Claude Code transcript - page 2/3

Rolls N mazes for --iters steps in float64. (State norms reach ~1e26 by step 200, so a
float32 norm computation overflows in its sum of squares and returns inf/nan — the
*elements* mostly stay in float32 range, so float32 eval is still valid; the first step
where max|s| itself exceeds float32 range is recorded.) Per step:

   Interpretation: with the input re-fed at every step, the state never needs to
   preserve the maze, and the dynamics are free to be non-stationary — the network can
   effectively count steps and schedule a solution that peaks at readout time. The
   plain net has no such luxury: everything it knows must persist in the state. (What
   "stable" means mechanistically turned out to be subtle — see *What the dynamics
   say* below.)

**Caveats.** Phase-3 checkpoints are best-validation selections from (for the plain
variant) unstable training, selected at T=20 — a criterion that does not reward long-T
stability. Bansal et al.'s stabilizing recipe also differs from plain random-T: their
incremental-progress loss supervises *intermediate* steps and truncates backprop, which
plausibly is the load-bearing ingredient rather than budget randomness per se — we test
only the latter. All runs are single-seed pending the replicate sweep.

## What the dynamics say

Rolling 50 mazes for 200 steps in float64 and tracking the state and the thresholded
output of each recurrent model (`analyze_dynamics.py`, `dynamics.json`):

| model | ‖Δs‖/‖s‖ @200 | ‖s‖ @200 | output flips/px/step @200 | solved @200 (13×13) |
|---|---|---|---|---|
| Plain, fixed T | 0.35 | ~10²⁷ | **0.000000** | 94.6% |
| Recall, fixed T | 0.40 | ~10²⁶ | 0.00024 | 23.2% |
| Plain, random T | 0.054 | ~10⁶ | 0.00071 | 24.0% |
| Recall, random T | 0.087 | ~10⁹ | 0.00237 | 7.8% |

Two dissociations worth noticing:

1. **No model converges to a fixed point** — every state keeps growing (the fixed-T
   models multiply their state norm by ~1.35 per step, forever). Yet outputs freeze:
   pixel flip rates fall to ~zero everywhere. The fixed-T plain net freezes on the
   *correct* answer — its state direction stabilizes while the magnitude explodes
   (a "sign-stable divergence"), so the thresholded logits stop changing. The
   overthinking models freeze too — on *wrong* answers, after an error-accumulating
   transient in the ~30–100 step window.
2. **State-space calm does not mean answer stability.** The random-budget models have
   by far the most sedate states (relative step change 0.05–0.09, norms 20 orders of
   magnitude smaller) and still lose the answer. Stability that matters is stability
   of the *readout*, and it is not purchased by slow dynamics.

(Numerical footnote: those ~10²⁶ norms mean a float32 *norm computation* overflows even
though the state elements themselves stay in float32 range — evaluation in float32 is
unaffected except recall+random-T at 33×33, whose elements genuinely exceed float32
around step 196.)

- Dynamics analysis (analyze_dynamics.py → dynamics.json): DONE. No fixed points
  anywhere — states grow ~1.35×/step (fixed-T) yet OUTPUTS freeze (flip rate→0).
  p2 plain freezes CORRECT (sign-stable divergence); overthinkers freeze WRONG after
  a ~30-100 step error-accumulating transient. Random-T models have calm states
  (Δs/s ~0.05) but still lose the answer → readout stability ≠ state stability.
  float32 NaN in v1 of the script was a norm-computation artifact (sum of squares),
  NOT state overflow — evals valid (exception: recall_rand c=16 elements overflow
  f32 ~step 196). REPORT section "What the dynamics say" has the table.
- Replicates: seed-1 reruns

**A training-stability contrast appeared immediately.** Under the random budget, the
*plain* net's seed-0 run trains violently: it reaches 99.7% validation solved, but
repeatedly collapses to ~0% within an epoch or two (it ended mid-collapse, which is
also how we learned to keep best-validation checkpoints). The *recall* net trains
smoothly under the same regime — no collapses, 92–100% from epoch 7 on. The severity is
seed-dependent — the seed-1 plain+random run only dips to ~0.84 rather than collapsing —
but the direction is consistent: with a random budget the plain net is the wobblier of
the two, plausibly because re-fed input gives the recall net a fixed anchor at every
step, while the plain net's only memory of the maze is a state that the variable-length
objective keeps perturbing.

