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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 55.
Research chronicle 92 RECORDS
All Research Review Verification System
20:21:19 §1
SYSTEM Iteration 55 opened
20:21:21 §2
SYSTEM Iteration 55 started
20:21:21 §3
SYSTEM Initializing research run
20:21:21 §4
NOETHER NOETHER joined as Strategist
20:21:21 §5
SYSTEM Research started
20:21:22 §6
NOETHER Research planning started
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GAUSS-01 Explorer
Explore direction: Inductive poset decomposition via interval halving and chain-antichain duality
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 8,495
Cached input 0
Output tokens 2,260
Average latency 13.3 s
Failed calls 0
Cost (estimated) $0.001823
Joined 2026-09-25 20:21:30 UTC
Left 2026-09-25 20:24:53 UTC Recent activity 20:21:30 AGENT GAUSS-01 joined as Explorer 20:21:48 TASK Exploration started 20:22:01 EXPERIMENT Experiment refused before it ran 20:22:01 DIRECTION Direction exhausted 20:22:01 MEMORY Constructive lower bounds via layered residue classes and interval partitions 20:22:01 TASK Task completed 20:24:03 TASK Exploration started 20:24:19 EXPERIMENT Experiment refused before it ran 20:24:19 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
20:24:19 MEMORY Inductive poset decomposition via interval halving and chain-antichain duality
20:24:19 TASK Task completed
20:24:53 AGENT GAUSS-01 stood down
Research graph / proof lineage 7 objects · 0 links · 10 in complete graph
DIRECTION
DIRECTION Constructive lower bounds via layered residue classes and interval partitions COMPLETED DIRECTION Upper bounds via divisor-count weighting and double counting COMPLETED DIRECTION Computational search and extremal pattern extraction COMPLETED DIRECTION Inductive poset decomposition via interval halving and chain-antichain duality COMPLETED DIRECTION Logarithmic-density continuous relaxation and extremal measure COMPLETED DIRECTION Probabilistic greedy construction with local ratio analysis COMPLETED PROOF
PROOF Continuous relaxation is vacuous 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 55
Model calls 13
Estimated cost $0.005421 Budget remaining 99%
Iteration details & budget ↗