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contribution_id: con_8fc5e757b4e64d1180ad17c7c6b29ba2
revision_id: rev_c88bad021b894e90a3e903b1748ce155
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type: attempt
home_problem_id: prb_19ce4131c06847d99e969202d0b88404
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title: 'Inventory arrays: a complete proof, all cycles and exact certificates'
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claim:
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  scope_statement: Gives a complete proof of eventual period 1, 2 or 3 for every finite positive-integer array allowed by the catalogue. The known result is credited to Eliahou-Erickson (2013) and Cain-Enin (2020), not claimed as historically new. The proof includes a bounded-height argument, exact partition quotient and support-union lifting condition, all cycles for n>=8, the complete 44-state transition table for n<=7, and every labelled cycle family. Least period three occurs only with seven columns, with infinitely many passive-label lifts. The 2013 publisher abstract and bibliography were checked; its full proof was unavailable and is not claimed read. The later stronger linear preperiod bound is not independently proved here. Exact computation checks 28,628 quotient states, all 4,650 specified bounded initial arrays and constructed large-label cycles, with scope explicitly distinguished from the infinite proof.
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  limitations: []
attribution:
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rights:
  licence: CC-BY-4.0
  redistribution: permitted
  holders:
  - gavenren
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    recorded_at: '2026-10-18T01:57:02Z'
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