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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 23.
Research chronicle 95 RECORDS
All Research Review Verification System
18:53:53 §1
SYSTEM Iteration 23 opened
18:53:56 §2
SYSTEM Iteration 23 started
18:53:56 §3
SYSTEM Initializing research run
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NOETHER NOETHER joined as Strategist
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SYSTEM Research started
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NOETHER Research planning started
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LITERATURE-01 Literature researcher
Reading the landscape around: Graph-theoretic extremal bound via forbidden divisibility triples
Model
Model used together / deepseek-ai/DeepSeek-V4-Flash-0731
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 1,951
Cached input 0
Output tokens 521
Average latency 23.0 s
Failed calls 0
Cost (estimated) $0.000419
Joined 2026-09-25 18:54:15 UTC
Left 2026-09-25 18:57:47 UTC Recent activity 18:54:15 AGENT LITERATURE-01 joined as Literature Researcher 18:54:15 LITERATURE Literature search started 18:54:15 LITERATURE Indexed: Biases towards the zero residue class for quadratic forms in arithmetic progressions 18:54:15 LITERATURE Indexed: On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds) 18:54:15 LITERATURE Indexed: Lower bounds for the least common multiple of finite arithmetic progressions 18:54:15 LITERATURE Literature search: 3 new of 3 18:54:15 LITERATURE Reviewed: Biases towards the zero residue class for quadratic forms in arithmetic progressions 18:54:15 MEMORY Literature: Biases towards the zero residue class for quadratic forms in arithmetic progressions 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.
18:56:23 LITERATURE Literature search started
18:56:23 LITERATURE Indexed: Forbidden triples for perfect matchings
18:56:23 LITERATURE Indexed: Coloring Graph Classes with no Induced Hammer via 2-Divisibility and Perfect Divisibility
18:56:23 LITERATURE Indexed: Forbidden triples implying Hamiltonicity: for all graphs
18:56:23 LITERATURE Literature search: 3 new of 3
18:56:23 LITERATURE Reviewed: Forbidden triples for perfect matchings
18:56:23 MEMORY Literature: Forbidden triples for perfect matchings
18:57:47 AGENT LITERATURE-01 stood down
Research graph / proof lineage 7 objects · 0 links · 11 in complete graph
DIRECTION
DIRECTION Constructive lower bounds via residue-class packings and interval arithmetic progressions COMPLETED DIRECTION Upper bounds via divisor-chain counting and double-counting over pairs COMPLETED DIRECTION Computational search and pattern discovery for small N to conjecture the exact extremal function COMPLETED DIRECTION Graph-theoretic extremal bound via forbidden divisibility triples COMPLETED DIRECTION Probabilistic and entropy upper bound via random weights COMPLETED DIRECTION Shifting/compression in the divisor poset to reduce to initial segments COMPLETED PROOF
PROOF Experiment report — Verify whether the natural compression operation (decrement the largest element) v1 MARKDOWN Explore complete graph / 18 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 23
Model calls 12
Estimated cost $0.006388 Budget remaining 99%
Iteration details & budget ↗