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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 48.
Research chronicle 93 RECORDS
All Research Review Verification System
20:03:41 §1
SYSTEM Iteration 48 opened
20:03:43 §2
SYSTEM Iteration 48 started
20:03:43 §3
SYSTEM Initializing research run
20:03:43 §4
NOETHER NOETHER joined as Strategist
20:03:43 §5
SYSTEM Research started
20:03:44 §6
NOETHER Research planning started
↑ 87 earlier records Research record
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GAUSS-01 Explorer
Explore direction: Multiplicative Dilates and Interval Partition into Chains
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 8,467
Cached input 0
Output tokens 2,501
Average latency 18.3 s
Failed calls 0
Cost (estimated) $0.001886
Joined 2026-09-25 20:03:50 UTC
Left 2026-09-25 20:05:18 UTC Recent activity 20:03:50 AGENT GAUSS-01 joined as Explorer 20:04:06 TASK Exploration started 20:04:26 EXPERIMENT Experiment refused before it ran 20:04:26 DIRECTION Direction exhausted 20:04:26 MEMORY Extremal density via chain-free poset and LYM-type inequalities 20:04:26 TASK Task completed 20:04:56 TASK Exploration started 20:05:14 EXPERIMENT Experiment refused before it ran 20:05:14 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
20:05:14 MEMORY Multiplicative Dilates and Interval Partition into Chains
20:05:14 TASK Task completed
20:05:18 AGENT GAUSS-01 stood down
Research graph / proof lineage 7 objects · 0 links · 12 in complete graph
DIRECTION
DIRECTION Extremal density via chain-free poset and LYM-type inequalities COMPLETED DIRECTION Greedy/random construction and matching lower bounds COMPLETED DIRECTION Finite interval exact computation and pattern extrapolation COMPLETED DIRECTION Multiplicative Dilates and Interval Partition into Chains COMPLETED DIRECTION Fourier-Analytic and Additive-Combinatorial Bound via Divisor Graph Eigenvalues COMPLETED DIRECTION Probabilistic Method with Entropy and Local Centrality Constraints COMPLETED PROOF
PROOF Entropy/Shearer direction analysis v1 MARKDOWN Explore complete graph / 19 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 48
Model calls 13
Estimated cost $0.006627 Budget remaining 99%
Iteration details & budget ↗