method: truncated-BPTT bias bound via rho(A) — contraction certifies tail-only gradients

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

PAPER.md

@@ -145,6 +145,15 @@ adapter + entrance-faded LoRA rank 8 on the band's first layers, loop-only
 via a global toggle), and **stack** arms (distill-warm + loop training;
 distilled adapter evaluated in loop mode).
 
+**Truncated backprop is certified by contraction.** Recurrent-regime arms
+train with tail-only BPTT (gradients through the last 4 iterations; the
+no-grad prefix stores no activations, so memory is constant in depth).
+The truncation bias scales as ρ(A)^(T−tail) — at ρ=0.3 the discarded terms
+are ≤1%, making the cheap estimator essentially exact; at ρ≥1 it is
+dominated by what it discards. Stability, fixed-point convergence, valid
+tail gradients, and the convergence-halting exit signal are all the same
+dial.
+
 **Inference.** Looped prompt states are causally independent of generated
 tokens: computed once at prefill, written into the KV cache by a hooked
 forward pass, generation native. Verified bit-identical to the slow path.
@@ -405,6 +414,18 @@ Mixture-of-Recursions (2507.10524); Relaxed Recursive Transformers
 (2410.20672); Coconut; pause tokens (Goyal et al.) — whose trained variant
 proved a genuine rival, not a strawman (§3.2, §3.5).
 
+**Saunshi et al. (2025)** argue looped transformers trade composition
+against memorization: looping buys iterative reasoning, not fact storage.
+Our results reproduce this axis *within one frozen model*: k>0 moves only
+the plan-dependent (compositional) slice, leaves recall-flavored MC
+benchmarks flat (§3.8), and the content-injecting distill arm — not the
+loop — is what nudges knowledge benchmarks up. Their looping-based
+regularization (loop harder on reasoning, relax for retrieval) has an
+inference-time analogue in our difficulty gate: route predicted
+plan-dependent prompts to k=4 and everything else to k=0, which is the
+exact base model. Retrofit looping makes the composition/memorization
+trade a *per-prompt routing decision* instead of a pretraining commitment.
+
 What remains distinct here: interpretability-derived placement with causal
 validation; a fully frozen base with bit-exact k=0 and zero-decode-cost KV
 write-in; the complete attribution ladder including compute-matched