Explicit cocycles for graph Hall algebras
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
For the reduced simplicial set whose simplices are subgraphs of a finite labelled undirected multigraph equipped with ordered partitions of their vertices, we construct explicit mod-two cohomology classes indexed by subgraphs. Vertex restriction pulls a supported top-dimensional cochain into a global cocycle. Permutation cycles pair with these classes as a Kronecker basis, while a vertex filtration bounds the total cohomology dimension using augmented Boolean order complexes. Alexander-Whitney multiplication recovers the graph Hall product. This gives a candidate proof of the finite-graph Hall/cohomology identification stated as Conjecture 1.1 by Toury. The construction concerns labelled subgraphs, not isomorphism classes of rooted forests. Loops and labelled parallel edges are allowed. Exact finite checks accompany the proof; priority and external mathematical assessment remain pending.
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Cite this paper
Lluis Eriksson (2026). Explicit cocycles for graph Hall algebras. AIRR.SCIENCE, ARR-2026-52NQWHR7R28GX9PX, v1. https://airr.science/papers/ARR-2026-52NQWHR7R28GX9PX/versions/v1/ Research preprint; ARR-2026-52NQWHR7R28GX9PX v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Version history
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Developed with AI assistance in one Codex chat: literature triage, proof construction, code and author-side checks. No independently authenticated model runtime or separate reviewer is claimed. Internal exact computations and visual PDF inspection were performed by the same assistant; no human mathematical verification, formal verification, external assessment or rating has been obtained. Prior work credited in manuscript. Lluis Eriksson is also AIRR founder/operator/responsible editor.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 1
- Range
- 4.50–4.50
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/1 benchmark matches. This is an evidence description, not a probability of correctness.
Only one distinct model; One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
OpenRouter · nvidia/nemotron-3-ultra-550b-a55b:free · not involved in manuscript · review context: unknown4.50
Recommendation: Major Revision · Material objections: 0
The manuscript proves Theorem 1, a canonical graded algebra isomorphism between the graph Hall algebra over F2 and the cohomology algebra of the simplicial set of ordered partitions of subgraphs. The proof constructs explicit cycles z_H and cocycles α_H via vertex restriction, shows they form a Kronecker basis using a vertex filtration that bounds Betti numbers by the number of subgraphs, and verifies the Alexander-Whitney product recovers the Hall multiplication. The argument is rigorous, complete for the stated setting (finite labelled undirected multigraphs with loops, F2 coefficients), and accompanied by finite computational checks. The result addresses Conjecture 1.1 of Toury (arXiv:2609.16849) but does not extend to isomorphism classes, other characteristics, or homotopy types. The author discloses a conflict as AIRR founder/editor; priority and external review are pending.
- Correctness confidence
- All lemmas and the main theorem are proved in detail; the filtration bound, cycle/cocycle construction, order independence, and Alexander-Whitney multiplication check are logically sound and correctly use F2 coefficients. No gaps or errors detected.
- Rigor
- Proofs are self-contained modulo cited simplicial model; explicit chain-level arguments, careful handling of normalization, degenerate simplices, and vertex restrictions; the long exact sequence of the filtration yields the Betti number bound, and Kronecker pairing confirms basis.
- Novelty
- Provides the first explicit cocycle representatives and a canonical isomorphism proving Toury's Conjecture 1.1 for finite labelled multigraphs over F2; the construction is new even though the underlying simplicial set is known from Bergner et al.
- Significance
- Resolves a concrete conjecture linking graph Hall algebras to cohomology of a 2-Segal simplicial set; the explicit basis and product formula may facilitate further computations and generalizations, but the result is specialized to F2 and labelled graphs.
- Reproducibility
- Mathematical proof is fully reproducible; companion Python scripts verify small examples but were developed in the same assistant context and are not independently reviewed.
Strengths
- Explicit cocycle construction with elementary cancellations over F2
- Clear filtration argument giving exact Betti number count
- Order independence proved by explicit cochain homotopy
- Alexander-Whitney product checked on cochains, matching Hall product exactly
- Scope and limitations honestly stated (F2, labelled graphs, no homotopy claim)
- Companion code for small-graph verification provided
Weaknesses
- Computational verification not independent (same assistant context)
- Cited conjecture (Toury) is from a preprint, not yet peer-reviewed
- Author is also AIRR founder/editor, creating a conflict of interest
- No bibliographic search establishing worldwide priority for the explicit cocycles
- Proof relies on the simplicial graph model of Bergner et al. without re-verifying simplicial identities
Potential errors
None reported.
Strong novelty candidates
- Explicit cocycle basis for H*(|X_G|; F2) indexed by subgraphs
- Canonical isomorphism Φ_G without choices of orderings
- Combinatorial interpretation of cup product as Hall product
Unresolved material objections
None reported.
arr:assessment:b51185b7-013b-4495-ac43-c11141891f7a · prompt ARR-ASSESS-1.0 · response SHA-256 ebd3a0e8815c3f7669c372afa012bced8341ee98cd9683334f702683c23b6ebb · canonical PDF SHA-256 ab5ddc46b216f47b8c3a8bb96ca81b37b979b138f69d6252dbdc483c5f3bc983
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Exact-version author-authorized Working paper only. Minimum release basis: clean scanner, coherent finite-graph contribution, explicit proof and finite controls, attributed antecedents and stated limitations. Scientific correctness and novelty remain subject to assessment. No independent human review, editorial acceptance or rating is claimed.