Validate exact initial FA erosion distribution

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diff --git a/notes/32_e0_distribution_validation.md b/notes/32_e0_distribution_validation.md
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+# Exact e0 Distribution Validation
+
+This note records the clean distributional validation for the actual FA initial
+operator erosion theorem.
+
+## Theorem Being Validated
+
+Use a one-hidden-layer MLP with Gaussian feedback:
+
+```text
+x -> W1 -> ReLU -> W2 -> output
+```
+
+For fixed forward weights, data, gates, and residual:
+
+```text
+speed_BP = ||g_output^BP||^2 + ||g_hidden^BP||^2
+speed_FA = ||g_output^BP||^2 + <g_hidden^BP, g_hidden^FA(B)>
+e0(B) = 1 - speed_FA / speed_BP
+```
+
+Because the hidden FA gradient is linear in Gaussian feedback `B`:
+
+```text
+<g_hidden^BP, g_hidden^FA(B)> = <C(W,r), B>
+```
+
+Therefore:
+
+```text
+<C(W,r), B> ~ Normal(0, sigma_B^2 ||C(W,r)||^2)
+```
+
+and:
+
+```text
+e0(B) ~ Normal(
+  1 - ||g_output^BP||^2 / speed_BP,
+  sigma_B^2 ||C(W,r)||^2 / speed_BP^2
+)
+```
+
+This is an exact conditional distribution for one-hidden-layer Gaussian FA at
+initialization. It does not involve training horizon `T`.
+
+## Experiment
+
+Script:
+
+```text
+scripts/actual_fa_initial_erosion_distribution.py
+```
+
+Run:
+
+```text
+python scripts/actual_fa_initial_erosion_distribution.py \
+  --widths 16 32 64 128 \
+  --init-seeds 3 \
+  --feedback-samples 4096 \
+  --torch-threads 8 \
+  --outdir outputs/actual_fa_initial_erosion_distribution
+```
+
+Outputs:
+
+```text
+outputs/actual_fa_initial_erosion_distribution/e0_distribution_theory_vs_empirical.png
+outputs/actual_fa_initial_erosion_distribution/e0_distribution_moment_calibration.png
+outputs/actual_fa_initial_erosion_distribution/e0_distribution_rows.csv
+```
+
+## Results
+
+Representative rows:
+
+| width | init | theory mean | empirical mean | theory std | empirical std | KS | p-value |
+|---:|---:|---:|---:|---:|---:|---:|---:|
+| 16 | 0 | 0.113830 | 0.113413 | 0.027228 | 0.027401 | 0.0150 | 0.314 |
+| 16 | 1 | 0.113705 | 0.113859 | 0.027261 | 0.027278 | 0.0103 | 0.778 |
+| 32 | 0 | 0.121730 | 0.121544 | 0.018952 | 0.018693 | 0.0114 | 0.654 |
+| 32 | 2 | 0.227193 | 0.227691 | 0.031148 | 0.030779 | 0.0166 | 0.206 |
+| 64 | 0 | 0.031155 | 0.031119 | 0.003652 | 0.003583 | 0.0118 | 0.609 |
+| 128 | 0 | 0.023004 | 0.022989 | 0.001630 | 0.001610 | 0.0135 | 0.441 |
+
+All tested panels show close agreement between theory density and empirical
+random-feedback histograms.
+
+## Paper Use
+
+This is the clean validation figure for the actual FA local erosion theorem.
+
+It should replace any tangent-hierarchy figure if the goal is:
+
+```text
+theory predicts a distribution
+large random-feedback sampling produces the same distribution
+```
+
+Suggested main-text wording:
+
+```text
+For one-hidden-layer Gaussian FA, the conditional distribution of the initial
+operator erosion is exactly Gaussian. Figure X compares this predicted density
+against 4096 random feedback draws per initialization across widths; the
+empirical histograms and theoretical densities coincide without fitted
+parameters.
+```
+
+This figure belongs to:
+
+```text
+Contribution 4: actual FA initial operator erosion theorem
+Contribution 5: distributional empirical validation
+```
+
+It does not require a training-time axis.
diff --git a/scripts/actual_fa_initial_erosion_distribution.py b/scripts/actual_fa_initial_erosion_distribution.py
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+#!/usr/bin/env python3
+"""Plot exact initial FA erosion distributions for one-hidden-layer MLPs.