    ap.add_argument("--iters", type=int, default=200)
    ap.add_argument("--out", default="dynamics.json")
    args = ap.parse_args()

    (ROOT / args.out).write_text(json.dumps(results))
    print("wrote", args.out)

2. **Test-time compute converts into problem size.** The plain recurrent net, trained
   only at T=20 on 13×13 mazes, solves 41% of 33×33 mazes when run for 100 steps —
   a 50× improvement over running it at its training budget (0.8%). Its curve is
   roughly stable after the peak (37% at T=200). *(The peak's magnitude turned out to
   be strongly seed-dependent — see Replication below; the qualitative
   thinking-longer-helps effect is what survives.)*

Randomizing the budget did not remove overthinking — it *spread* it. The recall net
still collapses past its trained range; and in this seed the plain net — the phase-2
extrapolation champion — loses both its long-horizon stability (24% at T=200 on
training-size mazes) and almost all of its size extrapolation (41% → 5% peak at 33×33).

The story these seed-0 numbers suggest: the plain fixed-budget net's stability was not
a default that recall lacked, but an *emergent* property — dynamics that arise when
(a) the state is the only carrier of maze information and (b) the training objective
never asks the dynamics to be anything other than "converged by step 20". Training on a
range of budgets replaces "converge" with "be right anywhere in [10, 30]", which is
satisfiable by transient dynamics that pass through correctness and move on. The
replication run (below) supports the *stability* half of this story but not the
extrapolation half.

only the latter.

## Replication across seeds

All four recurrent configs were re-run with seed 1 (different init *and* different
training mazes). Two summary statistics per run — long-horizon stability at training
size (13×13, solved @T=200) and extrapolation peak (33×33, best solved over T):

| config | 13×13 @200, s0 | s1 | 33×33 peak, s0 | s1 |
|---|---|---|---|---|
| Plain, fixed T | **94.6** | **98.0** | 41.2 | 6.8 |
| Recall, fixed T | 23.2 | 53.0 | 4.4 | 22.8 |
| Plain, random T | 24.0 | 41.4 | 5.4 | 13.6 |
| Recall, random T | 7.8 | 4.0 | 5.0 | 0.4 |

**What survives replication:**

- *Thinking longer helps beyond the training budget*: every recurrent run, both seeds,
  peaks on larger mazes at 30–100 steps — far past the ~20-step budget. The dial is real.
- *Plain + fixed budget is the only long-horizon-stable configuration*: ≥94% at T=200
  on training-size mazes in both seeds; every other config decays to 4–53%.
- *Recall overthinks at training size* in both seeds and both budget regimes; random
  budgets never fix it.

**What does not survive:**

- *The magnitude of extrapolation.* The seed-0 showpiece (41% of 33×33 mazes) falls to
  6.8% at seed 1; meanwhile seed-1 recall-fixed reaches 22.8% and seed-1 plain-random
  13.6%. With n=2 seeds and this variance, there is no defensible architecture ordering
  for extrapolation *peak* — including phase 3's "random budgets destroy extrapolation",
  which the seed-1 numbers contradict (13.6 > 6.8). What can be said: how far a run
  extrapolates is a lottery over training runs, while whether it holds its answer over
  long rollouts is not.
- *The violence of plain+random training instability*: catastrophic 0%-collapses at
  seed 0, only mild dips (to ~0.84) at seed 1. Direction (plain wobblier than recall
  under random budgets) is consistent.

