# Prespecified Experiment Protocol

Version 1.0 - frozen before running the new experiment suite.

## 1. Research questions

RQ1. Does the exact implementation satisfy the factorization, gap recursion,
relative-energy audit, trajectory decomposition, and regret identities to
numerical precision?

RQ2. When transition entropy and continuation-return dispersion are varied
independently, does the local entropic gap respond to return dispersion rather
than entropy alone?

RQ3. Does a teacher-forced finite causal Transformer recover the observational
action and event conditionals, and does a control readout using the learned
event kernel reduce decision error relative to the naive generative readout?

RQ4. Under finite tabular data, how do support coverage, maximum supported
transition error, value error, and true regret trade off as the support
threshold changes?

## 2. Experimental world

- Horizon: K = 4 rounds.
- Actions: 4.
- Events: 3.
- Complete trajectories: (4 x 3)^4 = 20,736.
- Action reference: uniform unless a sensitivity experiment says otherwise.
- Policy temperature: tau = 1 unless a sensitivity experiment says otherwise.
- Behavioral policy: stochastic and full support.
- Rewards and transition logits are generated from a recorded environment
  seed. Full history is the state, so the exact process is Markov.

## 3. Factorial intervention

Transition entropy and continuation-return dispersion are manipulated in a
2 x 2 design:

- transition entropy: low, high;
- event/continuation dispersion: low, high.

The entropy intervention changes only the temperature of the event kernel.
The dispersion intervention changes only the scale of event-dependent reward
and continuation features. All four cells reuse the same base random tables
within an environment seed.

## 4. Exact tests and rejection criteria

The implementation is rejected if any of the following exceeds 1e-10 in
float64 arithmetic:

- trajectory factorization normalization error;
- Proposition-B local recursion residual;
- relative-energy reconstruction residual;
- pathwise choice-event decomposition residual;
- conditional mean of event innovations;
- KL-regret identity residual for the behavioral policy.

## 5. Tabular estimation

Sample sizes: 100, 300, 1,000, 3,000, 10,000, and 30,000 trajectories.

- Conditional probabilities use symmetric Dirichlet smoothing alpha = 0.5.
- Reward observations receive zero-mean Gaussian noise with standard deviation
  0.10; estimates are conditional sample means with a zero prior.
- Primary transition metrics: occupancy-weighted TV and maximum TV on the
  declared supported set.
- Support thresholds: 1, 5, 10, 20, 50 observations per decision-state/action
  chance node.
- Primary decision metric: true KL-regularized regret in the original world.
- The coverage-restriction cost and within-gate estimation cost are reported
  separately.

## 6. Neural estimation

- Architecture: 2-layer causal Transformer, width 32, 4 heads, FFN width 128.
- Typed token vocabulary: BOS, four actions, three events.
- Position parity masks invalid action/event tokens.
- Optimizer: AdamW, learning rate 3e-3, batch size 512, 1,200 updates.
- Training seeds: 5 per environment seed and entropy condition in the full
  run; 1 in pilot. The fitted observational model is reused across the two
  dispersion cells because dispersion changes rewards only and leaves the
  complete training distribution unchanged.
- Evaluation enumerates every valid prefix and complete trajectory.
- Primary metrics: exact trajectory KL, event-kernel max TV on reachable
  prefixes, control-value error, and true regret of the extracted policy.

If PyTorch is unavailable, the neural experiment remains pending and the paper
must state that only exact/tabular evidence has been completed.

## 7. Replication structure

- Exact/factorial suite: 20 environment seeds.
- Tabular suite: 10 environment seeds x 5 data seeds.
- Neural suite: 5 environment seeds x 5 training seeds x 2 entropy conditions;
  each fitted model is evaluated under both reward-dispersion conditions. A
  smaller run may be labeled pilot only.
- Report mean, standard error, and 95% bootstrap confidence interval.

## 8. Interpretation rules

- A result can support consistency with a theorem but cannot prove a theorem.
- Weighted TV cannot substitute for the maximum supported TV in the bound.
- Near-zero post-correction correlation is environment-specific; the exact
  audit identity is assessed by reconstruction residual.
- Neural failure does not refute the function-space identity; it identifies
  approximation, optimization, or representation limits.
- No synthetic result is presented as causal evidence about human lives.

## 9. Primary figures

1. 2 x 2 factorial local gap and generative-policy regret.
2. Tabular sample-size curves for trajectory KL and transition TV.
3. Coverage-error-regret trade-off under support gating.
4. Neural training curves and final exact-prefix metrics.
5. Relative-energy audit and trajectory-attribution recovery.
