Claudecycles Revisited

claudescycles

An experiment in **AI-assisted mathematical replication with structured scaffolding**.

Knuth's ["Claude's Cycles"](claude-cycles.pdf) (2026) describes how Claude Opus 4.6, coached by Filip Stappers, discovered a construction decomposing the arcs of a directed graph `G_m` into three Hamiltonian cycles for odd `m`. The process was impressive — but Knuth also documented friction: Claude lost context on restarts, had to be repeatedly reminded to log its progress, and eventually "got stuck" on the even-`m` case.

This repo asks: **can better harness design fix those failure modes?** We used Codex CLI (GPT-5.2) with a structured execution harness ([AGENTS.md](AGENTS.md)) that enforces verifier-first development, mandatory logging, restart-safe memory, and machine-readable evidence. The goal was to replicate and extend the paper's results while testing whether scaffolding disciplines improve reliability and reproducibility over lightweight coaching.

For a comparison of the baseline against four strict-cleanroom runs (GPT-5.2, GPT-5.3-Codex, GPT-5.4, and Claude Opus 4.6 — each working without any reference to the paper), see **[COMPARISON.md](COMPARISON.md)**.

**What we found:**

• The harness eliminated context loss and documentation drift — every experiment has exact commands and reproducible artifacts.
• It did not produce independent discovery: the model accessed the reference paper after ~30 minutes of independent work (see Methodology for the full timeline).
• As an extension, CP-SAT found verified even-m decompositions for m=4,6,8 while the 02 Mar 2026 draft still left the general even-m case open; within our own runs, however, a general even-m construction remained elusive.

Addendum (2026-03-29): Knuth's 2026-03-16 Revision

On March 29, 2026 we fetched the current Stanford PDF and archived both source snapshots under `references/papers/`:

• `references/papers/claude-cycles-2026-03