## What the dynamics say

## Findings in one paragraph

Weight-tied recurrent nets really can trade test-time compute for problem size: trained
on 13×13 mazes with a 20-step budget, they solve substantially larger mazes only when
run well past that budget (peaks at 30–100 steps, both seeds), while fixed-depth CNNs
with 8–15× more parameters collapse outside the training size. *How far* that
extrapolation reaches is a lottery across training runs (41% vs 7% of 33×33 mazes for
identical configs at different seeds); *whether the answer survives long rollouts* is
not: in every run, only the plain (no-recall) net trained at a fixed budget holds its
answer at 200 steps — every other variant "overthinks" its way from ~100% to 4–53%,
and randomizing the training budget does not fix this (contradicting the intuition we
took from the deep-thinking literature, where input-recall is the stabilizer).
Mechanistically, no model converges to a fixed point: states grow ~1.35×/step forever,
outputs freeze anyway (sign-stable divergence) — the good model freezes on the right
answer, overthinkers freeze on wrong ones, and calm slow-growing states offer no
protection.

## Setup

Mechanistically, no model converges to a fixed point: the fixed-budget nets' states grow
~1.35×/step forever, yet the stable model's *output* freezes completely on the right
answer (a sign-stable divergence, both seeds); overthinkers either freeze on wrong
answers or keep churning; and the random-budget models' much calmer, slower-growing
states offer no protection at all.

The same probe on the seed-1 replicates (`dynamics_s1.json`) confirms the load-bearing
rows and refines the others. Dissociations worth noticing:

1. **No model converges to a fixed point** — every state keeps growing (the fixed-T
   models multiply their state norm by ~1.35 per step, forever). Yet the *readout* can
   be perfectly stable: the fixed-T plain net's pixel flip rate falls to exactly zero
   at every size, in both seeds, and at training size it freezes on the *correct*
   answer — its state direction stabilizes while the magnitude explodes (a "sign-stable
   divergence"), so the thresholded logits stop changing.
2. **Overthinkers fail in two ways.** Seed-0 recall freezes on *wrong* answers after an
   error-accumulating transient in the ~30–100 step window; seed-1 recall never settles
   at larger sizes (still flipping 1.5–1.7% of pixels per step at T=200). Frozen-wrong
   and churning are both compatible with the same solved-rate collapse.
3. **State-space calm does not mean answer stability.** In both seeds the random-budget
   models have by far the most sedate states (relative step change 0.01–0.09, norms
   ~20 orders of magnitude smaller) and still lose the answer. The stability that
   matters is stability of the readout, and it is not purchased by slow dynamics.

(Numerical footnote: ~10²⁶⁺ norms mean a float32 *norm computation* overflows even
though state elements themselves usually stay in float32 range, so float32 evaluation
is valid — with exceptions at large sizes/late steps where elements genuinely exceed
float32 (seed-0 recall+random-T at 33×33 from step ~196; seed-1 plain fixed-T at
21×21/33×33 from steps ~189/~159). Those runs' T=200 large-maze eval entries slightly
understate the model.)

- Replicates (seed 1, all 4 recurrent configs): DONE + evaled (no c=6 test/train
  overlap raised). VERDICT — replicates: thinking-past-budget helps (peaks 30-100
  steps both seeds); plain fixed-T uniquely long-horizon stable (94.6/98.0 @200 c=6);
  recall overthinks both seeds; randT never fixes overthinking. Does NOT replicate:
  extrapolation MAGNITUDE (c=16 peak: plain-fixed 41.2→6.8; recall-fixed 4.4→22.8;
  plain-rand 5.4→13.6 — orderings flip; "randT destroys extrapolation" was seed-0
  artifact, retracted in report); plain-rand training collapse severity (violent s0,
  mild dips s1; note s0 "reproduction" was same-seed determinism, not evidence).
  Dynamics on s1 (dynamics_s1.json): plain fixed-T = exploding norms + flips→0 at all
  sizes 2/2 seeds (freezes correct at c=6); s1 recall CHURNS at c≥10 (1.5%/step @200)
  vs s0 froze-wrong — both failure modes noted; randT calm states replicate + still
  fail. f32 element overflow at large size/late step for some runs (s1 plain c=10/16
  @189/159; s0 recall-rand c=16 @196) — footnoted in report.
- REPORT.md: results, replication section, dynamics section, TL;DR all written.
- REMAINING: final viz build (5 seed-0 models + seed-variance footer note), visual
  pass in browser (CPU now free), writeup polish, two subagent reviews (code +
  writeup) — David explicitly re-confirmed he wants the independent review. Leave
  preview open for David.

