related_work: unify anchoring across McLeish/Lys/ours; link 12B alpha miscalibration to their stability results

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

related_work/relevant_to_us.md

@@ -130,3 +130,31 @@ tiny trained merge on a frozen base + prompt-only latent planning with zero
 decode cost + a full attribution ladder + difficulty gating.** Neither paper
 touches any of the last three items; both strengthen the premise that band
 looping is a real mechanism rather than a curiosity.
+
+## One mechanism, three training regimes (anchoring)
+
+All three loop-entry designs are the same idea — **re-ground the looped
+state in the baseline forward pass** — differing only in how much training
+supports it:
+
+| regime | anchor form | training | outcome |
+|---|---|---|---|
+| Lys naive | none (α_state = 1) | zero | collapse (off-manifold drift) |
+| Lys regularized | fixed inference-time interpolation η·h⁽⁰⁾ + (1−η)·h⁽ᵗ⁾ | zero | modest, consistent gains |
+| **ours (Paper A)** | fixed mix (1−α)e + α·ŝ + trained zero-init MLP([e;ŝ]) | 1.6M params, frozen base | large gains at 2B-effective |
+| McLeish | e concatenated every iteration through a **learned** adapter | all params, 50B tokens | gains at scale; anchor constraint absorbed by retraining |
+
+Reading: how much off-manifold state the frozen band tolerates is bounded,
+and anchor strength must make up for whatever training does not provide.
+Lys's naive-looping collapse is the zero-training limit of the phenomenon;
+McLeish's full retraining is the opposite limit, where the manifold itself
+moves to accommodate the loop and explicit anchoring dissolves into learned
+injection. Our **12B α-miscalibration** (α=0.3 tuned at 2B; untrained loop
+craters 72.6→23.0 at 12B) is then not a quirk but the same stability law
+seen mid-spectrum: the tolerable loop share depends on the model, and a
+coefficient tuned on one substrate overdrives another. This connects
+directly to Lys's distribution-shift account and predicts the queued
+α∈{0.1, 0.15} 12B arm should recover much of the loss; their softmax
+auto-alignment (adaptive η, training-free) is the natural fallback if no
+fixed α transfers. Worth a paragraph in Paper A (design justification +
+12B analysis) and Paper B (stability mechanism).