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neural: grounding refinement + all five Layer-1.5 figures

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Grounding refinement (18 reps): forward-KL is the operative neural
collapse metric, not H or tail-survival. The RNN's smoothing keeps
spurious tail modes alive, so tail_truth_mass_alive is flat/non-monotone
in g and H stays ~0.8 of H*; only forward-KL falls monotonically (dry
2.08 -> g=0.2: 0.75, paired t up to 3.3). The sharp g* << 1 is an
exact-operator feature carried by the histogram bridge (0.047); the
trained RNN confirms the SIGN and softens the sharpness (half the KL gap
closes by g~0.04, but full recovery needs g~0.19). Blueprint 3.5's
directional claim holds; the pre-registered 95%-of-H*/tail falsifier is
not met because those are the wrong metrics for a smoothing model.

Robustness: a fully-degenerate RNN can emit only invalid codewords, so
measure_distribution now returns a terminal-collapse sentinel (fixation
on the dominant mode) instead of crashing a long sweep. Edge test added
(94 tests green).

Figures: plot_{bridge,collapse,grounding,architectures,recombination}.py,
each a pure function of its committed bundle, wired into `make figures`
(glob plot_*.py minus plot_E[1-6]/_*). bridge sits on the exact H_eq
curve (g*=0.047); recombination shows max-merge rising while mean-distill
stays flat; architectures shows the collapse/rescue signs across
histogram/GRU/MLP.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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Giorgio Gilestro 2026-07-05 08:14:19 +01:00
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"""`bridge` figure — the histogram bridge reproduces Layer-1 E2 exactly (the HARD GATE).
The histogram model reduces Layer 1.5 to Layer 1 (MLE histogram + multinomial resampling),
so running it through the *neural* runner must reproduce E2's grounding phase boundary and its
exact `H_eq` closed form. Recovering g*≈0.047 here (vs Layer-1's 0.048) is what licenses every
later trained-model result to be read against the analytic core.
Four panels: (A) H trajectories (g=0 collapses, g>0 plateau); (B) the phase boundary —
stationary H vs g on the exact `H_eq` curve, with the operational g* + bootstrap CI; (C) tail
survival vs g; (D) per-rarity-band survival (the m·p*_i≳1 threshold). Reads only the bundle.
Usage: python figures/plot_bridge.py [results/bridge]
"""
from __future__ import annotations
import sys
from pathlib import Path
import matplotlib.pyplot as plt
import numpy as np
sys.path.insert(0, str(Path(__file__).parent))
from _figlib import load_bundle, mean_ci, savefig # noqa: E402
sys.path.insert(0, str(Path(__file__).parents[1] / "src"))
from knowledge.analysis import critical_grounding, reduce_to_stationary # noqa: E402
from knowledge.metrics import heterozygosity # noqa: E402
from neural.config import SyntheticCfg # noqa: E402
from neural.synthetic import make_mode_truth # noqa: E402
def main(results_dir: str = "results/bridge") -> None:
df, cfg = load_bundle(results_dir)
n = cfg["dynamics"]["n"]
syn = SyntheticCfg(**cfg["synthetic"])
H_star = heterozygosity(make_mode_truth(syn).p_star)
def H_eq(m):
m = np.asarray(m, dtype=float)
return np.where(m <= 0, 0.0, H_star * m * (2 * n + m - 1) / (n + 2 * n * m + m * m))
