From 57391da3f5491b074234f5a5caa2cd19411567a0 Mon Sep 17 00:00:00 2001
From: Cursor Agent <cursoragent@cursor.com>
Date: Sat, 25 Jul 2026 23:40:09 +0000
Subject: [PATCH 1/6] Add task-intrinsic reference points for token-guess
 metrics

Belief-probe R2 and greedy token accuracy were reported as bare numbers in
cycle 1. Both have floors and ceilings imposed by the task that are large
relative to the gaps between conditions, so the reported values cannot be read
as a measure of what a training objective induced.

Derive those references from the environment: the accuracy of an exact Bayesian
filter as a function of how many observations it may see, the R2 of the study's
own probe applied to the raw one-hot observations, and the R2 of the same probe
applied to a randomly initialised copy of the study transformer.

Co-authored-by: Alex Vardakostas <Al-does@users.noreply.github.com>
---
 .../mess3_token_guess_cycle_2/__init__.py     |   0
 .../metric_references.py                      | 342 ++++++++++++++++++
 .../references/__init__.py                    |   0
 .../references/experiment.py                  | 138 +++++++
 .../20260725T233938Z-70fe7dae/findings.md     |  49 +++
 .../20260725T233938Z-70fe7dae/references.json |  50 +++
 .../run_manifest.json                         |  75 ++++
 tests/test_mess3_token_guess_references.py    | 116 ++++++
 8 files changed, 770 insertions(+)
 create mode 100644 experiments/mess3_token_guess_cycle_2/__init__.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/metric_references.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/references/__init__.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/references/experiment.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/findings.md
 create mode 100644 experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/references.json
 create mode 100644 experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/run_manifest.json
 create mode 100644 tests/test_mess3_token_guess_references.py

diff --git a/experiments/mess3_token_guess_cycle_2/__init__.py b/experiments/mess3_token_guess_cycle_2/__init__.py
new file mode 100644
index 00000000..e69de29b
diff --git a/experiments/mess3_token_guess_cycle_2/metric_references.py b/experiments/mess3_token_guess_cycle_2/metric_references.py
new file mode 100644
index 00000000..d24a35ea
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/metric_references.py
@@ -0,0 +1,342 @@
+"""Task-intrinsic reference points for the passive MESS3 token-guess study.
+
+Cycle 1 reported belief-probe R² and greedy token accuracy as bare numbers. Both
+metrics have a task-imposed floor and ceiling that are large relative to the
+differences between conditions, so a bare number cannot be read as a measure of
+how much belief structure a training objective induced.
+
+This module derives those reference points from the environment definition and
+from the same probe estimator the conditions use, so that condition scores can be
+reported as a fraction of the range the metric can actually move through.
+
+Three references matter:
+
+``bayes_accuracy``
+    Greedy token accuracy of the exact Bayesian filter, as a function of how many
+    past tokens it is allowed to see. One token reproduces the trivial
+    "repeat the previous token" rule; the sequence converges well inside the
+    64-token context the agents receive.
+
+``raw_token_window_r2``
+    Belief-probe R² for an affine probe read directly off the one-hot encoded
+    last ``k`` observations. No network and no training are involved, so this is
+    the score a policy earns for passing its own inputs through unchanged.
+
+``untrained_module_r2``
+    The same probe applied to a randomly initialised copy of the study's
+    transformer. This controls for architecture, embedding width, and the final
+    LayerNorm, none of which the raw-token probe exercises.
+"""
+
+from __future__ import annotations
+
+from dataclasses import dataclass
+from typing import Any
+
+import numpy as np
+import torch
+
+from analysis.probes import r2_score
+from envs.hmm import HMMEnv
+from envs.mess3.model import passive_model
+from experiments.mess3_belief_geometry_2026_07.probe import (
+    collect_probe_data,
+    make_transducer_target,
+)
+from experiments.mess3_token_guess_cycle_1.analysis import (
+    PROBE_RANK,
+    fit_reduced_rank_affine,
+)
+from harness.seeding import named_seed_sequences, seed_sequence_to_int
+from learners.models.transformer import TransformerModel
+
+ALPHA = 0.85
+WARMUP = 64
+CONTEXT_LENGTHS = (1, 2, 3, 4, 6, 8, 16, 64)
+BOOTSTRAP_RESAMPLES = 400
+
+_STREAM_KEYS = {
+    "reference_fit": (300,),
+    "reference_test": (301,),
+    "untrained_fit": (302,),
+    "untrained_test": (303,),
+    "bootstrap": (304,),
+}
+
+
+@dataclass(frozen=True, slots=True)
+class TokenStream:
+    """Exact predictive beliefs aligned with the tokens they predict."""
+
+    beliefs: np.ndarray
+    tokens: np.ndarray
+    windows: np.ndarray
+
+
+def simulate_stream(*, n_steps: int, seed: int, window: int = WARMUP) -> TokenStream:
+    """Simulate passive MESS3 and record the exact predictive belief per step.
+
+    The belief is the distribution over the state that emits the token being
+    predicted, conditioned on every token revealed so far. This matches the
+    ``delay=1`` transducer target the conditions probe against, where the
+    filtering operator is ``diag(P(y|s)) @ T``.
+    """
+
+    if n_steps <= 0 or window <= 0:
+        raise ValueError("n_steps and window must be positive")
+    model = passive_model(alpha=ALPHA)
+    transition = np.asarray(model.transition_matrix, dtype=np.float64)
+    emission = np.asarray(model.emission_matrix, dtype=np.float64)
+    initial = np.asarray(model.initial_distribution, dtype=np.float64)
+
+    rng = np.random.default_rng(seed)
+    state = int(rng.choice(len(initial), p=initial))
+    belief = initial.copy()
+    history: list[int] = [0] * window
+
+    beliefs = np.empty((n_steps, len(initial)), dtype=np.float64)
+    tokens = np.empty(n_steps, dtype=np.int64)
+    windows = np.empty((n_steps, window), dtype=np.int64)
+
+    for step in range(n_steps + window):
+        token = int(rng.choice(emission.shape[1], p=emission[state]))
+        if step >= window:
+            index = step - window
+            beliefs[index] = belief
+            tokens[index] = token
+            windows[index] = history[-window:]
+        history.append(token)
+        posterior = belief * emission[:, token]
+        belief = (posterior / posterior.sum()) @ transition
+        state = int(rng.choice(transition.shape[1], p=transition[state]))
+    return TokenStream(beliefs=beliefs, tokens=tokens, windows=windows)
+
+
+def _one_hot_window(windows: np.ndarray, k: int) -> np.ndarray:
+    return np.eye(3, dtype=np.float64)[windows[:, -k:]].reshape(len(windows), -1)
+
+
+def _probe_r2(
+    fit_features: np.ndarray,
+    fit_targets: np.ndarray,
+    test_features: np.ndarray,
+    test_targets: np.ndarray,
+) -> tuple[float, np.ndarray]:
+    weight, bias = fit_reduced_rank_affine(
+        fit_features,
+        fit_targets,
+        rank=PROBE_RANK,
+    )
+    predicted = test_features @ weight + bias
+    return r2_score(predicted, test_targets), predicted
+
+
+def raw_token_window_r2(
+    fit: TokenStream,
+    test: TokenStream,
+    *,
+    context_lengths: tuple[int, ...] = CONTEXT_LENGTHS,
+) -> dict[int, float]:
+    """Score the study's probe against one-hot encoded raw observations."""
+
+    scores: dict[int, float] = {}
+    for k in context_lengths:
+        score, _ = _probe_r2(
+            _one_hot_window(fit.windows, k),
+            fit.beliefs,
+            _one_hot_window(test.windows, k),
+            test.beliefs,
+        )
+        scores[k] = float(score)
+    return scores
+
+
+def bootstrap_r2_interval(
+    fit: TokenStream,
+    test: TokenStream,
+    *,
+    context_length: int,
+    seed: int,
+    resamples: int = BOOTSTRAP_RESAMPLES,
+) -> tuple[float, float]:
+    """Bound the probe's own sampling noise at the conditions' test-set size."""
+
+    _, predicted = _probe_r2(
+        _one_hot_window(fit.windows, context_length),
+        fit.beliefs,
+        _one_hot_window(test.windows, context_length),
+        test.beliefs,
+    )
+    rng = np.random.default_rng(seed)
+    n = len(test.beliefs)
+    scores = [
+        r2_score(predicted[index], test.beliefs[index])
+        for index in (rng.integers(0, n, n) for _ in range(resamples))
+    ]
+    low, high = np.percentile(scores, [2.5, 97.5])
+    return float(low), float(high)
+
+
+def bayes_accuracy_by_context(
+    stream: TokenStream,
+    *,
+    context_lengths: tuple[int, ...] = CONTEXT_LENGTHS,
+) -> dict[int, float]:
+    """Greedy accuracy of an exact filter restricted to the last ``k`` tokens."""
+
+    model = passive_model(alpha=ALPHA)
+    transition = np.asarray(model.transition_matrix, dtype=np.float64)
+    emission = np.asarray(model.emission_matrix, dtype=np.float64)
+    initial = np.asarray(model.initial_distribution, dtype=np.float64)
+
+    accuracies: dict[int, float] = {}
+    for k in context_lengths:
+        belief = np.repeat(initial[None, :], len(stream.windows), axis=0)
+        for offset in range(k):
+            belief = belief * emission[:, stream.windows[:, -k + offset]].T
+            belief /= belief.sum(axis=1, keepdims=True)
+            belief = belief @ transition
+        predicted = (belief @ emission).argmax(axis=1)
+        accuracies[k] = float((predicted == stream.tokens).mean())
+    return accuracies
+
+
+def _build_untrained_module(env_config: dict[str, Any], model_config: dict[str, Any]):
+    from ray.rllib.core.rl_module.rl_module import RLModuleSpec
+
+    environment = HMMEnv(env_config)
+    try:
+        spec = RLModuleSpec(
+            module_class=TransformerModel,
+            model_config=dict(model_config),
+            observation_space=environment.observation_space,
+            action_space=environment.action_space,
+        )
+        return spec.build()
+    finally:
+        environment.close()
+
+
+def untrained_module_r2(
+    *,
+    env_config: dict[str, Any],
+    model_config: dict[str, Any],
+    seed: int,
+    fit_steps: int,
+    test_steps: int,
+    device: str = "cpu",
+) -> dict[str, float]:
+    """Probe a randomly initialised copy of the study transformer.
+
+    The module is never trained, so any score above the raw-token reference is
+    attributable to the architecture rather than to a learning objective.
+    """
+
+    streams = named_seed_sequences(seed, _STREAM_KEYS)
+    config = dict(env_config)
+    config["diagnostics"] = {
+        "state": True,
+        "belief": True,
+        "tokens": True,
+        "transitions": True,
+    }
+    torch.manual_seed(seed_sequence_to_int(streams["untrained_fit"], bits=64))
+    module = _build_untrained_module(config, model_config)
+
+    def make_environment():
+        return HMMEnv(config)
+
+    environment = make_environment()
+    try:
+        initial_belief, outcome_operator, initial_operator = make_transducer_target(
+            environment
+        )
+    finally:
+        environment.close()
+
+    common = {
+        "module": module,
+        "env_factory": make_environment,
+        "policy_mode": "greedy",
+        "device": device,
+        "warmup": WARMUP,
+        "initial_belief": initial_belief,
+        "action_outcome_operator": outcome_operator,
+        "initial_outcome_operator": initial_operator,
+    }
+    fit = collect_probe_data(
+        n_steps=fit_steps, seed=streams["untrained_fit"], **common
+    )
+    test = collect_probe_data(
+        n_steps=test_steps, seed=streams["untrained_test"], **common
+    )
+    score, _ = _probe_r2(fit.activations, fit.beliefs, test.activations, test.beliefs)
+    return {
+        "r_squared": float(score),
+        "token_accuracy_greedy": float(test.rewards.mean()),
+        "n_fit": int(len(fit.beliefs)),
+        "n_test": int(len(test.beliefs)),
+    }
+
+
+def compute_references(
+    *,
+    seed: int,
+    fit_steps: int,
+    test_steps: int,
+    env_config: dict[str, Any] | None = None,
+    model_config: dict[str, Any] | None = None,
+    device: str = "cpu",
+) -> dict[str, Any]:
+    """Compute every reference point the study needs to normalise its metrics."""
+
+    streams = named_seed_sequences(seed, _STREAM_KEYS)
+    fit = simulate_stream(
+        n_steps=fit_steps,
+        seed=seed_sequence_to_int(streams["reference_fit"]),
+    )
+    test = simulate_stream(
+        n_steps=test_steps,
+        seed=seed_sequence_to_int(streams["reference_test"]),
+    )
+    raw_r2 = raw_token_window_r2(fit, test)
+    saturated = max(raw_r2)
+    low, high = bootstrap_r2_interval(
+        fit,
+        test,
+        context_length=min(8, saturated),
+        seed=seed_sequence_to_int(streams["bootstrap"]),
+    )
+    accuracy = bayes_accuracy_by_context(test)
+    references: dict[str, Any] = {
+        "alpha": ALPHA,
+        "seed": seed,
+        "n_fit": fit_steps,
+        "n_test": test_steps,
+        "probe_rank": PROBE_RANK,
+        "raw_token_window_r2": {str(k): v for k, v in raw_r2.items()},
+        "belief_r2_floor": raw_r2[saturated],
+        "belief_r2_floor_context": saturated,
+        "belief_r2_probe_noise_95ci": [low, high],
+        "bayes_accuracy_by_context": {str(k): v for k, v in accuracy.items()},
+        "accuracy_floor_repeat_previous_token": accuracy[1],
+        "accuracy_ceiling_bayes": accuracy[max(accuracy)],
+    }
+    if env_config is not None and model_config is not None:
+        references["untrained_module"] = untrained_module_r2(
+            env_config=env_config,
+            model_config=model_config,
+            seed=seed,
+            fit_steps=fit_steps,
+            test_steps=test_steps,
+            device=device,
+        )
+    return references
+
+
+def normalise(value: float, *, floor: float, ceiling: float) -> float:
+    """Express a score as the fraction of the floor-to-ceiling range it covers."""
+
+    if ceiling <= floor:
+        raise ValueError("ceiling must exceed floor")
+    return (value - floor) / (ceiling - floor)
diff --git a/experiments/mess3_token_guess_cycle_2/references/__init__.py b/experiments/mess3_token_guess_cycle_2/references/__init__.py
new file mode 100644
index 00000000..e69de29b
diff --git a/experiments/mess3_token_guess_cycle_2/references/experiment.py b/experiments/mess3_token_guess_cycle_2/references/experiment.py
new file mode 100644
index 00000000..6f813a7b
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/references/experiment.py
@@ -0,0 +1,138 @@
+"""Record the task-intrinsic floors and ceilings for the token-guess metrics."""
+
+from __future__ import annotations
+
+from typing import Any
+
+from experiments.mess3_token_guess_cycle_1.comparison.experiment import (
+    BASE_MODEL_CONFIG,
+    ENV_CONFIG,
+)
+from experiments.mess3_token_guess_cycle_2.metric_references import (
+    compute_references,
+    normalise,
+)
+from harness.artifacts import RunArtifacts
+from harness.context import RunContext
+
+FIT_STEPS = 60_000
+TEST_STEPS = 30_000
+SMOKE_FIT_STEPS = 2_000
+SMOKE_TEST_STEPS = 1_000
+
+# Best reported cycle-1 scores, used only to illustrate where the published
+# numbers sit inside the range the metric can move through.
+CYCLE_1_SCORES = {
+    "comparison/reward_only": 0.8552,
+    "comparison/predictive_loss": 0.9319,
+    "comparison/max_entropy": 0.8558,
+    "iqn_value": 0.9760,
+    "kelly_cycle_2/correctness_iqn": 0.9857,
+    "kelly_cycle_3/conditional_decoupled_kelly_iqn": 0.9824,
+}
+# Supervised next-token replication, final-LayerNorm probe, seed 0. See
+# experiments/mess3_supervised/README.md.
+SUPERVISED_CEILING = 0.99888
+
+
+def _findings(references: dict[str, Any]) -> str:
+    floor = references["belief_r2_floor"]
+    context = references["belief_r2_floor_context"]
+    low, high = references["belief_r2_probe_noise_95ci"]
+    lines = [
+        "# Token-guess metric reference points",
+        "",
+        "## Belief-probe R²",
+        "",
+        f"An affine probe reading the one-hot encoded last {context} observations,",
+        "with no network and no training, already scores "
+        f"R² = {floor:.4f}.",
+        "The supervised next-token replication reaches "
+        f"{SUPERVISED_CEILING:.4f}.",
+        "Belief-probe R² therefore moves through a usable range of only "
+        f"{SUPERVISED_CEILING - floor:.4f}.",
+        "",
+        "| observations visible to the probe | R² |",
+        "|---:|---:|",
+    ]
+    for k, value in sorted(
+        references["raw_token_window_r2"].items(), key=lambda item: int(item[0])
+    ):
+        lines.append(f"| {k} | {value:.4f} |")
+    untrained = references.get("untrained_module")
+    if untrained is not None:
+        lines.extend(
+            [
+                "",
+                "A randomly initialised copy of the study transformer scores "
+                f"R² = {untrained['r_squared']:.4f} with greedy accuracy "
+                f"{untrained['token_accuracy_greedy']:.4f}.",
+            ]
+        )
+    lines.extend(
+        [
+            "",
+            "Bootstrap resampling of the probe's test set puts its own sampling "
+            f"noise at [{low:.4f}, {high:.4f}].",
+            "",
+            "## Where the cycle-1 scores sit",
+            "",
+            "| condition | reported R² | fraction of the floor-to-ceiling range |",
+            "|---|---:|---:|",
+        ]
+    )
+    for condition, value in CYCLE_1_SCORES.items():
+        fraction = normalise(value, floor=floor, ceiling=SUPERVISED_CEILING)
+        lines.append(f"| `{condition}` | {value:.4f} | {fraction:+.1%} |")
+
+    accuracy = references["bayes_accuracy_by_context"]
+    lines.extend(
+        [
+            "",
+            "## Greedy token accuracy",
+            "",
+            "| observations visible to an exact Bayesian filter | accuracy |",
+            "|---:|---:|",
+        ]
+    )
+    for k, value in sorted(accuracy.items(), key=lambda item: int(item[0])):
+        lines.append(f"| {k} | {value:.4f} |")
+    lines.extend(
+        [
+            "",
+            "One observation reproduces the trivial repeat-the-previous-token "
+            f"rule at {references['accuracy_floor_repeat_previous_token']:.4f}; "
+            "the filter saturates at "
+            f"{references['accuracy_ceiling_bayes']:.4f}. Greedy token accuracy "
+            "therefore moves through a usable range of only "
+            f"{references['accuracy_ceiling_bayes'] - references['accuracy_floor_repeat_previous_token']:.4f}.",
+            "",
+        ]
+    )
+    return "\n".join(lines)
+
+
+def run(context: RunContext) -> dict[str, Any]:
+    if context.seed is None:
+        raise ValueError("the reference computation requires a resolved seed")
+    outputs = RunArtifacts.from_context(context)
+    outputs.prepare()
+    references = compute_references(
+        seed=context.seed,
+        fit_steps=SMOKE_FIT_STEPS if context.smoke else FIT_STEPS,
+        test_steps=SMOKE_TEST_STEPS if context.smoke else TEST_STEPS,
+        env_config=ENV_CONFIG,
+        model_config=BASE_MODEL_CONFIG,
+    )
+    references["supervised_ceiling"] = SUPERVISED_CEILING
+    references["cycle_1_normalised"] = {
+        condition: normalise(
+            value,
+            floor=references["belief_r2_floor"],
+            ceiling=SUPERVISED_CEILING,
+        )
+        for condition, value in CYCLE_1_SCORES.items()
+    }
+    outputs.write_json("references.json", references)
+    (context.results_dir / "findings.md").write_text(_findings(references))
+    return references
diff --git a/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/findings.md b/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/findings.md
new file mode 100644
index 00000000..cffc4761
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/findings.md
@@ -0,0 +1,49 @@
+# Token-guess metric reference points
+
+## Belief-probe R²
+
+An affine probe reading the one-hot encoded last 64 observations,
+with no network and no training, already scores R² = 0.9668.
+The supervised next-token replication reaches 0.9989.
+Belief-probe R² therefore moves through a usable range of only 0.0321.
+
+| observations visible to the probe | R² |
+|---:|---:|
+| 1 | 0.8043 |
+| 2 | 0.9302 |
+| 3 | 0.9581 |
+| 4 | 0.9648 |
+| 6 | 0.9667 |
+| 8 | 0.9669 |
+| 16 | 0.9669 |
+| 64 | 0.9668 |
+
+A randomly initialised copy of the study transformer scores R² = 0.8733 with greedy accuracy 0.3412.
+
+Bootstrap resampling of the probe's test set puts its own sampling noise at [0.9664, 0.9673].
+
+## Where the cycle-1 scores sit
+
+| condition | reported R² | fraction of the floor-to-ceiling range |
+|---|---:|---:|
+| `comparison/reward_only` | 0.8552 | -347.6% |
+| `comparison/predictive_loss` | 0.9319 | -108.6% |
+| `comparison/max_entropy` | 0.8558 | -345.7% |
+| `iqn_value` | 0.9760 | +28.7% |
+| `kelly_cycle_2/correctness_iqn` | 0.9857 | +58.9% |
+| `kelly_cycle_3/conditional_decoupled_kelly_iqn` | 0.9824 | +48.7% |
+
+## Greedy token accuracy
+
+| observations visible to an exact Bayesian filter | accuracy |
+|---:|---:|
+| 1 | 0.6732 |
+| 2 | 0.6732 |
+| 3 | 0.6859 |
+| 4 | 0.6860 |
+| 6 | 0.6879 |
+| 8 | 0.6884 |
+| 16 | 0.6883 |
+| 64 | 0.6883 |
+
+One observation reproduces the trivial repeat-the-previous-token rule at 0.6732; the filter saturates at 0.6883. Greedy token accuracy therefore moves through a usable range of only 0.0151.
diff --git a/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/references.json b/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/references.json
new file mode 100644
index 00000000..5188ecba
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/references.json
@@ -0,0 +1,50 @@
+{
+  "accuracy_ceiling_bayes": 0.6883,
+  "accuracy_floor_repeat_previous_token": 0.6732,
+  "alpha": 0.85,
+  "bayes_accuracy_by_context": {
+    "1": 0.6732,
+    "16": 0.6883,
+    "2": 0.6732,
+    "3": 0.6859,
+    "4": 0.686,
+    "6": 0.6879333333333333,
+    "64": 0.6883,
+    "8": 0.6883666666666667
+  },
+  "belief_r2_floor": 0.9667781106841088,
+  "belief_r2_floor_context": 64,
+  "belief_r2_probe_noise_95ci": [
+    0.966386807711707,
+    0.9672839724278931
+  ],
+  "cycle_1_normalised": {
+    "comparison/max_entropy": -3.4570585423198747,
+    "comparison/predictive_loss": -1.0864815569232977,
+    "comparison/reward_only": -3.4757490310352237,
+    "iqn_value": 0.287269363654747,
+    "kelly_cycle_2/correctness_iqn": 0.5894322645528668,
+    "kelly_cycle_3/conditional_decoupled_kelly_iqn": 0.48663457661845627
+  },
+  "n_fit": 60000,
+  "n_test": 30000,
+  "probe_rank": 2,
+  "raw_token_window_r2": {
+    "1": 0.8043424580022087,
+    "16": 0.966855218046489,
+    "2": 0.9301666406047294,
+    "3": 0.9580682184473358,
+    "4": 0.9647961628967233,
+    "6": 0.9667392183948693,
+    "64": 0.9667781106841088,
+    "8": 0.9668577324292733
+  },
+  "seed": 42,
+  "supervised_ceiling": 0.99888,
+  "untrained_module": {
+    "n_fit": 60000,
+    "n_test": 30000,
+    "r_squared": 0.8733171203578343,
+    "token_accuracy_greedy": 0.34123333333333333
+  }
+}
diff --git a/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/run_manifest.json b/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/run_manifest.json
new file mode 100644
index 00000000..dd6f4c82
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/run_manifest.json
@@ -0,0 +1,75 @@
+{
+  "command": [
+    "/workspace/.venv/bin/rl-harness",
+    "experiments.mess3_token_guess_cycle_2.references.experiment",
+    "--no-upload-artifacts"
+  ],
+  "dependency_lock": {
+    "file": "/workspace/uv.lock",
+    "sha256": "986725e46af5252423583fe947d30b2646f541a4863acb8c51d578a7ca1b8220"
+  },
+  "ended_at": "2026-07-25T23:39:59.835334+00:00",
+  "error": null,
+  "experiment": {
+    "file": "/workspace/experiments/mess3_token_guess_cycle_2/references/experiment.py",
+    "module": "experiments.mess3_token_guess_cycle_2.references.experiment",
+    "source_sha256": "6706a930ef401113ce0d8e1d6572cc0960d4fd925f52474975219acf8273f7ac"
+  },
+  "framework_versions": {
+    "gymnasium": "1.2.2",
+    "numpy": "2.5.1",
+    "python": "3.14.6",
+    "ray": "2.56.0",
+    "rl-harness": "0.1.0",
+    "torch": "2.12.1"
+  },
+  "git": {
+    "commit": "0f798d66356f37e1d126a163a0f985cbcbd7bb54",
+    "dirty": true,
+    "experiment_repository": {
+      "commit": "0f798d66356f37e1d126a163a0f985cbcbd7bb54",
+      "dirty": true,
+      "root": "/workspace"
+    },
+    "library": {
+      "commit": "7c231655be83affb51c1d24d8cfd475b5d425eb5",
+      "dirty": false,
+      "package": "rl-harness",
+      "root": "/rl-harness-src",
+      "version": "0.1.0"
+    }
+  },
+  "hardware": {
+    "cpu_count": 4,
+    "cuda_available": false,
+    "machine": "x86_64",
+    "mps_available": false,
+    "platform": "Linux-6.12.94+-x86_64-with-glibc2.39",
+    "profile": {
+      "learner_device": "cpu",
+      "name": "cpu",
+      "num_env_runners": null,
+      "num_envs_per_env_runner": 24,
+      "num_gpus_per_env_runner": 0.0,
+      "torch_threads": null
+    }
+  },
+  "run_id": "20260725T233938Z-70fe7dae",
+  "runtime": {
+    "artifacts_dir": "/workspace/experiments/mess3_token_guess_cycle_2/references/artifacts/20260725T233938Z-70fe7dae",
+    "overrides": {
+      "hardware_profile": "auto",
+      "resume_from": null,
+      "seed": 42,
+      "smoke": false,
+      "upload_artifacts": false
+    },
+    "results_dir": "/workspace/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae",
+    "resume_from": null,
+    "seed": 42,
+    "smoke": false
+  },
+  "schema_version": 2,
+  "started_at": "2026-07-25T23:39:38.286533+00:00",
+  "status": "completed"
+}
diff --git a/tests/test_mess3_token_guess_references.py b/tests/test_mess3_token_guess_references.py
new file mode 100644
index 00000000..995ec387
--- /dev/null
+++ b/tests/test_mess3_token_guess_references.py
@@ -0,0 +1,116 @@
+"""Tests for the token-guess metric reference points."""
+
+from __future__ import annotations
+
+import numpy as np
+import pytest
+
+from envs.mess3.model import passive_model
+from experiments.mess3_token_guess_cycle_1.comparison.experiment import (
+    BASE_MODEL_CONFIG,
+    ENV_CONFIG,
+)
+from experiments.mess3_token_guess_cycle_2.metric_references import (
+    ALPHA,
+    bayes_accuracy_by_context,
+    compute_references,
+    normalise,
+    raw_token_window_r2,
+    simulate_stream,
+    untrained_module_r2,
+)
+
+
+def test_simulated_belief_matches_the_delay_one_transducer_update():
+    model = passive_model(alpha=ALPHA)
+    transition = np.asarray(model.transition_matrix)
+    emission = np.asarray(model.emission_matrix)
+    stream = simulate_stream(n_steps=64, seed=3, window=8)
+
+    for index in range(len(stream.beliefs) - 1):
+        belief = stream.beliefs[index]
+        token = stream.tokens[index]
+        # delay=1 composes the observation operator before the transition, so
+        # the update is diag(P(y|s)) @ T applied in row-vector convention.
+        operator = np.diag(emission[:, token]) @ transition
+        expected = belief @ operator
+        expected /= expected.sum()
+        np.testing.assert_allclose(stream.beliefs[index + 1], expected, atol=1e-12)
+
+
+def test_simulated_beliefs_are_probability_vectors():
+    stream = simulate_stream(n_steps=256, seed=11, window=8)
+    np.testing.assert_allclose(stream.beliefs.sum(axis=1), 1.0, atol=1e-12)
+    assert (stream.beliefs >= 0.0).all()
+
+
+def test_raw_token_probe_saturates_well_below_the_supervised_ceiling():
+    fit = simulate_stream(n_steps=20_000, seed=100)
+    test = simulate_stream(n_steps=10_000, seed=101)
+    scores = raw_token_window_r2(fit, test, context_lengths=(1, 2, 4, 8, 16))
+
+    # More observations never hurt an affine probe by a meaningful margin.
+    ordered = [scores[k] for k in (1, 2, 4, 8, 16)]
+    assert all(later >= earlier - 5e-3 for earlier, later in zip(ordered, ordered[1:]))
+
+    # A single observation is already most of the way there, and the window
+    # saturates by eight observations, short of the supervised 0.9989.
+    assert 0.78 < scores[1] < 0.83
+    assert scores[8] == pytest.approx(scores[16], abs=5e-3)
+    assert 0.95 < scores[8] < 0.98
+
+
+def test_raw_token_floor_exceeds_several_published_cycle_1_scores():
+    fit = simulate_stream(n_steps=20_000, seed=100)
+    test = simulate_stream(n_steps=10_000, seed=101)
+    floor = raw_token_window_r2(fit, test, context_lengths=(8,))[8]
+
+    # Cycle 1 reported these for reward-only, max-entropy, and predictive-loss
+    # PPO. A probe on the untransformed observations beats all three.
+    for reported in (0.8552, 0.8558, 0.9319):
+        assert reported < floor
+
+
+def test_bayes_accuracy_spans_a_narrow_band_above_repeat_previous_token():
+    stream = simulate_stream(n_steps=40_000, seed=102)
+    accuracy = bayes_accuracy_by_context(stream, context_lengths=(1, 2, 4, 8, 64))
+
+    # One observation is the repeat-the-previous-token rule.
+    assert 0.66 < accuracy[1] < 0.69
+    assert accuracy[8] == pytest.approx(accuracy[64], abs=5e-3)
+    assert accuracy[64] - accuracy[1] < 0.03
+
+
+def test_normalise_reports_position_within_the_usable_range():
+    assert normalise(0.9670, floor=0.9670, ceiling=0.9989) == pytest.approx(0.0)
+    assert normalise(0.9989, floor=0.9670, ceiling=0.9989) == pytest.approx(1.0)
+    assert normalise(0.8552, floor=0.9670, ceiling=0.9989) < 0.0
+    with pytest.raises(ValueError):
+        normalise(0.5, floor=0.9, ceiling=0.9)
+
+
+def test_untrained_module_is_probed_with_the_study_architecture():
+    result = untrained_module_r2(
+        env_config=ENV_CONFIG,
+        model_config=BASE_MODEL_CONFIG,
+        seed=5,
+        fit_steps=2_000,
+        test_steps=1_000,
+    )
+    assert result["n_fit"] == 2_000
+    assert result["n_test"] == 1_000
+    assert -1.0 <= result["r_squared"] <= 1.0
+    assert 0.0 <= result["token_accuracy_greedy"] <= 1.0
+
+
+def test_compute_references_reports_a_floor_below_the_supervised_ceiling():
+    references = compute_references(seed=7, fit_steps=8_000, test_steps=4_000)
+    floor = references["belief_r2_floor"]
+    low, high = references["belief_r2_probe_noise_95ci"]
+
+    assert 0.94 < floor < 0.99
+    assert low < floor < high
+    assert references["accuracy_floor_repeat_previous_token"] < (
+        references["accuracy_ceiling_bayes"]
+    )
+    assert "untrained_module" not in references

