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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 53.
Research chronicle 92 RECORDS
All Research Review Verification System
20:16:42 §1
SYSTEM Iteration 53 opened
20:16:45 §2
SYSTEM Iteration 53 started
20:16:45 §3
SYSTEM Initializing research run
20:16:45 §4
NOETHER NOETHER joined as Strategist
20:16:45 §5
SYSTEM Research started
20:16:46 §6
NOETHER Research planning started
↑ 86 earlier records Research record
Agent record Close ×
NOETHER Strategist
Plan iteration 2
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 1,075
Cached input 517
Output tokens 870
Average latency 10.6 s
Failed calls 0
Cost (estimated) $0.000338
Joined 2026-09-25 20:16:45 UTC
Left 2026-09-25 20:19:14 UTC Recent activity 20:16:52 NOVELTY Novelty check: new 20:16:52 DIRECTION Direction proposed: Exact search via dynamic programming and ILP on small n 20:16:52 DIRECTION Direction assigned to GAUSS-01: Weighted divisor-count upper bound via harmonic sums 20:16:52 DIRECTION Direction assigned to GAUSS-02: Construction with large primes and near-prime intervals 20:16:52 DIRECTION Direction assigned to GAUSS-03: Exact search via dynamic programming and ILP on small n 20:16:52 TASK Task completed 20:18:09 TASK Research planning started 20:18:26 PLAN Research plan created Every figure here is aggregated from this agent’s recorded model calls, failed attempts included — a failed attempt is still an attempt.
07 Senior reviewer Not started On the roster; the workflow has not woken it in this iteration. 20:18:26
NOVELTY
Novelty check: new
20:18:26 DIRECTION Direction proposed: Entropy/density upper bound via random sampling and divisor-chain counting
20:18:26 MEMORY Novelty check: Structural decomposition by divisor poset layers with interval-specific chain constraints
20:18:26 NOVELTY Novelty check: partially explored
20:18:26 DIRECTION Direction proposed: Structural decomposition by divisor poset layers with interval-specific chain constraints
20:18:26 NOVELTY Novelty check: new
20:18:26 DIRECTION Direction proposed: Modular residue class construction via CRT and density packing
20:18:26 DIRECTION Direction assigned to GAUSS-01: Entropy/density upper bound via random sampling and divisor-chain counting
20:18:26 DIRECTION Direction assigned to GAUSS-02: Structural decomposition by divisor poset layers with interval-specific chain constraints
20:18:26 DIRECTION Direction assigned to GAUSS-03: Modular residue class construction via CRT and density packing
20:18:26 TASK Task completed
20:19:14 AGENT NOETHER stood down Mission
Runtime 9 h 14 min
Iteration 57 of 57
Directions 331
Hypotheses 17
Experiments 266
Rejected paths 64 Current direction No direction is open.
Verified 0 Lean-verified intermediate lemmas A verified lemma covers exactly the declaration the kernel accepted. The problem stays open until it is settled either way. Literature
Indexed 109
Highly relevant 1 Iteration 53
Model calls 12
Estimated cost $0.006474 Budget remaining 99%
Iteration details & budget ↗
Research graph / proof lineage 7 objects · 0 links · 10 in complete graph
DIRECTION
DIRECTION Weighted divisor-count upper bound via harmonic sums COMPLETED DIRECTION Construction with large primes and near-prime intervals COMPLETED DIRECTION Exact search via dynamic programming and ILP on small n COMPLETED DIRECTION Entropy/density upper bound via random sampling and divisor-chain counting COMPLETED DIRECTION Structural decomposition by divisor poset layers with interval-specific chain constraints COMPLETED DIRECTION Modular residue class construction via CRT and density packing COMPLETED PROOF
PROOF Experiment report — Verify that for the natural weight families w(a)=1, 1/a, 1/sqrt(a), 1/a^2, the C v1 MARKDOWN Explore complete graph / 17 objects →