abstract rewritten: fold in regime sweep (two independent dials, B-tie isolation, ceiling survival, output-orbit)

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

PAPER.md

@@ -11,41 +11,44 @@ https://git.draic.info/nils/jspace (public). Statistics:
 Interpretability work with an averaged-Jacobian lens ("J-lens") partitions a
 pretrained language model's depth into regimes, including a mid-depth
 *workspace* band that holds verbalizable, unspoken intermediate content. We
-retrofit recurrence onto this band in a **frozen** model: a 1.6M-parameter
+retrofit recurrence onto this band in a **frozen** gemma-4-E2B: a
-anchor-dominant merge adapter (0.03% of parameters) at the band entrance
+1.6M-parameter anchor-dominant merge adapter (0.03% of parameters) at the
-turns the non-self-map band into a stable fixed-point iteration, trained with
+band entrance turns the non-self-map band into a stable recurrence, trained
-self-generated, verifier-filtered supervision. Looping the workspace over the
+with self-generated, verifier-filtered supervision. Looping the workspace
-prompt ("latent planning") raises pass@1 on plan-dependent MBPP problems from
+over the prompt ("latent planning") raises pass@1 on plan-dependent MBPP
-5.5% to 37.5±5.5 over five seeds (best seed 43.6%), with zero visible tokens
+problems from 5.5% to 37.5±5.5% over five seeds — pooled across MBPP,
-and zero additional decode cost. The effect is real and highly reliable —
+HumanEval, and Rust/MultiPL-E, 4.2%→35.6% (McNemar p≈1.5e-10) — with zero
-pooled across MBPP, HumanEval, and Rust/MultiPL-E, the plan-dependent bucket
+visible tokens and zero decode cost. Placement is decisive, not convenient:
-moves from 4.2% to 35.6% (McNemar p≈1.5e-10) — and **placement is decisive,
+the gain exists only at the lens-identified boundary (L14), collapsing below
-not a convenience**: the gain appears only when the loop enters at the
+it, and KV-sharing structurally nulls entrances above it. Net of the
-lens-identified boundary (L14), collapsing at L13 and below and structurally
+untrained-merge perturbation floor (20.0%), the loop-specific effect
-nulling above. A substantial part of the gross effect is generic
+survives paired testing (p=0.007).
-perturbation — the untrained merge alone reaches 20.0% on the same bucket
+
-(it conditions on k=0 failure, so regression-to-mean contributes to any
+A two-part attribution program then bounds the mechanism. First, the
-intervention) — but the loop-specific net effect over that floor is real:
+content is *amortized, not computed*: recurrence-free plan-distillation
-paired on items, loop-vs-untrained-merge is 17–4 discordant (p=0.007).
+into the same adapter matches the loop, gains do not stack, and inference
+depth beyond k≈4 is flat — the state trajectory is an output-stable orbit,
+not a converging computation (half of prompts' states never converge at
+cos 0.9995 by k=8, with no difficulty gradient, so no convergence-based
+early exit falls out). Width rivals depth on code (trained pause registers:
+36.4%); recurrence is needed where state must evolve (GSM8K carry,
+Blocksworld planning). Second, a pre-registered regime sweep spanning
+unconstrained learned recurrence (Huginn-style), spectrally constrained
+state maps (Parcae-style), per-iteration weights (Bae-style), and learned
+anchor coefficients shows that **dynamical stability and substrate fidelity
+are independent dials**: spectral radius governs convergence only (an
+unconstrained map drifts to ρ≈4.5 with no fit benefit; a constrained one
+stays at ρ≈0.3 with no fit cost — both lose 17 points of easy-item
+accuracy), while fidelity is governed by fixed-point *location*, causally
+isolated to one design choice — tying the input map to the anchor's convex
+complement, B=(1−α)I. Per-iteration weights strand the gain at trained
+depths; every regime buys the same hard-bucket gain (36–46%); no regime
+exceeds the amortization ceiling at this budget. The hand-tuned recipe is
+thus the measured optimum of its design space, not a lucky point in it.
+At 12B the anchor coefficient must become state-dependent (3.8K parameters)
+to preserve the substrate — the one dial that is task- and scale-dependent.
+Details and exact numbers: §1 and §3.
 
-A complete attribution program then deflates the mechanism's mystique: the
-loop's content is *amortizable* (plan-distillation into the same adapter,
-recurrence-free, matches it; nothing stacks; looping distilled weights
-degrades them), width rivals depth (trained pause registers reach 36.4%),
-and verifier-assisted (oracle) sampling wins overall accuracy at matched
-compute — though the *deployable* selector loses that edge entirely. What
-survives is precise: the implant owns exactly the plan-dependent slice at
-zero visible tokens and zero decode cost, and its placement is dictated by
-the lens. Transfer dissociates by distance: near-distribution the trained
-and untrained implants tie (HumanEval); far from it (LiveCodeBench) the
-*untrained* merge significantly helps while the trained content
-significantly hurts — the learned content is task-local, the recurrence
-substrate is general. At 12B a
-constant merge coefficient destroys the substrate; a state-dependent
-coefficient (3.8K parameters) restores MBPP but not Blocksworld or GSM8K —
-the anchor coefficient is the stability dial that unifies this work with
-McLeish et al. (2511.07384) and Lys et al. (2602.14759), and it is task-
-and scale-dependent. Details and exact numbers: §1 and §3.
 
 ## 1. What this paper claims