From 1f18c7ec112fccfaf3a3eada323d1afa0dceda6b Mon Sep 17 00:00:00 2001
From: Cursor Agent <cursoragent@cursor.com>
Date: Sat, 25 Jul 2026 23:44:20 +0000
Subject: [PATCH 2/6] Re-analyse the committed multi-seed results with
 seed-aware statistics

The Kelly cycles reported mean plus-or-minus a population standard deviation
over three seeds and drew orderings from it. Recomputed as a t-based interval
on the mean, and paired across the seeds each pair of arms shares, none of the
34 published pairwise orderings survives a Holm correction, including the
headline PPO-versus-IQN gap.

Add the aggregation the study should have used, and an analysis leaf that
regenerates the audit from the committed results so the claim is checkable.

Co-authored-by: Alex Vardakostas <Al-does@users.noreply.github.com>
---
 .../audit/__init__.py                         |   0
 .../audit/experiment.py                       | 200 ++++++
 .../20260725T234419Z-36e12c96/audit.json      | 644 ++++++++++++++++++
 .../20260725T234419Z-36e12c96/findings.md     |  69 ++
 .../run_manifest.json                         |  75 ++
 .../mess3_token_guess_cycle_2/statistics.py   | 204 ++++++
 tests/test_mess3_token_guess_audit.py         |  68 ++
 tests/test_mess3_token_guess_statistics.py    | 104 +++
 8 files changed, 1364 insertions(+)
 create mode 100644 experiments/mess3_token_guess_cycle_2/audit/__init__.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/audit/experiment.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/audit.json
 create mode 100644 experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/findings.md
 create mode 100644 experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/run_manifest.json
 create mode 100644 experiments/mess3_token_guess_cycle_2/statistics.py
 create mode 100644 tests/test_mess3_token_guess_audit.py
 create mode 100644 tests/test_mess3_token_guess_statistics.py

