halting probe: output-stable orbit, not state fixed point; no free ACT at state level; paper claims softened

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

PAPER.md

@@ -81,7 +81,8 @@ not laws.)*
    without recurrence matches it; stacking buys nothing (loop-training a
    distill-warmed adapter: 34.5%, below distill alone; running the distilled
    adapter in loop mode: drops to 20.0%); deeper k at inference is flat
-   (k=8: 40.0%). The recurrence is a *training-time scaffold* that lets the
+   (k=8: 40.0%; output-stable despite residual state drift, §3.8). The
+recurrence is a *training-time scaffold* that lets the
    adapter find plan-shaped content — content that can equally be put there
    by distillation if plans are available.
 
@@ -346,8 +347,14 @@ plan-representation overlaps; task-specific training still dominates.
 ### 3.8 Mechanism, verification, deployment
 
 The trained loop takes a large first step (cos(s₁,s₀)=0.926 vs 0.977
-untrained) and converges bit-exactly by k≈3–4; accuracy and lens-sharpening
-plateau there. P(latent concept) under the J-lens at the band exit rises
+untrained); accuracy and lens-sharpening plateau by k≈3–4. A population
+probe (n=250, state-cosine threshold 0.9995) shows the plateau is
+*output-level*: half the prompts' states are still drifting at 1e-3–1e-4
+cosine scale at k=8 while generation is already depth-stable — an
+output-stable orbit rather than a literal state fixed point, with no
+difficulty gradient in state-convergence depth. Consequently,
+convergence-based early exit ("free ACT") does not fall out of the state
+trajectory; halting would need an output-level signal. P(latent concept) under the J-lens at the band exit rises
 0.015→0.13 across iterations (~8× the untrained hold) — the lens that placed
 the implant also renders its silent content inspectable. The STaR labels
 train a free difficulty gate (route predicted-hard to k=4, else k=0);