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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 17.
Research chronicle 93 RECORDS
All Research Review Verification System
18:23:31 §1
SYSTEM Iteration 17 opened
18:23:34 §2
SYSTEM Iteration 17 started
18:23:34 §3
SYSTEM Initializing research run
18:23:34 §4
NOETHER NOETHER joined as Strategist
18:23:34 §5
SYSTEM Research started
18:23:35 §6
NOETHER Research planning started
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GAUSS-02 Explorer
Explore direction: Lower-bound construction family from layered intervals with controlled multiple counts
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 7,794
Cached input 0
Output tokens 2,315
Average latency 29.9 s
Failed calls 0
Cost (estimated) $0.001740
Joined 2026-09-25 18:23:59 UTC
Left 2026-09-25 18:29:32 UTC Recent activity 18:23:59 AGENT GAUSS-02 joined as Explorer 18:24:21 TASK Exploration started 18:24:47 EXPERIMENT Experiment refused before it ran 18:24:47 DIRECTION Direction exhausted 18:24:47 MEMORY Poset chain decomposition and weighted Dilworth-type bound 18:24:47 TASK Task completed 18:27:33 TASK Exploration started 18:28:10 EXPERIMENT Experiment refused before it ran 18:28:10 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
18:28:10 MEMORY Lower-bound construction family from layered intervals with controlled multiple counts
18:28:10 TASK Task completed
18:29:32 AGENT GAUSS-02 stood down
Research graph / proof lineage 7 objects · 0 links · 15 in complete graph
DIRECTION
DIRECTION Probabilistic construction with random multiples and density optimization COMPLETED DIRECTION Poset chain decomposition and weighted Dilworth-type bound COMPLETED DIRECTION Exact computation for small n and induction with block decomposition COMPLETED DIRECTION Exact brute-force and layered-construction benchmark for [1,N] and short intervals COMPLETED DIRECTION Lower-bound construction family from layered intervals with controlled multiple counts COMPLETED DIRECTION Upper bound by grouping each element with its first two multiples and charging deletions COMPLETED PROOF
PROOF Exact maxima for [1,N], N<=24 v1 MARKDOWN Explore complete graph / 22 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 17
Model calls 13
Estimated cost $0.0486 Budget remaining 98%
Iteration details & budget ↗