## 1. Problem, Operational Scope, and Contributions

Training multi-layer causal Transformers without a global backward pass is a core problem in local credit assignment [1, 2, 3, 9]. A foundational methodological distinction separates **backward-free local credit assignment** from **approximate or manual backpropagation**: disabling automatic differentiation (`torch.no_grad()`) or replacing local analytical derivatives with finite-difference probes does not remove the backward phase if current-step downstream error vectors are multiplied by layer weight matrices in reverse order ($\mathbf{e}_{\text{in}} = \mathbf{e}_{\text{out}} W$) or relayed from downstream blocks to upstream blocks within the same training step.

In Chung's **Weight Maximization (WM) with eligibility traces** [1], each unit $i$ in stage $m$ injects a local forward exploration perturbation $\xi_{n, i}^{(m)}(t)$ during the forward pass of batch $t$, records a local pre-synaptic eligibility trace $Z_{i, :}^{(m)}(t)$, and updates at a subsequent step using the column weight-norm change $\sum_{j} W_{j, i}^{(m+1)} \Delta W_{j, i}^{(m+1)}$ of its outgoing receiver synapses $W_{:, i}^{(m+1)}$.

### 1.1 Operational Definition of Delayed Perturbation-Differential Column-Norm Credit and Pipeline Clocks

To adapt delayed WM to multi-token Transformer minibatches where receiver layers are updated by adaptive or spectrally orthogonalized optimizers (`AdamW` or `Pre-Polar Muon` [7, 8]), we state precisely what feedback our algorithm constructs, how its normalization is parameterized in code, and how pipeline clocks are ordered:

1. **Installed Receiver Displacement vs. Perturbation-Differential Plasticity Tensor:** At step $t$, each receiver layer $m+1$ updates its live weights $W^{(m+1)}(t) = W^{(m+1)}(t-1) + \delta W^{(m+1)}(t-1)$ via its optimizer from the clean batch gradient (or delayed WM pseudo-gradient). To evaluate the directional effect of stage $m$'s forward perturbation $\xi^{(m)}(t_{\text{orig}})$ on receiver $m+1$'s synaptic plasticity without confounding from other stages' noise or optimizer state history, our reward routine (`compute_stream_wm_rewards_chunked` in `exp26`; `compute_stream_wm_rewards_fast` in `exp32`–`exp35`, parameterized by `scale_factor`) constructs for each token stream $\mathcal{S}_g$ ($g \in \{1, \dots, G\}$, $|\mathcal{S}_g| = M_g = BT / G$) a **perturbation-differential plasticity tensor**
   $$
   \Delta \tilde{W}_{j, i}^{(m+1, g)}(t_{\text{orig}}; \text{scale\_factor}) := \frac{\text{scale\_factor}}{M_g} \sum_{n \in \mathcal{S}_g} u_{n, j}^{(m+1)}(t_{\text{orig}}) \, \Delta y_{n, i}^{(m)}(t_{\text{orig}}), \qquad \Delta y_{n, i}^{(m)} := y_{n, i}^{\text{pert}}(\sigma \xi^{(m)}) - y_{n, i}^{\text{clean}}(0),
   $$
   where $u_{n, :}^{(m+1)}(t_{\text{orig}})$ is the receiver's local post-synaptic modulation vector and $\Delta y_{n, :}^{(m)}(t_{\text{orig}})$ is the perturbed-minus-clean forward activation displacement at stage $m$'s output. At reward construction clock $t_{\text{constr}}$, unit $i$ contracts $\Delta \tilde{W}_{:, i}^{(m+1, g)}(t_{\text{orig}}; \text{scale\_factor})$ with its **live outgoing receiver column** $W_{:, i}^{(m+1)}(t_{\text{constr}})$:
   $$
   R_i^{(m, g)}(t_{\text{orig}}, t_{\text{constr}}; \text{scale\_factor}) = \sum_{j=1}^{d_{\text{next}}} W_{j, i}^{(m+1)}(t_{\text{constr}}) \, \Delta \tilde{W}_{j, i}^{(m+1, g)}(t_{\text{orig}}; \text{scale\_factor}).
   $$
   The parameter update then divides the stream-summed modulation–input product $\sum_{g=1}^G R_i^{(m, g)} Z_{i, :}^{(m, g)}$ by $N = BT$ (`-(mod_flat.t() @ x_flat) / BT`). Thus, $\Delta \tilde{W}^{(m+1, g)}$ is a local perturbation-differential Hebbian plasticity tensor rather than a decomposition of the installed optimizer displacement $\delta W^{(m+1)}$, and reading $W^{(m+1)}(t_{\text{constr}})$ at construction clock $t_{\text{constr}}$ makes the reward directly sensitive to any receiver weight update $\delta W^{(m+1)}(t-1) = W^{(m+1)}(t) - W^{(m+1)}(t-1)$ installed between origin $t_{\text{orig}}$ and construction $t_{\text{constr}}$.
2. **Three-Clock Pipeline Schedules:** For each stage $m \in \{1, 2, 3\}$, we track $(t_{\text{orig}}, t_{\text{constr}}, t_{\text{inst}})$ relative to installation step $t_{\text{inst}} = t$. Across `exp26`–`exp35`, three distinct clock schedules are evaluated:
   - **True Both-Stages Pre-Install (`both_stages_pre_install`, `exp26`–`exp30` and `exp35` Part B2):** `Stage 3`: $(t, t, t)$; `Stage 2`: $(t-1, t-1 \text{ [pre-Stage3-step]}, t)$ (reward constructed at $t-1$ on $W_{\text{lm}}(t-1)$ prior to Stage 3's batch-$t-1$ update, installed at $t$); `Stage 1`: $(t-2, t-1 \text{ [pre-Stage2-step]}, t)$ (reward constructed and cached at $t-1$ on $W_{\text{proj}}(t-1)$ prior to Stage 2's installation of its batch-$t-2$ update, then installed into Stage 1 at $t$).
   - **Stage-2 Pre-Install / Stage-1 Post-Install (`stage2_pre_install`, `run_exp33_partB.py`, `run_exp34.py`, `exp35` Part B1):** `Stage 3`: $(t, t, t)$; `Stage 2`: $(t-1, t-1 \text{ [pre-Stage3-step]}, t)$ (constructed at $t-1$ prior to Stage 3's update); `Stage 1`: $(t-2, t \text{ [post-Stage2-step]}, t)$ (constructed at $t$ on live $W_{\text{proj}}(t)$ after Stage 2 installed its batch-$t-2$ update at the end of $t-1$).
   - **Strict Post-Installation Construction (`strict_post_install`, `exp32`–`exp35`):** `Stage 3`: $(t, t, t)$; `Stage 2`: $(t-1, t \text{ [post-Stage3-step]}, t)$ (constructed at $t$ on $W_{\text{lm}}(t) = W_{\text{lm}}(t-1) + \delta W_{\text{lm}}(t-1)$ strictly *after* Stage 3 installed its batch-$t-1$ update); `Stage 1`: $(t-2, t \text{ [post-Stage2-step]}, t)$ (constructed at $t$ on $W_{\text{proj}}(t)$ strictly *after* Stage 2 installed its batch-$t-2$ update).
   *(Note: The `CausalAuditLedger` fields in our JSON logs record declared trainer configuration metadata summarizing the code path of each arm rather than runtime hardware profiler counters; in `exp33`–`exp35`, all `stage2_pre_install` ledgers are reconciled to record `(t-2, t [post-Stage2-step], t)`).*

