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 #12never observed
start → read_context P = 0.353 · sampled 0.0%start → think P = 0.353 · sampled 0.0%read_context → observe P = 0.500 · sampled 0.0%word_count → observe P = 0.647 · sampled 0.0%observe → think P = 0.273 · sampled 0.0%observe → verify P = 0.500 · sampled 0.0%think → word_count P = 0.647 · sampled 0.0%verify → ✓ pass P = 0.353 · sampled 0.0%verify → ✗ fail P = 0.353 · sampled 0.0%startstart visited in 0% of samplesread_contextact:read_context visited in 0% of samplesword_countact:word_count visited in 0% of samplesobserveobserve visited in 0% of samplesthinkthink 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· 7 folds · total surprise 5.2
fold 0read_context1.04 nats
fold 1observe0.69 nats
fold 2think1.30 nats
fold 3word_count0.44 nats
fold 4observe0.44 nats
fold 5verify0.69 nats
fold 6✓ pass0.65 nats
Real attempt #12 (passed)· 7 folds · total surprise 5.2
fold 0read_context1.04 nats
fold 1observe0.69 nats
fold 2think1.30 nats
fold 3word_count0.44 nats
fold 4observe0.44 nats
fold 5verify0.69 nats
fold 6✓ pass0.65 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).