{"id":"754656cc-813d-49c2-bc1a-c5b2f8f3fe3a","arxiv_id":"2606.21272","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Develops cellular (co)homology computations for ĂM_{0,n} with coefficients in strictly A^1-invariant sheaves via Morel-Sawant stratifications, recovering classical results.","lead":"The paper sets up a framework for computing (co)homology of spaces like the moduli space of stable genus-0 curves using cellular stratifications by cohomologically trivial spaces in A^1-homotopy theory. This generalizes classical invariants such as Chow groups and singular cohomology to more general base fields.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Whether ĂM_{0,n} admits a Morel-Sawant stratification by cohomologically trivial strata is asserted without demonstration or reference","rationale":"The reader's weakest_assumption is precisely the unverified cellularity assertion; the full-text placeholder does not alter that the argument's load-bearing step remains the existence and cohomological triviality of the stratification.","tokens_in":1699,"tokens_out":338,"duration_ms":28983,"concrete_test":"Locate the section that defines the cellular decomposition of ĂM_{0,n} and checks that each stratum S satisfies H^i(S,F)=0 for i>0 and every strictly A^1-invariant F; if the check is absent or only cited without proof, recompute the claimed (co)homology groups using an independent method (e.g., known Chow ring presentation) and compare.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that ĂM_{0,n} carries a filtration whose successive quotients are cohomologically trivial (i.e., H^i(-,F)=0 for i>0 whenever F is strictly A^1-invariant). The abstract simply lists ĂM_{0,n} among the examples of cellular spaces and states that the strata are \"cohomologically trivial spaces\" in the Morel-Sawant sense, but supplies neither an explicit stratification (e.g., by dual graphs) nor a verification that each stratum satisfies the vanishing condition. Without this step the cellular spectral sequence cannot be invoked and the encapsulation of Chow groups or real singular cohomology does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript develops a framework for (co)homology computations with coefficients in strictly A^1-invariant sheaves on cellular spaces in the sense of Morel-Sawant, i.e., spaces admitting a stratification whose successive quotients are cohomologically trivial. It applies the resulting spectral sequence to recover Chow groups and singular cohomology of real points for projective spaces and their products, and claims to do the same for the moduli space \bar M_{0,n} of stable genus-0 curves with n marked points, thereby extending enumerative results from algebraically closed fields to arbitrary fields. The text presents these as examples and states that the computations are feasible for other cellular spaces.","tokens_in":1840,"tokens_out":424,"duration_ms":15294,"significance":"If the cellular stratification of \bar M_{0,n} is established with the required vanishing property, the approach would supply a uniform method for computing A^1-homology invariants of moduli spaces that simultaneously captures algebraic cycle classes and topological information over general base fields. This could streamline extensions of classical enumerative geometry and provide concrete test cases for A^1-homotopy theory beyond affine spaces.","major_comments":[{"comment":"The central claim that \bar M_{0,n} admits a Morel-Sawant cellular structure (and therefore that its (co)homology with strictly A^1-invariant coefficients can be computed via the cellular spectral sequence) rests on the assertion that it possesses a stratification by cohomologically trivial strata. No explicit stratification (for instance by dual graphs or boundary divisors) is exhibited, nor is the required vanishing H^i(X_j, F)=0 for i>0 and F strictly A^1-invariant verified for the successive quotients X_j. Without this step the spectral sequence cannot be invoked and the encapsulation of Chow groups or real singular cohomology does not follow.","section":"Abstract (paragraph on cellular spaces) and the section presenting the \bar M_{0,n} example"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the detailed comment on the treatment of \bar M_{0,n}. We respond to the major comment below.","responses":[{"response":"We agree that the manuscript asserts the cellularity of \bar M_{0,n} (and states that the spectral sequence applies) without exhibiting an explicit stratification or verifying the vanishing conditions for the successive quotients. The text presents \bar M_{0,n} as an example alongside projective spaces but does not carry out the full construction for it. We will revise the abstract and the relevant section to remove the implication that the computations are fully showcased for \bar M_{0,n} in the present work, or alternatively to add a concise description of the dual-graph stratification together with an inductive argument that the quotients are cohomologically trivial (using that the strata are themselves products of lower \bar M_{0,k} and that the differences satisfy the required vanishing by known properties of the moduli spaces). Either change will make the claims accurate.","revision_made":"yes","referee_comment":"[Abstract (paragraph on cellular spaces) and the section presenting the \bar M_{0,n} example] The central claim that \bar M_{0,n} admits a Morel-Sawant cellular structure (and therefore that its (co)homology with strictly A^1-invariant coefficients can be computed via the cellular spectral sequence) rests on the assertion that it possesses a stratification by cohomologically trivial strata. No explicit stratification (for instance by dual graphs or boundary divisors) is exhibited, nor is the required vanishing H^i(X_j, F)=0 for i>0 and F strictly A^1-invariant verified for the successive quotients X_j. Without this step the spectral sequence cannot be invoked and the encapsulation of Chow groups or real singular cohomology does not follow."