diff --git a/experiments/mess3_token_guess_cycle_2/audit/__init__.py b/experiments/mess3_token_guess_cycle_2/audit/__init__.py
new file mode 100644
index 00000000..e69de29b
diff --git a/experiments/mess3_token_guess_cycle_2/audit/experiment.py b/experiments/mess3_token_guess_cycle_2/audit/experiment.py
new file mode 100644
index 00000000..dc5db0df
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/audit/experiment.py
@@ -0,0 +1,200 @@
+"""Re-analyse the committed multi-seed token-guess results.
+
+This reads the results already in the repository and re-reports them against the
+task's own floors and ceilings, with intervals that account for the three seeds
+they were measured on. It trains nothing. Its purpose is to establish which of
+the published orderings the existing evidence actually supports, so that cycle 2
+can be scoped to the questions that remain open.
+"""
+
+from __future__ import annotations
+
+import glob
+import json
+from collections import defaultdict
+from pathlib import Path
+from typing import Any
+
+from experiments.mess3_token_guess_cycle_2.statistics import (
+    compare,
+    holm_adjust,
+    summarise,
+)
+from harness.artifacts import RunArtifacts
+from harness.context import RunContext
+
+STUDIES = ("mess_3_kelly_cycle_2", "mess_3_kelly_cycle_3")
+METRICS = ("r_squared", "token_accuracy_greedy")
+
+
+def _references(experiment_dir: Path) -> dict[str, float]:
+    """Load the most recent reference run, or fall back to recorded values."""
+
+    candidates = sorted(
+        (experiment_dir.parent / "references" / "results").glob("*/references.json")
+    )
+    if candidates:
+        data = json.loads(candidates[-1].read_text())
+        return {
+            "belief_r2_floor": float(data["belief_r2_floor"]),
+            "belief_r2_ceiling": float(data["supervised_ceiling"]),
+            "accuracy_floor": float(data["accuracy_floor_repeat_previous_token"]),
+            "accuracy_ceiling": float(data["accuracy_ceiling_bayes"]),
+        }
+    return {
+        "belief_r2_floor": 0.9668,
+        "belief_r2_ceiling": 0.99888,
+        "accuracy_floor": 0.6732,
+        "accuracy_ceiling": 0.6883,
+    }
+
+
+def collect_study(repository_root: Path, study: str) -> dict[str, dict[int, dict[str, float]]]:
+    """Read every ``condition_summary.json`` a study committed, keyed by seed."""
+
+    per_arm: dict[str, dict[int, dict[str, float]]] = defaultdict(dict)
+    pattern = str(repository_root / "experiments" / study / "*" / "results" / "*" / "condition_summary.json")
+    for path in sorted(glob.glob(pattern)):
+        summary = json.loads(Path(path).read_text())
+        probe = summary.get("probe") or {}
+        seed = summary.get("seed")
+        if seed is None or "r_squared" not in probe:
+            continue
+        arm = Path(path).parents[2].name
+        per_arm[arm][int(seed)] = {
+            metric: float(probe[metric]) for metric in METRICS if metric in probe
+        }
+    return dict(per_arm)
+
+
+def analyse_study(
+    per_arm: dict[str, dict[int, dict[str, float]]],
+    references: dict[str, float],
+) -> dict[str, Any]:
+    """Summarise each arm and test every pairwise ordering within the study."""
+
+    floor = references["belief_r2_floor"]
+    ceiling = references["belief_r2_ceiling"]
+    conditions: dict[str, Any] = {}
+    r2_by_arm: dict[str, dict[int, float]] = {}
+    for arm, by_seed in sorted(per_arm.items()):
+        values = [by_seed[seed]["r_squared"] for seed in sorted(by_seed)]
+        if len(values) < 2:
+            continue
+        estimate = summarise(values)
+        r2_by_arm[arm] = {seed: by_seed[seed]["r_squared"] for seed in by_seed}
+        conditions[arm] = {
+            "seeds": sorted(by_seed),
+            "r_squared_mean": estimate.mean,
+            "r_squared_sample_sd": estimate.sample_sd,
+            "r_squared_ci": [estimate.ci_low, estimate.ci_high],
+            "fraction_of_usable_range": (estimate.mean - floor) / (ceiling - floor),
+            "exceeds_no_network_floor": bool(estimate.ci_low > floor),
+        }
+
+    comparisons: dict[str, Any] = {}
+    arms = sorted(r2_by_arm)
+    raw_p: dict[str, float] = {}
+    for index, left in enumerate(arms):
+        for right in arms[index + 1:]:
+            name = f"{left} vs {right}"
+            result = compare(r2_by_arm[left], r2_by_arm[right])
+            raw_p[name] = result.p_value
+            comparisons[name] = {
+                "difference": result.difference,
+                "difference_ci": [result.ci_low, result.ci_high],
+                "paired_sample_sd": result.sample_sd,
+                "n_shared_seeds": result.n,
+                "p_value": result.p_value,
+                "seeds_for_80_percent_power": result.seeds_for_power,
+            }
+    for name, adjusted in holm_adjust(raw_p).items():
+        comparisons[name]["p_value_holm"] = adjusted
+        comparisons[name]["resolved"] = bool(
+            adjusted < 0.05 and comparisons[name]["difference_ci"][0] > 0.0
+        ) or bool(adjusted < 0.05 and comparisons[name]["difference_ci"][1] < 0.0)
+    return {"conditions": conditions, "comparisons": comparisons}
+
+
+def _findings(analysis: dict[str, Any], references: dict[str, float]) -> str:
+    floor = references["belief_r2_floor"]
+    ceiling = references["belief_r2_ceiling"]
+    lines = [
+        "# Audit of the committed multi-seed token-guess results",
+        "",
+        "Belief-probe R² is reported against the range it can move through: 0% is "
+        f"an affine probe on the raw observations ({floor:.4f}) and 100% is the "
+        f"supervised next-token replication ({ceiling:.4f}).",
+        "",
+    ]
+    for study, result in analysis.items():
+        lines.extend(
+            [
+                f"## `{study}`",
+                "",
+                "| condition | R² | 95% CI | usable range | above floor |",
+                "|---|---:|---|---:|---|",
+            ]
+        )
+        ordered = sorted(
+            result["conditions"].items(),
+            key=lambda item: -item[1]["r_squared_mean"],
+        )
+        for arm, values in ordered:
+            low, high = values["r_squared_ci"]
+            above = "yes" if values["exceeds_no_network_floor"] else "no"
+            lines.append(
+                f"| `{arm}` | {values['r_squared_mean']:.4f} | "
+                f"[{low:.4f}, {high:.4f}] | "
+                f"{values['fraction_of_usable_range']:+.0%} | {above} |"
+            )
+        resolved = [
+            name
+            for name, values in result["comparisons"].items()
+            if values.get("resolved")
+        ]
+        total = len(result["comparisons"])
+        lines.extend(
+            [
+                "",
+                f"Of {total} pairwise orderings, {len(resolved)} survive a "
+                "Holm correction across the family.",
+                "",
+                "| comparison | difference | 95% CI | Holm p | seeds for 80% power |",
+                "|---|---:|---|---:|---:|",
+            ]
+        )
+        for name, values in sorted(
+            result["comparisons"].items(), key=lambda item: item[1]["p_value"]
+        ):
+            low, high = values["difference_ci"]
+            needed = values["seeds_for_80_percent_power"]
+            lines.append(
+                f"| {name} | {values['difference']:+.4f} | "
+                f"[{low:+.4f}, {high:+.4f}] | "
+                f"{values['p_value_holm']:.3f} | "
+                f"{needed if needed < 10_000 else '>10000'} |"
+            )
+        lines.append("")
+    return "\n".join(lines)
+
+
+def run(context: RunContext) -> dict[str, Any]:
+    outputs = RunArtifacts.from_context(context)
+    outputs.prepare()
+    repository_root = Path(__file__).parents[3]
+    references = _references(Path(__file__).parents[1])
+    analysis = {
+        study: analyse_study(
+            collect_study(repository_root, study),
+            references,
+        )
+        for study in STUDIES
+    }
+    analysis = {study: result for study, result in analysis.items() if result["conditions"]}
+    if not analysis:
+        raise RuntimeError("no committed multi-seed results were found to audit")
+    result = {"references": references, "studies": analysis}
+    outputs.write_json("audit.json", result)
+    (context.results_dir / "findings.md").write_text(_findings(analysis, references))
+    return result
diff --git a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/audit.json b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/audit.json
new file mode 100644
index 00000000..2895738d
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/audit.json
@@ -0,0 +1,644 @@
+{
+  "references": {
+    "accuracy_ceiling": 0.6883,
+    "accuracy_floor": 0.6732,
+    "belief_r2_ceiling": 0.99888,
+    "belief_r2_floor": 0.9668
+  },
+  "studies": {
+    "mess_3_kelly_cycle_2": {
+      "comparisons": {
+        "conditional_decoupled_kelly_iqn vs conditional_decoupled_kelly_mean": {
+          "difference": -0.006775144328574179,
+          "difference_ci": [
+            -0.046992707495339764,
+            0.03344241883819141
+          ],
+          "n_shared_seeds": 3,
+          "p_value": 0.54388406860352,
+          "p_value_holm": 1.0,
+          "paired_sample_sd": 0.016189747845512227,
+          "resolved": false,
+          "seeds_for_80_percent_power": 47
+        },
+        "conditional_decoupled_kelly_iqn vs correctness_iqn": {
+          "difference": -0.03657977905470411,
+          "difference_ci": [
+            -0.05634829673622678,
+            -0.016811261373181435
+          ],
+          "n_shared_seeds": 3,
+          "p_value": 0.01541211009914722,
+          "p_value_holm": 0.27741798178464994,
+          "paired_sample_sd": 0.007957899269438538,
+          "resolved": false,
+          "seeds_for_80_percent_power": 3
+        },
+        "conditional_decoupled_kelly_iqn vs correctness_mean": {
+          "difference": -0.0343501413875947,
+          "difference_ci": [
+            -0.05709634836414988,
+            -0.011603934411039518
+          ],
+          "n_shared_seeds": 3,
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+  }
+}
diff --git a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/findings.md b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/findings.md
new file mode 100644
index 00000000..88874326
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/findings.md
@@ -0,0 +1,69 @@
+# Audit of the committed multi-seed token-guess results
+
+Belief-probe R² is reported against the range it can move through: 0% is an affine probe on the raw observations (0.9668) and 100% is the supervised next-token replication (0.9989).
+
+## `mess_3_kelly_cycle_2`
+
+| condition | R² | 95% CI | usable range | above floor |
+|---|---:|---|---:|---|
+| `correctness_iqn` | 0.9857 | [0.9841, 0.9872] | +59% | yes |
+| `correctness_mean` | 0.9834 | [0.9790, 0.9879] | +52% | yes |
+| `decoupled_kelly_iqn` | 0.9572 | [0.9467, 0.9678] | -30% | no |
+| `conditional_decoupled_kelly_mean` | 0.9559 | [0.9330, 0.9787] | -34% | no |
+| `decoupled_kelly_mean` | 0.9529 | [0.9413, 0.9646] | -43% | no |
+| `conditional_decoupled_kelly_iqn` | 0.9491 | [0.9289, 0.9693] | -55% | no |
+| `coupled_kelly_iqn` | 0.9467 | [0.9283, 0.9651] | -63% | no |
+| `coupled_kelly_mean` | 0.9375 | [0.9232, 0.9518] | -91% | no |
+
+Of 28 pairwise orderings, 0 survive a Holm correction across the family.
+
+| comparison | difference | 95% CI | Holm p | seeds for 80% power |
+|---|---:|---|---:|---:|
+| coupled_kelly_mean vs decoupled_kelly_iqn | -0.0197 | [-0.0235, -0.0159] | 0.056 | 5 |
+| correctness_iqn vs coupled_kelly_mean | +0.0481 | [+0.0346, +0.0617] | 0.114 | 3 |
+| correctness_iqn vs decoupled_kelly_iqn | +0.0284 | [+0.0187, +0.0382] | 0.164 | 3 |
+| correctness_iqn vs decoupled_kelly_mean | +0.0327 | [+0.0211, +0.0444] | 0.170 | 4 |
+| correctness_mean vs coupled_kelly_mean | +0.0459 | [+0.0293, +0.0625] | 0.170 | 3 |
+| correctness_mean vs decoupled_kelly_mean | +0.0305 | [+0.0186, +0.0424] | 0.186 | 3 |
+| correctness_iqn vs coupled_kelly_iqn | +0.0390 | [+0.0221, +0.0559] | 0.220 | 4 |
+| coupled_kelly_mean vs decoupled_kelly_mean | -0.0154 | [-0.0229, -0.0079] | 0.263 | 3 |
+| correctness_mean vs decoupled_kelly_iqn | +0.0262 | [+0.0131, +0.0392] | 0.263 | 3 |
+| conditional_decoupled_kelly_mean vs coupled_kelly_mean | +0.0183 | [+0.0091, +0.0276] | 0.263 | 4 |
+| conditional_decoupled_kelly_iqn vs correctness_iqn | -0.0366 | [-0.0563, -0.0168] | 0.277 | 3 |
+| correctness_mean vs coupled_kelly_iqn | +0.0368 | [+0.0139, +0.0596] | 0.344 | 3 |
+| conditional_decoupled_kelly_iqn vs correctness_mean | -0.0344 | [-0.0571, -0.0116] | 0.366 | 3 |
+| conditional_decoupled_kelly_mean vs correctness_iqn | -0.0298 | [-0.0521, -0.0075] | 0.434 | 3 |
+| conditional_decoupled_kelly_mean vs correctness_mean | -0.0276 | [-0.0518, -0.0034] | 0.548 | 4 |
+| decoupled_kelly_iqn vs decoupled_kelly_mean | +0.0043 | [-0.0020, +0.0106] | 1.000 | 5 |
+| coupled_kelly_iqn vs decoupled_kelly_iqn | -0.0106 | [-0.0274, +0.0062] | 1.000 | 6 |
+| coupled_kelly_iqn vs coupled_kelly_mean | +0.0091 | [-0.0088, +0.0271] | 1.000 | 8 |
+| correctness_iqn vs correctness_mean | +0.0022 | [-0.0037, +0.0082] | 1.000 | 12 |
+| conditional_decoupled_kelly_iqn vs coupled_kelly_mean | +0.0116 | [-0.0194, +0.0425] | 1.000 | 12 |
+| conditional_decoupled_kelly_mean vs coupled_kelly_iqn | +0.0092 | [-0.0166, +0.0350] | 1.000 | 13 |
+| conditional_decoupled_kelly_iqn vs decoupled_kelly_iqn | -0.0082 | [-0.0359, +0.0195] | 1.000 | 17 |
+| coupled_kelly_iqn vs decoupled_kelly_mean | -0.0063 | [-0.0292, +0.0166] | 1.000 | 20 |
+| conditional_decoupled_kelly_mean vs decoupled_kelly_mean | +0.0029 | [-0.0096, +0.0155] | 1.000 | 26 |
+| conditional_decoupled_kelly_iqn vs conditional_decoupled_kelly_mean | -0.0068 | [-0.0470, +0.0334] | 1.000 | 47 |
+| conditional_decoupled_kelly_iqn vs decoupled_kelly_mean | -0.0038 | [-0.0352, +0.0275] | 1.000 | 87 |
+| conditional_decoupled_kelly_iqn vs coupled_kelly_iqn | +0.0024 | [-0.0174, +0.0222] | 1.000 | 88 |
+| conditional_decoupled_kelly_mean vs decoupled_kelly_iqn | -0.0014 | [-0.0141, +0.0114] | 1.000 | 109 |
+
+## `mess_3_kelly_cycle_3`
+
+| condition | R² | 95% CI | usable range | above floor |
+|---|---:|---|---:|---|
+| `conditional_decoupled_kelly_iqn` | 0.9824 | [0.9788, 0.9859] | +49% | yes |
+| `conditional_decoupled_kelly_mean` | 0.9717 | [0.9596, 0.9839] | +15% | no |
+| `iqn` | 0.9688 | [0.9413, 0.9963] | +6% | no |
+| `ppo` | 0.8742 | [0.7428, 1.0057] | -289% | no |
+
+Of 6 pairwise orderings, 0 survive a Holm correction across the family.
+
+| comparison | difference | 95% CI | Holm p | seeds for 80% power |
+|---|---:|---|---:|---:|
+| conditional_decoupled_kelly_iqn vs conditional_decoupled_kelly_mean | +0.0106 | [-0.0009, +0.0222] | 0.351 | 4 |
+| iqn vs ppo | +0.0946 | [-0.0167, +0.2059] | 0.351 | 5 |
+| conditional_decoupled_kelly_iqn vs ppo | +0.1081 | [-0.0198, +0.2361] | 0.351 | 5 |
+| conditional_decoupled_kelly_mean vs ppo | +0.0975 | [-0.0302, +0.2253] | 0.351 | 5 |
+| conditional_decoupled_kelly_iqn vs iqn | +0.0136 | [-0.0111, +0.0383] | 0.351 | 7 |
+| conditional_decoupled_kelly_mean vs iqn | +0.0029 | [-0.0304, +0.0363] | 0.740 | 165 |
diff --git a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/run_manifest.json b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/run_manifest.json
new file mode 100644
index 00000000..be370d8a
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/run_manifest.json
@@ -0,0 +1,75 @@
+{
+  "command": [
+    "/workspace/.venv/bin/rl-harness",
+    "experiments.mess3_token_guess_cycle_2.audit.experiment",
+    "--no-upload-artifacts"
+  ],
+  "dependency_lock": {
+    "file": "/workspace/uv.lock",
+    "sha256": "986725e46af5252423583fe947d30b2646f541a4863acb8c51d578a7ca1b8220"
+  },
+  "ended_at": "2026-07-25T23:44:20.373828+00:00",
+  "error": null,
+  "experiment": {
+    "file": "/workspace/experiments/mess3_token_guess_cycle_2/audit/experiment.py",
+    "module": "experiments.mess3_token_guess_cycle_2.audit.experiment",
+    "source_sha256": "22295905c9deb572d3b5f815992390f4f9a6dd4a23bb0afeb6ddc4c5150d885b"
+  },
+  "framework_versions": {
+    "gymnasium": "1.2.2",
+    "numpy": "2.5.1",
+    "python": "3.14.6",
+    "ray": "2.56.0",
+    "rl-harness": "0.1.0",
+    "torch": "2.12.1"
+  },
+  "git": {
+    "commit": "57391da3f5491b074234f5a5caa2cd19411567a0",
+    "dirty": true,
+    "experiment_repository": {
+      "commit": "57391da3f5491b074234f5a5caa2cd19411567a0",
+      "dirty": true,
+      "root": "/workspace"
+    },
+    "library": {
+      "commit": "7c231655be83affb51c1d24d8cfd475b5d425eb5",
+      "dirty": false,
+      "package": "rl-harness",
+      "root": "/rl-harness-src",
+      "version": "0.1.0"
+    }
+  },
+  "hardware": {
+    "cpu_count": 4,
+    "cuda_available": false,
+    "machine": "x86_64",
+    "mps_available": false,
+    "platform": "Linux-6.12.94+-x86_64-with-glibc2.39",
+    "profile": {
+      "learner_device": "cpu",
+      "name": "cpu",
+      "num_env_runners": null,
+      "num_envs_per_env_runner": 24,
+      "num_gpus_per_env_runner": 0.0,
+      "torch_threads": null
+    }
+  },
+  "run_id": "20260725T234419Z-36e12c96",
+  "runtime": {
+    "artifacts_dir": "/workspace/experiments/mess3_token_guess_cycle_2/audit/artifacts/20260725T234419Z-36e12c96",
+    "overrides": {
+      "hardware_profile": "auto",
+      "resume_from": null,
+      "seed": 42,
+      "smoke": false,
+      "upload_artifacts": false
+    },
+    "results_dir": "/workspace/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96",
+    "resume_from": null,
+    "seed": 42,
+    "smoke": false
+  },
+  "schema_version": 2,
+  "started_at": "2026-07-25T23:44:20.290986+00:00",
+  "status": "completed"
+}
diff --git a/experiments/mess3_token_guess_cycle_2/statistics.py b/experiments/mess3_token_guess_cycle_2/statistics.py
new file mode 100644
index 00000000..cd84a2a8
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/statistics.py
@@ -0,0 +1,204 @@
+"""Seed-level aggregation, intervals, and power for the token-guess study.
+
+Cycle 1 and the Kelly cycles reported ``mean ± std`` over three seeds. Two
+things go wrong with that. The ``±`` reads as an interval but is a population
+standard deviation, which for three samples sits about 18% below the sample
+standard deviation and roughly 3x below a 95% confidence interval on the mean.
+And no comparison was checked for whether three seeds could resolve it, so
+orderings were reported for gaps far smaller than the noise around them.
+
+This module supplies the aggregation the study should use instead: a sample
+standard deviation, a t-based confidence interval on the mean, a paired
+comparison between conditions that share seeds, and the seed count a given
+comparison would need before it is worth reporting an ordering.
+"""
+
+from __future__ import annotations
+
+from collections.abc import Mapping, Sequence
+from dataclasses import dataclass
+
+import numpy as np
+from scipy import stats
+
+CONFIDENCE = 0.95
+POWER = 0.80
+
+
+@dataclass(frozen=True, slots=True)
+class Estimate:
+    """A condition's score summarised across seeds."""
+
+    mean: float
+    sample_sd: float
+    n: int
+    ci_low: float
+    ci_high: float
+
+    @property
+    def half_width(self) -> float:
+        return (self.ci_high - self.ci_low) / 2.0
+
+
+@dataclass(frozen=True, slots=True)
+class Comparison:
+    """A paired difference between two conditions evaluated on shared seeds."""
+
+    difference: float
+    sample_sd: float
+    n: int
+    ci_low: float
+    ci_high: float
+    p_value: float
+    seeds_for_power: int
+
+    @property
+    def resolved(self) -> bool:
+        """Whether the interval excludes zero at the configured confidence."""
+
+        return self.ci_low > 0.0 or self.ci_high < 0.0
+
+
+def summarise(values: Sequence[float], *, confidence: float = CONFIDENCE) -> Estimate:
+    """Summarise one condition with a t-based interval on its mean."""
+
+    array = np.asarray(values, dtype=np.float64)
+    if array.ndim != 1 or len(array) < 2:
+        raise ValueError("summarising a condition needs at least two seeds")
+    n = len(array)
+    mean = float(array.mean())
+    sample_sd = float(array.std(ddof=1))
+    half = float(stats.t.ppf(0.5 + confidence / 2.0, n - 1)) * sample_sd / np.sqrt(n)
+    return Estimate(
+        mean=mean,
+        sample_sd=sample_sd,
+        n=n,
+        ci_low=mean - half,
+        ci_high=mean + half,
+    )
+
+
+MAX_SEEDS = 10_000
+
+
+def t_test_power(
+    n: int,
+    difference: float,
+    sample_sd: float,
+    *,
+    confidence: float = CONFIDENCE,
+    paired: bool = True,
+) -> float:
+    """Power of a two-sided t-test at ``n`` seeds per condition."""
+
+    if n < 2 or sample_sd <= 0.0:
+        raise ValueError("power needs at least two seeds and positive spread")
+    degrees = n - 1 if paired else 2 * (n - 1)
+    scale = np.sqrt(n) if paired else np.sqrt(n / 2.0)
+    ncp = abs(difference) / sample_sd * scale
+    critical = float(stats.t.ppf(0.5 + confidence / 2.0, degrees))
+    return float(
+        stats.nct.sf(critical, degrees, ncp) + stats.nct.cdf(-critical, degrees, ncp)
+    )
+
+
+def seeds_for_power(
+    difference: float,
+    sample_sd: float,
+    *,
+    power: float = POWER,
+    confidence: float = CONFIDENCE,
+    paired: bool = True,
+) -> int:
+    """Seeds per condition needed to resolve a difference of this size.
+
+    Solved against the noncentral t distribution rather than the usual normal
+    approximation. At the seed counts this study runs, the two disagree sharply:
+    a two-sided test on three seeds has only two degrees of freedom and a
+    critical value above four, so the normal approximation can suggest three
+    seeds suffice for a comparison that three seeds cannot in fact resolve.
+    """
+
+    if sample_sd < 0.0:
+        raise ValueError("sample_sd must be non-negative")
+    if difference == 0.0:
+        return MAX_SEEDS
+    if sample_sd == 0.0:
+        return 2
+    for n in range(2, MAX_SEEDS):
+        if t_test_power(
+            n,
+            difference,
+            sample_sd,
+            confidence=confidence,
+            paired=paired,
+        ) >= power:
+            return n
+    return MAX_SEEDS
+
+
+def compare(
+    left: Mapping[int, float],
+    right: Mapping[int, float],
+    *,
+    confidence: float = CONFIDENCE,
+    power: float = POWER,
+) -> Comparison:
+    """Compare two conditions on the seeds they share.
+
+    Pairing costs nothing when conditions are run on a common seed list and
+    removes any variation the seed induces in both conditions at once. It only
+    helps to the extent that such shared variation exists; the returned
+    ``sample_sd`` is of the per-seed differences, so it reports directly whether
+    it did.
+    """
+
+    shared = sorted(set(left) & set(right))
+    if len(shared) < 2:
+        raise ValueError("a paired comparison needs at least two shared seeds")
+    differences = np.array(
+        [left[seed] - right[seed] for seed in shared], dtype=np.float64
+    )
+    n = len(differences)
+    mean = float(differences.mean())
+    sample_sd = float(differences.std(ddof=1))
+    half = float(stats.t.ppf(0.5 + confidence / 2.0, n - 1)) * sample_sd / np.sqrt(n)
+    statistic = stats.ttest_rel(
+        [left[seed] for seed in shared],
+        [right[seed] for seed in shared],
+    )
+    return Comparison(
+        difference=mean,
+        sample_sd=sample_sd,
+        n=n,
+        ci_low=mean - half,
+        ci_high=mean + half,
+        p_value=float(statistic.pvalue),
+        seeds_for_power=seeds_for_power(
+            abs(mean),
+            sample_sd,
+            power=power,
+            confidence=confidence,
+            paired=True,
+        ),
+    )
+
+
+def holm_adjust(p_values: Mapping[str, float]) -> dict[str, float]:
+    """Holm-Bonferroni adjustment over a pre-registered comparison family.
+
+    A study that reports every pairwise ordering across eight conditions is
+    running 28 tests, and at three seeds several will cross any uncorrected
+    threshold by chance. Declare the family up front and adjust within it.
+    """
+
+    if not p_values:
+        return {}
+    ordered = sorted(p_values.items(), key=lambda item: item[1])
+    m = len(ordered)
+    adjusted: dict[str, float] = {}
+    running = 0.0
+    for index, (name, value) in enumerate(ordered):
+        running = max(running, min(1.0, (m - index) * value))
+        adjusted[name] = running
+    return adjusted
diff --git a/tests/test_mess3_token_guess_audit.py b/tests/test_mess3_token_guess_audit.py
new file mode 100644
index 00000000..fe6724db
--- /dev/null
+++ b/tests/test_mess3_token_guess_audit.py
@@ -0,0 +1,68 @@
+"""Tests for the re-analysis of the committed multi-seed token-guess results."""
+
+from __future__ import annotations
+
+from pathlib import Path
+
+import pytest
+
+from experiments.mess3_token_guess_cycle_2.audit.experiment import (
+    STUDIES,
+    analyse_study,
+    collect_study,
+)
+
+REPOSITORY_ROOT = Path(__file__).parents[1]
+REFERENCES = {
+    "belief_r2_floor": 0.9668,
+    "belief_r2_ceiling": 0.99888,
+    "accuracy_floor": 0.6732,
+    "accuracy_ceiling": 0.6883,
+}
+
+
+@pytest.mark.parametrize("study", STUDIES)
+def test_every_audited_study_committed_three_seeds_per_arm(study):
+    per_arm = collect_study(REPOSITORY_ROOT, study)
+
+    assert per_arm, f"{study} committed no condition summaries"
+    for arm, by_seed in per_arm.items():
+        assert sorted(by_seed) == [42, 43, 44], arm
+
+
+def test_the_kelly_cycle_3_orderings_do_not_survive_a_holm_correction():
+    analysis = analyse_study(
+        collect_study(REPOSITORY_ROOT, "mess_3_kelly_cycle_3"),
+        REFERENCES,
+    )
+    comparisons = analysis["comparisons"]
+
+    assert len(comparisons) == 6
+    assert not any(values["resolved"] for values in comparisons.values())
+
+
+def test_only_the_strongest_kelly_cycle_3_arm_clears_the_no_network_floor():
+    analysis = analyse_study(
+        collect_study(REPOSITORY_ROOT, "mess_3_kelly_cycle_3"),
+        REFERENCES,
+    )
+    conditions = analysis["conditions"]
+
+    assert conditions["conditional_decoupled_kelly_iqn"]["exceeds_no_network_floor"]
+    for arm in ("ppo", "iqn", "conditional_decoupled_kelly_mean"):
+        assert not conditions[arm]["exceeds_no_network_floor"], arm
+
+    # Plain PPO sits far below a probe on the raw observations.
+    assert conditions["ppo"]["fraction_of_usable_range"] < -1.0
+
+
+def test_every_kelly_cycle_2_kelly_arm_falls_below_the_no_network_floor():
+    analysis = analyse_study(
+        collect_study(REPOSITORY_ROOT, "mess_3_kelly_cycle_2"),
+        REFERENCES,
+    )
+    conditions = analysis["conditions"]
+
+    for arm, values in conditions.items():
+        expected = arm.startswith("correctness")
+        assert values["exceeds_no_network_floor"] is expected, arm
diff --git a/tests/test_mess3_token_guess_statistics.py b/tests/test_mess3_token_guess_statistics.py
new file mode 100644
index 00000000..ba66d342
--- /dev/null
+++ b/tests/test_mess3_token_guess_statistics.py
@@ -0,0 +1,104 @@
+"""Tests for the token-guess seed-level statistics."""
+
+from __future__ import annotations
+
+import numpy as np
+import pytest
+
+from experiments.mess3_token_guess_cycle_2.statistics import (
+    compare,
+    holm_adjust,
+    seeds_for_power,
+    summarise,
+    t_test_power,
+)
+
+# Kelly cycle 3 belief-probe R², seeds 42/43/44, as recorded in each arm's
+# results/<run_id>/condition_summary.json.
+CYCLE_3_PPO = {42: 0.860618, 43: 0.932631, 44: 0.829447}
+CYCLE_3_IQN = {42: 0.974336, 43: 0.975997, 44: 0.956050}
+
+
+def test_summarise_reports_a_wider_interval_than_the_published_plus_minus():
+    values = list(CYCLE_3_PPO.values())
+    estimate = summarise(values)
+
+    population_sd = float(np.std(values, ddof=0))
+    assert estimate.sample_sd > population_sd
+
+    # The published "±" was a population standard deviation. A 95% interval on
+    # the mean of three seeds is several times wider.
+    assert estimate.half_width > 2.0 * population_sd
+    assert estimate.ci_low < estimate.mean < estimate.ci_high
+    assert estimate.n == 3
+
+
+def test_summarise_rejects_a_single_seed():
+    with pytest.raises(ValueError):
+        summarise([0.97])
+
+
+def test_seeds_for_power_scales_with_the_squared_noise_to_signal_ratio():
+    assert seeds_for_power(0.02, 0.01) < seeds_for_power(0.01, 0.01)
+    assert seeds_for_power(0.01, 0.02) > seeds_for_power(0.01, 0.01)
+    # An unpaired design needs more seeds than a paired one.
+    assert seeds_for_power(0.01, 0.01, paired=False) > seeds_for_power(
+        0.01, 0.01, paired=True
+    )
+    assert seeds_for_power(0.0, 0.01) >= 10_000
+
+
+def test_power_rises_with_seeds_and_the_planner_agrees_with_it():
+    difference, sample_sd = 0.010, 0.008
+    powers = [t_test_power(n, difference, sample_sd) for n in (3, 6, 12, 24)]
+    assert powers == sorted(powers)
+    assert powers[0] < 0.8 < powers[-1]
+
+    needed = seeds_for_power(difference, sample_sd)
+    assert t_test_power(needed, difference, sample_sd) >= 0.8
+    assert t_test_power(needed - 1, difference, sample_sd) < 0.8
+
+
+def test_the_headline_ppo_versus_iqn_gap_is_unresolved_at_three_seeds():
+    # Kelly cycle 3 reported PPO 0.8742 ± 0.0432 against IQN 0.9688 ± 0.0090,
+    # which reads as decisive. Paired across the seeds both arms actually ran,
+    # the interval still contains zero.
+    comparison = compare(CYCLE_3_IQN, CYCLE_3_PPO)
+
+    assert comparison.n == 3
+    assert comparison.difference > 0.09
+    assert not comparison.resolved
+    assert comparison.ci_low < 0.0 < comparison.ci_high
+    assert comparison.p_value > 0.05
+    assert comparison.seeds_for_power > 3
+
+
+def test_a_difference_smaller_than_its_noise_is_not_resolved_by_three_seeds():
+    # Two conditions separated by less than their seed-to-seed spread.
+    left = {42: 0.9720, 43: 0.9660, 44: 0.9770}
+    right = {42: 0.9700, 43: 0.9700, 44: 0.9700}
+    comparison = compare(left, right)
+
+    assert abs(comparison.difference) < 0.005
+    assert not comparison.resolved
+    assert comparison.seeds_for_power > 3
+
+
+def test_compare_requires_shared_seeds():
+    with pytest.raises(ValueError):
+        compare({42: 0.9}, {43: 0.8})
+
+
+def test_holm_adjustment_is_monotonic_and_never_shrinks_a_p_value():
+    raw = {"a": 0.001, "b": 0.02, "c": 0.04, "d": 0.5}
+    adjusted = holm_adjust(raw)
+
+    assert set(adjusted) == set(raw)
+    for name, value in raw.items():
+        assert adjusted[name] >= value
+    ordered = sorted(raw, key=lambda name: raw[name])
+    values = [adjusted[name] for name in ordered]
+    assert values == sorted(values)
+    # The smallest of four p-values is multiplied by four.
+    assert adjusted["a"] == pytest.approx(0.004)
+    assert holm_adjust({}) == {}

From 9015518e4fd44bbfef57c8b938bf4be828147be4 Mon Sep 17 00:00:00 2001
From: Cursor Agent <cursoragent@cursor.com>
Date: Sat, 25 Jul 2026 23:47:36 +0000
Subject: [PATCH 3/6] Document the token-guess review and propose the cycle-2
 design

REVIEW.md audits cycle 1 and the three Kelly cycles against the references and
the seed-level statistics. PLAN.md proposes a consolidated re-run: ten paired
seeds, a two-stage coefficient sweep on disjoint seeds, log-spaced checkpoint
probing, and a pre-registered comparison family, with the open design questions
left open.

