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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
↑ 87 earlier records 17:26:06 §88
GAUSS-02 GAUSS-02 stood down
17:26:06 §89
GAUSS-03 GAUSS-03 stood down
17:26:06 §90
LITERATURE-01 LITERATURE-01 stood down
17:26:06 §91
ARES ARES stood down
17:26:06 §92
SYSTEM Research run finished
17:26:06 §93
SYSTEM Iteration 9 completed
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 → Research record
Agent record Close ×
Mission
Runtime 8 h 13 min
Iteration 34 of 34
Directions 193
Hypotheses 16
Experiments 143
Rejected paths 59 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 73
Highly relevant 1 Iteration 9
Model calls 12
Estimated cost $0.0485 Budget remaining 98%
Iteration details & budget ↗
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 1,480
Cached input 0
Output tokens 1,092
Average latency 15.6 s
Failed calls 0
Cost (estimated) $0.000513
Joined 2026-09-25 17:00:50 UTC
Left 2026-09-25 17:26:06 UTC Recent activity 17:01:09 DIRECTION Direction proposed: Density increment / Fourier-analytic approach on divisor graph 17:01:09 NOVELTY Novelty check: new 17:01:09 DIRECTION Direction proposed: Probabilistic construction and lower bound via random subsets 17:01:09 DIRECTION Direction assigned to GAUSS-01: Extremal hypergraph via 3-uniform divisibility hypergraph 17:01:09 DIRECTION Direction assigned to GAUSS-02: Density increment / Fourier-analytic approach on divisor graph 17:01:09 DIRECTION Direction assigned to GAUSS-03: Probabilistic construction and lower bound via random subsets 17:01:09 TASK Task completed 17:24:57 TASK Research planning started 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.
17:25:12 PLAN Research plan created
17:25:12 NOVELTY Novelty check: new
17:25:12 DIRECTION Direction proposed: Chain-decomposition upper bound via 2- and 3-smooth chains
17:25:12 NOVELTY Novelty check: new
17:25:12 DIRECTION Direction proposed: Explicit layered construction: (N/3, N] plus controlled small elements
17:25:12 NOVELTY Novelty check: new
17:25:12 DIRECTION Direction proposed: Targeted literature search for this Erdős-type problem
17:25:12 DIRECTION Direction assigned to GAUSS-01: Chain-decomposition upper bound via 2- and 3-smooth chains
17:25:12 DIRECTION Direction assigned to GAUSS-02: Explicit layered construction: (N/3, N] plus controlled small elements
17:25:12 DIRECTION Direction assigned to GAUSS-03: Targeted literature search for this Erdős-type problem
17:25:12 TASK Task completed
17:26:06 AGENT NOETHER stood down