### 1.2 Summary of Contributions

Across 150 three-seed training runs (`51` three-seed arms across `exp26`–`exp35`, Seeds `46, 47, 48`) on ClimbMix `d6` ($L=6, D=384, V=8,192$, $360$ steps $= 0.737\text{M}$ tokens) and Native `50,332,176` ($L=8, D=512, V=8,192, T=2,048$, $100$ steps $= 13.1\text{M}$ tokens), where every backward-free run executes inside `with torch.no_grad():` (`assert not torch.is_grad_enabled()`) with zero reverse matrix pullbacks, zero VJP helpers, and zero same-step relays, we establish four results within the short-horizon developmental regime tested:

- **Reconciled Stream-Partitioned Estimator Analysis & Scale-Normalized Controls (Proposition 1, `exp26`, `exp35`):** We reconcile the exact implemented contraction $\widehat{G}_{i, :}^{\text{code}}(G; \text{scale\_factor}) = \frac{1}{N}\sum_{g=1}^G R_i^{(m, g)}(\text{scale\_factor}) Z_{i, :}^{(m, g)} = \frac{\text{scale\_factor}}{M_g} \widehat{G}_{i, :}^{\text{norm}}(G)$ across both the default code convention `scale_factor = 1` (whose conditional mean scales as $1/M_g$ and covariance as $1/M_g^2$, giving mean $0.5$ on $N=4, G=2, M_g=2, a_n=x_n=1$) and the scale-normalized convention `scale_factor = M_g` (whose conditional mean $\frac{1}{N}\sum_{n=1}^N a_{n, i} x_n$ is invariant to $M_g$, giving $1.0$). We further distinguish **separable channels** (Stage 1: `mlp.c_fc, attn.c_v`, **Proposition 1(a–b)**, where first-order variance vanishes at $M_g = 1$) from **non-separable `RMSNorm` interfaces** (Stage 2: `x_final, x_0`, **Proposition 1(c)**), deriving the exact single-token ($M_g=1$) cross-coordinate variance $\frac{a_{n, i}^2 x_{n, i}^2 (D - x_{n, i}^2)}{D^2 \text{rms}(z_n)^2} \|h_n\|_2^2$ (giving $0.064$ at $D=2, z=(1, 2)$). Controlled 3-seed `scale_factor = M_g` runs (`exp35`) confirm that holding the conditional mean invariant to $M_g$ leaves `d6` learning curves virtually unchanged (`2.6097 -> 2.3361 -> 2.3008 -> 2.2338` for `scale_factor = M_g` vs. `2.5939 -> 2.3364 -> 2.2966 -> 2.2338` for `scale_factor = 1` across $G \in \{1, 64, 256, 2048\}$; and `2.2539 ± 0.0013` vs. `2.2541 ± 0.0013` at $G=64$ under `Pre-Polar Muon`), establishing that variance reduction rather than raw step-scale changes drives the stream-partitioning gain.
- **Forward Causal Attention-Filtered Value Eligibility Trace (Proposition 2, `exp28`):** Because $v(b, s) = W_v x_{\text{attn\_in}}(b, s)$ commutes with causal attention over $s \le \tau$, computing the causal attention-filtered pre-synaptic trace $\bar{x}_{\text{attn\_in}}^{(h)}(b, \tau) = \sum_{s \le \tau} A_{b, h, \tau, s} x_{\text{attn\_in}}(b, s)$ during the forward pass and perturbing post-attention value outputs lifts single-step `attn.c_v` cosine alignment **$8.2\times$ from `+0.1104` to `+0.9069`**. Pairing $Z^{(2)}(t-1)$ with independent current-batch rewards $R^{(2)}(t)$ (`mismatched_origin_control`, `exp28`) degrades `d6` from `2.2338 ± 0.0102` to **`2.7608 ± 0.0056` DEV BPB (`-0.5270` BPB contrast)**.
