TiedAlphaAdapter: learned per-dim alpha with tied B (anchored by construction); pre-registration item 14

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

results-loop/PROTOCOL_UNIFIED.md

@@ -163,3 +163,20 @@ merge level (35.7/39.3/42.9% at k=2/4/8 — a one-item-per-depth-doubling
 crawl that at k=8 reaches what the contractive merge reaches at k=4,
 never approaching the amortization ceiling from above). 4x parameters
 bought nothing. Depth-monotone computation did not emerge at this budget.
+
+14. **Tied-alpha arm (pre-registered 2026-07-15, before training).**
+    TiedAlphaAdapter: x = (1−a)⊙e + a⊙ŝ + MLP([e;ŝ]), a = σ(â) per-dim
+    learned, init a=0.3 everywhere (bit-equal to MergeAdapter at step 0,
+    verified). B tied to (1−a): convex combination keeps the LTI fixed
+    point on the e–ŝ segment (substrate-anchored by construction),
+    ρ = max(a) < 1 guaranteed, +d≈1.5K params. Standard curriculum,
+    s0 = band(e), e400, eval ks 0,2,4,8 on 250 items. This is the one
+    untested cell combining parcae's learnable decay with the merge's
+    anchoring. Predictions: (a) substrate fidelity preserved (easy ≈
+    merge's 88%, unlike both rec arms' ~70%) because anchoring, not
+    ρ, controls fidelity; (b) hard-bucket at merge level (no significant
+    gain — per-dim constant α is not where capability lives, per the
+    adaptive-α E2B result); (c) learned a drifts slightly DOWN from 0.3
+    (as in parcae). If (a) holds while rec arms failed it, the
+    fixed-point-location dial is causally isolated: same learnable-decay
+    freedom, only the tie to (1−a) differs from parcae.