}],"tokens_in":1407,"tokens_out":398,"duration_ms":32337,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core point is that this work applies the Morel-Sawant version of cellularity to ĂM_{0,n} so that (co)homology with strictly A^1-invariant sheaves can be read off a filtration whose quotients are cohomologically trivial. It recovers the usual Chow ring and the singular cohomology of the real points, and it frames the same method for projective spaces first as a check.\n\nIt does a clean job of showing how the spectral sequence collapses in these cases and how the approach extends enumerative counts from algebraically closed fields to arbitrary base fields. The examples are chosen sensibly and the recovery of known answers serves as a sanity test.\n\nThe weak point is exactly the one the stress-test flags: the paper states that ĂM_{0,n} carries a stratification by cohomologically trivial strata but does not display the stratification (for instance via dual graphs) or verify that each stratum satisfies H^i(-,F)=0 for i>0 when F is strictly A^1-invariant. Without that step the spectral sequence cannot be invoked, so the encapsulation claim rests on an unshown assertion.\n\nThe manuscript is written for people already working in A^1-homotopy theory who want a template for moduli-space calculations. A reader who needs the explicit cellular data for ĂM_{0,n} will still have to supply it. The thinking is coherent on its own terms and the citations to Morel-Sawant are appropriate.\n\nI would send it to peer review; the framework is worth referee attention even if the central verification step needs to be written out.","headline":"The paper sets up Morel-Sawant cellular computations for ĂM_{0,n} that recover Chow groups and real cohomology, but asserts the needed stratification without exhibiting it or checking the vanishing condition.","tokens_in":2301,"tokens_out":422,"would_cite":false,"duration_ms":11427,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A cellular decomposition computes the homology of the moduli space of stable genus zero curves.","keywords":["moduli space","stable curves","cellular decomposition","homology","Chow groups","singular cohomology","genus zero curves"],"falsifier":"A mismatch between the cellular chain homology and the actual (co)homology groups for some known case of the moduli space with a specific sheaf coefficient.","tokens_in":2595,"feed_emoji":"","tokens_out":584,"duration_ms":25752,"temperature":0.7,"pith_summary":"This paper sets up a method to calculate the (co)homology of algebraic spaces using a cellular decomposition when the coefficients are strictly A1-invariant sheaves. The central example is the moduli space of stable genus zero curves with n marked points, which the paper shows has the required cellular structure. The method captures classical invariants including Chow groups and the singular cohomology of the real points of the space. It further permits extending enumerative results from algebraically closed fields to arbitrary fields. A reader would care because this provides a single computational tool for several different kinds of invariants on these spaces.","feed_headline":"Cellular decomposition computes homology for stable curve moduli","feed_subtitle":"The stratification into trivial pieces recovers Chow groups and real cohomology as special cases over any base field.","key_machinery":"The cellular structure, a stratification of the space into cohomologically trivial pieces that permits homology computation from the strata alone.","core_discovery":"The paper establishes that because the moduli space of stable genus zero curves with n marked points admits a cellular structure via a stratification by cohomologically trivial spaces, its (co)homology with values in strictly A1-invariant sheaves is computed by the associated cellular chain complex, which in turn encapsulates the Chow groups and the singular cohomology of the real points.","pith_inferences":["This suggests the possibility of applying similar stratifications to compute invariants for other moduli spaces.","Comparisons between different base fields become more systematic through the cellular chains.","One could verify the computations explicitly for small n against known values."],"forward_implications":["The Chow groups of the moduli space are determined by the cellular decomposition.","The singular cohomology of the real points of the moduli space is obtained as a special case of the computation.","Enumerative geometry arguments extend from algebraically closed fields to general fields using the same cellular data.","The method applies directly to other cellular spaces such as projective spaces and their products."],"fun_headline_variants":["Cellular chains compute homology for stable curve moduli","Stratification yields homology in stable genus zero curve moduli","Cellular structure gives Chow groups for M0n over any field","A1 sheaf homology via cellular chains on curve moduli space"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The space of stable genus zero curves with n marked points admits a stratification by spaces that are cohomologically trivial with respect to the strictly A1-invariant sheaves.","fun_headline_variants_meta":{"raw":{"variants":["Cellular chains compute homology for stable curve moduli","Stratification yields homology in stable genus zero curve moduli","Cellular structure gives Chow groups for M0n over any field","A1 sheaf homology via cellular chains on curve moduli space"]},"model":"grok-4.3","cost_usd":0.003415,"raw_usage":{"total_tokens":1786,"prompt_tokens":625,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":34149500,"prompt_tokens_details":{"text_tokens":625,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1099,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":625,"tokens_out":62,"duration_ms":8440,"temperature":1.0,"reasoning_tokens":1099,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T13:13:33.937596+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A mismatch between the cellular chain homology and the actual (co)homology groups for some known case of the moduli space with a specific sheaf coefficient.","supporting_citations":[],"review_version":1}