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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 37.
Research chronicle 95 RECORDS
All Research Review Verification System
19:35:33 §1
SYSTEM Iteration 37 opened
19:35:35 §2
SYSTEM Iteration 37 started
19:35:35 §3
SYSTEM Initializing research run
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NOETHER NOETHER joined as Strategist
19:35:35 §5
SYSTEM Research started
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NOETHER Research planning started
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GAUSS-03 Explorer
Explore direction: Independence number of divisor graph via spectral/LP bounds
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 7,945
Cached input 0
Output tokens 2,447
Average latency 18.0 s
Failed calls 0
Cost (estimated) $0.001798
Joined 2026-09-25 19:35:46 UTC
Left 2026-09-25 19:38:35 UTC Recent activity 19:35:46 AGENT GAUSS-03 joined as Explorer 19:36:08 TASK Exploration started 19:36:34 EXPERIMENT Experiment refused before it ran 19:36:34 DIRECTION Direction exhausted 19:36:34 MEMORY Computational search and pattern extraction for small n 19:36:34 TASK Task completed 19:37:51 TASK Exploration started 19:38:05 EXPERIMENT Experiment refused before it ran 19:38:05 DIRECTION 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. Direction exhausted
19:38:05 MEMORY Independence number of divisor graph via spectral/LP bounds
19:38:05 TASK Task completed
19:38:35 AGENT GAUSS-03 stood down
Research graph / proof lineage 7 objects · 0 links · 14 in complete graph
DIRECTION
DIRECTION Poset chain-decomposition and Ahlswede–Zhang bound for [1,n] COMPLETED DIRECTION Lower-bound constructions via multiplicative dilates and prime-gap layers COMPLETED DIRECTION Computational search and pattern extraction for small n COMPLETED DIRECTION Entropy and local-density upper bounds via random sampling COMPLETED DIRECTION Recursive block construction with coprime layers COMPLETED DIRECTION Independence number of divisor graph via spectral/LP bounds COMPLETED PROOF
PROOF Greedy recursive block selection results v1 MARKDOWN Explore complete graph / 21 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 37
Model calls 14
Estimated cost $0.007667 Budget remaining 99%
Iteration details & budget ↗