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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
AGENTS / 6 6 agents · 0 working
Frontier availability Showing iteration 9.
Research chronicle 93 RECORDS
All Research Review Verification System
17:00:48 §1
SYSTEM Iteration 9 opened
17:00:50 §2
SYSTEM Iteration 9 started
17:00:50 §3
SYSTEM Initializing research run
17:00:50 §4
NOETHER NOETHER joined as Strategist
17:00:50 §5
SYSTEM Research started
17:00:51 §6
NOETHER Research planning started
↑ 87 earlier records Research record
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GAUSS-03 Explorer
Explore direction: Targeted literature search for this Erdős-type problem
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 6,684
Cached input 0
Output tokens 2,235
Average latency 29.2 s
Failed calls 0
Cost (estimated) $0.001562
Joined 2026-09-25 17:01:09 UTC
Left 2026-09-25 17:26:06 UTC Recent activity 17:01:09 AGENT GAUSS-03 joined as Explorer 17:01:28 TASK Exploration started 17:01:55 EXPERIMENT Experiment refused before it ran 17:01:55 DIRECTION Direction exhausted 17:01:55 MEMORY Probabilistic construction and lower bound via random subsets 17:01:55 TASK Task completed 17:25:30 TASK Exploration started 17:26:06 EXPERIMENT Experiment refused before it ran 17:26:06 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
17:26:06 MEMORY Targeted literature search for this Erdős-type problem
17:26:06 TASK Task completed
17:26:06 AGENT GAUSS-03 stood down
Research graph / proof lineage 7 objects · 0 links · 12 in complete graph
DIRECTION
DIRECTION Extremal hypergraph via 3-uniform divisibility hypergraph COMPLETED DIRECTION Density increment / Fourier-analytic approach on divisor graph COMPLETED DIRECTION Probabilistic construction and lower bound via random subsets COMPLETED DIRECTION Chain-decomposition upper bound via 2- and 3-smooth chains COMPLETED DIRECTION Explicit layered construction: (N/3, N] plus controlled small elements COMPLETED DIRECTION Targeted literature search for this Erdős-type problem COMPLETED PROOF
PROOF Experiment report — Verify by exact search that f(N)=ceil(N/2) for all N≤200, where f(N) is the max 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 9
Model calls 12
Estimated cost $0.0485 Budget remaining 98%
Iteration details & budget ↗