versions of the filtered-spectrum model are constructed, the same antidiagonal formula would give odd-primary slices of $\\mathbb{C}$-motivic analogues, including $\\mathrm{mmf}$ at odd primes.","Our inference: since $\\Gamma_\\star(X)\\simeq \\nu X$ for bounded below $X$ with even $MU$-homology, these slice statements can be read as statements about synthetic spectra, and a synthetic proof might bypass the d\\'ecalage comparison used here.","Our inference: the weight-by-weight identification of the slice spectral sequence with the Adams-Novikov spectral sequence suggests that slice filtrations of $\\Gamma_\\star(X)$ could be used as a bookkeeping device for classical stable homotopy computations, turning motivic weight into a new grading in large stem calculations."],"forward_implications":["For $X=\\mathrm{tmf}$, the formula gives all effective slices of $\\mathrm{mmf}$ in closed form, and the same formula reproduces the previously known slices of the spheres, $MGL$, $kq$, and $kgl$ in the $2$-complete cellular $\\mathbb{C}$-motivic category.","The effective slice spectral sequence of $\\Gamma_\\star(X)$ and the classical Adams-Novikov spectral sequence of $X$ carry the same information: $d_r(x)=\\tau^r y$ in the slice spectral sequence exactly when $d_{2r+1}(x)=y$ in the Adams-Novikov spectral sequence.","The connective and very effective slices of $\\Gamma_\\star(X)$ also have explicit filtered-spectrum descriptions; in particular the Betti realization of the $q$-th very effective cover is the classical $2q$-connective cover of $X$.","For Landweber exact classical spectra $X$, the effective and very effective covers of $\\Gamma_\\star(X)$ coincide, extending a known agreement result to all objects of this form."],"supporting_citations":[{"why":"Supplies the equivalence $\\mathrm{SH}_{\\mathbb{C}}\\simeq \\mathrm{Mod}_{\\Gamma_\\star(S)}$ that turns $\\Gamma_\\star(X)$ into a $\\mathbb{C}$-motivic spectrum.","marker":"[GIKR22]"},{"why":"Provides the cellular filtrations and the comparison showing the effective filtration agrees with the full $\\mathbb{C}$-motivic one.","marker":"[Hea19]"},{"why":"Shows even $MU$-homology kills $d_2$ differentials in the Adams-Novikov spectral sequence, the key hypothesis behind the closed slice formula.","marker":"[Rav86, Proposition 4.4.2]"},{"why":"Compares d\\'ecalage spectral sequences with truncated Adams-Novikov spectral sequences in the proof of the slice spectral sequence theorem.","marker":"[Lev15, Proposition 6.3]"},{"why":"Restates the d\\'ecalage comparison in the filtered-spectrum language used by the paper.","marker":"[vN25, Theorem 2.80]"},{"why":"Identifies the $\\mathbb{C}$-motivic analogues of $S$, $ko$, $tmf$, $MU$, $ku$, and Eilenberg-MacLane spectra, making them instances of the theorem.","marker":"[CQ21, Corollary 7.7]"},{"why":"Introduces the effective filtration and the slice conjectures that the paper reproduces for the sphere, $MGL$, and $M\\mathbb{Z}$.","marker":"[Voe02a]"},{"why":"Verifies that $\\mathrm{tmf}$ has even $MU$-homology, so the main theorem applies to $\\mathrm{mmf}$.","marker":"[Mat16, Corollary 5.2]"}],"fun_headline_variants":["Slice formula yields motivic slices from Adams-Novikov E2","Motivic tmf slices come from Adams-Novikov E2","Slice formula recovers Voevodsky's conjectures","Adams-Novikov E2 page computes motivic slices","From Adams-Novikov E2 to motivic slices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results rest on identifying the filtered-spectrum model with the $2$-complete cellular $\\mathbb{C}$-motivic category and on assuming the connective and very effective filtrations computed in that model are the genuine motivic filtrations; the paper proves this agreement for the effective filtration only.","fun_headline_variants_meta":{"raw":{"variants":["Slice formula yields motivic slices from Adams-Novikov E2","Motivic tmf slices come from Adams-Novikov E2","Slice formula recovers Voevodsky's conjectures","Adams-Novikov E2 page computes motivic slices","From Adams-Novikov E2 to motivic slices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000607,"raw_usage":{"total_tokens":2911,"prompt_tokens":1107,"completion_tokens":1804,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":723,"completion_tokens_details":{"reasoning_tokens":1722}},"tokens_in":723,"tokens_out":1804,"duration_ms":11943,"temperature":1.0,"reasoning_tokens":1722,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:39:34.757898+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $X=\\mathrm{tmf}$ and let $q=1$. The theorem predicts that $s_1(\\mathrm{mmf})$ is the wedge of $\\Sigma^{s,1}M E^{s,f}_2$ over all $s+f=2$, so its homotopy groups should be free $\\mathbb{Z}[\\tau]$-modules on exactly the classes of the $s+f=2$ antidiagonal of the Adams-Novikov $E_2$-page of $\\mathrm{tmf}$. A reader could compute $\\pi_{*,*}(s_1(\\mathrm{mmf}))$ from the filtered-spectrum description of the slice in Proposition 7.1 and the known $\\mathrm{tmf}$ Adams-Novikov chart; any missing class, extra $\\tau$-torsion, or non-split extension would falsify the wedge decomposition.","supporting_citations":[],"review_version":2}