schema_version: 1
contribution_id: con_ac00e604494c4651b594bd8d70588680
revision_id: rev_3850b505b85840f6980d4ab0610f34c9
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type: attempt
home_problem_id: prb_8776481bd1fd4c26858ec39911a03b31
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  statement_revision_id: stm_32426acdcfe14d9699388ec2dd314529
  scope: whole_statement
title: 'Lucas residue completeness: classification, constructive lifting and exact certificates'
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claim:
  type: literature_report
  scope_statement: Answers the stated classification affirmatively with attribution to the published 2013 Avila-Chen result, corroborated by Lang-Lang. Necessity explicitly invokes Burr's established Fibonacci-completeness classification, which is not reproved here. Direct sufficiency proofs cover the five exceptional moduli and every power of three. The constructive algorithm finds n < 8*3^(K-1) for any target A modulo 3^K, for every K >= 1. The selected indices n mod 8 in {1,2,7} inside this range represent each residue exactly once; the least Lucas state period is proved to be 8*3^(K-1). Modulus one is handled separately, and an example explains the failure of a naive CRT-completeness argument. Exact verification scans every Lucas state cycle for 2 <= m <= 5000 (4,999 moduli, 7,147,243 states), checks all 9,840 targets for K=1..8 by independent fast doubling, and checks 24 larger examples through K=128. These finite checks do not replace the infinite proofs. No historical novelty or new open-problem breakthrough is claimed.
  assumptions: []
  limitations: []
attribution:
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    github_id: 206281466
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rights:
  licence: CC-BY-4.0
  redistribution: permitted
  holders:
  - gavenren
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    recorded_at: '2026-10-18T01:15:26Z'
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