$("foot").textContent = `Generated __DATE__ · models trained on 20,000 mazes of size 13×13 · `
  + `solved-rate curves from 500 unseen mazes per size · recurrent training budget ${TRAIN_ITERS} steps `
  + `(random budget: 10–30) · one training seed shown — extrapolation peaks vary strongly across seeds `
  + `(see REPORT.md), the stability/overthinking contrast does not`;

  html, body { margin: 0; }
  html { background: #f9f9f7; color-scheme: light dark; }
  @media (prefers-color-scheme: dark) { html { background: #0d0d0d; } }

```
ml_venv\Scripts\python.exe run_phase2.py          # 4 main models (resumable driver)
ml_venv\Scripts\python.exe run_phase3.py          # 2 random-budget models
ml_venv\Scripts\python.exe run_replicates.py      # 4 seed-1 replicates
ml_venv\Scripts\python.exe eval.py --run <name>   # per-run size × iters sweep
ml_venv\Scripts\python.exe analyze_dynamics.py --runs <r1,r2,...> [--out f.json]
ml_venv\Scripts\python.exe viz_export.py --runs p2_fixed_d16,p2_rec_plain_i20,p2_rec_recall_i20,p3_rec_plain_rand,p3_rec_recall_rand
ml_venv\Scripts\python.exe build_viewer.py --out maze_viewer.html
```

Caveat on checkpoint selection: phase-2 runs predate best-checkpoint saving and are
evaluated at their (converged, stable) final epoch; phase-3 and replicate runs are
evaluated at their best-validation epoch (selected at T=20 on 13×13). Training-size
@T=20 numbers are therefore mildly optimistic for the latter group; the long-horizon
and extrapolation comparisons are unaffected in direction (the selection criterion
rewards neither).

Training-size
@T=20 numbers are therefore mildly optimistic for the latter group, and the
last-vs-best asymmetry between the groups is a residual confound — though the
selection criterion (val solved at T=20) rewards neither long-horizon stability nor
extrapolation, so it should not manufacture those contrasts.

1. **No model converges to a fixed point** — every state keeps growing (the fixed-T
   models multiply their state norm by ~1.26–1.38 per step, forever). Yet the *readout*
   can be essentially stable: the fixed-T plain net's pixel flip rate falls to exactly
   zero at every size in seed 1, and at 13×13 and 33×33 in seed 0 (the seed-0 21×21
   probe still flips ~0.02% of pixels per step at T=200). At training size — the
   load-bearing case — it freezes on the *correct* answer in both seeds: its state
   direction stabilizes while the magnitude explodes (a "sign-stable divergence"), so
   the thresholded logits stop changing.

3. **State-space calm does not mean answer stability.** In both seeds the random-budget
   models have by far the most sedate states (relative step change 0.01–0.09, norms
   ~20 orders of magnitude smaller — the exception being seed-0 recall+random at 33×33,
   whose state blows up after ~step 150, see footnote) and still lose the answer. The
   stability that matters is stability of the readout, and it is not purchased by slow
   dynamics.

(Note the recurrent nets have **roughly 8–15× fewer parameters** than the CNNs they
beat — capacity is in iteration, not in weights.)

target mask on **every pixel**. (Pixel accuracy is uninformative — ~87–90% of pixels
are background depending on size; the all-zeros predictor gets ~0.87 at training size.)

Weight-tied recurrent nets really can trade test-time compute for problem size: trained
on 13×13 mazes with a 20-step budget, they solve substantially larger mazes only when
run past that budget (peaks at 30–100 steps, both seeds), while fixed-depth CNNs with
roughly 8–15× more parameters collapse outside the training size. *How far* that
extrapolation reaches is a lottery across training runs (41% vs 7% of 33×33 mazes for
identical configs at different seeds); *whether the answer survives long rollouts* was
not, in our runs: in every one, only the plain (no-recall) net trained at a fixed
budget holds its answer at 200 steps — every other variant "overthinks" its way from
~100% to 4–53%, and randomizing the training budget does not fix this (contradicting
the intuition we took from the deep-thinking literature, where input-recall is the
stabilizer). Mechanistically, no model converges to a fixed point: the fixed-budget
nets' states grow ~1.3×/step forever, yet the stable model's *output* freezes on the
right answer (a sign-stable divergence, both seeds); overthinkers either freeze on
wrong answers or keep churning; and the random-budget models' much calmer,
slower-growing states offer no protection at all.

