# Latent workspace recursion via a trained merge layer

A design note on one idea that came out of the J-lens work: **can we loop the
workspace band of a pretrained model to spend more compute per token — and if
so, how do we make it actually improve reasoning rather than just idle?**

Reproduction results this builds on are in [`RESULTS.md`](RESULTS.md); the
schematic is [`results/jlens_diagram.png`](results/jlens_diagram.png); a
distilled worked/failed/pitfalls ledger is in [`LESSONS.md`](LESSONS.md). All
experiments below are on `google/gemma-4-E2B-it` (35 layers, d=1536) on a DGX
Spark.

---

## 1. Why the workspace is a candidate for looping

The J-lens analysis gives every layer a readout of "what verbalizable concept
is here." Across models that partitions depth into four regimes:

| regime | E2B layers | what the J-lens reads |
|---|---|---|
| transduction | 0–5 | noise / surface lexical form |
| sensor | 6–13 | the current input token (echo ↑26%) |
| **workspace** | **14–30** | **unspoken abstract content ('spider', 'big'); held & broadcast** |
| motor | 31–34 | the next output token (→68%) |

Two measured properties make the workspace look like a recurrent variable that
is currently unrolled across distinct layers rather than iterated:

- **It is slow / persistent.** Top-10 J-lens content overlaps ~0.25 (Jaccard)
  between adjacent token positions — the workspace barely changes token to
  token.
- **It holds unspoken intermediates.** On "the animal that spins webs has how
  many legs?" the lens surfaces **spider** mid-network before the model emits
  **8**; on multi-step arithmetic it stages `21 → 42 → 49` across layers.

If that band is effectively approximating "iterate toward a settled abstract
representation," then **looping it should be a way to add sequential compute
without adding parameters** — the premise behind Universal Transformers,
Deep Equilibrium Models, and recent recurrent-depth LLMs (Huginn, Ouro,
Mixture-of-Recursions).

Workspace-band parameter mass (decoder only), for reference:

| model | band | params in band | active/token |
|---|---|---|---|
| E2B | L14–30 | 1.08 B (58%) | 1.08 B |
| 12B | L36–45 | 2.26 B (21%) | 2.26 B |
| 26B-MoE | L20–27 | 6.53 B (27%) | ≈0.82 B |

---

## 2. What we tried, and what actually happens

Probe: inject a residual at the band entrance, run the band, feed the result
back, and track (a) convergence `|Δ|/|·|`, (b) the J-lens 'spider' concept,
(c) the emitted answer. Baseline answer is `8`.

### 2a. Naive band loop — fails (ill-posed)
Feed L30's output straight back into L14. **Collapses in one iteration**
(`8 → )`). Reason: the band is **not a self-map** — its output lives 17 layers
of accumulation downstream of its input (larger norm, "settled" content), so
re-injecting it double-counts the band's contribution and leaves the input
distribution L14 expects.

### 2b. Anchored `H ← H0 + bandΔ(H)` — also fails
Merging the fixed pre-band residual `H0` as an additive anchor does not help
on its own: `bandΔ` is a large forward displacement, so `H0 + bandΔ` lands
back in post-band space and diverges (`|ΔH|/|H| ≈ 2` per step).

### 2c. Single tied layer `h ← h + β·Δ_L(h)` — well-posed but not useful
One layer *is* a genuine self-map (same residual point in and out). Result:
**stable, converges** (`|ΔH|` decreases monotonically), and **low-pass damping
extends it** — β=0.5 at L26 keeps `8` through ~6 loops vs. immediate decay at
β=1.0. But a **frozen** layer has no useful fixed point: pushed far enough it
relaxes to a degenerate attractor (`Set`, `number`, …) and the J-space concept
**never sharpens**. Confirms the Relaxed-Recursive-Transformer lesson: layers
not trained to be looped only tolerate it, they don't benefit.

### 2d. Merge layer between L13 and L14 — the fix (structurally)
Insert a merge that reconciles the looped-back post-band state `s` with the
fixed pre-band anchor `e = L13 output`:

```
L14_in = (1−α)·e + α·(s renormalized to |e|)
```

Untrained, anchor-dominant (**α=0.3**):

| k | `|Δs|/|s|` | `|s|/|e|` | answer | P(ans) |
|---|---|---|---|---|
| 0 | 0.00 | 2.23 | `8` | 1.000 |
| 1 | 0.29 | 2.11 | `8` | 1.000 |
| 5 | 0.12 | 2.09 | `8` | 1.000 |
| 10 | 0.10 | 2.09 | `8` | 1.000 |

The band is now a **well-posed, stable, answer-preserving recurrence**:
converges to a fixed point (`|Δs|` 0.29→0.10), norm controlled, `8` intact
through 11+ loops — versus collapse-in-one-step for the naive loop.

