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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 56.
Research chronicle 96 RECORDS
All Research Review Verification System
20:24:53 §1
SYSTEM Iteration 56 opened
20:24:55 §2
SYSTEM Iteration 56 started
20:24:55 §3
SYSTEM Initializing research run
20:24:55 §4
NOETHER NOETHER joined as Strategist
20:24:55 §5
SYSTEM Research started
20:24:56 §6
NOETHER Research planning started
↑ 90 earlier records Research record
Agent record Close ×
Research graph / proof lineage 7 objects · 0 links · 11 in complete graph
DIRECTION
DIRECTION Probabilistic extremal bound via divisor-count concentration COMPLETED DIRECTION Constructive family via prime-power intervals COMPLETED DIRECTION Computational search and structural conjecture on small intervals COMPLETED DIRECTION Weighted pair-counting upper bound via divisor-sum weights COMPLETED DIRECTION 2-adic valuation decomposition and bipartite matching bound COMPLETED DIRECTION Asymptotic upper bound via divisor function and covering by multiples COMPLETED PROOF
PROOF LP bound vs ceil(N/2) v1 MARKDOWN Explore complete graph / 18 objects →
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 1,071
Cached input 517
Output tokens 1,349
Average latency 22.3 s
Failed calls 0
Cost (estimated) $0.000471
Joined 2026-09-25 20:24:55 UTC
Left 2026-09-25 20:28:34 UTC Recent activity 20:25:11 DIRECTION Direction proposed: Computational search and structural conjecture on small intervals 20:25:11 DIRECTION Direction assigned to GAUSS-01: Probabilistic extremal bound via divisor-count concentration 20:25:11 DIRECTION Direction assigned to GAUSS-02: Constructive family via prime-power intervals 20:25:11 DIRECTION Direction assigned to GAUSS-03: Computational search and structural conjecture on small intervals 20:25:11 TASK Task completed 20:26:18 TASK Research planning started 20:26:49 PLAN Research plan created 20:26:49 MEMORY Novelty check: Weighted pair-counting upper bound via divisor-sum weights 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.
20:26:49 NOVELTY Novelty check: partially explored
20:26:49 DIRECTION Direction proposed: Weighted pair-counting upper bound via divisor-sum weights
20:26:49 MEMORY Novelty check: 2-adic valuation decomposition and bipartite matching bound
20:26:49 NOVELTY Novelty check: partially explored
20:26:49 DIRECTION Direction proposed: 2-adic valuation decomposition and bipartite matching bound
20:26:49 NOVELTY Novelty check: new
20:26:49 DIRECTION Direction proposed: Asymptotic upper bound via divisor function and covering by multiples
20:26:49 DIRECTION Direction assigned to GAUSS-01: Weighted pair-counting upper bound via divisor-sum weights
20:26:49 DIRECTION Direction assigned to GAUSS-02: 2-adic valuation decomposition and bipartite matching bound
20:26:49 DIRECTION Direction assigned to GAUSS-03: Asymptotic upper bound via divisor function and covering by multiples
20:26:49 TASK Task completed
20:28:34 AGENT NOETHER 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 56
Model calls 13
Estimated cost $0.005336 Budget remaining 99%
Iteration details & budget ↗