diff --git a/results-loop/fig_phase.png b/results-loop/fig_phase.png
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Binary files /dev/null and b/results-loop/fig_phase.png differ
diff --git a/scripts/fig_phase.py b/scripts/fig_phase.py
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--- /dev/null
+++ b/scripts/fig_phase.py
@@ -0,0 +1,72 @@
+"""Phase diagram of frozen-band recurrence: dynamics dial vs fidelity.
+
+x: spectral radius rho(A) of the trained state map (log scale)
+y: substrate fidelity = easy-bucket pass@1 at k=4 (free-running generation)
+label: hard-bucket pass@1 at k=4 (the gain every regime buys)
+All points: same frozen E2B band, same data, same 250-item eval, e400.
+"""
+from pathlib import Path
+
+import matplotlib
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt
+
+OUT = Path(__file__).resolve().parent.parent / "results-loop"
+
+#            name                     rho    easy   hard   color
+#            name                  rho(plot x)  easy   hard   color  dy
+PTS = [
+    ("untrained α-merge\n(training-free)", 0.38, 0.885, 0.143, "#8a8f98", 16),
+    ("trained merge\n(anchored B, curriculum)", 0.29, 0.885, 0.464, "#2b6cb0", -60),
+    ("Parcae rec\n(ρ<1 enforced, learned B)", 0.292, 0.713, 0.429, "#2f855a", -58),
+    ("unconstrained rec\n(learned A,B)", 4.5, 0.697, 0.393, "#c53030", 16),
+]
+# untrained merge x offset for visibility (true rho = 0.30)
+
+fig, ax = plt.subplots(figsize=(7.6, 5.2))
+ax.set_xscale("log")
+ax.axvspan(1.0, 20, color="#fff5f5", zorder=0)
+ax.axvline(1.0, color="#c53030", lw=1.2, ls="--")
+ax.text(1.06, 0.99, "ρ = 1  stability boundary", rotation=90, fontsize=8,
+        color="#c53030", va="top")
+ax.text(3.1, 0.615, "norm projection converts\nexplosion → stationary churn\n"
+        "(training survives, substrate pays)", fontsize=8, color="#c53030",
+        ha="center")
+
+for name, rho, easy, hard, c, dy in PTS:
+    ax.scatter(rho, easy, s=90 + 900 * hard, color=c, alpha=0.85, zorder=3,
+               edgecolor="white", linewidth=1.5)
+    ax.annotate(f"{name}\nhard {hard:.0%}",
+                (rho, easy), textcoords="offset points", xytext=(0, dy),
+                ha="center", fontsize=8.5, color=c)
+
+ax.annotate("", xy=(0.278, 0.735), xytext=(0.288, 0.862),
+            arrowprops=dict(arrowstyle="->", color="#2f855a", lw=1.3))
+ax.text(0.42, 0.79, "learned B + no curriculum:\nfixed point leaves the\n"
+        "substrate manifold", fontsize=8, color="#2f855a", ha="left")
+
+ax.set_xlim(0.2, 12)
+ax.set_ylim(0.58, 1.0)
+ax.set_xlabel("ρ(A) — spectral radius of the trained state map (dynamics dial)")
+ax.set_ylabel("substrate fidelity — easy-bucket pass@1 at k=4")
+ax.set_title("Frozen-band recurrence phase diagram: stability ≠ fidelity\n"
+             "(marker size ∝ hard-bucket gain; every regime buys the same "
+             "~40–46%, only one keeps the substrate)", fontsize=10.5)
+ax.grid(True, color="#ececec", lw=0.7, which="both")
+ax.set_axisbelow(True)
+for s in ("top", "right"):
+    ax.spines[s].set_visible(False)
+
+fig.text(0.13, -0.06,
+         "Same frozen gemma-4-E2B band (L14–30), same MBPP data, same "
+         "250-item eval, e400 checkpoints. ρ from checkpoints (power "
+         "iteration / diag max). Dynamics (x) and fixed-point location "
+         "(y) are independent dials: contraction guarantees convergence "
+         "and certified tail gradients, but only anchored content — fixed "
+         "B=(1−α)I, zero-init correction, difficulty→depth curriculum — "
+         "keeps the fixed point substrate-preserving.", fontsize=7.5,
+         color="#555", wrap=True)
+fig.tight_layout()
+fig.savefig(OUT / "fig_phase.png", dpi=140, facecolor="white",
+            bbox_inches="tight")
+print("wrote", OUT / "fig_phase.png")