{"id":"251fff15-67bd-4cbd-bad7-aa4f4406d986","arxiv_id":"2605.30298","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The mod-2 cohomology algebra of moduli stacks of fixed-rank, fixed-degree real vector bundles on type I real algebraic curves is determined by characteristic classes induced from complex Atiyah-Bott classes.","lead":"The paper determines the mod-2 cohomology algebra of moduli stacks of real vector bundles of fixed rank and degree on type I real algebraic curves. It expresses that algebra via characteristic classes induced from the classical complex Atiyah-Bott classes, extending a core tool of algebraic geometry to the real setting.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Abstract-only review leaves the comparison-map bridge from complex Atiyah-Bott classes to the real mod-2 algebra uncheckable; no further load-bearing flaw can be isolated.","rationale":"The Reader correctly identified the sole load-bearing assumption that can be read off the abstract: that the induced Atiyah-Bott classes present the full real mod-2 algebra. Because the full text is unavailable, no finer-grained concern (incorrect spectral-sequence page, incomplete set of relations, failure for certain ranks/degrees, etc.) can be isolated. The honest non-finding is therefore that the Reader's UNVERDICTED / LOW-confidence assessment already captures the situation; no adjustment of the verdict is warranted. The concrete test above is the minimal next step that would convert the abstract claim into a checkable mathematical statement.","tokens_in":1791,"tokens_out":538,"duration_ms":5243,"concrete_test":"Obtain the full text and extract the statement of the comparison morphism (or the diagram relating the real moduli stack to the complex Atiyah-Bott stack) together with the theorem that claims the induced classes generate. Independently verify, for the lowest non-trivial case (rank 2, degree 0 on a real curve of genus 1 or 2), that the resulting algebra matches a direct computation of the mod-2 cohomology of the real moduli stack via an equivariant spectral sequence or known results on real Grassmannians; any mismatch or missing generator would falsify the determination claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the mod-2 cohomology algebra of the moduli stacks of real vector bundles of fixed rank and degree on a type-I real algebraic curve is determined by characteristic classes induced from the complex Atiyah-Bott classes. The load-bearing step is therefore that the natural comparison maps (real-to-complex or fixed-locus maps) pull back enough generators and relations to present the full real algebra. With only the abstract available, neither the construction of those maps, the spectral-sequence or equivariant-cohomology argument that would show they generate, nor any explicit presentation of the resulting algebra can be examined. Consequently the structural bridge asserted in the abstract cannot be stress-tested for hidden assumptions (e.g., freeness of the action, vanishing of higher differentials, or completeness of the induced characteristic classes). This is precisely the gap already flagged by the Reader; no additional, more precise technical vulnerability can be identified from the given material.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript studies the moduli stacks of real vector bundles of fixed rank and degree on a type I real algebraic curve. Its central claim is that the mod-2 cohomology algebra of these stacks is determined by characteristic classes induced from the complex Atiyah–Bott classes via natural comparison maps from the complex moduli setting.","tokens_in":1973,"tokens_out":603,"duration_ms":10815,"significance":"If the determination is correct and complete, the result would give an explicit algebraic presentation of the mod-2 cohomology of these real moduli stacks, extending the classical Atiyah–Bott theory from the complex to the real (type I) setting. That would be a natural and useful contribution to the topology of moduli stacks in real algebraic geometry, provided the comparison maps and generation/relation arguments are fully established.","major_comments":[{"comment":"Only the abstract is available for this review. The load-bearing claim—that characteristic classes induced from the complex Atiyah–Bott classes via natural comparison maps present the full mod-2 cohomology algebra of the real moduli stacks—cannot be checked. Construction of the comparison maps, any spectral-sequence or equivariant-cohomology argument establishing generation and relations, and the resulting algebra presentation are all unavailable. A proper technical assessment of soundness is therefore impossible on the given material.","section":null},{"comment":"Abstract: