Chain-of-thought folds

Each reasoning step folds one idea into the next. Real attempts define which folds are likely; a Markov chain Monte Carlo sampler then explores every chain of thought the agent could take, the way origami configuration spaces are sampled.

Fold graph

current samplereal attempt #38never observed
start → user P = 0.647 · sampled 0.0%ask_arkher → observe P = 0.647 · sampled 0.0%observe → think P = 0.647 · sampled 0.0%think → ✓ pass P = 0.647 · sampled 0.0%user → verify P = 0.647 · sampled 0.0%verify → ask_arkher P = 0.647 · sampled 0.0%startstart visited in 0% of samplesask_arkheract:ask_arkher visited in 0% of samplesobserveobserve visited in 0% of samplesthinkthink visited in 0% of samplesuseruser visited in 0% of samplesverifyverify visited in 0% of samples✓ passpass visited in 0% of samples✗ failfail visited in 0% of samples

Folded chains

Current MCMC sample· 6 folds · total surprise 2.5
fold 0user0.44 nats
fold 1verify0.44 nats
fold 2ask_arkher0.44 nats
fold 3observe0.44 nats
fold 4think0.44 nats
fold 5✓ pass0.35 nats
Real attempt #38 (passed)· 6 folds · total surprise 2.5
fold 0user0.44 nats
fold 1verify0.44 nats
fold 2ask_arkher0.44 nats
fold 3observe0.44 nats
fold 4think0.44 nats
fold 5✓ pass0.35 nats

Crease angle = surprise of that fold (−log P). Sharp creases are unusual jumps from one idea to the next; mountain and valley folds alternate, keeping the strip flat.

Most visited chains

Run the sampler to estimate which chains of thought dominate.

How it works

  • States are kinds of steps: think, act with a tool, observe a result, get verified.
  • Folds are transitions between them, with probabilities learned from the 2 real attempts (smoothed, so unseen folds stay possible).
  • Energy of a chain = total surprise of its folds, plus β if it ends in failure. Chains are sampled with probability ∝ e−E/T.
  • Metropolis–Hastings proposes refolding, inserting or removing a step, or flipping the outcome, and accepts with probability min(1, e−ΔE/T × proposal ratio).
  • Inspired by Hikaru Kuribayashi's MCMC sampling of origami and linkage configuration spaces (Regeneron ISEF 2026).