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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 25.
Research chronicle 92 RECORDS
All Research Review Verification System
↑ 86 earlier records 19:02:20 §87
GAUSS-01 GAUSS-01 stood down
19:02:20 §88
GAUSS-02 GAUSS-02 stood down
19:02:20 §89
GAUSS-03 GAUSS-03 stood down
19:02:20 §90
LITERATURE-01 LITERATURE-01 stood down
19:02:20 §91
SYSTEM Research run finished
19:02:20 §92
SYSTEM Iteration 25 completed
Research graph / proof lineage 7 objects · 0 links · 10 in complete graph
DIRECTION
DIRECTION Probabilistic construction with random sparse multiples COMPLETED DIRECTION Poset chain decomposition and Ahlswede–Zhang upper bound COMPLETED DIRECTION Exact computation for small n and induction pattern COMPLETED DIRECTION Density increment via divisibility hypergraph containers COMPLETED DIRECTION Modular decomposition by largest odd divisor COMPLETED DIRECTION Constructive extremal family via numbers with k prime factors COMPLETED PROOF
PROOF Experiment report — Verify that for N up to 10^6, the family of numbers with exactly k prime factors v1 MARKDOWN Explore complete graph / 17 objects → Research record
Agent record Close ×
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 25
Model calls 12
Estimated cost $0.005948 Budget remaining 99%
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 1,075
Cached input 0
Output tokens 1,046
Average latency 12.4 s
Failed calls 0
Cost (estimated) $0.000443
Joined 2026-09-25 19:00:04 UTC
Left 2026-09-25 19:02:20 UTC Recent activity 19:00:17 DIRECTION Direction proposed: Poset chain decomposition and Ahlswede–Zhang upper bound 19:00:17 NOVELTY Novelty check: new 19:00:17 DIRECTION Direction proposed: Exact computation for small n and induction pattern 19:00:17 DIRECTION Direction assigned to GAUSS-01: Probabilistic construction with random sparse multiples 19:00:17 DIRECTION Direction assigned to GAUSS-02: Poset chain decomposition and Ahlswede–Zhang upper bound 19:00:17 DIRECTION Direction assigned to GAUSS-03: Exact computation for small n and induction pattern 19:00:17 TASK Task completed 19:00:54 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. 19:01:09
PLAN
Research plan created
19:01:09 NOVELTY Novelty check: new
19:01:09 DIRECTION Direction proposed: Density increment via divisibility hypergraph containers
19:01:09 NOVELTY Novelty check: new
19:01:09 DIRECTION Direction proposed: Modular decomposition by largest odd divisor
19:01:09 NOVELTY Novelty check: new
19:01:09 DIRECTION Direction proposed: Constructive extremal family via numbers with k prime factors
19:01:09 DIRECTION Direction assigned to GAUSS-01: Density increment via divisibility hypergraph containers
19:01:09 DIRECTION Direction assigned to GAUSS-02: Modular decomposition by largest odd divisor
19:01:09 DIRECTION Direction assigned to GAUSS-03: Constructive extremal family via numbers with k prime factors
19:01:09 TASK Task completed
19:02:20 AGENT NOETHER stood down