the asserted structural bridge (induced classes suffice as generators and relations for the real algebra) is the sole content of the determination claim. Without the full text one cannot test hidden assumptions such as freeness of actions, vanishing of higher differentials, or completeness of the induced characteristic classes. This gap is load-bearing for the central claim and cannot be resolved from the abstract alone.","section":null}],"minor_comments":[{"comment":"Abstract: the phrase “type I real algebraic curve” is used without a brief parenthetical reminder of the standard definition (real curve whose real locus is nonempty and of maximal topological type, or the equivalent fixed-locus condition under the anti-holomorphic involution). A short clarification would help non-specialist readers.","section":null},{"comment":"Abstract: “determined \\ldots in terms of characteristic classes induced from the complex Atiyah–Bott classes” leaves open whether the result is a full presentation (generators and relations) or only a generation statement. The full paper should make the precise algebraic claim explicit in the introduction.","section":null}],"recommendation":"uncertain","confidential_remarks":"Full text was not supplied; only the abstract appears in the review package. I therefore cannot issue a substantive accept/revise/reject recommendation. Once the complete manuscript is available the comparison-map and generation arguments should be the primary focus of a second reading. Scope appears appropriate for a pure-math algebraic-geometry/topology journal, but novelty and correctness cannot be judged from the abstract alone."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is an abstract-only look at a pure algebraic-geometry note, so we are flying half-blind. The punchline is a structure theorem: the mod-2 cohomology algebra of the moduli stacks of real vector bundles of fixed rank and degree on a type-I real algebraic curve is determined by characteristic classes induced from the classical complex Atiyah-Bott classes.\n\nWhat is new is the real-side determination itself. Complex Atiyah-Bott is classical; carrying the generators and relations over to the real moduli stacks via natural comparison maps, and claiming that those induced classes present the full mod-2 algebra, is a concrete extension of that program. If the proofs hold, it organizes invariants for people working on real curves and real gauge theory. Circularity burden looks low—this is the usual pure-math computation style, not parameter fitting.\n\nThe soft spot is exactly the one the stress-test flags, and it is forced by the missing text rather than by any visible crack in the claim. The load-bearing step is that the comparison maps (real-to-complex or fixed-locus) pull back enough generators and relations. Without the constructions, spectral-sequence arguments, or an explicit algebra presentation, we cannot check freeness assumptions, vanishing of differentials, or completeness of the induced classes. That is a genuine information gap, not a manufactured flaw. Nothing in the abstract suggests the result is forced by a normalization trick or invents entities.\n\nWho it is for: specialists in moduli of real bundles, equivariant cohomology, and the topology of real algebraic curves. A serious referee in that circle should see the full paper; the claim is important enough inside the subfield and formally grounded enough in the classical theory to deserve referee time rather than a desk reject. I would not cite it yet—there is no text—and I would not bring an abstract to reading group, but I would accept it for peer review once the manuscript is in hand. Treat the Reader’s low soundness score as “cannot assess,” not as “probably wrong.”","headline":"Abstract-only structure theorem for mod-2 cohomology of real moduli stacks; solid-looking extension of Atiyah-Bott, but the comparison-map bridge is uncheckable without the paper.","tokens_in":2579,"tokens_out":517,"would_cite":false,"duration_ms":4385,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H60","14D23","14P25","55N91"],"pacs":[],"model":"grok-4.5","headline":"The mod-2 cohomology of real moduli stacks of fixed-rank, fixed-degree vector bundles on type I real curves is fixed by classes pulled back from the complex Atiyah-Bott generators.","keywords":["moduli stacks","real vector bundles","type I real curves","mod-2 cohomology","Atiyah-Bott classes","characteristic classes","real algebraic curves"],"falsifier":"An explicit calculation, for a low-rank or low-genus type I curve, of the mod-2 cohomology ring of the real moduli stack that produces a class or relation not accounted for by the induced Atiyah-Bott characteristic