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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
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NOETHER NOETHER joined as Strategist
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SYSTEM Research started
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NOETHER Research planning started
↑ 86 earlier records Research record
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LITERATURE-01 Literature researcher
Reading the landscape around: Entropy/density upper bound via random sampling and divisor-chain counting
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 1,867
Cached input 936
Output tokens 542
Average latency 5.1 s
Failed calls 0
Cost (estimated) $0.000310
Joined 2026-09-25 20:16:52 UTC
Left 2026-09-25 20:19:14 UTC Recent activity 20:16:52 AGENT LITERATURE-01 joined as Literature Researcher 20:16:52 LITERATURE Literature search started 20:16:52 LITERATURE Literature search: 3 new of 3 20:16:52 LITERATURE Reviewed: A remark on divisor-weighted sums 20:16:52 MEMORY Literature: A remark on divisor-weighted sums 20:18:26 LITERATURE Literature search started 20:18:26 LITERATURE Indexed: Counting cycles by random sampling : theory and practice 20:18:26 LITERATURE Indexed: A Fast Algorithm for Estimating Two-Dimensional Sample Entropy Based on an Upper Confidence Bound and Monte Carlo Sampling 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
LITERATURE
Literature search: 3 new of 3
20:18:26 LITERATURE Reviewed: An Upper Bound on the Divisor Counting Function
20:18:26 MEMORY Literature: An Upper Bound on the Divisor Counting Function
20:19:14 AGENT LITERATURE-01 stood down
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 → 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 ↗