- *Plain + fixed budget is the only long-horizon-stable configuration we trained*:
  ≥94% at T=200 on training-size mazes in both seeds; every other config decays to
  4–53%.

The *recall* net trains
smoothly under the same regime — no collapses, 92–100% from epoch 7 (0-based) on.

| Recall, random T | 7.8 | 4.0 | 5.0 | 2.6 |

    # data streams live at 1000+seed (train), 2000+seed (val), 3000 (test, eval.py);
    # large seeds would alias one stream onto another
    assert 0 <= args.seed < 1000, "seed must be < 1000 to keep data streams disjoint"
    torch.manual_seed(args.seed)
    out = ROOT / "runs" / args.out

    if evals and len(evals) == len(data["runs"]):
        data["eval"] = evals
    cells = {json.loads((ROOT / "runs" / r["id"] / "log.json").read_text())["config"]["cells"]
             for r in data["runs"]}
    assert len(cells) == 1, f"runs trained at different sizes: {cells}"
    data["trainCells"] = cells.pop()

- Final viz: maze_viewer.html built & verified (canvas pixel colors, slider, charts,
  tooltip values vs eval.json, dark mode — incl. fixing body-bg var scoping bug).
  Note: browser-pane SCREENSHOTS time out persistently this session (not CPU-related);
  all visual verification done via javascript_tool pixel sampling.
- REVIEWS: BOTH DONE (independent subagents).
  - Writeup review: 0 critical; ~90 recomputed numbers all matched; 1 major
    (overgeneralized "flip rate exactly zero at every size both seeds" — false at
    s0 c=10, ~0.02%/step residual) + 6 minor wording items. ALL FIXED in REPORT.md.
  - Code review: verdict sound; empirically verified maze tree property, hash
    disjointness (incl. val↔test = 0 overlap both seeds), uint8-vs-logit threshold
    equivalence, sweep-indexing bitwise-identical to fresh forwards, eval.json
    reproduces from checkpoints. 1 major: replication-table cell recall-rand s1
    c=16 peak was 0.4, truth 2.6@30 — FIXED. Minor suggestions implemented:
    train.py seed<1000 assert, build_viewer.py reads trainCells from configs.
    Not implemented (noted, no contamination found): test excl. train∪val by
    construction; unordered start/goal hash.
- ROUND COMPLETE pending David's sign-off. Deliverables: REPORT.md,
  maze_viewer.html (self-contained; open in any browser), preview at
  http://localhost:8731/maze_viewer.html while server runs.

[auto-memory content redacted for sharing]

torch.set_num_threads(int(os.environ.get("MAZE_TORCH_THREADS", "12")))

import argparse
import json
import os
import time
from pathlib import Path

Requirements: Python 3.10+, `torch` (CPU is fine), `numpy`. Everything below runs on a
desktop CPU; the longest driver is ~1.5 h (thread count via `MAZE_TORCH_THREADS`,
default 12).

```
python run_phase2.py          # 4 main models (resumable driver)
python run_phase3.py          # 2 random-budget models
python run_replicates.py      # 4 seed-1 replicates
python eval.py --run <name>   # per-run size × iters sweep
python analyze_dynamics.py --runs <r1,r2,...> [--out f.json]
python viz_export.py --runs p2_fixed_d16,p2_rec_plain_i20,p2_rec_recall_i20,p3_rec_plain_rand,p3_rec_recall_rand
python build_viewer.py --out maze_viewer.html
```

Notes: drivers run models sequentially and skip finished runs, so they can simply be
re-launched after an interruption. Thread count: `MAZE_TORCH_THREADS` (default 12).

## Independent review

Two independent model instances audited this work after completion; their findings were
applied before release.