g_values = sorted(df["g"].unique())
last = int(cfg["generations"] * 0.8)
stat = df[df["generation"] >= last]
fig, axes = plt.subplots(2, 2, figsize=(13, 9))
colors = plt.cm.viridis(np.linspace(0, 0.9, len(g_values)))
# Panel A: H trajectories, one line per g.
ax = axes[0, 0]
for g, c in zip(g_values, colors):
s = df[df["g"] == g].groupby("generation")["heterozygosity"].mean()
ax.plot(s.index, s.values, color=c, label=f"g={g:g}")
ax.axhline(H_star, ls=":", color="gray", lw=1)
ax.set(xlabel="generation", ylabel="heterozygosity $H$",
title="Trajectories: g=0 collapses, g>0 plateau")
ax.legend(frameon=False, fontsize=8, ncol=2)
# Panel B: stationary H vs g on the exact H_eq curve + operational g* with bootstrap CI.
ax = axes[0, 1]
st = reduce_to_stationary(stat, value_col="heterozygosity", last_frac=1.0)
gg, Hm, Hci = mean_ci(stat, "g", "heterozygosity")
m_of_g = stat.groupby("g")["m"].first().to_numpy()
nz = gg > 0
ax.errorbar(gg[nz], Hm[nz], yerr=Hci[nz], fmt="o", color="#1f77b4", capsize=3, zorder=3,
label="neural histogram runner")
ax.plot(gg[~nz], Hm[~nz], "o", mfc="white", mec="#1f77b4", zorder=3)
m_grid = np.linspace(0, m_of_g.max(), 400)
ax.plot(m_grid / (n + m_grid), H_eq(m_grid), "k--", zorder=2, label=r"exact $H_{eq}$ (Layer 1)")
ax.axhline(H_star, ls=":", color="gray", lw=1, label="$H^*$ (truth)")
r = critical_grounding(st, H_star=H_star, frac=0.95, seed=7)
ax.axvspan(r["ci_low"], r["ci_high"], color="#d62728", alpha=0.15)
ax.axvline(r["g_star"], color="#d62728", lw=1.2,
label=f"$g^*$={r['g_star']:.3f} (95% CI [{r['ci_low']:.3f},{r['ci_high']:.3f}])")
ax.set(xlabel="grounding fraction $g=m/(n+m)$", ylabel="stationary $H$",
title=r"Bridge reproduces the exact $H_{eq}$ and $g^\star$")
ax.legend(frameon=False, fontsize=8)
# Panel C: tail survival vs g (item-count and truth-mass).
ax = axes[1, 0]
ig, Im, Ici = mean_ci(stat, "g", "tail_frac_alive")
mg, Mm, Mci = mean_ci(stat, "g", "tail_truth_mass_alive")
ax.errorbar(ig, Im, yerr=Ici, fmt="s-", color="#d62728", capsize=3, label="tail items alive")
ax.errorbar(mg, Mm, yerr=Mci, fmt="o-", color="#9467bd", capsize=3, label="tail truth-mass alive")
ax.set(xlabel="grounding fraction $g$", ylabel="fraction of tail retained",
title="Tail survival rises with g (deep tail lags)")
ax.legend(frameon=False, fontsize=9)
# Panel D: per-rarity-band survival across g.
ax = axes[1, 1]
band_cols = sorted(c for c in df.columns if c.startswith("band") and c.endswith("_alive"))
band_colors = plt.cm.plasma(np.linspace(0.1, 0.85, len(band_cols)))
for col, c in zip(band_cols, band_colors):
s = stat.groupby("g")[col].mean()
depth = col.replace("band", "").replace("_alive", "")
lab = f"band {depth}" + (" (deepest)" if col == band_cols[-1] else
" (shallowest)" if depth == "0" else "")
ax.plot(s.index, s.values, "-o", color=c, ms=4, label=lab)
ax.set(xlabel="grounding fraction $g$", ylabel="fraction of band alive",
title=r"Per-rarity band: the $m\,p^*_i\gtrsim1$ threshold")
ax.legend(frameon=False, fontsize=8)
fig.suptitle("bridge — the histogram model reproduces Layer-1 E2 through the neural runner "
f"($g^*$={r['g_star']:.3f} vs Layer-1 0.048; HARD GATE passed)", y=1.0, fontsize=13)
fig.tight_layout()
savefig(fig, results_dir, "bridge")
if __name__ == "__main__":
main(*sys.argv[1:])