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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 10.
Research chronicle 94 RECORDS
All Research Review Verification System
17:26:06 §1
SYSTEM Iteration 10 opened
17:26:09 §2
SYSTEM Iteration 10 started
17:26:09 §3
SYSTEM Initializing research run
17:26:09 §4
NOETHER NOETHER joined as Strategist
17:26:09 §5
SYSTEM Research started
17:26:10 §6
NOETHER Research planning started
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GAUSS-02 Explorer
Explore direction: Construction: verify (N/3, N] and add lower layers for density above 2/3
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 3
Input tokens 9,969
Cached input 3,315
Output tokens 4,579
Average latency 24.4 s
Failed calls 1
Cost (estimated) $0.002314
Joined 2026-09-25 17:26:27 UTC
Left 2026-09-25 18:00:49 UTC Recent activity 17:26:27 AGENT GAUSS-02 joined as Explorer 17:26:49 TASK Exploration started 17:27:47 EXPERIMENT Experiment refused before it ran 17:27:47 DIRECTION Direction exhausted 17:27:47 MEMORY Explicit construction via residue classes and interval packing 17:27:47 TASK Task completed 18:00:25 TASK Exploration started 18:00:44 EXPERIMENT Experiment refused before it ran 18:00:44 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
18:00:44 MEMORY Construction: verify (N/3, N] and add lower layers for density above 2/3
18:00:44 TASK Task completed
18:00:49 AGENT GAUSS-02 stood down 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 10
Model calls 13
Estimated cost $0.0524 Budget remaining 97%
Iteration details & budget ↗
Research graph / proof lineage 7 objects · 0 links · 9 in complete graph
DIRECTION
DIRECTION Probabilistic upper bound via divisor-count concentration COMPLETED DIRECTION Explicit construction via residue classes and interval packing COMPLETED DIRECTION Computational search and pattern extrapolation for exact small-n values COMPLETED DIRECTION Exact solver for f(N) and interval variants via ILP/SAT COMPLETED DIRECTION Construction: verify (N/3, N] and add lower layers for density above 2/3 COMPLETED DIRECTION Upper bound via chain decomposition into m·2^i·3^j families and literature search COMPLETED PROOF
PROOF Chain-decomposition bound check v1 MARKDOWN Explore complete graph / 16 objects →