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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.
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
11:41:48 §189
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 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
Indexed 18
Highly relevant 0 Iteration 7 No frontier agent engaged
Model calls 44
Estimated cost $0.0449 Budget remaining 98%
Iteration details & budget ↗
TURING Critic
Review every hypothesis of this iteration
Model
Model used together / meta-llama/Llama-3.3-70B-Instruct-Turbo
Routing profile independent_critic
Usage
Model calls 2
Input tokens 1,918
Cached input 0
Output tokens 290
Average latency 2.9 s
Failed calls 0
Cost (estimated) $0.002296
Joined 2026-09-25 11:36:55 UTC
Left 2026-09-25 11:41:48 UTC Recent activity 11:36:55 AGENT TURING joined as Critic 11:37:10 TASK Critical review started 11:37:15 REVIEW Hypothesis H-001 reviewed 11:37:15 MEMORY Review of H-001 11:37:15 GATE No hypothesis passed the workflow gate 11:37:15 TASK Task completed 11:38:23 TASK Critical review started 11:38:27 REVIEW Hypothesis H-002 reviewed 11:38:27 MEMORY Review of H-002 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.
11:38:27 GATE No hypothesis passed the workflow gate
11:38:27 TASK Task completed
11:41:48 AGENT TURING stood down