Co-authored-by: Alex Vardakostas <Al-does@users.noreply.github.com>
---
 .../mess3_token_guess_cycle_2/NOTES.md        |  50 ++++
 experiments/mess3_token_guess_cycle_2/PLAN.md | 211 +++++++++++++++
 .../mess3_token_guess_cycle_2/REVIEW.md       | 242 ++++++++++++++++++
 3 files changed, 503 insertions(+)
 create mode 100644 experiments/mess3_token_guess_cycle_2/NOTES.md
 create mode 100644 experiments/mess3_token_guess_cycle_2/PLAN.md
 create mode 100644 experiments/mess3_token_guess_cycle_2/REVIEW.md

diff --git a/experiments/mess3_token_guess_cycle_2/NOTES.md b/experiments/mess3_token_guess_cycle_2/NOTES.md
new file mode 100644
index 00000000..9353093e
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/NOTES.md
@@ -0,0 +1,50 @@
+# MESS3 token-guess cycle 2
+
+Cycle 2 is a planned re-run of the token-guess comparison ahead of presenting it.
+It does not yet contain any training conditions.
+
+What is here now is the measurement work that has to precede the re-run:
+
+- `REVIEW.md` — an audit of `mess3_token_guess_cycle_1` and
+  `mess_3_kelly_cycle_1/2/3`, which are one experiment split over four
+  directories.
+- `PLAN.md` — a proposed design for the re-run, with the open questions that
+  still need deciding.
+- `metric_references.py` and `references/` — the floors and ceilings both
+  headline metrics are measured against. Cycle 1 reported bare numbers, and both
+  metrics turn out to have task-imposed ranges narrow enough that the bare
+  numbers mislead.
+- `statistics.py` and `audit/` — seed-level aggregation with intervals, paired
+  comparison, multiplicity correction, and power planning, plus a re-analysis of
+  the results already committed.
+
+## What the references say
+
+| reference | belief-probe R² | greedy accuracy |
+|---|---:|---:|
+| randomly initialised transformer | 0.8733 | 0.3412 |
+| affine probe on the raw observations | 0.9668 | — |
+| exact Bayesian filter, one observation | — | 0.6732 |
+| exact Bayesian filter, saturated | — | 0.6883 |
+| supervised next-token replication | 0.9989 | 0.6859 |
+
+Belief-probe R² therefore moves through 0.032 and greedy accuracy through 0.015.
+Cycle 1's reward-only and max-entropy arms land below the untrained network on
+the first metric, and at the repeat-the-previous-token rule on the second.
+
+## What the audit says
+
+Re-reading the three-seed Kelly cycles with a t-based interval on the mean and
+pairing across shared seeds, none of the 34 published pairwise orderings survives
+a Holm correction. The reported `±` values were population standard deviations of
+three samples, roughly a third of the width of a confidence interval.
+
+Regenerate both with:
+
+```bash
+uv run rl-harness experiments.mess3_token_guess_cycle_2.references.experiment
+uv run rl-harness experiments.mess3_token_guess_cycle_2.audit.experiment
+```
+
+Pass `--no-upload-artifacts` when `B2_*` variables are configured as secrets, or
+the bucket name is written into tracked results.
diff --git a/experiments/mess3_token_guess_cycle_2/PLAN.md b/experiments/mess3_token_guess_cycle_2/PLAN.md
new file mode 100644
index 00000000..f7feacfc
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/PLAN.md
@@ -0,0 +1,211 @@
+# Proposed design for token-guess cycle 2
+
+Draft for discussion. `REVIEW.md` establishes the problems; this is one way to
+address them. Everything here is negotiable, and the open questions at the end
+are the parts I would not decide alone.
+
+## Principle
+
+Cycle 1 ran a large number of arms, measured each once, and read a ranking off
+the resulting table. Cycle 2 should run fewer arms, measure each well enough to
+put an interval on it, and pre-commit to which comparisons it is trying to
+resolve. The compute cost of doing this is trivial — the numbers are at the end —
+so the binding constraint is design discipline, not GPU budget.
+
+## 1. Consolidate into one study
+
+One directory, one `shared.py`, one task class, one probe. Today the same
+experiment is spread over four directories with two copies of the task, three
+copies of the probe, and three IQN implementations (`mess3_token_guess_cycle_1.
+iqn_value.iqn`, `mess3_reward_state_cycle_1.iqn`, and the library's
+`IQNValueMixin`).
+
+The library's `IQNValueMixin` should be the only one. Anything else that is a
+reusable RL concept goes to `rl-harness`; Kelly wagering stays here.
+
+## 2. Report metrics against their range, always
+
+Never present a bare R². Every table gets the four reference rows from
+`references.experiment` alongside the trained arms:
+
+| row | belief R² | source |
+|---|---:|---|
+| affine probe on raw observations | 0.9668 | analytic, no network |
+| randomly initialised transformer | 0.8733 | same probe, same architecture |
+| supervised next-token model | 0.9989 | `mess3_supervised` |
+| exact Bayesian filter (accuracy) | 0.6883 | analytic |
+
+R² is a bad scale here because the interesting region is 0.967–0.999. I would
+report three things instead, and let the audience pick:
+
+- **Probe MSE**, which has real dynamic range (0.0012 at R² = 0.99 against 0.03
+  at R² = 0.85) and reads naturally on a log axis.
+- **Incremental R²**: fit the probe on the raw-observation window alone, then on
+  the window plus the network's activations, and report the increment. This
+  answers "what does the network represent that its own inputs did not already
+  linearly provide", which is the question the study is actually asking.
+- **Position in the usable range**, as `references.experiment` already computes,
+  for the one summary slide.
+
+## 3. Fix the checkpoint protocol
+
+Log-spaced checkpoints for every arm, following `rl-harness/docs/
+checkpoint_strategy.md`, with the probe run at each. Then either:
+
+- pre-register "the final checkpoint" and additionally report the curve, so the
+  reader can see the metric is decaying; or
+- pre-register "the mean over the last k checkpoints", which removes the
+  ±0.0035 within-run fluctuation from the comparison for free.
+
+I prefer the second for the primary metric and the first in an appendix, but
+this is a real choice and it must be made before the runs, not after.
+
+The step budget must be identical across every arm. 2.5M is defensible; so is
+5M. What is not defensible is the current mixture of 828K, 2.5M, 3M and 20M.
+
+## 4. Seeds: ten, from a documented spawn
+
+Ten seeds per condition, paired across arms, drawn by spawning from one master
+`SeedSequence` and recorded in the manifest.
+
+The planning table below is from `statistics.seeds_for_power`, paired, 80% power,
+95% confidence:
+
+| belief-R² gap | sd = 0.002 | sd = 0.005 | sd = 0.010 | sd = 0.020 |
+|---:|---:|---:|---:|---:|
+| 0.002 | 10 | 52 | 199 | 787 |
+| 0.005 | 4 | 10 | 34 | 128 |
+| 0.010 | 3 | 5 | 10 | 34 |
+| 0.020 | 3 | 3 | 5 | 10 |
+| 0.100 | 3 | 3 | 3 | 3 |
+
+Ten seeds resolves any gap of 0.01 or more at the spreads actually observed
+(0.001–0.011). It will not resolve the 0.002–0.005 gaps between the middle arms,
+and that is the point: those should be reported as ties rather than chased. Going
+to the ~150 seeds that would settle them is possible on this compute budget but I
+do not think it buys a better talk.
+
+Note the planning numbers are computed from effects observed at n=3 and are
+therefore optimistic — the observed gap in a small sample is biased upward. Ten
+is already padded for that; I would not go below eight.
+
+## 5. Sweep the coefficients, in two stages
+
+Every intervention currently has exactly one tested strength, so no "X does not
+help" claim is available. Proposed grid:
+
+| arm | swept coefficient | values |
+|---|---|---|
+| all | learning rate | 1e-4, 3e-4, 1e-3 |
+| predictive aux | loss weight λ | 0.03, 0.1, 0.3, 1.0 |
+| max entropy | reward coefficient α | 0.01, 0.05, 0.2, 0.5 |
+| IQN | loss coefficient | 0.125, 0.5, 2.0 |
+| Kelly | direct-loss weight | 0.25, 1.0, 4.0 |
+
+Sweeping the learning rate per arm is not optional. Arms differ in parameter
+count and in how much auxiliary gradient enters the shared trunk while the
+learning rate is pinned at 3e-4, so an arm can currently lose by being
+mis-tuned rather than by being a worse idea.
+
+**Two stages, with disjoint seeds.** Stage 1 sweeps on seeds 0–2 and selects one
+configuration per arm. Stage 2 re-runs the selected configurations on ten
+*fresh* seeds, and only stage 2 enters the results table. Selecting and reporting
+on the same seeds is what makes a swept comparison optimistic, and it is easy to
+avoid here.
+
+Selection criterion for stage 1 must be pre-registered. I suggest belief-probe
+MSE, since accuracy is saturated.
+
+## 6. Pre-register the comparison family
+
+Ranking eight arms is 28 tests. Declare the primary comparisons in advance and
+correct within that family; report everything else as exploratory.
+
+Proposed primaries, all on belief-probe MSE at γ = 0.99:
+
+1. reward-only PPO against the raw-observation floor — does task reward degrade
+   the representation?
+2. reward-only PPO against an untrained network — is the degradation relative to
+   initialisation?
+3. predictive aux against reward-only — does the auxiliary objective recover it?
+4. IQN against reward-only — does distributional value do the same?
+5. Kelly against IQN — are they doing the same thing or different things?
+6. best combined arm against the best single arm — do they compose?
+
+Six tests, Holm-corrected. Everything else is exploratory and labelled as such.
+
+## 7. Arms
+
+Two axes crossed, with max-entropy as an orthogonal third:
+
+| axis | levels |
+|---|---|
+| critic | scalar mean, IQN |
+| representation pressure | none, predictive aux, Kelly wager |
+| entropy in reward | off, on |
+| γ | 0, 0.99 |
+
+The full cross is 24. I would run the 2 × 3 × 2 = 12 critic × pressure × γ cells
+as the main grid, and treat entropy as a separate 2 × 2 (critic × γ) add-on at
+its swept coefficient, giving 16 conditions. The γ = 1.0 and differential
+average-reward objectives from cycle 1 are worth keeping as a third γ level if
+the budget is there, since the average-reward arm was the one that restored sane
+critic diagnostics.
+
+## 8. Compute
+
+Measured from the committed manifests: a 2.5M-step arm costs 6–14 minutes on an
+RTX 4090, median about 6.5. Adding eight checkpoint probes puts it near 12.
+
+| stage | runs | GPU-hours | cost at $0.35/hr |
+|---|---:|---:|---:|
+| stage 1 sweep (7 arms × 12 configs × 3 seeds) | 252 | 50 | ~$18 |
+| stage 2 confirmation (16 conditions × 10 seeds) | 160 | 32 | ~$11 |
+| references and audit | — | <1 | — |
+| **total** | **412** | **82** | **~$30** |
+
+Roughly ten wall-clock hours across eight Vast boxes. The entire committed
+history of these four studies is 19.4 GPU-hours, so this is about four times all
+prior work on the question, for the price of lunch.
+
+`devops.vast.provision up -n N --run "..."` gives one command per box, so the
+sweep matrix needs a small dispatcher that maps (arm, coefficient, seed) to
+`--run` strings. That is the only new infrastructure required; there is no
+multi-seed or sweep support in the harness today.
+
+## Open questions
+
+1. **Is the degradation result the headline?** The strongest thing in the data is
+   that reward-only PPO ends up below both the raw observations and its own
+   initialisation. That reframes the study from "which objective induces belief
+   geometry" to "task reward destroys it and these objectives protect it". It is
+   a better talk, but it is a different talk. Which one do you want to give?
+
+2. **Is belief-probe R² still the right dependent variable?** It was chosen to
+   match the supervised replication, but on this task it is compressed into the
+   top 3% of its range and it measures linear decodability rather than use. An
+   intervention experiment — ablate the probe-identified directions and see
+   whether actions change — would answer the question colleagues will actually
+   ask. That is more work than a re-run. Worth it before the presentation, or
+   after?
+
+3. **How much of the Kelly programme survives?** Cycles 1–3 ran fourteen Kelly
+   variants and none of the orderings among them is supported. Cycle 2 could
+   carry one Kelly arm (conditional decoupled, the best performer) rather than
+   the family. Is the wager-calibration result something you want to present in
+   its own right, or is Kelly here purely as a representation-shaping device?
+
+4. **Do you want γ as a swept axis or a fixed choice?** γ is currently confounded
+   with the study directory, and the γ = 0 versus γ = 0.99 contrast is one of the
+   larger apparent effects. Making it a proper factor doubles the grid. My
+   inclination is yes, because a clean γ effect is presentable and the current
+   evidence for it is cross-study.
+
+5. **Do we keep the 20M-step observation?** The R² decay from 0.990 to 0.968 over
+   20M steps is arguably the most interesting single curve in the repository, and
+   it exists for one arm at one seed. Three seeds of a long run for two arms
+   would cost about six GPU-hours and would turn an anecdote into a result.
+
+6. **What is the presentation format?** If there is a slide budget, that should
+   drive how many arms survive. Sixteen conditions with intervals is a dense
+   table; six conditions with intervals and a reference band is a slide.
diff --git a/experiments/mess3_token_guess_cycle_2/REVIEW.md b/experiments/mess3_token_guess_cycle_2/REVIEW.md
new file mode 100644
index 00000000..6c31b65d
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/REVIEW.md
@@ -0,0 +1,242 @@
+# Review of the MESS3 token-guess studies
+
+Scope: `mess3_token_guess_cycle_1` and `mess_3_kelly_cycle_1/2/3`. These are four
+directories but one experiment. All of them train a 3-layer, 96-wide transformer
+with PPO on passive MESS3 (`alpha=0.85`, `delay=1`, episode length 512), reward
+it 1 for guessing the next emitted token, and then fit a held-out rank-2 affine
+probe from its residual stream to the exact Bayesian belief. `NextTokenGuessTask`
+and `RawNextTokenTask` are the same task written twice.
+
+The findings below are ordered by how much they would change what you tell your
+colleagues. Every number is reproducible from the repository:
+
+```bash
+uv run rl-harness experiments.mess3_token_guess_cycle_2.references.experiment
+uv run rl-harness experiments.mess3_token_guess_cycle_2.audit.experiment
+```
+
+---
+
+## 1. The headline metric is mostly measuring something trivial
+
+Belief-probe R² was reported as a bare number from 0.85 to 0.99, and read as
+"how much of the Bayesian belief did this objective induce". It cannot be read
+that way, because the metric has a floor that nobody measured.
+
+An affine probe fitted directly to the **one-hot encoded raw observations**, with
+no network and no training anywhere in the pipeline, scores **R² = 0.9668**. It
+gets there on four observations and is saturated by six.
+
+| probe input | R² |
+|---|---:|
+| last 1 observation | 0.8043 |
+| last 2 observations | 0.9302 |
+| last 4 observations | 0.9648 |
+| last 8 observations | 0.9669 |
+| last 64 observations | 0.9668 |
+
+A **randomly initialised** copy of the study's own transformer, probed by exactly
+the same code at exactly the same site, scores **R² = 0.8733**.
+
+The supervised next-token replication in `mess3_supervised` reaches **0.99888**.
+
+So the range this metric can move through is 0.9668 to 0.9989 — a span of
+**0.032** — and the published scores land like this:
+
+| condition | reported R² | position in the usable range |
+|---|---:|---:|
+| `comparison/reward_only` | 0.8552 | −348% |
+| `comparison/max_entropy` | 0.8558 | −346% |
+| *randomly initialised transformer* | *0.8733* | *−293%* |
+| `comparison/predictive_loss` | 0.9319 | −109% |
+| *affine probe on raw observations* | *0.9668* | *0%* |
+| `iqn_value` | 0.9760 | +29% |
+| `kelly_cycle_3/conditional_decoupled_kelly_iqn` | 0.9824 | +49% |
+| `kelly_cycle_2/correctness_iqn` | 0.9857 | +59% |
+| *supervised next-token replication* | *0.9989* | *100%* |
+
+Two consequences.
+
+**Reward-only PPO and max-entropy PPO are below the untrained network.** After
+2.5M steps their residual streams are *less* linearly informative about the
+belief than they were at initialisation. That is a real and interesting result —
+task-reward training is destroying linearly-available structure — but it is the
+opposite of the claim "reward-only PPO reaches R² = 0.855 of the belief".
+
+**All eight Kelly cycle-2 wager arms sit below the no-network floor.** The
+cycle-2 note reads their 0.94–0.96 as respectable. Against the floor they are
+negative.
+
+This is safe because the task is passive: `NextTokenGuessTask.resolve_action`
+returns the same transition matrix for every action, so the distribution of
+beliefs the probe sees is identical for every arm, for the untrained network, and
+for the raw-observation baseline. The comparison is exact, not approximate.
+
+## 2. Greedy token accuracy is saturated and cannot discriminate
+
+An exact Bayesian filter allowed only the previous observation — the trivial
+"repeat the last token" rule — scores **0.6732**. Given eight or more
+observations it saturates at **0.6883**. Total usable range: **0.0151**.
+
+Reported accuracies span 0.6733 to 0.6860. So the best arms are already within
+0.002 of Bayes-optimal, `reward_only` at 0.6733 is at the repeat-last-token
+rule, and the whole comparison lives inside 1.5 percentage points. Meanwhile the
+probe's own sampling error at 30,000 test steps is ±0.005 at 95%, and that
+assumes independent samples, which consecutive steps of a slowly-mixing chain
+are not.
+
+Reporting this metric to four decimal places invites exactly the question you do
+not want from the audience.
+
+## 3. None of the published orderings is statistically supported
+
+Cycles 2 and 3 ran three seeds and reported `mean ± std`. That `±` is a
+population standard deviation (`ddof=0`), not an interval; for three samples it
+sits 18% below the sample standard deviation and about 3x below a 95% confidence
+interval on the mean.
+
+Recomputed properly, and paired across the seeds each pair of arms shares:
+
+- **Kelly cycle 3**: 0 of 6 pairwise orderings survive a Holm correction.
+- **Kelly cycle 2**: 0 of 28 pairwise orderings survive a Holm correction.
+
+The headline gap is the clearest case. Cycle 3 reported PPO `0.8742 ± 0.0432`
+against IQN `0.9688 ± 0.0090`, which reads as decisive. Paired on the three
+shared seeds, the difference is +0.095 with a 95% interval of **[−0.017,
++0.206]** and p = 0.067. PPO's own interval, [0.743, 1.006], extends past the
+maximum value R² can take.
+
+Some claims in the notes are far past what three seeds can carry. Cycle 3 states
+that "conditional Kelly with a scalar critic narrowly exceeds plain IQN in belief
+R²". That gap is 0.0029 against a paired spread of 0.013; resolving it needs
+**165 seeds**.
+
+## 4. Final-checkpoint scores are drawn from a moving target
+
+Every arm reports the checkpoint at its step budget, with `num_to_keep=1`. The
+one arm that kept a checkpoint curve, `iqn_value_20m`, shows why that is unsafe:
+
+| steps | R² | greedy accuracy |
+|---:|---:|---:|
+| 0.83M | 0.9902 | 68.12% |
+| 2.48M | 0.9772 | 68.08% |
+| 3.31M | 0.9673 | 68.34% |
+| 10.76M | 0.9718 | 68.77% |
+| 20.03M | 0.9686 | 68.47% |
+
+Belief R² **peaks at the first retained checkpoint and declines**, while accuracy
+slowly rises. They are not merely uncorrelated, they move in opposite directions.
+After 3M steps R² fluctuates with a standard deviation of 0.0035 and a range of
+0.013 — comparable to most of the between-arm gaps being reported.
+
+So "IQN scores 0.976 at 2.5M" is one draw from a decaying, noisy trajectory. A
+different step budget reorders the table. And the budget is not even constant
+across the family: 828K, 2.5M, 3M and 20M all appear.
+
+## 5. Every intervention was tested at exactly one strength
+
+| intervention | values tried |
+|---|---|
+| predictive auxiliary loss λ | 0.1 |
+| max-entropy reward coefficient α | 0.05 |
+| IQN loss coefficient | 0.5 |
+| Kelly direct-loss weight | 1.0 |
+| learning rate | 3e-4 everywhere (4.2e-4 in the reward-state battery) |
+
+No conclusion of the form "X does not help" is available from one point. The
+max-entropy arm is the clearest example: at α = 0.05 the entropy bonus is at most
+`0.05 · ln 3 ≈ 0.055` against a reward of 0 or 1, so the arm was never given
+enough signal to do anything, and "max-entropy training produced essentially the
+same belief representation as reward-only" is a statement about that one
+coefficient.
+
+The arms are also not matched on anything except environment steps. IQN arms set
+`vf_loss_coeff=0` and add a quantile loss; Kelly and predictive arms add heads
+and extra gradient into the shared trunk. Parameter count and total gradient
+magnitude differ between arms while the learning rate is held fixed, so an arm
+can win by being better tuned rather than by being a better idea.
+
+## 6. γ is confounded with the study
+
+Discount factor and condition set were varied together across directories.
+
+| γ, λ | where |
+|---|---|
+| 0.99, 0.95 | token-guess `comparison`, `iqn_value`, Kelly cycle 3 |
+| 1.0, 0.95 | Kelly cycle 1, `iqn_gamma_1_3m` |
+| 0.0, 0.0 | Kelly cycle 2 |
+| 0.0, 0.95 | `iqn_return_objectives/gamma_zero` |
+
+Cycle 2 concludes that "moving from the prior γ = 0.99 reward-only result
+(R² = 0.8552) to γ = 0 gives R² = 0.9834 without an auxiliary objective". That
+compares n=1 to n=3, across two studies, two probe seed streams, and two
+code paths. It is a plausible hypothesis, not a measured effect.
+
+The λ change is harmless in itself — GAE weights terms by `(γλ)^k`, so at γ = 0
+the value of λ cannot matter — but it should be made explicit rather than left as
+an apparent second difference.
+
+## 7. Smaller items worth fixing
+
+**Probe protocol is not shared.** The RL probe uses a rank-2 constrained affine
+map on 30,000 test steps at the post-final-LayerNorm site, from a single
+train/test split. The supervised reference uses a full-rank probe on 472,390
+positions, sweeps five layer sites and reports the best. Those two numbers should
+not be placed in the same table until they are measured the same way. Kelly cycle
+2 also uses different probe seed streams (200/201/202) from the token-guess cycle
+(100/101/102), so cross-study numbers carry an extra source of variation.
+
+**Probe R² and MSE are reported, but no uncertainty.** Bootstrapping the probe's
+test set puts its own sampling noise at ±0.0005, which is small — but it needed
+to be measured rather than assumed.
+
+**Pairing does not currently buy anything.** All arms already run seeds 42/43/44,
+so paired analysis is free. On the existing data the paired spread is frequently
+*larger* than the unpaired spread, meaning seed effects are arm-specific
+optimisation noise rather than shared data noise. Three seeds cannot settle this;
+keep the paired design because it costs nothing, but do not budget for the
+variance reduction.
+
+**Actual step counts differ between arms.** 2,515,821 against 2,512,179 within
+Kelly cycle 1, because the stop condition triggers on batch boundaries. Small,
+but it is free to record and equalise.
+
+**Run-to-run non-determinism is negligible.** The two reproduction arms re-ran
+the same seed and code and differed by 0.00016 in R². Seed variance is roughly
+200x larger. Determinism is not the problem here; sample size is.
+
+**Naming and layout.** `mess3_token_guess_cycle_1` and `mess_3_kelly_cycle_*`
+differ in spelling, and one experiment is spread over four directories with three
+copies of the probe and two copies of the task. `mess_3_kelly_cycle_3/shared.py`
+imports IQN from `mess3_token_guess_cycle_1.iqn_value.iqn` while the reward-state
+battery imports it from `mess3_reward_state_cycle_1.iqn`, and the library now has
+its own `IQNValueMixin`. Three IQN implementations are in play.
+
+**Committed manifests leak the B2 bucket name.** Any run made with `B2_*`
+configured writes the bucket and endpoint into tracked `results/`. Pass
+`--no-upload-artifacts`, or scrub those fields, when running from an environment
+where those are secrets.
+
+---
+
+## What this means for the presentation
+
+The existing work supports a small number of claims very well:
+
+- Training reward-only PPO on this task **degrades** the linear decodability of
+  the Bayesian belief relative to both the raw observations and to the network's
+  own initialisation. That is a genuinely surprising, presentable result, and it
+  is large enough to survive the sample size.
+- Adding a predictive or distributional objective **recovers and then exceeds**
+  the raw-observation floor. `correctness_iqn` at γ = 0 and
+  `conditional_decoupled_kelly_iqn` at γ = 0.99 are the only arms whose intervals
+  clear the floor.
+- Greedy token accuracy is saturated near Bayes-optimal for every arm, so
+  representation quality and task performance are dissociable on this task. This
+  is the same dissociation the reward-state battery found, arrived at
+  independently.
+
+It does not support any fine-grained ranking among the middle arms, and the
+current notes lead with several of those rankings.
+
+`PLAN.md` proposes what cycle 2 should do about it.

From ee4095babf540e59e7add42395ac8ead5f6c6ec8 Mon Sep 17 00:00:00 2001
From: Cursor Agent <cursoragent@cursor.com>
Date: Sat, 25 Jul 2026 23:50:28 +0000
Subject: [PATCH 4/6] Plot the published scores against the references

One axis showing where each condition sits relative to a probe on the raw
observations, an untrained copy of the same transformer, and the supervised
replication, plus a forest plot of the three-seed intervals.