+
+For a fixed forward initialization and residual, one-hidden-layer FA has
+
+    speed_FA = output_speed + <C, B>
+
+with Gaussian feedback B. Therefore the one-step erosion
+
+    e0 = 1 - speed_FA / speed_BP
+
+is exactly Gaussian conditional on the forward weights and data.
+"""
+
+from __future__ import annotations
+
+import argparse
+import csv
+import math
+from dataclasses import asdict, dataclass
+from pathlib import Path
+
+import matplotlib.pyplot as plt
+import numpy as np
+import torch
+from scipy import stats
+
+
+Tensor = torch.Tensor
+
+
+@dataclass(frozen=True)
+class DistributionRow:
+    width: int
+    init_seed: int
+    feedback_samples: int
+    bp_speed: float
+    output_speed: float
+    hidden_bp_speed: float
+    theory_mean: float
+    theory_std: float
+    empirical_mean: float
+    empirical_std: float
+    ks_stat: float
+    ks_pvalue: float
+    mean_error: float
+    std_error: float
+
+
+def parse_args() -> argparse.Namespace:
+    parser = argparse.ArgumentParser(description="Exact e0 distribution validation.")
+    parser.add_argument("--input-dim", type=int, default=16)
+    parser.add_argument("--output-dim", type=int, default=4)
+    parser.add_argument("--widths", type=int, nargs="+", default=[16, 32, 64, 128])
+    parser.add_argument("--train-samples", type=int, default=128)
+    parser.add_argument("--init-seeds", type=int, default=3)
+    parser.add_argument("--feedback-samples", type=int, default=4096)
+    parser.add_argument("--data-seed", type=int, default=123)
+    parser.add_argument(
+        "--feedback-scale",
+        choices=["relu", "fan-in", "unit"],
+        default="relu",
+    )
+    parser.add_argument("--torch-threads", type=int, default=8)
+    parser.add_argument(
+        "--outdir",
+        type=Path,
+        default=Path("outputs/actual_fa_initial_erosion_distribution"),
+    )
+    return parser.parse_args()
+
+
+def feedback_scale(rows: int, mode: str) -> float:
+    if mode == "relu":
+        return math.sqrt(2.0 / rows)
+    if mode == "fan-in":
+        return math.sqrt(1.0 / rows)
+    if mode == "unit":
+        return 1.0
+    raise ValueError(f"Unknown feedback scale: {mode}")
+
+
+def make_data(samples: int, input_dim: int, output_dim: int, seed: int) -> tuple[Tensor, Tensor]:
+    generator = torch.Generator(device="cpu")
+    generator.manual_seed(seed)
+    x = torch.randn(samples, input_dim, generator=generator, dtype=torch.float64)
+    y = torch.randn(samples, output_dim, generator=generator, dtype=torch.float64)
+    return x, y
+
+
+def init_weights(input_dim: int, output_dim: int, width: int, seed: int) -> tuple[Tensor, Tensor]:
+    generator = torch.Generator(device="cpu")
+    generator.manual_seed(seed)
+    w1 = torch.randn(width, input_dim, generator=generator, dtype=torch.float64) * math.sqrt(2.0 / input_dim)
+    w2 = torch.randn(output_dim, width, generator=generator, dtype=torch.float64) / math.sqrt(width)
+    return w1, w2
+
+
+def forward(w1: Tensor, w2: Tensor, x: Tensor) -> tuple[Tensor, Tensor, Tensor]:
+    z = x @ w1.T
+    h = torch.relu(z)
+    out = h @ w2.T
+    return z, h, out
+
+
+def theory_terms(
+    w1: Tensor,
+    w2: Tensor,
+    x: Tensor,
+    y: Tensor,
+    scale: float,
+) -> tuple[float, float, float, float, float, Tensor]:
+    z, h, out = forward(w1, w2, x)
+    batch = x.shape[0]
+    delta_out = (out - y) / batch
+    gate = (z > 0).to(torch.float64)
+
+    grad_out = delta_out.T @ h
+    delta_hidden_bp = (delta_out @ w2) * gate
+    grad_hidden_bp = delta_hidden_bp.T @ x
+
+    output_speed = float(torch.sum(grad_out * grad_out))
+    hidden_bp_speed = float(torch.sum(grad_hidden_bp * grad_hidden_bp))
+    bp_speed = output_speed + hidden_bp_speed
+
+    # g_hidden_FA[a, d] = sum_c B[a, c] S[a, c, d]
