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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 3.
Research chronicle 117 RECORDS
All Research Review Verification System
↑ 111 earlier records 11:20:42 §112
GAUSS-02 GAUSS-02 stood down
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LITERATURE-01 LITERATURE-01 stood down
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LITERATURE-02 LITERATURE-02 stood down
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TURING TURING stood down
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SYSTEM Research run finished
11:20:42 §117
SYSTEM Iteration 3 completed
Research graph / proof lineage 9 objects · 3 links · 8 in complete graph
DIRECTION
DIRECTION Poset‑Chain Decomposition and Dilworth‑type Upper Bounds REJECTED DIRECTION Probabilistic Construction via Lovász Local Lemma REJECTED DIRECTION Arithmetic Construction Using Large Prime Factors and Sieve Estimates REJECTED DIRECTION Smooth‑Number Lattice Decomposition REJECTED DIRECTION 3‑Uniform Hypergraph Turán Method COMPLETED DIRECTION Exact Computation and Pattern Extraction REJECTED HYPOTHESIS
HYPOTHESIS H-001 REJECTED REVIEW
REVIEW Review (needs_work) NEEDS WORK PROOF
PROOF Greedy sizes for selected N v1 MARKDOWN Explore complete graph / 17 objects → Research record
Agent record Close ×
Mission
Runtime 5 h 32 min
Iteration 7 of 7
Directions 33
Hypotheses 7
Experiments 3
Rejected paths 34 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 3 No frontier agent engaged
Model calls 19
Estimated cost $0.0158 Budget remaining 99%
Iteration details & budget ↗
LITERATURE-01 Literature researcher
Reading the landscape around: Smooth‑Number Lattice Decomposition
Model
Model used together / openai/gpt-oss-120b
Routing profile cheap_reasoning
Usage
Model calls 2
Input tokens 2,039
Cached input 138
Output tokens 1,169
Average latency 6.2 s
Failed calls 0
Cost (estimated) $0.001008
Joined 2026-09-25 11:18:41 UTC
Left 2026-09-25 11:20:42 UTC Recent activity 11:18:41 AGENT LITERATURE-01 joined as Literature Researcher 11:18:41 LITERATURE Literature search started 11:18:41 LITERATURE Indexed: Decomposition Theory for Lattices without Chain Conditions 11:18:41 LITERATURE Indexed: Effective Versions of the Chain Decomposition Theorem 11:18:41 LITERATURE Indexed: The Numbers of Shared Upper Bounds Determine a Poset 11:18:41 LITERATURE Literature search: 3 new of 3 11:18:41 LITERATURE Reviewed: Decomposition Theory for Lattices without Chain Conditions 11:18:41 MEMORY Literature: Decomposition Theory for Lattices without Chain Conditions Every figure here is aggregated from this agent’s recorded model calls, failed attempts included — a failed attempt is still an attempt.
08 Research director Not started On the roster; the workflow has not woken it in this iteration. 09 Senior reviewer Not started On the roster; the workflow has not woken it in this iteration. 11:19:51
LITERATURE
Literature search started
11:19:51 LITERATURE Indexed: Nuclide Number Density Prediction in the Lattice Physics Calculation Based on Dynamic Mode Decomposition
11:19:51 LITERATURE Indexed: On the number field sieve: polynomial selection and smooth elements in number fields
11:19:51 LITERATURE Indexed: Self-Attractive Hartree Decomposition: Partitioning Electron Density into Smooth Localized Fragments
11:19:51 LITERATURE Literature search: 3 new of 3
11:19:51 LITERATURE Reviewed: Nuclide Number Density Prediction in the Lattice Physics Calculation Based on Dynamic Mode Decomposition
11:19:51 MEMORY Literature: Nuclide Number Density Prediction in the Lattice Physics Calculation Based on Dynamic Mode Decomposition
11:20:42 AGENT LITERATURE-01 stood down