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How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab
Showing iteration 7, with earlier context.
Research chronicle 189 RECORDS
All Research Review Verification System
↑ 183 earlier records 11:41:48 §184
NOETHER Task completed
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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
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SYSTEM Iteration 7 completed
Research graph / proof lineage 18 objects · 3 links · 8 in complete graph
DIRECTION
DIRECTION Poset & Chain Decomposition Bounds REJECTED DIRECTION Probabilistic Construction via Lovász Local Lemma REJECTED DIRECTION Greedy/Sieve Method with Harmonic Estimates REJECTED DIRECTION Divisor‑Graph Turán Approach COMPLETED DIRECTION Primitive‑Set Extension via Prime‑Factor Constraints REJECTED DIRECTION 3‑Uniform Hypergraph Turán / Dependent Random Choice COMPLETED DIRECTION Spectral bound on the divisor graph with out‑degree constraint REJECTED DIRECTION Mixed‑factor construction with large‑prime anchors REJECTED DIRECTION Exact ILP for moderate N and pattern extraction REJECTED DIRECTION Entropy‑Based Upper Bound via Shearer’s Inequality COMPLETED DIRECTION Largest‑Prime‑Factor Partition and Smooth‑Number Estimates REJECTED DIRECTION Exact Extremal Computation via SAT/SMT with Symmetry Breaking REJECTED DIRECTION Exponent‑vector antichain via Sperner theory REJECTED DIRECTION Adaptation of multiplicative Sidon set results REJECTED DIRECTION Recursive block decomposition and symmetry‑reduced integer programming REJECTED HYPOTHESIS
HYPOTHESIS H-001 REJECTED REVIEW
REVIEW Review (needs_work) NEEDS WORK PROOF
PROOF Experiment report — Verify that for all N ≤ 2000 the greedy set S_N satisfies |S_N| ≥ N / (3*log(N)) v1 MARKDOWN Explore complete graph / 26 objects → Research record
Agent record Close ×
Mission
Runtime 2 h 33 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
Indexed 18
Highly relevant 0 Iteration 7 No frontier agent engaged
Model calls 44
Estimated cost $0.0449 Budget remaining 98%
Iteration details & budget ↗
LITERATURE-01 Literature researcher
Reading the landscape around: Exponent‑vector antichain via Sperner theory
Model
Model used together / openai/gpt-oss-120b
Routing profile cheap_reasoning
Usage
Model calls 5
Input tokens 5,598
Cached input 345
Output tokens 2,821
Average latency 4.3 s
Failed calls 0
Cost (estimated) $0.002532
Joined 2026-09-25 11:36:16 UTC
Left 2026-09-25 11:41:48 UTC Recent activity 11:38:50 LITERATURE Indexed: An upper bound for the minimum degree of a graph 11:38:50 LITERATURE Indexed: Number of irreducible factors and degree in divisor graph of Zp[x,n] 11:38:50 LITERATURE Literature search: 3 new of 3 11:38:50 LITERATURE Reviewed: Degree Distance of Zero-Divisor Graph Г[Z_n ] 11:38:50 MEMORY Literature: Degree Distance of Zero-Divisor Graph Г[Z_n ] 11:39:45 LITERATURE Literature search started 11:39:45 LITERATURE Indexed: Refinements and Generalizations of the Shannon Lower Bound via Extensions of the Kraft Inequality 11:39:45 LITERATURE Indexed: Shearer’s inequality and infimum rule for Shannon entropy and topological entropy 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.
11:39:45 LITERATURE Indexed: New Upper Bound and Lower Bound for Degree-Based Network Entropy
11:39:45 LITERATURE Literature search: 3 new of 3
11:39:45 LITERATURE Reviewed: Refinements and Generalizations of the Shannon Lower Bound via Extensions of the Kraft Inequality
11:39:45 MEMORY Literature: Refinements and Generalizations of the Shannon Lower Bound via Extensions of the Kraft Inequality
11:40:42 LITERATURE Literature search started
11:40:42 LITERATURE Indexed: Algebraic methods in Sperner theory
11:40:42 LITERATURE Indexed: The flow-theoretic approach in Sperner theory
11:40:42 LITERATURE Indexed: Optimal Error-Detecting Codes for General Asymmetric Channels via Sperner Theory
11:40:42 LITERATURE Literature search: 3 new of 3
11:40:42 LITERATURE Reviewed: Algebraic methods in Sperner theory
11:40:42 MEMORY Literature: Algebraic methods in Sperner theory
11:41:48 AGENT LITERATURE-01 stood down