+    # C[a, c] = sum_d g_hidden_BP[a, d] S[a, c, d]
+    # Use einsum over samples n and input dimension d:
+    # S[a,c,d] = sum_n gate[n,a] * x[n,d] * delta_out[n,c]
+    s_tensor = torch.einsum("na,nd,nc->acd", gate, x, delta_out)
+    coeff = torch.einsum("ad,acd->ac", grad_hidden_bp, s_tensor)
+    z_std = scale * float(torch.linalg.norm(coeff))
+    theory_mean = 1.0 - output_speed / bp_speed
+    theory_std = z_std / bp_speed
+    return bp_speed, output_speed, hidden_bp_speed, theory_mean, theory_std, coeff
+
+
+def sample_erosions(
+    bp_speed: float,
+    output_speed: float,
+    coeff: Tensor,
+    scale: float,
+    samples: int,
+    seed: int,
+) -> np.ndarray:
+    generator = torch.Generator(device="cpu")
+    generator.manual_seed(seed)
+    values = []
+    for _ in range(samples):
+        feedback = torch.randn(
+            coeff.shape,
+            generator=generator,
+            dtype=torch.float64,
+        ) * scale
+        hidden_mixed = float(torch.sum(feedback * coeff))
+        speed_fa = output_speed + hidden_mixed
+        values.append(1.0 - speed_fa / bp_speed)
+    return np.array(values, dtype=np.float64)
+
+
+def run_one(args: argparse.Namespace, x: Tensor, y: Tensor, width: int, init_seed: int) -> tuple[DistributionRow, np.ndarray]:
+    w1, w2 = init_weights(args.input_dim, args.output_dim, width, init_seed)
+    scale = feedback_scale(width, args.feedback_scale)
+    bp_speed, output_speed, hidden_bp_speed, theory_mean, theory_std, coeff = theory_terms(
+        w1,
+        w2,
+        x,
+        y,
+        scale,
+    )
+    erosions = sample_erosions(
+        bp_speed,
+        output_speed,
+        coeff,
+        scale,
+        args.feedback_samples,
+        seed=1_000_000 + 10_000 * init_seed + width,
+    )
+    standardized = (erosions - theory_mean) / theory_std
+    ks = stats.kstest(standardized, "norm")
+    row = DistributionRow(
+        width=width,
+        init_seed=init_seed,
+        feedback_samples=args.feedback_samples,
+        bp_speed=bp_speed,
+        output_speed=output_speed,
+        hidden_bp_speed=hidden_bp_speed,
+        theory_mean=theory_mean,
+        theory_std=theory_std,
+        empirical_mean=float(np.mean(erosions)),
+        empirical_std=float(np.std(erosions, ddof=1)),
+        ks_stat=float(ks.statistic),
+        ks_pvalue=float(ks.pvalue),
+        mean_error=float(np.mean(erosions) - theory_mean),
+        std_error=float(np.std(erosions, ddof=1) - theory_std),
+    )
+    return row, erosions
+
+
+def write_rows(path: Path, rows: list[DistributionRow]) -> None:
+    path.parent.mkdir(parents=True, exist_ok=True)
+    with path.open("w", newline="") as handle:
+        writer = csv.DictWriter(handle, fieldnames=list(DistributionRow.__annotations__.keys()))
+        writer.writeheader()
+        for row in rows:
+            writer.writerow(asdict(row))
+
+
+def plot_overlay(
+    rows: list[DistributionRow],
+    samples: dict[tuple[int, int], np.ndarray],
+    outdir: Path,
+) -> Path:
+    selected: list[DistributionRow] = []
+    for width in sorted({row.width for row in rows}):
+        width_rows = [row for row in rows if row.width == width]
+        selected.append(width_rows[0])
+
+    cols = 2
+    panel_rows = int(math.ceil(len(selected) / cols))
+    fig, axes = plt.subplots(panel_rows, cols, figsize=(11.0, 4.2 * panel_rows), dpi=180, squeeze=False)
+    for ax in axes.ravel():
+        ax.axis("off")
+
+    for ax, row in zip(axes.ravel(), selected):
+        ax.axis("on")
+        values = samples[(row.width, row.init_seed)]
+        lo = min(float(np.quantile(values, 0.001)), row.theory_mean - 4.0 * row.theory_std)
+        hi = max(float(np.quantile(values, 0.999)), row.theory_mean + 4.0 * row.theory_std)
+        grid = np.linspace(lo, hi, 500)