Caveats: it only stabilizes when the **anchor dominates** (α≥0.6 drifts to
garbage), and it **holds rather than sharpens** — the 'spider' J-space concept
fluctuates (~0.01–0.04) rather than growing. Untrained, the merge buys a
stable substrate, not extra reasoning.

**This is the DEQ / Huginn "input-injection" structure**: a fixed input
(anchor) merged with an iterated state makes the core a self-map. Our merge is
that injection, done by hand.

---

## 3. The proposal: train *only* the merge layer

The merge layer converts the ill-posed loop into a stable, answer-preserving
substrate, and it is the **single small component to train** — a little
MLP / LoRA at L13→L14 — to make iterations *improve* the answer, with L14–30
and the rest of the model frozen. This is the cheapest possible "train the
loop in."

```
        e (L13 out, fixed anchor)
                │
  s_k ──▶ [ merge adapter θ ] ──▶ L14 … L30 ──▶ s_{k+1}   (× k loops)
                                        │
                                    L31 … L34 ─▶ logits (answer)
```

### Objective (self-contained, no bigger teacher)
Latent chain-of-thought distillation (Coconut / STaR-style):
1. Let the **frozen base model** produce CoT answers; keep only traces whose
   final answer is **correct** (rejection sampling).
2. Train the adapter so the **latent looped** forward — k loops, *no CoT
   emitted* — reaches that correct answer. Loss = cross-entropy on answer
   tokens only; adapter is the sole trainable tensor; band unrolled k times.

The supervision is answers the model can reach *with* CoT but not *without* —
so loop depth substitutes for reasoning tokens, and everything stays
in-distribution for the frozen base.

### Training data
- **GSM8K** (already cached: `openai/gsm8k`) — multi-step arithmetic; single
  verifiable numeric answer; exactly the `21→42→49` staging we observed.
- **Synthetic multi-hop templates** (the spider family) — controllable hop
  count for a difficulty curriculum, and known intermediates so the J-lens can
  check whether looping sharpens them. Optionally real 2-hop QA
  (2WikiMultiHop / StrategyQA) for naturalness.
- **Avoid**: raw pretraining text (no reasoning signal → re-teaches
  preservation), pure 1-pass self-distillation (target = no-loop answer, loop
  can't beat it), bigger-model distillation (conflates "loop helps" with
  "bigger helps").

### The design point that makes or breaks it
Force the loop to be **used**: train across variable k with a
**difficulty→depth curriculum** (easy items solvable at k=1, hard items only
at k>1). Otherwise the adapter learns to solve everything at k=1 and ignores
the recurrence — the "limited evidence of latent CoT" failure the
recurrent-depth interpretability papers report.

### Evaluation (closes the loop with our tooling)
- **Accuracy vs. k** on held-out two-hop templates + GSM8K test. Success =
  the curve **rises with k** — the thing that flatly did *not* happen
  untrained.
- **J-lens sharpening**: does the intermediate concept (spider, arithmetic
  partial) grow across loops after training, where it only fluctuated before?

---

## 4. Results of the trained run

The §3 experiment was run (July 2026, ~25 min on the GB10): 1,024 GSM8K train
items labeled by the frozen model (direct pass vs CoT pass → 165 easy / 416
hard / 443 dropped as unreachable; baselines 16% direct, 53% CoT), then 800
steps of adapter-only training (1.6M params, batch 8, CE on answer tokens)
with the curriculum k=1 easy / k=2 mixed / k=4 hard-only. Eval: greedy decode
through the looped forward on 256 held-out test items (frozen-model baselines
there: 11.3% direct, 55.9% CoT), trained adapter vs the untrained α-merge.