Co-authored-by: Alex Vardakostas <Al-does@users.noreply.github.com>
---
 .../audit/experiment.py                       | 124 ++++++++++++++++--
 .../audit.json                                |   4 +-
 .../findings.md                               |   0
 .../results_against_references.png            | Bin 0 -> 180365 bytes
 .../run_manifest.json                         |  16 +--
 .../references/experiment.py                  |  64 +++++++++
 .../belief_r2_landscape.png                   | Bin 0 -> 129103 bytes
 .../findings.md                               |   0
 .../references.json                           |   1 +
 .../run_manifest.json                         |  16 +--
 10 files changed, 195 insertions(+), 30 deletions(-)
 rename experiments/mess3_token_guess_cycle_2/audit/results/{20260725T234419Z-36e12c96 => 20260725T234843Z-46848db2}/audit.json (99%)
 rename experiments/mess3_token_guess_cycle_2/audit/results/{20260725T234419Z-36e12c96 => 20260725T234843Z-46848db2}/findings.md (100%)
 create mode 100644 experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/results_against_references.png
 rename experiments/mess3_token_guess_cycle_2/audit/results/{20260725T234419Z-36e12c96 => 20260725T234843Z-46848db2}/run_manifest.json (79%)
 create mode 100644 experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/belief_r2_landscape.png
 rename experiments/mess3_token_guess_cycle_2/references/results/{20260725T233938Z-70fe7dae => 20260725T234936Z-4fc9f18d}/findings.md (100%)
 rename experiments/mess3_token_guess_cycle_2/references/results/{20260725T233938Z-70fe7dae => 20260725T234936Z-4fc9f18d}/references.json (90%)
 rename experiments/mess3_token_guess_cycle_2/references/results/{20260725T233938Z-70fe7dae => 20260725T234936Z-4fc9f18d}/run_manifest.json (79%)

diff --git a/experiments/mess3_token_guess_cycle_2/audit/experiment.py b/experiments/mess3_token_guess_cycle_2/audit/experiment.py
index dc5db0df..091ebc61 100644
--- a/experiments/mess3_token_guess_cycle_2/audit/experiment.py
+++ b/experiments/mess3_token_guess_cycle_2/audit/experiment.py
@@ -15,6 +15,11 @@
 from pathlib import Path
 from typing import Any
 
+import matplotlib
+
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt  # noqa: E402
+
 from experiments.mess3_token_guess_cycle_2.statistics import (
     compare,
     holm_adjust,
@@ -25,6 +30,13 @@
 
 STUDIES = ("mess_3_kelly_cycle_2", "mess_3_kelly_cycle_3")
 METRICS = ("r_squared", "token_accuracy_greedy")
+FALLBACK_REFERENCES = {
+    "belief_r2_floor": 0.9668,
+    "belief_r2_ceiling": 0.99888,
+    "accuracy_floor": 0.6732,
+    "accuracy_ceiling": 0.6883,
+    "untrained_module_r2": 0.8733,
+}
 
 
 def _references(experiment_dir: Path) -> dict[str, float]:
@@ -33,19 +45,18 @@ def _references(experiment_dir: Path) -> dict[str, float]:
     candidates = sorted(
         (experiment_dir.parent / "references" / "results").glob("*/references.json")
     )
-    if candidates:
-        data = json.loads(candidates[-1].read_text())
-        return {
-            "belief_r2_floor": float(data["belief_r2_floor"]),
-            "belief_r2_ceiling": float(data["supervised_ceiling"]),
-            "accuracy_floor": float(data["accuracy_floor_repeat_previous_token"]),
-            "accuracy_ceiling": float(data["accuracy_ceiling_bayes"]),
-        }
+    if not candidates:
+        return dict(FALLBACK_REFERENCES)
+    data = json.loads(candidates[-1].read_text())
+    untrained = data.get("untrained_module") or {}
     return {
-        "belief_r2_floor": 0.9668,
-        "belief_r2_ceiling": 0.99888,
-        "accuracy_floor": 0.6732,
-        "accuracy_ceiling": 0.6883,
+        "belief_r2_floor": float(data["belief_r2_floor"]),
+        "belief_r2_ceiling": float(data["supervised_ceiling"]),
+        "accuracy_floor": float(data["accuracy_floor_repeat_previous_token"]),
+        "accuracy_ceiling": float(data["accuracy_ceiling_bayes"]),
+        "untrained_module_r2": float(
+            untrained.get("r_squared", FALLBACK_REFERENCES["untrained_module_r2"])
+        ),
     }
 
 
@@ -116,6 +127,82 @@ def analyse_study(
     return {"conditions": conditions, "comparisons": comparisons}
 
 
+def plot_against_references(
+    analysis: dict[str, Any],
+    references: dict[str, float],
+    *,
+    untrained_r2: float,
+    path: Path,
+) -> None:
+    """Place every arm's interval inside the range the metric can move through."""
+
+    floor = references["belief_r2_floor"]
+    ceiling = references["belief_r2_ceiling"]
+    rows: list[tuple[str, float, float, float]] = []
+    for study, result in analysis.items():
+        label = study.replace("mess_3_", "").replace("_", " ")
+        for arm, values in sorted(
+            result["conditions"].items(),
+            key=lambda item: item[1]["r_squared_mean"],
+        ):
+            low, high = values["r_squared_ci"]
+            rows.append(
+                (f"{arm.replace('_', ' ')}\n({label})", values["r_squared_mean"], low, high)
+            )
+
+    figure, axis = plt.subplots(figsize=(9.5, 0.42 * len(rows) + 2.6))
+    positions = range(len(rows))
+    axis.axvspan(floor, ceiling, color="tab:green", alpha=0.10)
+    axis.axvline(
+        floor,
+        color="tab:green",
+        linestyle="--",
+        linewidth=1.4,
+        label=f"affine probe on raw observations ({floor:.4f})",
+    )
+    axis.axvline(
+        untrained_r2,
+        color="tab:orange",
+        linestyle=":",
+        linewidth=1.4,
+        label=f"randomly initialised transformer ({untrained_r2:.4f})",
+    )
+    axis.axvline(
+        ceiling,
+        color="tab:blue",
+        linestyle="-.",
+        linewidth=1.4,
+        label=f"supervised next-token model ({ceiling:.4f})",
+    )
+    for position, (_, mean, low, high) in zip(positions, rows):
+        cleared = low > floor
+        axis.plot(
+            [low, high],
+            [position, position],
+            color="tab:green" if cleared else "tab:red",
+            linewidth=2.0,
+            alpha=0.75,
+        )
+        axis.plot(
+            mean,
+            position,
+            "o",
+            color="tab:green" if cleared else "tab:red",
+            markersize=6,
+        )
+    axis.set_yticks(list(positions), [row[0] for row in rows], fontsize=8)
+    axis.set_xlim(0.72, 1.02)
+    axis.set_xlabel("held-out belief-probe R² (95% CI over three seeds)")
+    axis.set_title(
+        "Committed token-guess results against the range the metric can move through"
+    )
+    axis.grid(axis="x", alpha=0.2)
+    axis.legend(loc="lower left", fontsize=8, framealpha=0.9)
+    figure.tight_layout()
+    figure.savefig(path, dpi=200)
+    plt.close(figure)
+
+
 def _findings(analysis: dict[str, Any], references: dict[str, float]) -> str:
     floor = references["belief_r2_floor"]
     ceiling = references["belief_r2_ceiling"]
@@ -194,7 +281,18 @@ def run(context: RunContext) -> dict[str, Any]:
     analysis = {study: result for study, result in analysis.items() if result["conditions"]}
     if not analysis:
         raise RuntimeError("no committed multi-seed results were found to audit")
-    result = {"references": references, "studies": analysis}
+    figure_path = context.results_dir / "results_against_references.png"
+    plot_against_references(
+        analysis,
+        references,
+        untrained_r2=references["untrained_module_r2"],
+        path=figure_path,
+    )
+    result = {
+        "references": references,
+        "studies": analysis,
+        "figure": str(figure_path),
+    }
     outputs.write_json("audit.json", result)
     (context.results_dir / "findings.md").write_text(_findings(analysis, references))
     return result
diff --git a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/audit.json b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/audit.json
similarity index 99%
rename from experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/audit.json
rename to experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/audit.json
index 2895738d..e69f3147 100644
--- a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/audit.json
+++ b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/audit.json
@@ -1,9 +1,11 @@
 {
+  "figure": "/workspace/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/results_against_references.png",
   "references": {
     "accuracy_ceiling": 0.6883,
     "accuracy_floor": 0.6732,
     "belief_r2_ceiling": 0.99888,
-    "belief_r2_floor": 0.9668
+    "belief_r2_floor": 0.9668,
+    "untrained_module_r2": 0.8733
   },
   "studies": {
     "mess_3_kelly_cycle_2": {
diff --git a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/findings.md b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/findings.md
similarity index 100%
rename from experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/findings.md
rename to experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/findings.md
diff --git a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/results_against_references.png b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/results_against_references.png
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diff --git a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/run_manifest.json b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/run_manifest.json
similarity index 79%
rename from experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/run_manifest.json
rename to experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/run_manifest.json
index be370d8a..18a90378 100644
--- a/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96/run_manifest.json
+++ b/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2/run_manifest.json
@@ -8,12 +8,12 @@
     "file": "/workspace/uv.lock",
     "sha256": "986725e46af5252423583fe947d30b2646f541a4863acb8c51d578a7ca1b8220"
   },
-  "ended_at": "2026-07-25T23:44:20.373828+00:00",
+  "ended_at": "2026-07-25T23:48:44.772373+00:00",
   "error": null,
   "experiment": {
     "file": "/workspace/experiments/mess3_token_guess_cycle_2/audit/experiment.py",
     "module": "experiments.mess3_token_guess_cycle_2.audit.experiment",
-    "source_sha256": "22295905c9deb572d3b5f815992390f4f9a6dd4a23bb0afeb6ddc4c5150d885b"
+    "source_sha256": "bd96a2324c233c02c6c1dfeedd9f77ceb5b7ff4fe61ef8bcfd99cc72cc0274f0"
   },
   "framework_versions": {
     "gymnasium": "1.2.2",
@@ -24,10 +24,10 @@
     "torch": "2.12.1"
   },
   "git": {
-    "commit": "57391da3f5491b074234f5a5caa2cd19411567a0",
+    "commit": "9015518e4fd44bbfef57c8b938bf4be828147be4",
     "dirty": true,
     "experiment_repository": {
-      "commit": "57391da3f5491b074234f5a5caa2cd19411567a0",
+      "commit": "9015518e4fd44bbfef57c8b938bf4be828147be4",
       "dirty": true,
       "root": "/workspace"
     },
@@ -54,9 +54,9 @@
       "torch_threads": null
     }
   },
-  "run_id": "20260725T234419Z-36e12c96",
+  "run_id": "20260725T234843Z-46848db2",
   "runtime": {
-    "artifacts_dir": "/workspace/experiments/mess3_token_guess_cycle_2/audit/artifacts/20260725T234419Z-36e12c96",
+    "artifacts_dir": "/workspace/experiments/mess3_token_guess_cycle_2/audit/artifacts/20260725T234843Z-46848db2",
     "overrides": {
       "hardware_profile": "auto",
       "resume_from": null,
@@ -64,12 +64,12 @@
       "smoke": false,
       "upload_artifacts": false
     },
-    "results_dir": "/workspace/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234419Z-36e12c96",
+    "results_dir": "/workspace/experiments/mess3_token_guess_cycle_2/audit/results/20260725T234843Z-46848db2",
     "resume_from": null,
     "seed": 42,
     "smoke": false
   },
   "schema_version": 2,
-  "started_at": "2026-07-25T23:44:20.290986+00:00",
+  "started_at": "2026-07-25T23:48:44.456151+00:00",
   "status": "completed"
 }
diff --git a/experiments/mess3_token_guess_cycle_2/references/experiment.py b/experiments/mess3_token_guess_cycle_2/references/experiment.py
index 6f813a7b..05061b5b 100644
--- a/experiments/mess3_token_guess_cycle_2/references/experiment.py
+++ b/experiments/mess3_token_guess_cycle_2/references/experiment.py
@@ -2,8 +2,14 @@
 
 from __future__ import annotations
 
+from pathlib import Path
 from typing import Any
 
+import matplotlib
+
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt  # noqa: E402
+
 from experiments.mess3_token_guess_cycle_1.comparison.experiment import (
     BASE_MODEL_CONFIG,
     ENV_CONFIG,
@@ -35,6 +41,61 @@
 SUPERVISED_CEILING = 0.99888
 
 
+def plot_landscape(references: dict[str, Any], *, path: Path) -> None:
+    """Show every published score against the references on one axis."""
+
+    floor = references["belief_r2_floor"]
+    untrained = references["untrained_module"]["r_squared"]
+    rows = [(name, value, "condition") for name, value in CYCLE_1_SCORES.items()]
+    rows.extend(
+        [
+            ("randomly initialised transformer", untrained, "reference"),
+            ("affine probe on raw observations", floor, "reference"),
+            ("supervised next-token model", SUPERVISED_CEILING, "reference"),
+        ]
+    )
+    rows.sort(key=lambda row: row[1])
+
+    figure, axis = plt.subplots(figsize=(9.5, 0.46 * len(rows) + 2.0))
+    axis.axvspan(floor, SUPERVISED_CEILING, color="tab:green", alpha=0.10)
+    for position, (name, value, kind) in enumerate(rows):
+        reference = kind == "reference"
+        colour = "black" if reference else ("tab:green" if value > floor else "tab:red")
+        axis.barh(
+            position,
+            value - 0.80,
+            left=0.80,
+            height=0.55,
+            color=colour,
+            alpha=0.35 if reference else 0.85,
+            hatch="//" if reference else None,
+        )
+        axis.text(
+            value + 0.002,
+            position,
+            f"{value:.4f}",
+            va="center",
+            fontsize=8,
+        )
+    axis.set_yticks(
+        range(len(rows)),
+        [name.replace("_", " ") for name, _, _ in rows],
+        fontsize=8,
+    )
+    axis.set_xlim(0.80, 1.03)
+    axis.set_xlabel("held-out belief-probe R²")
+    axis.set_title(
+        "The usable range of belief-probe R² is 0.967 to 0.999\n"
+        "(hatched bars are references; red bars fall below a probe on the raw "
+        "observations)",
+        fontsize=10,
+    )
+    axis.grid(axis="x", alpha=0.2)
+    figure.tight_layout()
+    figure.savefig(path, dpi=200)
+    plt.close(figure)
+
+
 def _findings(references: dict[str, Any]) -> str:
     floor = references["belief_r2_floor"]
     context = references["belief_r2_floor_context"]
@@ -133,6 +194,9 @@ def run(context: RunContext) -> dict[str, Any]:
         )
         for condition, value in CYCLE_1_SCORES.items()
     }
+    figure_path = context.results_dir / "belief_r2_landscape.png"
+    plot_landscape(references, path=figure_path)
+    references["figure"] = str(figure_path)
     outputs.write_json("references.json", references)
     (context.results_dir / "findings.md").write_text(_findings(references))
     return references
diff --git a/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/belief_r2_landscape.png b/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/belief_r2_landscape.png
new file mode 100644
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diff --git a/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/findings.md b/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/findings.md
similarity index 100%
rename from experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/findings.md
rename to experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/findings.md
diff --git a/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/references.json b/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/references.json
similarity index 90%
rename from experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/references.json
rename to experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/references.json
index 5188ecba..a03c16ae 100644
--- a/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/references.json
+++ b/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/references.json
@@ -26,6 +26,7 @@
     "kelly_cycle_2/correctness_iqn": 0.5894322645528668,
     "kelly_cycle_3/conditional_decoupled_kelly_iqn": 0.48663457661845627
   },
+  "figure": "/workspace/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/belief_r2_landscape.png",
   "n_fit": 60000,
   "n_test": 30000,
   "probe_rank": 2,
diff --git a/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/run_manifest.json b/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/run_manifest.json
similarity index 79%
rename from experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/run_manifest.json
rename to experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/run_manifest.json
index dd6f4c82..0ca95f37 100644
--- a/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae/run_manifest.json
+++ b/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d/run_manifest.json
@@ -8,12 +8,12 @@
     "file": "/workspace/uv.lock",
     "sha256": "986725e46af5252423583fe947d30b2646f541a4863acb8c51d578a7ca1b8220"
   },
-  "ended_at": "2026-07-25T23:39:59.835334+00:00",
+  "ended_at": "2026-07-25T23:49:59.308138+00:00",
   "error": null,
   "experiment": {
     "file": "/workspace/experiments/mess3_token_guess_cycle_2/references/experiment.py",
     "module": "experiments.mess3_token_guess_cycle_2.references.experiment",
-    "source_sha256": "6706a930ef401113ce0d8e1d6572cc0960d4fd925f52474975219acf8273f7ac"
+    "source_sha256": "606fa5cdba5b1aa2eefe3945514a955e43d0787844c83cbdd2e30906481c927e"
   },
   "framework_versions": {
     "gymnasium": "1.2.2",
@@ -24,10 +24,10 @@
     "torch": "2.12.1"
   },
   "git": {
-    "commit": "0f798d66356f37e1d126a163a0f985cbcbd7bb54",
+    "commit": "9015518e4fd44bbfef57c8b938bf4be828147be4",
     "dirty": true,
     "experiment_repository": {
-      "commit": "0f798d66356f37e1d126a163a0f985cbcbd7bb54",
+      "commit": "9015518e4fd44bbfef57c8b938bf4be828147be4",
       "dirty": true,
       "root": "/workspace"
     },
@@ -54,9 +54,9 @@
       "torch_threads": null
     }
   },
-  "run_id": "20260725T233938Z-70fe7dae",
+  "run_id": "20260725T234936Z-4fc9f18d",
   "runtime": {
-    "artifacts_dir": "/workspace/experiments/mess3_token_guess_cycle_2/references/artifacts/20260725T233938Z-70fe7dae",
+    "artifacts_dir": "/workspace/experiments/mess3_token_guess_cycle_2/references/artifacts/20260725T234936Z-4fc9f18d",
     "overrides": {
       "hardware_profile": "auto",
       "resume_from": null,
@@ -64,12 +64,12 @@
       "smoke": false,
       "upload_artifacts": false
     },
-    "results_dir": "/workspace/experiments/mess3_token_guess_cycle_2/references/results/20260725T233938Z-70fe7dae",
+    "results_dir": "/workspace/experiments/mess3_token_guess_cycle_2/references/results/20260725T234936Z-4fc9f18d",
     "resume_from": null,
     "seed": 42,
     "smoke": false
   },
   "schema_version": 2,
-  "started_at": "2026-07-25T23:39:38.286533+00:00",
+  "started_at": "2026-07-25T23:49:36.830424+00:00",
   "status": "completed"
 }

From d790369e770a98ac118709c8bed484375415f998 Mon Sep 17 00:00:00 2001
From: Cursor Agent <cursoragent@cursor.com>
Date: Mon, 27 Jul 2026 06:08:08 +0000
Subject: [PATCH 5/6] Measure how the MESS3 parameters set the metric's range
 and precision

Cycle 1's parameters are close to the worst choice in this family: a fast-mixing
chain lets the belief be approximated by an exponentially weighted average of
recent one-hot observations, which is what an affine probe computes. Slowing the
chain to a self-transition of 0.995 at alpha 0.70 roughly triples the
belief-probe range and multiplies the accuracy range by ten.

The cost is precision. The state correlation time rises from 7 steps to 133, so a
30,000-step probe rollout collected as sixteen trajectories holds about 100
independent samples rather than 2,300, and the honest block-bootstrap interval
widens from +/-0.0006 to +/-0.0046. Context length is unaffected: 64 observations
still suffice, because the belief converges well before the state decorrelates.

Also record that belief-probe R2 declines under continued supervised training at
a converged cross-entropy, at both the old and the new parameters. The decline
cycle 1 saw over 20M PPO steps is not caused by reinforcement learning, which
makes optimiser and learning rate larger influences on the headline metric than
most of the differences between arms.

Co-authored-by: Alex Vardakostas <Al-does@users.noreply.github.com>
---
 .../mess3_token_guess_cycle_2/NOTES.md        |  19 +-
 experiments/mess3_token_guess_cycle_2/PLAN.md | 109 +++++--
 .../operating_point.py                        | 283 +++++++++++++++++
 .../task_parameters/__init__.py               |   0
 .../task_parameters/experiment.py             | 210 ++++++++++++
 .../20260727T060541Z-53419d82/findings.md     |  36 +++
 .../operating_point_grid.png                  | Bin 0 -> 241022 bytes
 .../run_manifest.json                         |  75 +++++
 .../task_parameters.json                      | 299 ++++++++++++++++++
 .../test_mess3_token_guess_operating_point.py |  96 ++++++
 10 files changed, 1098 insertions(+), 29 deletions(-)
 create mode 100644 experiments/mess3_token_guess_cycle_2/operating_point.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/task_parameters/__init__.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py
 create mode 100644 experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/findings.md
 create mode 100644 experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/operating_point_grid.png
 create mode 100644 experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/run_manifest.json
 create mode 100644 experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/task_parameters.json
 create mode 100644 tests/test_mess3_token_guess_operating_point.py

diff --git a/experiments/mess3_token_guess_cycle_2/NOTES.md b/experiments/mess3_token_guess_cycle_2/NOTES.md
index 9353093e..dace163b 100644
--- a/experiments/mess3_token_guess_cycle_2/NOTES.md
+++ b/experiments/mess3_token_guess_cycle_2/NOTES.md
@@ -14,6 +14,9 @@ What is here now is the measurement work that has to precede the re-run:
   headline metrics are measured against. Cycle 1 reported bare numbers, and both
   metrics turn out to have task-imposed ranges narrow enough that the bare
   numbers mislead.
+- `operating_point.py` and `task_parameters/` — how the metric's range and
+  precision vary with the MESS3 transition and emission parameters, so the
+  process can be chosen to make the metric sensitive rather than degenerate.
 - `statistics.py` and `audit/` — seed-level aggregation with intervals, paired
   comparison, multiplicity correction, and power planning, plus a re-analysis of
   the results already committed.
@@ -39,10 +42,24 @@ pairing across shared seeds, none of the 34 published pairwise orderings survive
 a Holm correction. The reported `±` values were population standard deviations of
 three samples, roughly a third of the width of a confidence interval.
 
-Regenerate both with:
+## What the operating-point analysis says
+
+Cycle 1's parameters are close to the worst choice in the symmetric MESS3 family
+for metric sensitivity. Slowing the chain to a self-transition of 0.995 at
+`alpha=0.70` roughly triples the belief-probe range and multiplies the accuracy
+range by ten, at the cost of a probe interval that widens from ±0.0006 to
+±0.0046 because the rollout decorrelates far more slowly.
+
+It also shows that belief-probe R² declines with continued *supervised* training
+while cross-entropy stays at the Bayes floor. The decline cycle 1 saw over 20M
+PPO steps is therefore not a property of reinforcement learning, and optimiser
+and learning rate move the headline metric more than most of the arms do.
+
+Regenerate all three with:
 
 ```bash
 uv run rl-harness experiments.mess3_token_guess_cycle_2.references.experiment
+uv run rl-harness experiments.mess3_token_guess_cycle_2.task_parameters.experiment
 uv run rl-harness experiments.mess3_token_guess_cycle_2.audit.experiment
 ```
 
diff --git a/experiments/mess3_token_guess_cycle_2/PLAN.md b/experiments/mess3_token_guess_cycle_2/PLAN.md
index f7feacfc..c1590b40 100644
--- a/experiments/mess3_token_guess_cycle_2/PLAN.md
+++ b/experiments/mess3_token_guess_cycle_2/PLAN.md
@@ -23,6 +23,41 @@ iqn_value.iqn`, `mess3_reward_state_cycle_1.iqn`, and the library's
 The library's `IQNValueMixin` should be the only one. Anything else that is a
 reusable RL concept goes to `rl-harness`; Kelly wagering stays here.
 
+## 1b. Move the operating point, and pay for it in probe size
+
+Cycle 1 ran at `alpha=0.85` with a self-transition of 0.9, which is close to the
+worst choice in this family for metric sensitivity: a fast-mixing chain lets the
+belief be approximated by an exponentially weighted average of recent one-hot
+observations, which is exactly what an affine probe computes. Slowing the chain
+breaks that approximation.
+
+| point | α | p | R² floor | R² range | accuracy range | probe ±95% at 30k steps |
+|---|---:|---:|---:|---:|---:|---:|
+| cycle 1 | 0.85 | 0.900 | 0.9671 | 0.033 | 0.015 | 0.0006 |
+| candidate A | 0.70 | 0.990 | 0.8967 | 0.103 | 0.134 | 0.0037 |
+| candidate B | 0.60 | 0.995 | 0.8782 | 0.122 | 0.136 | 0.0061 |
+| candidate C | 0.70 | 0.995 | 0.9061 | 0.094 | 0.145 | 0.0046 |
+
+Candidate C is my recommendation: about three times the belief-probe range and
+ten times the accuracy range, while keeping the task clearly learnable at a
+Bayes-optimal accuracy of 0.68.
+
+The trade is precision. Slowing the chain raises the state correlation time from
+7 steps to 133, so a 30,000-step probe rollout collected the way
+`collect_probe_data` collects one — sixteen parallel trajectories — contains
+roughly 100 independent samples rather than 2,300. The honest block-bootstrap
+interval on belief R² widens from ±0.0006 to ±0.0046, and on greedy accuracy to
+about ±0.09.
+
+That is fixable and cheap, but only if it is noticed: **raise the probe to around
+500,000 test steps**. Rollouts cost seconds, so this is the least expensive fix in
+the plan and the easiest one to leave out.
+
+Two consequences follow. Episode length should rise from 512 so that the reset
+back to a uniform belief is a smaller fraction of each trajectory. And at
+γ = 0.99 the effective horizon of about 100 steps is now shorter than the state's
+persistence, which is worth stating when γ is discussed.
+
 ## 2. Report metrics against their range, always
 
 Never present a bare R². Every table gets the four reference rows from
@@ -47,18 +82,18 @@ report three things instead, and let the audience pick:
 - **Position in the usable range**, as `references.experiment` already computes,
   for the one summary slide.
 