- **Code audit** (adversarial review of the pipeline; empirically re-verified the maze
  generator's tree property on generated mazes, hash disjointness of all cached splits
  including val↔test, equivalence of the viewer's uint8 threshold with the eval
  criterion, bitwise identity of the one-rollout iteration sweep with independent
  forward passes, and exact reproduction of eval.json entries from shipped
  checkpoints). Verdict: sound. One major finding — a transcription error in the
  replication table (2.6 was written as 0.4) — plus minor robustness suggestions;
  fixed/implemented.
- **Writeup audit** (recomputed ~90 numbers across all tables and prose from the raw
  JSONs; checked argmaxes, epoch trajectories, retraction consistency, and literature
  framing). Verdict: publishable-as-honest after fixes. Zero table errors found; one
  major finding — an overgeneralized flip-rate claim, contradicted at one seed/size by
  our own dynamics data — plus six wording-level items; all fixed.

Residual limitations remain as stated: n=2 seeds, checkpoint-selection asymmetry
between phase 2 and phase 3 (disclosed below), and no test of Bansal et al.'s
incremental-progress loss.

## Reproduction details

Caveat on checkpoint selection: phase-2 runs predate best-checkpoint saving and are

## Independent review

*(This section comes last in reading order — see the checkpoint caveat under
Reproduction details below first if you are re-deriving numbers.)*

Two independent model instances audited this work after completion; their findings were
applied before release.

Caveat on checkpoint selection: phase-2 runs predate best-checkpoint saving and are
evaluated at their (converged, stable) final epoch; phase-3 and replicate runs are
evaluated at their best-validation epoch (selected at T=20 on 13×13). Training-size
@T=20 numbers are therefore mildly optimistic for the latter group, and the
last-vs-best asymmetry between the groups is a residual confound — though the
selection criterion (val solved at T=20) rewards neither long-horizon stability nor
extrapolation, so it should not manufacture those contrasts.

## Independent review

Two independent model instances audited this work after completion; their findings were
applied before release.

- **Code audit** (adversarial review of the pipeline; empirically re-verified the maze
  generator's tree property on generated mazes, hash disjointness of all cached splits
  including val↔test, equivalence of the viewer's uint8 threshold with the eval
  criterion, bitwise identity of the one-rollout iteration sweep with independent
  forward passes, and exact reproduction of eval.json entries from shipped
  checkpoints). Verdict: sound. One major finding — a transcription error in the
  replication table (2.6 was written as 0.4) — plus minor robustness suggestions;
  fixed/implemented.
- **Writeup audit** (recomputed ~90 numbers across all tables and prose from the raw
  JSONs; checked argmaxes, epoch trajectories, retraction consistency, and literature
  framing). Verdict: publishable-as-honest after fixes. Zero table errors found; one
  major finding — an overgeneralized flip-rate claim, contradicted at one seed/size by
  our own dynamics data — plus six wording-level items; all fixed.

Residual limitations remain as stated: n=2 seeds, checkpoint-selection asymmetry
between phase 2 and phase 3 (disclosed above), and no test of Bansal et al.'s
incremental-progress loss.

# regenerable artifacts and caches
data/
logs/
viz_data.json
_preview.html
__pycache__/

# throwaway pipeline-sanity runs and superseded checkpoints (logs of *_v1* runs are
# kept — they are the training-instability evidence cited in REPORT.md)
runs/sanity_*/
runs/*_v1*/model.pt

Code, experiments, and writeup by Claude (Anthropic), as an exercise in autonomous ML
research; overall direction and light feedback by the repository owner, David P.
Reichert. Independently audited by two further model instances (code + writeup); audit
scope and outcomes are summarized at the end of `REPORT.md`.

**👉 Live demo: [maze_viewer.html](https://davidpreichert.github.io/maze-thinking/maze_viewer.html)**
(via GitHub Pages) — or open the file [`maze_viewer.html`](maze_viewer.html) locally in
any browser (self-contained, no dependencies, ~1.7 MB — download the raw file).

*Free-choice round of the automating-ML-research experiment, 2026-07-12. Status: final
(twice independently audited — see Independent review at the end).*

*Authorship: code, experiments, analysis, and text by Claude (Anthropic); overall
direction and light feedback by David P. Reichert.*

[auto-memory content redacted for sharing]