classes.","tokens_in":2660,"feed_emoji":"📐","tokens_out":751,"duration_ms":6897,"temperature":0.7,"pith_summary":"The paper studies the moduli stacks that classify real vector bundles of fixed rank and degree on a real algebraic curve of type I, and claims that their mod-2 cohomology algebras can be written down completely in terms of characteristic classes induced from the classical Atiyah-Bott generators of the corresponding complex moduli stacks. A type I real curve is a complex curve equipped with a real structure whose fixed locus is non-empty, so the real moduli problem sits over a non-empty real locus. The authors argue that the natural comparison maps from the complex setting produce enough characteristic classes to present the entire mod-2 algebra of the real stack. If the claim holds, the topology of these real moduli stacks becomes as computable as the well-studied complex case, at least with coefficients in F2.","feed_headline":"Real moduli stacks of vector bundles get their mod-2 cohomology from complex Atiyah-Bott c","feed_subtitle":"On type I real curves the full algebra is presented by characteristic classes pulled back from the complex side","key_machinery":"Characteristic classes on the real moduli stack induced from the complex Atiyah-Bott generators; these classes, together with the comparison maps that produce them, are asserted to generate and relate the full mod-2 cohomology algebra.","core_discovery":"The mod-2 cohomology algebra of the moduli stack of real vector bundles of fixed rank and degree on a type I real algebraic curve is completely determined by characteristic classes induced from the complex Atiyah-Bott classes via the natural comparison maps between the complex and real moduli problems.","pith_inferences":["Because the determination is only mod 2, integral or odd-primary cohomology of the same real stacks may require additional generators not visible from the complex Atiyah-Bott classes.","Type II real curves (empty real locus) are excluded; their moduli stacks may have genuinely different mod-2 cohomology that cannot be read from the same comparison.","An algorithmic presentation of the induced classes would turn the abstract determination into concrete ring presentations for each fixed rank, degree and genus."],"forward_implications":["The mod-2 cohomology rings of these real moduli stacks become explicit once the induced characteristic classes and their relations are written down.","Any topological invariant of the real stack that can be read from mod-2 cohomology (Stiefel-Whitney numbers, etc.) is determined by the same classes.","Comparisons between real and complex moduli stacks of vector bundles reduce, at the level of mod-2 cohomology, to the behaviour of the comparison maps on Atiyah-Bott generators.","The same presentation technique may apply to other real moduli problems whose complex counterparts are already controlled by known characteristic classes."],"fun_headline_variants":["Mod-2 cohomology of real bundle moduli stacks from Atiyah-Bott classes","Type I real curves: real moduli cohomology via complex characteristic classes","Real vector bundle stacks get mod-2 cohomology from complex Atiyah-Bott","Characteristic classes fix mod-2 cohomology of real moduli stacks on type I curves","Real moduli stacks of bundles: mod-2 algebra from complex Atiyah-Bott pullbacks"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The induced characteristic classes coming from the complex Atiyah-Bott generators already present the entire mod-2 cohomology algebra of the real moduli stacks.","fun_headline_variants_meta":{"raw":{"variants":["Mod-2 cohomology of real bundle moduli stacks from Atiyah-Bott classes","Type I real curves: real moduli cohomology via complex characteristic classes","Real vector bundle stacks get mod-2 cohomology from complex Atiyah-Bott","Characteristic classes fix mod-2 cohomology of real moduli stacks on type I curves","Real moduli stacks of bundles: mod-2 algebra from complex Atiyah-Bott pullbacks"]},"model":"grok-4.5","effort":"low","cost_usd":0.005802,"raw_usage":{"total_tokens":1373,"prompt_tokens":561,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":58020000,"prompt_tokens_details":{"text_tokens":561,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":728,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":561,"tokens_out":84,"duration_ms":5601,"temperature":1.0,"reasoning_tokens":728,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T15:37:38.361362+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit calculation, for a low-rank or low-genus type I curve, of the mod-2 cohomology ring of the real moduli stack that produces a class or relation not accounted for by the induced Atiyah-Bott characteristic classes.","supporting_citations":[],"review_version":2}