![accuracy vs k and J-lens sharpening](results-loop/loop_eval.png)

| k | untrained all | trained all | untrained hard | trained hard |
|---|---|---|---|---|
| 0 | .117 | .117 | .008 | .008 |
| 1 | .082 | .039 | .016 | .016 |
| 2 | .094 | .090 | .039 | **.063** |
| 4 | .090 | .082 | .047 | **.063** |
| 8 | .086 | .082 | .039 | .055 |

**What worked (the pre-registered J-lens criterion).** After training, loops
*sharpen* the latent concept instead of merely holding it: P('spider') under
the lens at L30 rises 0.015 → 0.107 → 0.130 across k=0→4, ~8× the untrained
control (which stays at 0.013–0.033, as in probe 2d), with the answer intact
(P(8)≈1.0) and the recurrence stable through k=8. Training the merge changed
the loop from a preserver into an amplifier.

**What partially worked.** On the hard (CoT-only) bucket, accuracy rises with
depth: 0.8% at k=0 → 6.3% at k=2–4 trained, vs 3.9–4.7% untrained. Depth buys
a small number of answers the model cannot produce in one pass — but it is
8/127 items, not a CoT replacement.

**What failed.** Overall accuracy does *not* rise with k: pushing free-running
generation through the loop damages easy items (97% → 21–35% trained, → 41–52%
even untrained), which swamps the hard-bucket gain; k=0 remains best overall.
The teacher-forced val looked much better than free-run decode — an
exposure-bias gap: training only ever saw gold answer tokens, generation feeds
back its own. Note the damage is mostly there in the untrained loop path too,
so it is a property of the substrate, not something training introduced.

**Obvious next steps.** (1) Free-run/scheduled-sampling training to close the
exposure-bias gap; (2) a k=0-preservation loss on easy items so looping never
costs known answers; (3) per-token adaptive depth (MoR-style) instead of
uniform k; (4) more data at grade-school difficulty (Orca-Math) and a
contamination-free eval with a difficulty dial (GSM-Symbolic variants mapped
to target k); (5) bigger adapter / more steps — 800 steps on 517 items is a
first probe, not a training run.

---

## 5. Latent planning on code (MBPP) — first overall win

Transposing to coding with one structural change: the loop applies **only to
the prompt span** (`loop_mask`) — the model "plans silently" while reading the
problem, then writes code through the plain path (generated tokens never loop,
so the §4 exposure-bias failure is absent by construction). Since causality
makes the looped prompt states constant across token steps, they are computed
once and frozen; generation then runs at native speed with a KV cache
(`generate_frozen_prompt`, verified bit-identical to the recomputing path).

Data: MBPP (974 basic Python problems, 3 asserts each). Frozen-model labeling:
direct pass@1 52-57%; plan-first 63% reachable (**caution**: with a 380-token
budget the plan pass scored 17% — "planning hurts" was purely truncation; at
700 tokens planning adds ~11 points). Labels: ~260 easy / 66 hard (plan-only)
/ rest drop. Training: same recipe, supervision = the model's own verified
passing code, CE on code tokens, curriculum k=1 easy / 2 mixed / 4 hard; the
38-item hard pool overfits past ~step 300 (evaluated the step-399 snapshot).

![MBPP pass@1 vs k](results-loop/loop_eval_code.png)

| k | untrained all | trained all | untrained hard | trained hard | trained easy |
|---|---|---|---|---|---|
| 0 | .488 | .488 | .036 | .036 | .984 |
| 1 | .472 | .488 | .179 | .429 | .852 |
| 2 | .476 | **.520** | .179 | .357 | .926 |
| 4 | .472 | .512 | .143 | **.464** | .885 |

- **Overall pass@1 beats the no-loop baseline** (52.0% vs 48.8% at k=2) — the
  criterion the GSM8K run failed. The untrained loop never does (≤47.6%).
- **Planning-dependent problems: 3.6% → 46.4%** at k=4 (26/56 items; McNemar
  p<1e-5). Silent loops recover ~46% of what explicit written planning
  achieves. Training matters: the untrained substrate reaches only 14-18%.
- The **easy-item dip** (98→89%) is substrate-inherent (untrained dips
  equally); training partially repairs it (92.6% vs 88.5% at k=2). Fix on the
  board: a per-prompt **gate** (probe on the k=0 workspace state predicting
  "will looping help?" — the STaR labels are its free supervision; a perfect
  gate scores ~86% on this mix).