+        density = stats.norm.pdf(grid, loc=row.theory_mean, scale=row.theory_std)
+        ax.hist(values, bins=70, density=True, color="#c65f16", alpha=0.42, label="empirical samples")
+        ax.plot(grid, density, color="#174f91", linewidth=2.4, label="theory Gaussian")
+        ax.axvline(row.theory_mean, color="#174f91", linewidth=1.4, linestyle="--")
+        ax.axvline(row.empirical_mean, color="#a84708", linewidth=1.4, linestyle=":")
+        ax.set_title(
+            f"width={row.width}, init={row.init_seed} | "
+            f"KS={row.ks_stat:.3f}, p={row.ks_pvalue:.2f}"
+        )
+        ax.set_xlabel("initial FA operator erosion e0")
+        ax.set_ylabel("density")
+        ax.legend(fontsize=8)
+        ax.grid(alpha=0.16)
+
+    fig.suptitle("Exact e0 distribution: theory density vs random-feedback samples", y=1.01)
+    fig.tight_layout()
+    path = outdir / "e0_distribution_theory_vs_empirical.png"
+    fig.savefig(path, bbox_inches="tight")
+    plt.close(fig)
+    return path
+
+
+def plot_moments(rows: list[DistributionRow], outdir: Path) -> Path:
+    fig, axes = plt.subplots(1, 2, figsize=(11.2, 4.8), dpi=180)
+    widths = sorted({row.width for row in rows})
+    for width in widths:
+        sub = [row for row in rows if row.width == width]
+        axes[0].scatter(
+            [row.theory_mean for row in sub],
+            [row.empirical_mean for row in sub],
+            s=34,
+            alpha=0.72,
+            label=f"w={width}",
+        )
+        axes[1].scatter(
+            [row.theory_std for row in sub],
+            [row.empirical_std for row in sub],
+            s=34,
+            alpha=0.72,
+            label=f"w={width}",
+        )
+    for ax, title in zip(axes, ["mean", "standard deviation"]):
+        values = []
+        if title == "mean":
+            values = [v for row in rows for v in (row.theory_mean, row.empirical_mean)]
+        else:
+            values = [v for row in rows for v in (row.theory_std, row.empirical_std)]
+        lo, hi = min(values), max(values)
+        pad = 0.04 * (hi - lo + 1e-12)
+        ax.plot([lo - pad, hi + pad], [lo - pad, hi + pad], color="black", linewidth=1.0)
+        ax.set_title(f"theory vs empirical {title}")
+        ax.set_xlabel("theory")
+        ax.set_ylabel("empirical")
+        ax.grid(alpha=0.16)
+    axes[0].legend(fontsize=8, ncols=2)
+    fig.tight_layout()
+    path = outdir / "e0_distribution_moment_calibration.png"
+    fig.savefig(path, bbox_inches="tight")
+    plt.close(fig)
+    return path
+
+
+def main() -> None:
+    args = parse_args()
+    torch.set_num_threads(args.torch_threads)
+    x, y = make_data(args.train_samples, args.input_dim, args.output_dim, args.data_seed)
+    rows: list[DistributionRow] = []
+    sample_map: dict[tuple[int, int], np.ndarray] = {}
+    for width in args.widths:
+        for init_seed in range(args.init_seeds):
+            row, erosions = run_one(args, x, y, width, init_seed)
+            rows.append(row)
+            sample_map[(width, init_seed)] = erosions
+            print(
+                f"width={width} init={init_seed}: "
+                f"mean theory={row.theory_mean:.6f} empirical={row.empirical_mean:.6f}; "
+                f"std theory={row.theory_std:.6f} empirical={row.empirical_std:.6f}; "
+                f"KS={row.ks_stat:.4f} p={row.ks_pvalue:.3g}",
+                flush=True,
+            )
+
+    args.outdir.mkdir(parents=True, exist_ok=True)
+    csv_path = args.outdir / "e0_distribution_rows.csv"
+    write_rows(csv_path, rows)
+    paths = [
+        plot_overlay(rows, sample_map, args.outdir),
+        plot_moments(rows, args.outdir),
+    ]
+    print(f"rows: {csv_path}")
+    for path in paths:
+        print(f"plot: {path}")
+
+
+if __name__ == "__main__":
+    main()