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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
↑ 86 earlier records 18:23:31 §87
GAUSS-02 GAUSS-02 stood down
18:23:31 §88
GAUSS-03 GAUSS-03 stood down
18:23:31 §89
LITERATURE-01 LITERATURE-01 stood down
18:23:31 §90
ARES ARES stood down
18:23:31 §91
SYSTEM Research run finished
18:23:31 §92
SYSTEM Iteration 16 completed
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 → Research record
Agent record Close ×
Mission
Runtime 9 h 10 min
Iteration 55 of 55
Directions 325
Hypotheses 17
Experiments 258
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 The corpus is not published by the engine.
Iteration 16
Model calls 11
Estimated cost $0.0507 Budget remaining 97%
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 1,503
Cached input 517
Output tokens 1,048
Average latency 20.5 s
Failed calls 0
Cost (estimated) $0.000447
Joined 2026-09-25 18:20:17 UTC
Left 2026-09-25 18:23:31 UTC Recent activity 18:20:29 NOVELTY Novelty check: new 18:20:29 DIRECTION Direction proposed: Computational search and extremal pattern extraction 18:20:29 DIRECTION Direction assigned to GAUSS-01: Random subset construction with divisor-free triples 18:20:29 DIRECTION Direction assigned to GAUSS-02: Poset rank-layer and chain-decomposition upper bound 18:20:29 DIRECTION Direction assigned to GAUSS-03: Computational search and extremal pattern extraction 18:20:29 TASK Task completed 18:21:56 TASK Research planning started 18:22:28 PLAN Research plan created 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. 18:22:28
NOVELTY
Novelty check: new
18:22:28 DIRECTION Direction proposed: Exhaustive/ILP computation of f(N) for N<=40 and density benchmark
18:22:28 NOVELTY Novelty check: new
18:22:28 DIRECTION Direction proposed: Lower-bound construction: augment (N/3,N] with small elements
18:22:28 MEMORY Novelty check: Upper bound via multiplicative-chain decompositions and Lean lemma
18:22:28 NOVELTY Novelty check: partially explored
18:22:28 DIRECTION Direction proposed: Upper bound via multiplicative-chain decompositions and Lean lemma
18:22:28 DIRECTION Direction assigned to GAUSS-01: Exhaustive/ILP computation of f(N) for N<=40 and density benchmark
18:22:28 DIRECTION Direction assigned to GAUSS-02: Lower-bound construction: augment (N/3,N] with small elements
18:22:28 DIRECTION Direction assigned to GAUSS-03: Upper bound via multiplicative-chain decompositions and Lean lemma
18:22:28 TASK Task completed
18:23:31 AGENT NOETHER stood down