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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 54.
Research chronicle 93 RECORDS
All Research Review Verification System
20:19:14 §1
SYSTEM Iteration 54 opened
20:19:15 §2
SYSTEM Iteration 54 started
20:19:15 §3
SYSTEM Initializing research run
20:19:15 §4
NOETHER NOETHER joined as Strategist
20:19:15 §5
SYSTEM Research started
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NOETHER Research planning started
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GAUSS-02 Explorer
Explore direction: Residue-class and dyadic block decomposition
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 8,154
Cached input 0
Output tokens 1,966
Average latency 13.3 s
Failed calls 0
Cost (estimated) $0.001692
Joined 2026-09-25 20:19:25 UTC
Left 2026-09-25 20:21:19 UTC Recent activity 20:19:25 AGENT GAUSS-02 joined as Explorer 20:19:42 TASK Exploration started 20:19:54 EXPERIMENT Experiment refused before it ran 20:19:54 DIRECTION Direction exhausted 20:19:54 MEMORY Constructive lower bound via multiples of a large prime or near-n intervals 20:19:54 TASK Task completed 20:20:53 TASK Exploration started 20:21:11 EXPERIMENT Experiment refused before it ran 20:21:11 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
20:21:11 MEMORY Residue-class and dyadic block decomposition
20:21:11 TASK Task completed
20:21:19 AGENT GAUSS-02 stood down
Research graph / proof lineage 7 objects · 0 links · 9 in complete graph
DIRECTION
DIRECTION Density/entropy upper bound via random sampling and divisor graph COMPLETED DIRECTION Constructive lower bound via multiples of a large prime or near-n intervals COMPLETED DIRECTION Exact computation and pattern detection for small n COMPLETED DIRECTION Poset chain-decomposition and LP dual upper bounds COMPLETED DIRECTION Residue-class and dyadic block decomposition COMPLETED DIRECTION Second-moment and divisor-sum analytic bounds COMPLETED PROOF
PROOF Exact f(N) for N=1..40 v1 MARKDOWN Explore complete graph / 16 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 54
Model calls 13
Estimated cost $0.006917 Budget remaining 99%
Iteration details & budget ↗