ParcaeAdapter: rho(A)<1 by construction (ZOH negative-diag), rho logging in rec arms, pre-registration item 12

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

results-loop/PROTOCOL_UNIFIED.md

@@ -117,3 +117,19 @@ number for the unified adapter exists at time of writing.
     Known risk, stated in advance: 600 steps may be far too little for
     this regime (McLeish et al. use ~50B tokens); a null here bounds the
     cheap-retrofit budget only, not the regime.
+
+12. **Parcae-constrained recurrent arm (pre-registered 2026-07-15, before
+    training; Prairie et al. 2026 parameterization).** Same as item 11 but
+    A = exp(−Δt·exp(a)) diagonal → ρ(A) < 1 by construction; init exactly
+    the α=0.3 merge (verified bit-equal at init). ρ(A) logged every 10
+    steps in BOTH arms. Theory-derived predictions, stated in advance:
+    (a) contraction ⇒ fixed point is a function of e ⇒ the Parcae arm
+    SATURATES in k (no depth-monotone gain) and its converged performance
+    is amortizable — if so, our deflationary result is a corollary of
+    ρ<1, and our observed k≈3–4 convergence is the geometric rate 0.3^k;
+    (b) the UNCONSTRAINED item-11 arm either drifts toward ρ≥1 (watch the
+    ρ log: divergent runs should show ρ≥1 before loss spikes) or, if it
+    gains monotone depth-performance, does so with ρ near 1 — the edge of
+    stability is where genuine iteration must live. Either outcome
+    formalizes "the anchor coefficient is the stability dial" as
+    "the anchor coefficient is the spectral radius".