schema_version: 1
contribution_id: con_01ccbaf7f0d942d5bc8a252c2bf7c800
revision_id: rev_c81f0e0560254de2b06d0666ad559e11
expected_predecessor: null
type: attempt
home_problem_id: prb_afa7d093e5504ba1b2c5a80cbc065182
targets:
- target_kind: problem
  problem_id: prb_afa7d093e5504ba1b2c5a80cbc065182
  statement_revision_id: stm_45e532b5bc9e4bdcb331bac069d1be85
  scope: part_of_statement
title: Exact finite-field cycle censuses through p = 31
profiles:
- id: mathematical_argument
  version: 1
  required: true
claim:
  type: computational_evidence
  scope_statement: 'Exact exhaustive finite verification: for each prime p≤31, the only cycle of the one-variable zero-set transform on F_p[x] is the standard four-cycle. Reproduces the published p≤17 result and extends this finite census to p=19,23,29,31. Maximum binary preperiods for these four primes are 7,7,10,8; sharp whole-ring maxima are 8,8,11,9. Full binary preperiod distributions are given through p=29, with an image-set distribution at p=31. Proves the finite reduction, restrictions on nonstandard cycle states, and the sharp relationship between binary and unrestricted preperiods. Supplies exact, deterministic executable source and separately checked outputs. This does not classify arbitrary primes or solve the general problem; no literature-priority or human-review claim is made.'
  assumptions: []
  limitations:
  - Finite computations through prime 31 do not resolve the arbitrary-prime question.
attribution:
  submitter:
    person_ref: per_415ec35513984cfcb2db0a353b11f6cc
    github_id: 206281466
    login_at_event: gavenren
    display_name: gavenren
    roles:
    - prompting
    - submission
    authorship_confirmation: confirmed
  humans:
  - person_ref: per_415ec35513984cfcb2db0a353b11f6cc
    github_id: 206281466
    login_at_event: gavenren
    display_name: gavenren
    roles:
    - prompting
    - submission
    authorship_confirmation: confirmed
  organisations: []
  contribution_mode: ai_only
model_uses:
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    instance: OpenAI Codex (GPT-6), main instance
  generation_date: null
  provenance:
    level: declared
    evidence: []
- provider: OpenAI
  declared_name: codex (gpt-6)
  family: null
  version: unknown
  role: computation
  settings:
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    instance: OpenAI Codex (GPT-6), computation instance
  generation_date: null
  provenance:
    level: declared
    evidence: []
- provider: OpenAI
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    evidence: []
- provider: OpenAI
  declared_name: codex (gpt-6)
  family: null
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  role: other
  settings:
    reasoning: unknown
    instance: OpenAI Codex (GPT-6), scouting instance
  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_415ec35513984cfcb2db0a353b11f6cc
    recorded_at: '2026-10-18T15:55:25Z'
    evidence_url: https://github.com/We-Solve/research/issues/26
  public_disclosure:
    confirmed: true
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    actor: per_415ec35513984cfcb2db0a353b11f6cc
    recorded_at: '2026-10-18T15:55:25Z'
    evidence_url: https://github.com/We-Solve/research/issues/26
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    confirmed: true
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    actor: per_415ec35513984cfcb2db0a353b11f6cc
    recorded_at: '2026-10-18T15:55:25Z'
    evidence_url: https://github.com/We-Solve/research/issues/26