### Rung-2 design note: depth-graded band unfreezing

**Optimizer note (from McLeish et al., arXiv:2511.07384):** use **Muon**, not
AdamW, when unfreezing the band — they report AdamW loss-spikes to NaN when
training through recurrence; adapter-only training didn't hit this, band-LoRA
likely will. Also adopt their FLOPs convention for recurrent models
(FLOPs = (6N₁+2N₂)D, N₁ = params with grads, N₂ = forward-only) in any
compute-matched comparison. **Adaptive-α candidate (from Lys et al.,
arXiv:2602.14759):** their training-free softmax auto-alignment interpolation
is a cheap ablation against fixed α — directly relevant to the 12B α
miscalibration.

When the band itself is unfrozen (per-iteration LoRA), unfreeze
**entrance-faded**: full trainability at L14 decaying to frozen by ~L22
(e.g. LoRA α × max(0, 1−(ℓ−14)/8)). Rationale: the only novel inputs in the
system are what the first band layers see (merged fed-back states); the
band's *exit* distribution must stay on-manifold for the frozen motor/suffix
consumers, and the k-fold application of the band amplifies any trained
change — a depth-decaying profile is a trust region. Risk: if sharpening is
implemented by late-band broadcasts, capping their plasticity caps gains.
Ablation triplet (shares everything but the LoRA mask): uniform /
entrance-faded / exit-faded (falsification control — parity would mean
placement is irrelevant and only capacity matters). Composes with, but is
run before, per-iteration strength fading (Relaxed-Recursive axis).

## 6. Loop dynamics & cross-token carry probes

**Dynamics** ([`results-loop/loop_dynamics.png`](results-loop/loop_dynamics.png)):
the trained within-token loop is a true fixed-point iteration — a large first
step (cos(s₁,s₀)=0.926 vs 0.977 untrained), then bit-exact convergence by
k≈3-4 (cos=1.000), norm settled ~1.7% above the untrained level. Convergence
depth coincides with where accuracy and J-lens sharpening plateau; k>4 is a
literal no-op.

**Cross-token carry** (`probe_carry.py`): instead of k loops per token, carry
the band output across token steps — one band pass per token seeded with the
previous step's state (the amortized loop; cf. Feedback Transformer).
Untrained: stable and coherent over 60 free-running tokens (all three seeding
modes — last-position / mean / recency-weighted — with mean/EMA marginally
cleaner), only a mild norm drift (+12%/60 tokens). With the §4-trained
adapter: the 8× sharpening **transfers** to the carried regime on short
answers (P('spider') 0.015→0.12 in ~2 token steps) but long free-running text
collapses — the adapter never saw its own continuations. Conclusion: the
carry loop must be trained *as* the carry loop (sequential unroll on its own
trajectory; planned objective: fixed-point distillation — train one carried
band pass to land where the within-token k-loop converges — plus STaR
rollouts).

---

## 7. Status & related work

**Status.** Probes 2a–2d, the trained-merge experiments (§3–5), the dynamics
analysis and carry probes (§6) are run and reproducible on E2B:
`scripts/loop_common.py` (band re-run machinery, bit-exact; `loop_mask`;
`generate_frozen_prompt` fast path), `prep_star_data.py` / `prep_mbpp.py` (+
`prep_mbpp_fix.py`), `train_merge.py` / `train_merge_code.py`, `eval_loop.py`
/ `eval_loop_code.py`, `probe_carry.py`, plotting scripts; data, adapters and
eval JSONs in `results-loop/`. Toward a paper: needs seeds/full test sets +
CIs, the pause-token (equal-FLOPs) baseline, a second model scale (12B), a
contamination-free eval (GSM-Symbolic), and the per-prompt gate; the
cross-token carry (§6) is the successor experiment.

**Related work.** Universal Transformers (adaptive-depth recurrence); Deep
Equilibrium Models (solve for the fixed point directly); Geiping et al.
[recurrent-depth latent reasoning / "Huginn"](https://arxiv.org/abs/2502.05171)
(prelude→looped core→coda with input injection); [Mixture-of-Recursions](https://arxiv.org/abs/2507.10524)
(per-token learned recursion depth); [Relaxed Recursive Transformers](https://arxiv.org/abs/2410.20672)
(layer-tying + per-loop LoRA); Coconut (latent chain-of-thought). The
novel piece here is using an **interpretability signal (the J-lens)** both to
*choose the band to loop* and to *measure whether looping deepens computation*,
and training **only a merge adapter** on top of a frozen pretrained model.