-## 3. Fix the checkpoint protocol
+## 3. Treat training duration as an axis, not a hyperparameter
 
 Log-spaced checkpoints for every arm, following `rl-harness/docs/
-checkpoint_strategy.md`, with the probe run at each. Then either:
-
-- pre-register "the final checkpoint" and additionally report the curve, so the
-  reader can see the metric is decaying; or
-- pre-register "the mean over the last k checkpoints", which removes the
-  ±0.0035 within-run fluctuation from the comparison for free.
+checkpoint_strategy.md`, with the probe run at each.
 
-I prefer the second for the primary metric and the first in an appendix, but
-this is a real choice and it must be made before the runs, not after.
+Duration should not be tuned, because belief-probe R² is not monotonic in it and
+the decline is now known to be optimiser drift rather than anything about the
+objective. Picking the step budget after seeing the results would be choosing the
+ranking. So: pre-register the budget, pre-register whether the primary statistic
+is the final checkpoint or the mean over the last k, and report the whole curve
+either way. I prefer the mean over the last k, which removes the ±0.0035
+within-run fluctuation for free.
 
 The step budget must be identical across every arm. 2.5M is defensible; so is
 5M. What is not defensible is the current mixture of 828K, 2.5M, 3M and 20M.
@@ -89,23 +124,41 @@ Note the planning numbers are computed from effects observed at n=3 and are
 therefore optimistic — the observed gap in a small sample is biased upward. Ten
 is already padded for that; I would not go below eight.
 
-## 5. Sweep the coefficients, in two stages
+## 5. Sweep the optimiser and the coefficients, in two stages
 
-Every intervention currently has exactly one tested strength, so no "X does not
-help" claim is available. Proposed grid:
+Only four things are worth sweeping, and the first one matters more than the
+arms do. See `task_parameters/` for the evidence.
 
-| arm | swept coefficient | values |
+| priority | swept | values |
 |---|---|---|
-| all | learning rate | 1e-4, 3e-4, 1e-3 |
-| predictive aux | loss weight λ | 0.03, 0.1, 0.3, 1.0 |
-| max entropy | reward coefficient α | 0.01, 0.05, 0.2, 0.5 |
-| IQN | loss coefficient | 0.125, 0.5, 2.0 |
-| Kelly | direct-loss weight | 0.25, 1.0, 4.0 |
-
-Sweeping the learning rate per arm is not optional. Arms differ in parameter
-count and in how much auxiliary gradient enters the shared trunk while the
-learning rate is pinned at 3e-4, so an arm can currently lose by being
-mis-tuned rather than by being a worse idea.
+| 1 | optimiser | AdamW, Muon |
+| 1 | learning rate | 1e-4, 3e-4, 1e-3 |
+| 2 | predictive aux weight λ | 0.03, 0.1, 0.3, 1.0 |
+| 2 | max-entropy reward coefficient α | 0.01, 0.05, 0.2, 0.5 |
+| 2 | IQN loss coefficient | 0.125, 0.5, 2.0 |
+| 2 | Kelly direct-loss weight | 0.25, 1.0, 4.0 |
+
+**Optimiser and learning rate move the headline metric more than the arms do.**
+Training the study architecture on next-token prediction and probing it as it
+goes, belief-probe R² *falls* with continued training while cross-entropy sits at
+the Bayes floor — 0.9567 to 0.9318 over 6,000 AdamW steps at the cycle-1
+parameters, and 0.9487 to 0.9367 at the slower candidate. That is supervised
+training, so the decline over 20M PPO steps in cycle 1 is not an artefact of
+reinforcement learning. `mess3_supervised` independently found SGD at 0.9979
+against Muon at 0.9843 with identical next-token accuracy, a 0.014 swing from the
+optimiser alone that is larger than most of the between-arm gaps cycle 2 is
+trying to resolve. An arm comparison run at one fixed optimiser and learning rate
+is partly measuring optimiser drift.
+
+Sweeping the learning rate per arm is separately necessary because arms differ in
+parameter count and in how much auxiliary gradient enters the shared trunk, so an
+arm can lose by being mis-tuned rather than by being a worse idea.
+
+**Do not sweep:** context length (64 already suffices at every candidate operating
+point — the belief saturates by 32 observations), GAE λ (irrelevant at γ = 0, low
+leverage at γ = 0.99), batch size, minibatch size, epoch count, clip parameter,
+`d_model`, or `n_layers`. Fix these and record them. γ is a scientific factor,
+not a nuisance hyperparameter; keep it in the design rather than tuning it away.
 
 **Two stages, with disjoint seeds.** Stage 1 sweeps on seeds 0–2 and selects one
 configuration per arm. Stage 2 re-runs the selected configurations on ten
@@ -159,13 +212,13 @@ RTX 4090, median about 6.5. Adding eight checkpoint probes puts it near 12.
 
 | stage | runs | GPU-hours | cost at $0.35/hr |
 |---|---:|---:|---:|
-| stage 1 sweep (7 arms × 12 configs × 3 seeds) | 252 | 50 | ~$18 |
+| stage 1 sweep (7 arms × 6 optimiser/lr × 3 coefficient × 3 seeds) | 378 | 76 | ~$27 |
 | stage 2 confirmation (16 conditions × 10 seeds) | 160 | 32 | ~$11 |
-| references and audit | — | <1 | — |
-| **total** | **412** | **82** | **~$30** |
+| references, operating point, audit | — | <1 | — |
+| **total** | **538** | **108** | **~$38** |
 
-Roughly ten wall-clock hours across eight Vast boxes. The entire committed
-history of these four studies is 19.4 GPU-hours, so this is about four times all
+Roughly fourteen wall-clock hours across eight Vast boxes. The entire committed
+history of these four studies is 19.4 GPU-hours, so this is about five times all
 prior work on the question, for the price of lunch.
 
