schema_version: 1
contribution_id: con_2e87a1096c3e4ea6af06481d543964da
revision_id: rev_e8ad946218f24e28a43b83dfa4aed584
expected_predecessor: null
type: attempt
home_problem_id: prb_74eebac275fe494c8b594b660936f94c
targets:
- target_kind: problem
  problem_id: prb_74eebac275fe494c8b594b660936f94c
  statement_revision_id: stm_51b822516bbb4c399e1f144aa38acdd8
  scope: whole_statement
title: 'Equality in the matroid circumference bound: a 2026 resolution and explicit equality families'
profiles:
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claim:
  type: literature_report
  scope_statement: The June 15, 2026 author-hosted submitted preprint of Manoel Lemos and James Oxley answers the main exact-equality question affirmatively; its classification theorem is credited and used, with the four-element boundary proved separately. This submission does not claim a new general classification proof or journal acceptance. It independently proves explicit binary representations, every-subset rank formulas, and complete circuit/cocircuit descriptions for M_(s,t)=P_t(S_s(AG(3,2))), for all positive integers s,t. Every member is a connected nongraphic exact-equality case. S_2(AG(3,2)) is a simple rank-12 sixteen-element counterexample to the catalogue's parallel-only associated variant. The original Open Problem Garden wording broadly excludes trivial modifications and is not refuted by this observation. All connected nongraphic equality cases are not classified here. Exact computations enumerate 131,456 subsets for four small examples and check all 3,600 predicted circuits and 232 cocircuits of M_(2,2), with larger finite sampling limits stated explicitly.
  assumptions: []
  limitations:
  - General literature conclusions retain the cited preprint status and exact scope; journal publication is not asserted.
attribution:
  submitter:
    person_ref: per_09e7780cd1ac4ea68d2bfd052d96b9ca
    github_id: 206281466
    login_at_event: gavenren
    display_name: gavenren
    roles:
    - prompting
    - submission
    authorship_confirmation: confirmed
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    github_id: 206281466
    login_at_event: gavenren
    display_name: gavenren
    roles:
    - prompting
    - submission
    authorship_confirmation: confirmed
  organisations: []
  contribution_mode: ai_only
model_uses:
- provider: OpenAI
  declared_name: codex (gpt-6)
  family: null
  version: unknown
  role: generated_draft
  settings:
    reasoning: unknown
    instance: OpenAI Codex (GPT-6)
  generation_date: null
  provenance:
    level: declared
    evidence: []
lineage: []
artifacts:
- role: manuscript
  location: in_repository
  path: files/attempt.md
  media_type: text/markdown
- role: supplementary
  location: in_repository
  path: files/disclosures.md
  media_type: text/markdown
rights:
  licence: CC-BY-4.0
  redistribution: permitted
  holders:
  - gavenren
declarations:
  rights:
    confirmed: true
    source: form_field
    actor: per_09e7780cd1ac4ea68d2bfd052d96b9ca
    recorded_at: '2026-10-18T00:55:06Z'
    evidence_url: https://github.com/We-Solve/research/issues/7
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    confirmed: true
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    actor: per_09e7780cd1ac4ea68d2bfd052d96b9ca
    recorded_at: '2026-10-18T00:55:06Z'
    evidence_url: https://github.com/We-Solve/research/issues/7
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    confirmed: true
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    recorded_at: '2026-10-18T00:55:06Z'
    evidence_url: https://github.com/We-Solve/research/issues/7