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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 16.
Research chronicle 92 RECORDS
All Research Review Verification System
18:20:14 §1
SYSTEM Iteration 16 opened
18:20:17 §2
SYSTEM Iteration 16 started
18:20:17 §3
SYSTEM Initializing research run
18:20:17 §4
NOETHER NOETHER joined as Strategist
18:20:17 §5
SYSTEM Research started
18:20:18 §6
NOETHER Research planning started
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GAUSS-03 Explorer
Explore direction: Upper bound via multiplicative-chain decompositions and Lean lemma
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 7,440
Cached input 0
Output tokens 2,967
Average latency 36.9 s
Failed calls 0
Cost (estimated) $0.001872
Joined 2026-09-25 18:20:29 UTC
Left 2026-09-25 18:23:31 UTC Recent activity 18:20:29 AGENT GAUSS-03 joined as Explorer 18:20:47 TASK Exploration started 18:21:30 EXPERIMENT Experiment refused before it ran 18:21:30 DIRECTION Direction exhausted 18:21:30 MEMORY Computational search and extremal pattern extraction 18:21:30 TASK Task completed 18:22:48 TASK Exploration started 18:23:23 EXPERIMENT Experiment refused before it ran 18:23:23 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
18:23:23 MEMORY Upper bound via multiplicative-chain decompositions and Lean lemma
18:23:23 TASK Task completed
18:23:31 AGENT GAUSS-03 stood down
Research graph / proof lineage 7 objects · 0 links · 10 in complete graph
DIRECTION
DIRECTION Random subset construction with divisor-free triples COMPLETED DIRECTION Poset rank-layer and chain-decomposition upper bound COMPLETED DIRECTION Computational search and extremal pattern extraction COMPLETED DIRECTION Exhaustive/ILP computation of f(N) for N<=40 and density benchmark COMPLETED DIRECTION Lower-bound construction: augment (N/3,N] with small elements COMPLETED DIRECTION Upper bound via multiplicative-chain decompositions and Lean lemma COMPLETED PROOF
PROOF Experiment report — Verify that for all N<=200, the maximum f(N) of a subset of {1..N} with no eleme v1 MARKDOWN Explore complete graph / 17 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 16
Model calls 11
Estimated cost $0.0507 Budget remaining 97%
Iteration details & budget ↗