 `devops.vast.provision up -n N --run "..."` gives one command per box, so the
diff --git a/experiments/mess3_token_guess_cycle_2/operating_point.py b/experiments/mess3_token_guess_cycle_2/operating_point.py
new file mode 100644
index 00000000..512957cd
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/operating_point.py
@@ -0,0 +1,283 @@
+"""Choosing MESS3 transition and emission parameters for metric sensitivity.
+
+`REVIEW.md` found that at the parameters cycle 1 used (`alpha=0.85`, self-transition
+0.9) an affine probe on the raw observations already scores R² = 0.967, leaving
+the belief-probe metric only 0.03 to move through. The natural fix is to change
+the process so that the belief is a less linear function of the observation
+history.
+
+Two things have to be checked before committing to a new operating point, and
+they pull in opposite directions.
+
+Lowering the floor means slowing the chain, because a fast-mixing chain lets the
+belief be approximated by an exponentially weighted average of recent one-hot
+observations, which is exactly what an affine probe computes. But slowing the
+chain also raises the autocorrelation of any rollout drawn from it, and the probe
+is estimated from a rollout. Past some point the metric gains range and loses
+precision at the same rate, and nothing is won.
+
+This module measures both, plus the context length the belief actually requires,
+so the operating point can be chosen rather than guessed.
+"""
+
+from __future__ import annotations
+
+from dataclasses import dataclass
+
+import numpy as np
+
+from experiments.mess3_token_guess_cycle_1.analysis import PROBE_RANK
+from experiments.mess3_token_guess_cycle_2.metric_references import (
+    _probe_r2,
+    TokenStream,
+)
+
+N_CHAINS = 4_000
+# `collect_probe_data` steps this many environments in parallel, so a probe
+# rollout is this many trajectories rather than that many independent draws.
+PROBE_CHAINS = 16
+BURN_IN = 600
+FLOOR_WINDOWS = (4, 8, 16, 32, 64)
+CONTEXT_LENGTHS = (8, 16, 32, 64, 128)
+BLOCK = 512
+RESAMPLES = 300
+
+
+@dataclass(frozen=True, slots=True)
+class OperatingPoint:
+    """One symmetric MESS3 process, described by its two free parameters."""
+
+    alpha: float
+    self_transition: float
+
+    @property
+    def transition_matrix(self) -> np.ndarray:
+        matrix = np.full((3, 3), (1.0 - self.self_transition) / 2.0)
+        np.fill_diagonal(matrix, self.self_transition)
+        return matrix
+
+    @property
+    def emission_matrix(self) -> np.ndarray:
+        matrix = np.full((3, 3), (1.0 - self.alpha) / 2.0)
+        np.fill_diagonal(matrix, self.alpha)
+        return matrix
+
+    @property
+    def state_correlation_time(self) -> float:
+        """1 / (1 - second eigenvalue), the timescale the hidden state persists."""
+
+        return 1.0 / (1.0 - (3.0 * self.self_transition - 1.0) / 2.0)
+
+
+def simulate_parallel(
+    point: OperatingPoint,
+    *,
+    n_steps: int,
+    seed: int,
+    window: int = max(FLOOR_WINDOWS),
+    n_chains: int = N_CHAINS,
+) -> TokenStream:
+    """Draw from ``n_chains`` chains stepped together, after burning each in.
+
+    The chain count controls how correlated the returned samples are, which is a
+    property of the collection scheme rather than of the process. Use the default
+    for estimating a population quantity like the floor, and ``PROBE_CHAINS`` when
+    estimating how precise a real probe rollout would be.
+    """
+
+    transition = point.transition_matrix
+    emission = point.emission_matrix
+    stationary = np.full(3, 1.0 / 3.0)
+    rng = np.random.default_rng(seed)
+    cumulative_emission = emission.cumsum(1)
+    cumulative_transition = transition.cumsum(1)
+
+    state = rng.choice(3, size=n_chains, p=stationary)
+    belief = np.repeat(stationary[None, :], n_chains, axis=0)
+    history = np.zeros((n_chains, window), dtype=np.int64)
+
+    beliefs, tokens, windows = [], [], []
+    per_chain = int(np.ceil(n_steps / n_chains))
+    for step in range(BURN_IN + per_chain):
+        token = (rng.random(n_chains)[:, None] > cumulative_emission[state]).sum(1)
+        if step >= BURN_IN:
+            beliefs.append(belief.copy())
+            tokens.append(token.copy())
+            windows.append(history.copy())
+        history = np.concatenate([history[:, 1:], token[:, None]], axis=1)
+        posterior = belief * emission[:, token].T
+        belief = (posterior / posterior.sum(1, keepdims=True)) @ transition
+        state = (rng.random(n_chains)[:, None] > cumulative_transition[state]).sum(1)
+    # Order by chain so that contiguous slices are contiguous in time, which the
+    # block bootstrap relies on.
+    return TokenStream(
+        beliefs=np.concatenate(beliefs).reshape(-1, n_chains, 3).transpose(1, 0, 2).reshape(-1, 3)[:n_steps],
+        tokens=np.concatenate(tokens).reshape(-1, n_chains).T.reshape(-1)[:n_steps],
+        windows=np.concatenate(windows).reshape(-1, n_chains, window).transpose(1, 0, 2).reshape(-1, window)[:n_steps],
+    )
+
+
+def _one_hot(windows: np.ndarray, k: int) -> np.ndarray:
+    return np.eye(3, dtype=np.float64)[windows[:, -k:]].reshape(len(windows), -1)
+
+
+def belief_r2_floor(fit: TokenStream, test: TokenStream) -> float:
+    """The best affine readout of the raw observations, over window lengths."""
+
+    return max(
+        float(_probe_r2(_one_hot(fit.windows, k), fit.beliefs, _one_hot(test.windows, k), test.beliefs)[0])
+        for k in FLOOR_WINDOWS
+    )
+
+
+def accuracy_bounds(point: OperatingPoint, test: TokenStream) -> tuple[float, float]:
+    """Bayes-optimal accuracy, and the repeat-the-previous-observation rule."""
+
+    emission = point.emission_matrix
+    ceiling = float((( test.beliefs @ emission).argmax(1) == test.tokens).mean())
+    belief = np.repeat(np.full(3, 1.0 / 3.0)[None, :], len(test.windows), axis=0)
+    belief = belief * emission[:, test.windows[:, -1]].T
+    belief /= belief.sum(1, keepdims=True)
+    belief = belief @ point.transition_matrix
+    floor = float(((belief @ emission).argmax(1) == test.tokens).mean())
+    return floor, ceiling
+
+
+def context_requirement(
+    point: OperatingPoint,
+    test: TokenStream,
+    *,
+    context_lengths: tuple[int, ...] = CONTEXT_LENGTHS,
+) -> dict[int, float]:
+    """Best belief R² available to a model limited to ``k`` observations.
+
+    The exact posterior given the last ``k`` observations under a stationary
+    prior is what a ``k``-context model can compute at best, so this is an
+    architecture ceiling that no training objective can exceed.
+    """
+
+    transition = point.transition_matrix
+    emission = point.emission_matrix
+    scores: dict[int, float] = {}
+    for k in context_lengths:
+        if k > test.windows.shape[1]:
+            continue
+        belief = np.repeat(np.full(3, 1.0 / 3.0)[None, :], len(test.windows), axis=0)
+        for offset in range(k):
+            belief = belief * emission[:, test.windows[:, -k + offset]].T
+            belief /= belief.sum(1, keepdims=True)
+            belief = belief @ transition
+        residual = ((belief - test.beliefs) ** 2).sum()
+        total = ((test.beliefs - test.beliefs.mean(0)) ** 2).sum()
+        scores[k] = float(1.0 - residual / total)
+    return scores
+
+
+def integrated_autocorrelation(point: OperatingPoint, *, n_steps: int, seed: int) -> float:
+    """Sum the belief autocorrelation along a single chain.
+
+    Rollouts are drawn as trajectories, so this is what divides the nominal probe
+    size to give the number of independent samples it really contains.
+    """
+
+    transition = point.transition_matrix
+    emission = point.emission_matrix
+    cumulative_emission = emission.cumsum(1)
+    cumulative_transition = transition.cumsum(1)
+    rng = np.random.default_rng(seed)
+    stationary = np.full(3, 1.0 / 3.0)
+    state = int(rng.choice(3, p=stationary))
+    belief = stationary.copy()
+    series = np.empty(n_steps)
+    for step in range(n_steps + BURN_IN):
+        token = int((rng.random() > cumulative_emission[state]).sum())
+        if step >= BURN_IN:
+            series[step - BURN_IN] = belief[0]
+        posterior = belief * emission[:, token]
+        belief = (posterior / posterior.sum()) @ transition
+        state = int((rng.random() > cumulative_transition[state]).sum())
+
+    centred = series - series.mean()
+    variance = centred.var()
+    tau = 1.0
+    for lag in range(1, min(4_000, n_steps // 4)):
+        value = float((centred[:-lag] * centred[lag:]).mean() / variance)
+        if value <= 0.0:
+            break
+        tau += 2.0 * value
+    return tau
+
+
+def block_bootstrap_interval(
+    fit: TokenStream,
+    test: TokenStream,
+    *,
+    window: int,
+    seed: int,
+    block: int = BLOCK,
+    resamples: int = RESAMPLES,
+) -> tuple[float, float]:
+    """Resample contiguous blocks, which an i.i.d. bootstrap would understate."""
+
+    _, predicted = _probe_r2(
+        _one_hot(fit.windows, window),
+        fit.beliefs,
+        _one_hot(test.windows, window),
+        test.beliefs,
+    )
+    rng = np.random.default_rng(seed)
+    n = len(test.beliefs)
+    n_blocks = max(1, n // block)
+    scores = []
+    for _ in range(resamples):
+        starts = rng.integers(0, max(1, n - block), n_blocks)
+        index = (starts[:, None] + np.arange(block)[None, :]).ravel()
+        index = index[index < n]
+        residual = ((predicted[index] - test.beliefs[index]) ** 2).sum()
+        total = (
+            (test.beliefs[index] - test.beliefs[index].mean(0)) ** 2
+        ).sum()
+        scores.append(1.0 - residual / total)
+    low, high = np.percentile(scores, [2.5, 97.5])
+    return float(low), float(high)
+
+
+def evaluate(
+    point: OperatingPoint,
+    *,
+    fit_steps: int,
+    test_steps: int,
+    seed: int,
+) -> dict[str, float | dict[int, float]]:
+    """Score one candidate operating point on range and on precision."""
+
+    fit = simulate_parallel(point, n_steps=fit_steps, seed=seed)
+    test = simulate_parallel(point, n_steps=test_steps, seed=seed + 1)
+    floor = belief_r2_floor(fit, test)
+    accuracy_floor, accuracy_ceiling = accuracy_bounds(point, test)
+    tau = integrated_autocorrelation(point, n_steps=min(120_000, 40 * test_steps), seed=seed + 2)
+    # Estimate the probe's precision from a rollout collected the way the study
+    # collects one, not from the near-independent sample used above.
+    probe_fit = simulate_parallel(
+        point, n_steps=fit_steps, seed=seed + 4, n_chains=PROBE_CHAINS
+    )
+    probe_test = simulate_parallel(
+        point, n_steps=test_steps, seed=seed + 5, n_chains=PROBE_CHAINS
+    )
+    low, high = block_bootstrap_interval(probe_fit, probe_test, window=32, seed=seed + 3)
+    return {
+        "alpha": point.alpha,
+        "self_transition": point.self_transition,
+        "state_correlation_time": point.state_correlation_time,
+        "belief_r2_floor": floor,
+        "belief_r2_range": 1.0 - floor,
+        "accuracy_floor": accuracy_floor,
+        "accuracy_ceiling": accuracy_ceiling,
+        "accuracy_range": accuracy_ceiling - accuracy_floor,
+        "context_requirement": context_requirement(point, test),
+        "integrated_autocorrelation": tau,
+        "effective_sample_size": test_steps / tau,
+        "probe_block_bootstrap_ci": [low, high],
+        "probe_ci_half_width": (high - low) / 2.0,
+        "probe_rank": PROBE_RANK,
+    }
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/__init__.py b/experiments/mess3_token_guess_cycle_2/task_parameters/__init__.py
new file mode 100644
index 00000000..e69de29b
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py b/experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py
new file mode 100644
index 00000000..2aeebe55
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py
@@ -0,0 +1,210 @@
+"""Choose the MESS3 operating point for cycle 2."""
+
+from __future__ import annotations
+
+from typing import Any
+
+import matplotlib
+
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt  # noqa: E402
+
+from experiments.mess3_token_guess_cycle_2.operating_point import (
+    OperatingPoint,
+    accuracy_bounds,
+    belief_r2_floor,
+    evaluate,
+    simulate_parallel,
+)
+from harness.artifacts import RunArtifacts
+from harness.context import RunContext
+
+FIT_STEPS = 60_000
+TEST_STEPS = 30_000
+SMOKE_FIT_STEPS = 8_000
+SMOKE_TEST_STEPS = 4_000
+
+GRID_ALPHAS = (0.5, 0.6, 0.7, 0.85, 0.95)
+GRID_TRANSITIONS = (0.9, 0.97, 0.99, 0.995)
+SMOKE_ALPHAS = (0.7, 0.85)
+SMOKE_TRANSITIONS = (0.9, 0.995)
+
+CANDIDATES = {
+    "cycle_1": OperatingPoint(alpha=0.85, self_transition=0.90),
+    "candidate_a": OperatingPoint(alpha=0.70, self_transition=0.99),
+    "candidate_b": OperatingPoint(alpha=0.60, self_transition=0.995),
+    "candidate_c": OperatingPoint(alpha=0.70, self_transition=0.995),
+}
+
+# Measured by training the study architecture (3 layers, d_model 96, context 64)
+# on next-token prediction to the Bayes cross-entropy, then probing it. Recorded
+# rather than recomputed because it takes about twenty minutes per point on CPU.
+SUPERVISED_PROBE_R2 = {
+    "cycle_1": {1_500: 0.9567, 3_000: 0.9532, 4_500: 0.9434, 6_000: 0.9318},
+    "candidate_c": {1_500: 0.9487, 3_000: 0.9478, 4_500: 0.9439, 6_000: 0.9367},
+}
+
+
+def _plot_grid(grid: dict[str, Any], *, path) -> None:
+    alphas = sorted({entry["alpha"] for entry in grid.values()})
+    transitions = sorted({entry["self_transition"] for entry in grid.values()})
+    figure, axes = plt.subplots(1, 2, figsize=(11.5, 4.4))
+    for axis, key, title in (
+        (axes[0], "belief_r2_range", "belief-probe R² range (1 − floor)"),
+        (axes[1], "accuracy_range", "greedy accuracy range (Bayes − repeat-last)"),
+    ):
+        for transition in transitions:
+            values = [
+                grid[f"{alpha}_{transition}"][key]
+                for alpha in alphas
+                if f"{alpha}_{transition}" in grid
+            ]
+            axis.plot(alphas[: len(values)], values, "o-", label=f"p = {transition}")
+        axis.set_xlabel("emission concentration α")
+        axis.set_title(title, fontsize=10)
+        axis.grid(alpha=0.2)
+        axis.legend(fontsize=8)
+    axes[0].set_ylabel("usable range of the metric")
+    figure.suptitle(
+        "Both metrics are near-degenerate at the parameters cycle 1 used "
+        "(α = 0.85, p = 0.9)",
+        fontsize=11,
+    )
+    figure.tight_layout()
+    figure.savefig(path, dpi=200)
+    plt.close(figure)
+
+
+def _findings(result: dict[str, Any]) -> str:
+    lines = [
+        "# Choosing the MESS3 operating point",
+        "",
+        "## Candidates",
+        "",
+        "`range` is what the belief-probe metric can move through; `ESS` is how "
+        "many independent samples a 30,000-step probe rollout actually contains, "
+        "given how slowly the chain mixes.",
+        "",
+        "| point | α | p | τ | R² floor | R² range | acc range | ESS | probe ±95% |",
+        "|---|---:|---:|---:|---:|---:|---:|---:|---:|",
+    ]
+    for name, entry in result["candidates"].items():
+        lines.append(
+            f"| `{name}` | {entry['alpha']:.2f} | {entry['self_transition']:.3f} | "
+            f"{entry['state_correlation_time']:.0f} | "
+            f"{entry['belief_r2_floor']:.4f} | {entry['belief_r2_range']:.4f} | "
+            f"{entry['accuracy_range']:.4f} | "
+            f"{entry['effective_sample_size']:.0f} | "
+            f"{entry['probe_ci_half_width']:.4f} |"
+        )
+    lines.extend(
+        [
+            "",
+            "## Context length",
+            "",
+            "Best belief R² available to a model that can see only the last k "
+            "observations. No objective can beat this.",
+            "",
+            "| point | " + " | ".join(
+                f"k={k}"
+                for k in sorted(
+                    next(iter(result["candidates"].values()))["context_requirement"]
+                )
+            ) + " |",
+            "|---" * (1 + len(next(iter(result["candidates"].values()))["context_requirement"])) + "|",
+        ]
+    )
+    for name, entry in result["candidates"].items():
+        row = " | ".join(
+            f"{value:.4f}" for _, value in sorted(entry["context_requirement"].items())
+        )
+        lines.append(f"| `{name}` | {row} |")
+    lines.extend(
+        [
+            "",
+            "The study's context length of 64 is sufficient at every candidate, so "
+            "it does not need sweeping. The belief converges much faster than the "
+            "hidden state does, because each observation is informative enough to "
+            "wash out the prior well before the state decorrelates.",
+            "",
+            "## Supervised ceiling",
+            "",
+            "Training the study architecture on next-token prediction to the Bayes "
+            "cross-entropy, then probing it:",
+            "",
+            "| point | 1.5k steps | 3k | 4.5k | 6k |",
+            "|---|---:|---:|---:|---:|",
+        ]
+    )
+    for name, curve in SUPERVISED_PROBE_R2.items():
+        row = " | ".join(f"{curve[step]:.4f}" for step in sorted(curve))
+        lines.append(f"| `{name}` | {row} |")
+    lines.extend(
+        [
+            "",
+            "Belief-probe R² falls with continued training at both points while "
+            "cross-entropy stays at the Bayes floor. This is supervised training, "
+            "so the decline cycle 1 saw over 20M PPO steps is not caused by "
+            "reinforcement learning. It is optimiser-driven drift in a "
+            "representation the task no longer constrains, which makes learning "
+            "rate, optimiser, and training duration larger influences on the "
+            "headline metric than most of the differences between arms.",
+            "",
+        ]
+    )
+    return "\n".join(lines)
+
+
+def run(context: RunContext) -> dict[str, Any]:
+    if context.seed is None:
+        raise ValueError("choosing an operating point requires a resolved seed")
+    outputs = RunArtifacts.from_context(context)
+    outputs.prepare()
+    fit_steps = SMOKE_FIT_STEPS if context.smoke else FIT_STEPS
+    test_steps = SMOKE_TEST_STEPS if context.smoke else TEST_STEPS
+    alphas = SMOKE_ALPHAS if context.smoke else GRID_ALPHAS
+    transitions = SMOKE_TRANSITIONS if context.smoke else GRID_TRANSITIONS
+
+    grid: dict[str, Any] = {}
+    for alpha in alphas:
+        for transition in transitions:
+            point = OperatingPoint(alpha=alpha, self_transition=transition)
+            fit = simulate_parallel(point, n_steps=fit_steps, seed=context.seed)
+            test = simulate_parallel(point, n_steps=test_steps, seed=context.seed + 1)
+            floor = belief_r2_floor(fit, test)
+            accuracy_floor, accuracy_ceiling = accuracy_bounds(point, test)
+            grid[f"{alpha}_{transition}"] = {
+                "alpha": alpha,
+                "self_transition": transition,
+                "belief_r2_floor": floor,
+                "belief_r2_range": 1.0 - floor,
+                "accuracy_floor": accuracy_floor,
+                "accuracy_ceiling": accuracy_ceiling,
+                "accuracy_range": accuracy_ceiling - accuracy_floor,
+            }
+
+    candidates = {
+        name: evaluate(
+            point,
+            fit_steps=fit_steps,
+            test_steps=test_steps,
+            seed=context.seed,
+        )
+        for name, point in CANDIDATES.items()
+    }
+    figure_path = context.results_dir / "operating_point_grid.png"
+    _plot_grid(grid, path=figure_path)
+    result = {
+        "seed": context.seed,
+        "smoke": context.smoke,
+        "grid": grid,
+        "candidates": candidates,
+        "supervised_probe_r2": {
+            name: {str(step): value for step, value in curve.items()}
+            for name, curve in SUPERVISED_PROBE_R2.items()
+        },
+        "figure": str(figure_path),
+    }
+    outputs.write_json("task_parameters.json", result)
+    (context.results_dir / "findings.md").write_text(_findings(result))
+    return result
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/findings.md b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/findings.md
new file mode 100644
index 00000000..0891160c
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/findings.md
@@ -0,0 +1,36 @@
+# Choosing the MESS3 operating point
+
+## Candidates
+
+`range` is what the belief-probe metric can move through; `ESS` is how many independent samples a 30,000-step probe rollout actually contains, given how slowly the chain mixes.
+
+| point | α | p | τ | R² floor | R² range | acc range | ESS | probe ±95% |
+|---|---:|---:|---:|---:|---:|---:|---:|---:|
+| `cycle_1` | 0.85 | 0.900 | 7 | 0.9671 | 0.0329 | 0.0151 | 2324 | 0.0006 |
+| `candidate_a` | 0.70 | 0.990 | 67 | 0.8967 | 0.1033 | 0.1337 | 198 | 0.0037 |
+| `candidate_b` | 0.60 | 0.995 | 133 | 0.8782 | 0.1218 | 0.1362 | 104 | 0.0061 |
+| `candidate_c` | 0.70 | 0.995 | 133 | 0.9061 | 0.0939 | 0.1447 | 104 | 0.0046 |
+
+## Context length
+
+Best belief R² available to a model that can see only the last k observations. No objective can beat this.
+
+| point | k=8 | k=16 | k=32 | k=64 |
+|---|---|---|---|---|
+| `cycle_1` | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
+| `candidate_a` | 0.9418 | 0.9969 | 1.0000 | 1.0000 |
+| `candidate_b` | 0.7623 | 0.9431 | 0.9965 | 1.0000 |
+| `candidate_c` | 0.9232 | 0.9951 | 1.0000 | 1.0000 |
+
+The study's context length of 64 is sufficient at every candidate, so it does not need sweeping. The belief converges much faster than the hidden state does, because each observation is informative enough to wash out the prior well before the state decorrelates.
+
+## Supervised ceiling
+
+Training the study architecture on next-token prediction to the Bayes cross-entropy, then probing it:
+
+| point | 1.5k steps | 3k | 4.5k | 6k |
+|---|---:|---:|---:|---:|
+| `cycle_1` | 0.9567 | 0.9532 | 0.9434 | 0.9318 |
+| `candidate_c` | 0.9487 | 0.9478 | 0.9439 | 0.9367 |
+
+Belief-probe R² falls with continued training at both points while cross-entropy stays at the Bayes floor. This is supervised training, so the decline cycle 1 saw over 20M PPO steps is not caused by reinforcement learning. It is optimiser-driven drift in a representation the task no longer constrains, which makes learning rate, optimiser, and training duration larger influences on the headline metric than most of the differences between arms.
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/operating_point_grid.png b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/operating_point_grid.png
new file mode 100644
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new file mode 100644
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+++ b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/run_manifest.json
@@ -0,0 +1,75 @@
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diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/task_parameters.json b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/task_parameters.json
new file mode 100644
index 00000000..24a8c63a
--- /dev/null
+++ b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/task_parameters.json
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+      "context_requirement": {
+        "16": 0.995128172467821,
+        "32": 0.9999892025679498,
+        "64": 1.0,
+        "8": 0.923215622292938
+      },
+      "effective_sample_size": 104.08060835175199,
+      "integrated_autocorrelation": 288.23813076314525,
+      "probe_block_bootstrap_ci": [
+        0.9020596893487307,
+        0.9111640239783987
+      ],
+      "probe_ci_half_width": 0.004552167314833988,
+      "probe_rank": 2,
+      "self_transition": 0.995,
+      "state_correlation_time": 133.33333333333223
+    },
+    "cycle_1": {
+      "accuracy_ceiling": 0.6882666666666667,
+      "accuracy_floor": 0.6731333333333334,
+      "accuracy_range": 0.015133333333333332,
+      "alpha": 0.85,
+      "belief_r2_floor": 0.9671181729028167,
+      "belief_r2_range": 0.03288182709718335,
+      "context_requirement": {
+        "16": 0.9999999999275502,
+        "32": 1.0,
+        "64": 1.0,
+        "8": 0.9999795066409225
+      },
+      "effective_sample_size": 2324.268140252257,
+      "integrated_autocorrelation": 12.907288741971072,
+      "probe_block_bootstrap_ci": [
+        0.9663334399107788,
+        0.967472758102929
+      ],
+      "probe_ci_half_width": 0.0005696590960750925,
+      "probe_rank": 2,
+      "self_transition": 0.9,
+      "state_correlation_time": 6.6666666666666705
+    }
+  },
+  "figure": "/workspace/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/operating_point_grid.png",
+  "grid": {
+    "0.5_0.9": {
+      "accuracy_ceiling": 0.3857333333333333,
+      "accuracy_floor": 0.3646333333333333,
+      "accuracy_range": 0.021100000000000008,
+      "alpha": 0.5,
+      "belief_r2_floor": 0.9533460163985678,
+      "belief_r2_range": 0.04665398360143225,
+      "self_transition": 0.9
+    },
+    "0.5_0.97": {
+      "accuracy_ceiling": 0.4261333333333333,
+      "accuracy_floor": 0.3727666666666667,
+      "accuracy_range": 0.05336666666666662,
+      "alpha": 0.5,
+      "belief_r2_floor": 0.8948456505926992,
+      "belief_r2_range": 0.10515434940730084,
+      "self_transition": 0.97
+    },
+    "0.5_0.99": {
+      "accuracy_ceiling": 0.45253333333333334,
+      "accuracy_floor": 0.37233333333333335,
+      "accuracy_range": 0.0802,
+      "alpha": 0.5,
+      "belief_r2_floor": 0.8570197263850013,
+      "belief_r2_range": 0.14298027361499865,
+      "self_transition": 0.99
+    },
+    "0.5_0.995": {
+      "accuracy_ceiling": 0.47273333333333334,
+      "accuracy_floor": 0.3746333333333333,
+      "accuracy_range": 0.09810000000000002,
+      "alpha": 0.5,
+      "belief_r2_floor": 0.8477653930238022,
+      "belief_r2_range": 0.15223460697619784,
+      "self_transition": 0.995
+    },
+    "0.6_0.9": {
+      "accuracy_ceiling": 0.4498,
+      "accuracy_floor": 0.41983333333333334,
+      "accuracy_range": 0.02996666666666664,
+      "alpha": 0.6,
+      "belief_r2_floor": 0.9394608905059447,
+      "belief_r2_range": 0.060539109494055277,
+      "self_transition": 0.9
+    },
+    "0.6_0.97": {
+      "accuracy_ceiling": 0.5214,
+      "accuracy_floor": 0.43806666666666666,
+      "accuracy_range": 0.08333333333333331,
+      "alpha": 0.6,
+      "belief_r2_floor": 0.8848229869875442,
+      "belief_r2_range": 0.11517701301245575,
+      "self_transition": 0.97
+    },
+    "0.6_0.99": {
+      "accuracy_ceiling": 0.5583666666666667,
+      "accuracy_floor": 0.4363,
+      "accuracy_range": 0.12206666666666666,
+      "alpha": 0.6,
+      "belief_r2_floor": 0.8723894114642732,
+      "belief_r2_range": 0.12761058853572682,
+      "self_transition": 0.99
+    },
+    "0.6_0.995": {
+      "accuracy_ceiling": 0.5779333333333333,
+      "accuracy_floor": 0.4417333333333333,
+      "accuracy_range": 0.1362,
+      "alpha": 0.6,
+      "belief_r2_floor": 0.878248921449702,
+      "belief_r2_range": 0.12175107855029799,
+      "self_transition": 0.995
+    },
+    "0.7_0.9": {
+      "accuracy_ceiling": 0.5384333333333333,
+      "accuracy_floor": 0.5033333333333333,
+      "accuracy_range": 0.03510000000000002,
+      "alpha": 0.7,
+      "belief_r2_floor": 0.9409244989416722,
+      "belief_r2_range": 0.05907550105832782,
+      "self_transition": 0.9
+    },
+    "0.7_0.97": {
+      "accuracy_ceiling": 0.6201333333333333,
+      "accuracy_floor": 0.5280666666666667,
+      "accuracy_range": 0.09206666666666663,
+      "alpha": 0.7,
+      "belief_r2_floor": 0.8988869558203207,
+      "belief_r2_range": 0.10111304417967926,
+      "self_transition": 0.97
+    },
+    "0.7_0.99": {
+      "accuracy_ceiling": 0.6639666666666667,
+      "accuracy_floor": 0.5302666666666667,
+      "accuracy_range": 0.13370000000000004,
+      "alpha": 0.7,
+      "belief_r2_floor": 0.8967379018599352,
+      "belief_r2_range": 0.10326209814006482,
+      "self_transition": 0.99
+    },
+    "0.7_0.995": {
+      "accuracy_ceiling": 0.6837666666666666,
+      "accuracy_floor": 0.5390666666666667,
+      "accuracy_range": 0.14469999999999994,
+      "alpha": 0.7,
+      "belief_r2_floor": 0.906136201034249,
+      "belief_r2_range": 0.09386379896575103,
+      "self_transition": 0.995
+    },
+    "0.85_0.9": {
+      "accuracy_ceiling": 0.6882666666666667,
+      "accuracy_floor": 0.6731333333333334,
+      "accuracy_range": 0.015133333333333332,
+      "alpha": 0.85,
+      "belief_r2_floor": 0.9671181729028167,
+      "belief_r2_range": 0.03288182709718335,
+      "self_transition": 0.9
+    },
+    "0.85_0.97": {
+      "accuracy_ceiling": 0.7830333333333334,
+      "accuracy_floor": 0.7164666666666667,
+      "accuracy_range": 0.06656666666666666,
+      "alpha": 0.85,
+      "belief_r2_floor": 0.9378868344834231,
+      "belief_r2_range": 0.06211316551657686,
+      "self_transition": 0.97
+    },
+    "0.85_0.99": {
+      "accuracy_ceiling": 0.8216,
+      "accuracy_floor": 0.7285333333333334,
+      "accuracy_range": 0.09306666666666663,
+      "alpha": 0.85,
+      "belief_r2_floor": 0.937784849207427,
+      "belief_r2_range": 0.062215150792573004,
+      "self_transition": 0.99
+    },
+    "0.85_0.995": {
+      "accuracy_ceiling": 0.8370333333333333,
+      "accuracy_floor": 0.7351666666666666,
+      "accuracy_range": 0.10186666666666666,
+      "alpha": 0.85,
+      "belief_r2_floor": 0.9442063541758701,
+      "belief_r2_range": 0.055793645824129934,
+      "self_transition": 0.995
+    },
+    "0.95_0.9": {
+      "accuracy_ceiling": 0.8206,
+      "accuracy_floor": 0.8206,
+      "accuracy_range": 0.0,
+      "alpha": 0.95,
+      "belief_r2_floor": 0.9948367481576547,
+      "belief_r2_range": 0.00516325184234534,
+      "self_transition": 0.9
+    },
+    "0.95_0.97": {
+      "accuracy_ceiling": 0.8931333333333333,
+      "accuracy_floor": 0.8796,
+      "accuracy_range": 0.013533333333333286,
+      "alpha": 0.95,
+      "belief_r2_floor": 0.9810292859935144,
+      "belief_r2_range": 0.018970714006485645,
+      "self_transition": 0.97
+    },
+    "0.95_0.99": {
+      "accuracy_ceiling": 0.9311666666666667,
+      "accuracy_floor": 0.8981666666666667,
+      "accuracy_range": 0.03300000000000003,
+      "alpha": 0.95,
+      "belief_r2_floor": 0.974744097401559,
+      "belief_r2_range": 0.025255902598441016,
+      "self_transition": 0.99
+    },
+    "0.95_0.995": {
+      "accuracy_ceiling": 0.9419,
+      "accuracy_floor": 0.9030333333333334,
+      "accuracy_range": 0.038866666666666605,
+      "alpha": 0.95,
+      "belief_r2_floor": 0.9756246450262978,
+      "belief_r2_range": 0.024375354973702246,
+      "self_transition": 0.995
+    }
+  },
+  "seed": 42,
+  "smoke": false,
+  "supervised_probe_r2": {
+    "candidate_c": {
+      "1500": 0.9487,
+      "3000": 0.9478,
+      "4500": 0.9439,
+      "6000": 0.9367
+    },
+    "cycle_1": {
+      "1500": 0.9567,
+      "3000": 0.9532,
+      "4500": 0.9434,
+      "6000": 0.9318
+    }
+  }
+}
diff --git a/tests/test_mess3_token_guess_operating_point.py b/tests/test_mess3_token_guess_operating_point.py
new file mode 100644
index 00000000..2f968706
--- /dev/null
+++ b/tests/test_mess3_token_guess_operating_point.py
@@ -0,0 +1,96 @@
+"""Tests for choosing the MESS3 operating point."""
+
+from __future__ import annotations
+
+import numpy as np
+import pytest
+
+from experiments.mess3_token_guess_cycle_2.operating_point import (
+    PROBE_CHAINS,
+    OperatingPoint,
+    accuracy_bounds,
+    belief_r2_floor,
+    block_bootstrap_interval,
+    context_requirement,
+    integrated_autocorrelation,
+    simulate_parallel,
+)
+
+CYCLE_1 = OperatingPoint(alpha=0.85, self_transition=0.90)
+SLOW = OperatingPoint(alpha=0.70, self_transition=0.995)
+
+
+def _streams(point, fit=20_000, test=10_000, seed=7):
+    return (
+        simulate_parallel(point, n_steps=fit, seed=seed),
+        simulate_parallel(point, n_steps=test, seed=seed + 1),
+    )
+
+
+def test_operating_point_builds_valid_stochastic_matrices():
+    for point in (CYCLE_1, SLOW):
+        for matrix in (point.transition_matrix, point.emission_matrix):
+            np.testing.assert_allclose(matrix.sum(axis=1), 1.0)
+            assert (matrix >= 0.0).all()
+    assert CYCLE_1.transition_matrix[0, 0] == pytest.approx(0.90)
+    assert CYCLE_1.emission_matrix[1, 1] == pytest.approx(0.85)
+
+
+def test_slower_chains_persist_for_longer():
+    assert SLOW.state_correlation_time > 10 * CYCLE_1.state_correlation_time
+    assert CYCLE_1.state_correlation_time == pytest.approx(1 / (1 - 0.85), rel=1e-6)
+
+
+def test_slowing_the_chain_lowers_the_no_network_floor():
+    cycle_1_floor = belief_r2_floor(*_streams(CYCLE_1))
+    slow_floor = belief_r2_floor(*_streams(SLOW))
+
+    # Cycle 1's parameters leave the metric almost no room.
+    assert cycle_1_floor > 0.95
+    # The slower chain roughly triples what the metric can resolve.
+    assert (1 - slow_floor) > 2.0 * (1 - cycle_1_floor)
+
+
+def test_slowing_the_chain_also_widens_the_accuracy_range():
+    _, cycle_1_test = _streams(CYCLE_1)
+    _, slow_test = _streams(SLOW)
+    cycle_1_low, cycle_1_high = accuracy_bounds(CYCLE_1, cycle_1_test)
+    slow_low, slow_high = accuracy_bounds(SLOW, slow_test)
+
+    assert cycle_1_high - cycle_1_low < 0.03
+    assert slow_high - slow_low > 3.0 * (cycle_1_high - cycle_1_low)
+    # The task has to stay learnable: well above chance.
+    assert slow_high > 0.5
+
+
+def test_a_64_observation_context_suffices_at_every_candidate():
+    for point in (CYCLE_1, SLOW):
+        _, test = _streams(point)
+        scores = context_requirement(point, test, context_lengths=(8, 32, 64))
+        assert scores[64] > 0.999, point
+        assert scores[64] >= scores[32] >= scores[8]
+
+
+def test_slowing_the_chain_costs_independent_probe_samples():
+    fast = integrated_autocorrelation(CYCLE_1, n_steps=60_000, seed=3)
+    slow = integrated_autocorrelation(SLOW, n_steps=60_000, seed=3)
+
+    assert fast < 30.0
+    # The precision cost is what a wider metric is traded against.
+    assert slow > 5.0 * fast
+
+
+def test_a_realistically_collected_probe_has_a_much_wider_interval():
+    # Sixteen long trajectories is how `collect_probe_data` gathers a rollout.
+    # Four thousand short ones is nearly independent sampling, and flatters the
+    # interval by hiding the correlation a real probe actually incurs.
+    def interval(n_chains):
+        fit = simulate_parallel(SLOW, n_steps=20_000, seed=7, n_chains=n_chains)
+        test = simulate_parallel(SLOW, n_steps=10_000, seed=8, n_chains=n_chains)
+        low, high = block_bootstrap_interval(
+            fit, test, window=32, seed=5, resamples=120
+        )
+        return (high - low) / 2.0
+
+    assert interval(PROBE_CHAINS) > 2.0 * interval(4_000)
+    assert interval(PROBE_CHAINS) > 0.005

From 7a9cab7d9e3edde99287abb80ec3e5dbcf56c22a Mon Sep 17 00:00:00 2001
From: Cursor Agent <cursoragent@cursor.com>
Date: Mon, 27 Jul 2026 06:57:48 +0000
Subject: [PATCH 6/6] Size the context length and the probe from measurement

CausalTransformerEncoder bands attention at every layer, so the receptive field
is n_layers * context_len. Confirmed by perturbing inputs at increasing distance:
three layers at context_len 64 reach exactly 192 observations, while the belief
needs about 32. Dropping to context_len 16 leaves a 50% margin, costs nothing
measurable in available belief R2 at any candidate operating point, and makes a
learner step 3.7x faster, a cached rollout step 3.4x faster, and a whole PPO run
1.6x faster at matched environment steps.

Also replace the probe-precision argument with direct replication. Running the
identical probe 60 times on fresh rollouts gives a spread of 0.0025 at 30,000
test steps on the slower chain, against 0.0004 at the cycle-1 parameters, and
falls to 0.0005 by 500,000 steps. This confirms the moving-block bootstrap, which
predicted 0.0047 against a measured 0.0047, and confirms that the i.i.d.
bootstrap used in REVIEW.md understates the error whenever the chain is slow.

Co-authored-by: Alex Vardakostas <Al-does@users.noreply.github.com>
---
 .../mess3_token_guess_cycle_2/NOTES.md        |  11 ++
 experiments/mess3_token_guess_cycle_2/PLAN.md | 102 +++++++++++++----
 .../operating_point.py                        | 107 ++++++++++++++++++
 .../task_parameters/experiment.py             |  24 +++-
 .../findings.md                               |  11 +-
 .../operating_point_grid.png                  | Bin
 .../run_manifest.json                         |  16 +--
 .../task_parameters.json                      |  10 +-
 .../test_mess3_token_guess_operating_point.py |  56 +++++++++
 9 files changed, 304 insertions(+), 33 deletions(-)
 rename experiments/mess3_token_guess_cycle_2/task_parameters/results/{20260727T060541Z-53419d82 => 20260727T065519Z-ef24bfec}/findings.md (70%)
 rename experiments/mess3_token_guess_cycle_2/task_parameters/results/{20260727T060541Z-53419d82 => 20260727T065519Z-ef24bfec}/operating_point_grid.png (100%)
 rename experiments/mess3_token_guess_cycle_2/task_parameters/results/{20260727T060541Z-53419d82 => 20260727T065519Z-ef24bfec}/run_manifest.json (79%)
 rename experiments/mess3_token_guess_cycle_2/task_parameters/results/{20260727T060541Z-53419d82 => 20260727T065519Z-ef24bfec}/task_parameters.json (95%)

diff --git a/experiments/mess3_token_guess_cycle_2/NOTES.md b/experiments/mess3_token_guess_cycle_2/NOTES.md
index dace163b..6f792656 100644
--- a/experiments/mess3_token_guess_cycle_2/NOTES.md
+++ b/experiments/mess3_token_guess_cycle_2/NOTES.md
@@ -55,6 +55,17 @@ while cross-entropy stays at the Bayes floor. The decline cycle 1 saw over 20M
 PPO steps is therefore not a property of reinforcement learning, and optimiser
 and learning rate move the headline metric more than most of the arms do.
 
+Two operational consequences:
+
+- `CausalTransformerEncoder` bands attention at every layer, so `context_len=64`
+  over three layers reaches 192 observations while the belief needs about 32.
+  Dropping to `context_len=16` costs nothing measurable and makes a learner step
+  3.7x faster, a cached rollout step 3.4x faster, and a whole PPO run 1.6x
+  faster.
+- A 30,000-step probe of the slower chain has a measurement error of ±0.0025,
+  the same size as the smallest seed-to-seed spreads. Raise the probe to about
+  500,000 test steps, where it falls to ±0.0005.
+
 Regenerate all three with:
 
 ```bash
diff --git a/experiments/mess3_token_guess_cycle_2/PLAN.md b/experiments/mess3_token_guess_cycle_2/PLAN.md
index c1590b40..cb46fca8 100644
--- a/experiments/mess3_token_guess_cycle_2/PLAN.md
+++ b/experiments/mess3_token_guess_cycle_2/PLAN.md
@@ -42,16 +42,42 @@ Candidate C is my recommendation: about three times the belief-probe range and
 ten times the accuracy range, while keeping the task clearly learnable at a
 Bayes-optimal accuracy of 0.68.
 
-The trade is precision. Slowing the chain raises the state correlation time from
-7 steps to 133, so a 30,000-step probe rollout collected the way
-`collect_probe_data` collects one — sixteen parallel trajectories — contains
-roughly 100 independent samples rather than 2,300. The honest block-bootstrap
-interval on belief R² widens from ±0.0006 to ±0.0046, and on greedy accuracy to
-about ±0.09.
+The trade is precision, and it is worth being concrete about why.
 
-That is fixable and cheap, but only if it is noticed: **raise the probe to around
-500,000 test steps**. Rollouts cost seconds, so this is the least expensive fix in
-the plan and the easiest one to leave out.
+The probe number is an estimate, computed from a finite rollout. Two rollouts of
+the same checkpoint give two slightly different answers, and the spread between
+them is the metric's measurement error. That error depends on how many
+*independent* samples the rollout holds, which is not the same as how many steps
+it has. On a slow chain the hidden state sits still for hundreds of steps, so
+consecutive samples are near-duplicates and a long rollout can carry very little
+new information.
+
+Measured by direct replication — running the identical probe 60 times on fresh
+rollouts and taking the standard deviation of the answers:
+
+| operating point | 30k steps | 120k steps | 500k steps |
+|---|---:|---:|---:|
+| cycle 1 | 0.0004 | 0.0002 | 0.0001 |
+| candidate C | 0.0025 | 0.0012 | 0.0005 |
+
+At 30,000 steps the slow chain's measurement error is 0.0025, which is the same
+size as the smallest seed-to-seed spreads observed in the Kelly cycles. Probe
+noise would then be indistinguishable from seed variance, and adding seeds would
+not fix it.
+
+**Raise the probe to around 500,000 test steps**, where the measurement error
+falls to 0.0005 and is safely below anything the design needs to resolve. Probe
+rollouts are inference only and cost seconds, so this is the cheapest fix in the
+plan and the easiest one to omit by accident.
+
+Two notes on method. The i.i.d. bootstrap used in `REVIEW.md` understates this
+error, because resampling individual steps pretends they are independent; a
+moving-block bootstrap predicted ±0.0046 at 30k on candidate C and replication
+measured ±0.0047. And the simulation behind these numbers does not include the
+512-step episode resets the real environment applies, which partially
+decorrelate the rollout, so treat 0.0025 as an upper bound and confirm it once
+against the real probe by running it repeatedly on a single fixed checkpoint
+with different probe seeds.
 
 Two consequences follow. Episode length should rise from 512 so that the reset
 back to a uniform belief is a smaller fraction of each trajectory. And at
@@ -154,11 +180,46 @@ Sweeping the learning rate per arm is separately necessary because arms differ i
 parameter count and in how much auxiliary gradient enters the shared trunk, so an
 arm can lose by being mis-tuned rather than by being a worse idea.
 
-**Do not sweep:** context length (64 already suffices at every candidate operating
-point — the belief saturates by 32 observations), GAE λ (irrelevant at γ = 0, low
-leverage at γ = 0.99), batch size, minibatch size, epoch count, clip parameter,
-`d_model`, or `n_layers`. Fix these and record them. γ is a scientific factor,
-not a nuisance hyperparameter; keep it in the design rather than tuning it away.
+**Do not sweep:** context length (see below — shrink it once and fix it), GAE λ
+(irrelevant at γ = 0, low leverage at γ = 0.99), batch size, minibatch size, epoch
+count, clip parameter, `d_model`, or `n_layers`. Fix these and record them. γ is a
+scientific factor, not a nuisance hyperparameter; keep it in the design rather
+than tuning it away.
+
+## 5b. Shrink the context length once, then leave it alone
+
+`CausalTransformerEncoder` applies its causal band at *every layer*, so the
+receptive field is `n_layers × context_len`. Confirmed by perturbing inputs at
+increasing distance and watching when the output stops changing: three layers at
+`context_len=64` reach exactly 192 observations. The belief needs about 32.
+
+The cost is paid twice, because the learner recomputes over
+`lookback + max_seq_len = 192 + 32 = 224` positions to produce 32 useful
+embeddings, and the cached rollout path attends over the same band.
+
+| context_len | receptive field | learner step | rollout step | belief R² available |
+|---:|---:|---:|---:|---:|
+| 64 | 192 | 305 ms | 15.9 ms | 1.00000 |
+| 32 | 96 | 146 ms (2.1x) | 6.9 ms (2.3x) | 1.00000 |
+| 16 | 48 | 82 ms (3.7x) | 4.7 ms (3.4x) | 1.00000 |
+| 12 | 36 | 70 ms (4.3x) | 4.6 ms (3.5x) | 0.99997 |
+| 8 | 24 | 57 ms (5.4x) | 3.9 ms (4.0x) | 0.99966 |
+
+Belief R² is at candidate C; candidate B needs more, reaching 0.99983 only at a
+receptive field of 48.
+
+**Use `context_len=16`.** It costs nothing measurable in available belief R² at
+any candidate operating point, leaves a 50% margin over what the belief needs,
+and keeps the choice valid if α or p move later. `context_len=12` is defensible
+if candidate C is locked in, but the extra saving is small.
+
+`context_len` affects compute only, not capacity — positions are encoded with
+RoPE, so the parameter count does not change.
+
+End-to-end this is worth **1.6x**, measured with a real PPO loop at matched
+environment steps. It is less than the 3.4–3.7x component figures because
+environment stepping is pure NumPy and unaffected. The split differs on a 4090
+with 16 env runners, so measure it once there before re-budgeting.
 
 **Two stages, with disjoint seeds.** Stage 1 sweeps on seeds 0–2 and selects one
 configuration per arm. Stage 2 re-runs the selected configurations on ten
@@ -210,15 +271,18 @@ critic diagnostics.
 Measured from the committed manifests: a 2.5M-step arm costs 6–14 minutes on an
 RTX 4090, median about 6.5. Adding eight checkpoint probes puts it near 12.
 
+At `context_len=16` a run costs about 1.6x less than the measured 6–14 minutes,
+so budget roughly 7 minutes per run including checkpoint probes.
+
 | stage | runs | GPU-hours | cost at $0.35/hr |
 |---|---:|---:|---:|
-| stage 1 sweep (7 arms × 6 optimiser/lr × 3 coefficient × 3 seeds) | 378 | 76 | ~$27 |
-| stage 2 confirmation (16 conditions × 10 seeds) | 160 | 32 | ~$11 |
+| stage 1 sweep (7 arms × 6 optimiser/lr × 3 coefficient × 3 seeds) | 378 | 47 | ~$17 |
+| stage 2 confirmation (16 conditions × 10 seeds) | 160 | 20 | ~$7 |
 | references, operating point, audit | — | <1 | — |
-| **total** | **538** | **108** | **~$38** |
+| **total** | **538** | **67** | **~$24** |
 
-Roughly fourteen wall-clock hours across eight Vast boxes. The entire committed
-history of these four studies is 19.4 GPU-hours, so this is about five times all
+Roughly nine wall-clock hours across eight Vast boxes. The entire committed
+history of these four studies is 19.4 GPU-hours, so this is about three times all
 prior work on the question, for the price of lunch.
 
 `devops.vast.provision up -n N --run "..."` gives one command per box, so the
diff --git a/experiments/mess3_token_guess_cycle_2/operating_point.py b/experiments/mess3_token_guess_cycle_2/operating_point.py
index 512957cd..c343fac1 100644
--- a/experiments/mess3_token_guess_cycle_2/operating_point.py
+++ b/experiments/mess3_token_guess_cycle_2/operating_point.py
@@ -173,6 +173,54 @@ def context_requirement(
     return scores
 
 
+def receptive_field(*, n_layers: int, context_len: int) -> int:
+    """Observations the model can actually reach back to.
+
+    ``CausalTransformerEncoder`` applies a causal band of ``context_len`` at every
+    layer, so stacking them multiplies the reach. The study's ``context_len=64``
+    over three layers is a 192-observation receptive field, not 64.
+    """
+
+    if n_layers <= 0 or context_len <= 0:
+        raise ValueError("n_layers and context_len must be positive")
+    return n_layers * context_len
+
+
+def smallest_sufficient_context_len(
+    point: OperatingPoint,
+    test: TokenStream,
+    *,
+    n_layers: int = 3,
+    tolerance: float = 1e-3,
+    candidates: tuple[int, ...] = (4, 8, 12, 16, 20),
+) -> int:
+    """Smallest ``context_len`` that still reaches the architecture ceiling.
+
+    Compute in both the learner and the rollout path scales with the receptive
+    field, so anything beyond what the belief needs is paid for and discarded.
+
+    Only candidates whose receptive field fits inside the observation window
+    recorded on ``test`` can be checked, so widen that window before widening
+    these candidates.
+    """
+
+    testable = [
+        context_len
+        for context_len in sorted(candidates)
+        if receptive_field(n_layers=n_layers, context_len=context_len)
+        <= test.windows.shape[1]
+    ]
+    if not testable:
+        raise ValueError("no candidate fits inside the recorded observation window")
+    for context_len in testable:
+        field = receptive_field(n_layers=n_layers, context_len=context_len)
+        if context_requirement(point, test, context_lengths=(field,))[field] >= (
+            1.0 - tolerance
+        ):
+            return context_len
+    return testable[-1]
+
+
 def integrated_autocorrelation(point: OperatingPoint, *, n_steps: int, seed: int) -> float:
     """Sum the belief autocorrelation along a single chain.
 
@@ -242,6 +290,60 @@ def block_bootstrap_interval(
     return float(low), float(high)
 
 
+def replicate_probe_r2(
+    point: OperatingPoint,
+    *,
+    fit_steps: int,
+    test_steps: int,
+    replicates: int,
+    seed: int,
+    window: int = 32,
+) -> np.ndarray:
+    """Re-run the whole probe on fresh rollouts and return every answer.
+
+    This measures the metric's precision by direct replication rather than by
+    resampling one rollout, so it needs no assumption about the correlation
+    structure and can be used to check that a bootstrap is not lying.
+    """
+
+    scores = []
+    for index in range(replicates):
+        offset = seed + 1_000 * index
+        fit = simulate_parallel(
+            point, n_steps=fit_steps, seed=offset, n_chains=PROBE_CHAINS
+        )
+        test = simulate_parallel(
+            point, n_steps=test_steps, seed=offset + 500_000, n_chains=PROBE_CHAINS
+        )
+        score, _ = _probe_r2(
+            _one_hot(fit.windows, window),
+            fit.beliefs,
+            _one_hot(test.windows, window),
+            test.beliefs,
+        )
+        scores.append(float(score))
+    return np.array(scores)
+
+
+def probe_steps_for_spread(
+    measured_spread: float,
+    *,
+    measured_steps: int,
+    target_spread: float,
+) -> int:
+    """Probe steps needed for a target spread, assuming the usual 1/sqrt(n).
+
+    Verified against replication at 30k, 120k and 500k steps, where the observed
+    spread does fall as the square root of the step count.
+    """
+
+    if measured_spread <= 0.0 or target_spread <= 0.0:
+        raise ValueError("spreads must be positive")
+    if target_spread >= measured_spread:
+        return measured_steps
+    return int(np.ceil(measured_steps * (measured_spread / target_spread) ** 2))
+
+
 def evaluate(
     point: OperatingPoint,
     *,
@@ -265,6 +367,7 @@ def evaluate(
         point, n_steps=test_steps, seed=seed + 5, n_chains=PROBE_CHAINS
     )
     low, high = block_bootstrap_interval(probe_fit, probe_test, window=32, seed=seed + 3)
+    smallest_context = smallest_sufficient_context_len(point, test)
     return {
         "alpha": point.alpha,
         "self_transition": point.self_transition,
@@ -275,6 +378,10 @@ def evaluate(
         "accuracy_ceiling": accuracy_ceiling,
         "accuracy_range": accuracy_ceiling - accuracy_floor,
         "context_requirement": context_requirement(point, test),
+        "smallest_sufficient_context_len": smallest_context,
+        "smallest_sufficient_receptive_field": receptive_field(
+            n_layers=3, context_len=smallest_context
+        ),
         "integrated_autocorrelation": tau,
         "effective_sample_size": test_steps / tau,
         "probe_block_bootstrap_ci": [low, high],
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py b/experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py
index 2aeebe55..8aed5d99 100644
--- a/experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py
+++ b/experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py
@@ -122,10 +122,26 @@ def _findings(result: dict[str, Any]) -> str:
     lines.extend(
         [
             "",
-            "The study's context length of 64 is sufficient at every candidate, so "
-            "it does not need sweeping. The belief converges much faster than the "
-            "hidden state does, because each observation is informative enough to "
-            "wash out the prior well before the state decorrelates.",
+            "`CausalTransformerEncoder` applies its causal band at every layer, so "
+            "the receptive field is `n_layers * context_len`. The study's "
+            "`context_len=64` over three layers reaches 192 observations, roughly "
+            "six times what the belief needs. Compute in both the learner and the "
+            "cached rollout path scales with that reach.",
+            "",
+            "| point | smallest sufficient context_len | receptive field |",
+            "|---|---:|---:|",
+        ]
+        + [
+            f"| `{name}` | {entry['smallest_sufficient_context_len']} | "
+            f"{entry['smallest_sufficient_receptive_field']} |"
+            for name, entry in result["candidates"].items()
+        ]
+        + [
+            "",
+            "Measured on this repository's model: dropping `context_len` from 64 to "
+            "16 makes a learner step 3.7x faster and a cached rollout step 3.4x "
+            "faster, for an end-to-end PPO speed-up of 1.6x once environment "
+            "stepping, which does not change, is included.",
             "",
             "## Supervised ceiling",
             "",
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/findings.md b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/findings.md
similarity index 70%
rename from experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/findings.md
rename to experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/findings.md
index 0891160c..06436553 100644
--- a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/findings.md
+++ b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/findings.md
@@ -22,7 +22,16 @@ Best belief R² available to a model that can see only the last k observations.
 | `candidate_b` | 0.7623 | 0.9431 | 0.9965 | 1.0000 |
 | `candidate_c` | 0.9232 | 0.9951 | 1.0000 | 1.0000 |
 
-The study's context length of 64 is sufficient at every candidate, so it does not need sweeping. The belief converges much faster than the hidden state does, because each observation is informative enough to wash out the prior well before the state decorrelates.
+`CausalTransformerEncoder` applies its causal band at every layer, so the receptive field is `n_layers * context_len`. The study's `context_len=64` over three layers reaches 192 observations, roughly six times what the belief needs. Compute in both the learner and the cached rollout path scales with that reach.
+
+| point | smallest sufficient context_len | receptive field |
+|---|---:|---:|
+| `cycle_1` | 4 | 12 |
+| `candidate_a` | 8 | 24 |
+| `candidate_b` | 16 | 48 |
+| `candidate_c` | 8 | 24 |
+
+Measured on this repository's model: dropping `context_len` from 64 to 16 makes a learner step 3.7x faster and a cached rollout step 3.4x faster, for an end-to-end PPO speed-up of 1.6x once environment stepping, which does not change, is included.
 
 ## Supervised ceiling
 
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/operating_point_grid.png b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/operating_point_grid.png
similarity index 100%
rename from experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/operating_point_grid.png
rename to experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/operating_point_grid.png
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/run_manifest.json b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/run_manifest.json
similarity index 79%
rename from experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/run_manifest.json
rename to experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/run_manifest.json
index c267c9c8..eb0a4aad 100644
--- a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/run_manifest.json
+++ b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/run_manifest.json
@@ -8,12 +8,12 @@
     "file": "/workspace/uv.lock",
     "sha256": "986725e46af5252423583fe947d30b2646f541a4863acb8c51d578a7ca1b8220"
   },
-  "ended_at": "2026-07-27T06:06:20.805347+00:00",
+  "ended_at": "2026-07-27T06:55:55.380651+00:00",
   "error": null,
   "experiment": {
     "file": "/workspace/experiments/mess3_token_guess_cycle_2/task_parameters/experiment.py",
     "module": "experiments.mess3_token_guess_cycle_2.task_parameters.experiment",
-    "source_sha256": "c59ebbce428c97c674639bb1d7eaf5b0eeb38a12ddaee52dcf8366aae864e02d"
+    "source_sha256": "0de4d683e821de377b3cfa9bd0036b37ea6217262f0a43632d152204aac507b3"
   },
   "framework_versions": {
     "gymnasium": "1.2.2",
@@ -24,10 +24,10 @@
     "torch": "2.12.1"
   },
   "git": {
-    "commit": "ee4095babf540e59e7add42395ac8ead5f6c6ec8",
+    "commit": "d790369e770a98ac118709c8bed484375415f998",
     "dirty": true,
     "experiment_repository": {
-      "commit": "ee4095babf540e59e7add42395ac8ead5f6c6ec8",
+      "commit": "d790369e770a98ac118709c8bed484375415f998",
       "dirty": true,
       "root": "/workspace"
     },
@@ -54,9 +54,9 @@
       "torch_threads": null
     }
   },
-  "run_id": "20260727T060541Z-53419d82",
+  "run_id": "20260727T065519Z-ef24bfec",
   "runtime": {
-    "artifacts_dir": "/workspace/experiments/mess3_token_guess_cycle_2/task_parameters/artifacts/20260727T060541Z-53419d82",
+    "artifacts_dir": "/workspace/experiments/mess3_token_guess_cycle_2/task_parameters/artifacts/20260727T065519Z-ef24bfec",
     "overrides": {
       "hardware_profile": "auto",
       "resume_from": null,
@@ -64,12 +64,12 @@
       "smoke": false,
       "upload_artifacts": false
     },
-    "results_dir": "/workspace/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82",
+    "results_dir": "/workspace/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec",
     "resume_from": null,
     "seed": 42,
     "smoke": false
   },
   "schema_version": 2,
-  "started_at": "2026-07-27T06:05:42.001317+00:00",
+  "started_at": "2026-07-27T06:55:19.583001+00:00",
   "status": "completed"
 }
diff --git a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/task_parameters.json b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/task_parameters.json
similarity index 95%
rename from experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/task_parameters.json
rename to experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/task_parameters.json
index 24a8c63a..9f735818 100644
--- a/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/task_parameters.json
+++ b/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/task_parameters.json
@@ -22,6 +22,8 @@
       "probe_ci_half_width": 0.0036841656877789064,
       "probe_rank": 2,
       "self_transition": 0.99,
+      "smallest_sufficient_context_len": 8,
+      "smallest_sufficient_receptive_field": 24,
       "state_correlation_time": 66.66666666666612
     },
     "candidate_b": {
@@ -46,6 +48,8 @@
       "probe_ci_half_width": 0.006106949676209283,
       "probe_rank": 2,
       "self_transition": 0.995,
+      "smallest_sufficient_context_len": 16,
+      "smallest_sufficient_receptive_field": 48,
       "state_correlation_time": 133.33333333333223
     },
     "candidate_c": {
@@ -70,6 +74,8 @@
       "probe_ci_half_width": 0.004552167314833988,
       "probe_rank": 2,
       "self_transition": 0.995,
+      "smallest_sufficient_context_len": 8,
+      "smallest_sufficient_receptive_field": 24,
       "state_correlation_time": 133.33333333333223
     },
     "cycle_1": {
@@ -94,10 +100,12 @@
       "probe_ci_half_width": 0.0005696590960750925,
       "probe_rank": 2,
       "self_transition": 0.9,
+      "smallest_sufficient_context_len": 4,
+      "smallest_sufficient_receptive_field": 12,
       "state_correlation_time": 6.6666666666666705
     }
   },
-  "figure": "/workspace/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T060541Z-53419d82/operating_point_grid.png",
+  "figure": "/workspace/experiments/mess3_token_guess_cycle_2/task_parameters/results/20260727T065519Z-ef24bfec/operating_point_grid.png",
   "grid": {
     "0.5_0.9": {
       "accuracy_ceiling": 0.3857333333333333,
diff --git a/tests/test_mess3_token_guess_operating_point.py b/tests/test_mess3_token_guess_operating_point.py
index 2f968706..ecbcc730 100644
--- a/tests/test_mess3_token_guess_operating_point.py
+++ b/tests/test_mess3_token_guess_operating_point.py
@@ -13,7 +13,11 @@
     block_bootstrap_interval,
     context_requirement,
     integrated_autocorrelation,
+    probe_steps_for_spread,
+    receptive_field,
+    replicate_probe_r2,
     simulate_parallel,
+    smallest_sufficient_context_len,
 )
 
 CYCLE_1 = OperatingPoint(alpha=0.85, self_transition=0.90)
@@ -71,6 +75,58 @@ def test_a_64_observation_context_suffices_at_every_candidate():
         assert scores[64] >= scores[32] >= scores[8]
 
 
+def test_receptive_field_multiplies_the_band_by_the_layer_count():
+    # CausalTransformerEncoder applies the band at every layer, so the study's
+    # context_len of 64 over three layers reaches 192 observations.
+    assert receptive_field(n_layers=3, context_len=64) == 192
+    assert receptive_field(n_layers=3, context_len=16) == 48
+    assert receptive_field(n_layers=1, context_len=16) == 16
+    for bad in ({"n_layers": 0, "context_len": 8}, {"n_layers": 3, "context_len": 0}):
+        with pytest.raises(ValueError):
+            receptive_field(**bad)
+
+
+def test_the_study_context_len_is_far_larger_than_the_belief_needs():
+    for point in (CYCLE_1, SLOW):
+        _, test = _streams(point)
+        smallest = smallest_sufficient_context_len(point, test)
+        assert smallest <= 16, point
+        # Well inside the 192-observation reach the study currently pays for.
+        assert receptive_field(n_layers=3, context_len=smallest) <= 48
+
+
+def test_smallest_sufficient_context_len_needs_a_wide_enough_window():
+    _, test = _streams(SLOW)
+    narrow = simulate_parallel(SLOW, n_steps=2_000, seed=3, window=8)
+    assert smallest_sufficient_context_len(SLOW, test) >= 4
+    with pytest.raises(ValueError):
+        smallest_sufficient_context_len(SLOW, narrow, candidates=(16, 20))
+
+
+def test_replication_confirms_the_slow_chain_is_the_noisier_probe():
+    fast = replicate_probe_r2(
+        CYCLE_1, fit_steps=20_000, test_steps=10_000, replicates=8, seed=1
+    )
+    slow = replicate_probe_r2(
+        SLOW, fit_steps=20_000, test_steps=10_000, replicates=8, seed=1
+    )
+    assert slow.std(ddof=1) > 3.0 * fast.std(ddof=1)
+
+
+def test_probe_steps_for_spread_follows_the_square_root_law():
+    # Measured by replication: +/-0.0025 at 30,000 steps on the slow chain.
+    assert probe_steps_for_spread(
+        0.0025, measured_steps=30_000, target_spread=0.0005
+    ) == pytest.approx(750_000, rel=0.01)
+    # Never asks for fewer steps than were already run.
+    assert (
+        probe_steps_for_spread(0.0005, measured_steps=30_000, target_spread=0.002)
+        == 30_000
+    )
+    with pytest.raises(ValueError):
+        probe_steps_for_spread(0.001, measured_steps=30_000, target_spread=0.0)
+
+
 def test_slowing_the_chain_costs_independent_probe_samples():
     fast = integrated_autocorrelation(CYCLE_1, n_steps=60_000, seed=3)
     slow = integrated_autocorrelation(SLOW, n_steps=60_000, seed=3)