- **Diagonal Skip-Routing Surrogate, Data-Dependent Kurtosis Bound, & Complete 16-Cell + 4-Cell Clock/Optimizer Factorials (Propositions 3 & 4, `exp31`–`exp35`):** Routing lexical credit to `wte` via the diagonal skip surrogate $W_{\text{skip}} = \alpha_0 I_D$ (**Proposition 3**) cuts `d6` rule time by **$27.4\%$ (`205.5 -> 149.2` ms/step)** with identical accuracy (`2.1579 ± 0.0004` vs. `2.1577 ± 0.0004`). For Strict Post-Installation Construction, **Proposition 4** decomposes $\delta W_{\text{lm}}(t-1)$ into weight decay, momentum history $\beta_1 m^{(t-2)}$, and current-batch gradient $(1-\beta_1) g^{(t-1)}$. Under row-isotropic preconditioning, reward-side projection $P_{x_{\text{final}}(n)}$ annihilates the $O(1)$ radial self-update vector $b_{\text{rad}}(n)$ identically, leaving an oblique `RMSNorm`-Jacobian coupling residual bounded by the **data-dependent coordinate fourth moment** $\frac{\mathbb{E}[\|P_x b\|_2^2]}{\|b_{\text{rad}}\|_2^2} \le \frac{\kappa_4(x_{\text{final}}(n))}{D}$ where $\kappa_4(x) := \frac{1}{D}\sum_{i=1}^D x_i^4$. We show both that unrestricted sparse vectors $x = \sqrt{D/5}(1, 2, 0, \dots, 0)$ have non-vanishing ratio $0.0576$ ($\kappa_4(x)/D = 0.68$) for all $D \ge 2$, and that on trained ClimbMix `d6` states $\kappa_4(x_{\text{final}})$ has token mean $2.9398$ (`exp35` Part 0), yielding bound $\kappa_4(x_{\text{final}})/D = 0.007656$ and empirical ratio $0.007458$. Across our **complete 16-cell $2 \times 2 \times 2 \times 2$ Stage-2 Clock $\times$ Optimizer factorial (`exp34` + `exp35` Part B1)** and **4-cell True Both-Stages Pre-Install comparison (`exp35` Part B2)**:
  1. **Isolating Stage-1 vs. Stage-2 Clock Transitions (`exp35` Part B2):** Moving Stage 1 from `(t-2, t-1 [pre-Stage2], t)` (`both_stages_pre_install`) to `(t-2, t [post-Stage2], t)` (`stage2_pre_install`) shifts `d6` DEV BPB by only **`+0.0013` to `+0.0024` BPB** across all 4 `entrywise` cells (`2.1341 -> 2.1360` at `b05, 1e-8`; `2.1195 ± 0.0005 -> 2.1217 ± 0.0006` at `b05, 1e-4`), consistent with Stiefel-orthogonalized `Pre-Polar Muon` inducing only a `4.04%` (`cos = 0.999238`) Stage-1 reward drift. By contrast, moving Stage 2 from `(t-1, t-1, t)` to `(t-1, t [post-Stage3], t)` at `(entrywise, b05, 1e-8)` incurs a **`+0.0219 ± 0.0020` BPB penalty** (`2.1360 -> 2.1579`; joint 3-stage shift `+0.0237 ± 0.0018` BPB from `2.1341 -> 2.1579`).
  2. **Explaining Row-Isotropic vs. Entrywise Behavior (`Proposition 4`):** Across all 4 `row_iso` cells of the 16-cell cube, Stage-2 Pre-Install and Strict Post-Install agree to within **`0.0003` BPB** (`2.1577` vs. `2.1580`; `2.1662` vs. `2.1659`; `2.1296` vs. `2.1298`; `2.1335` vs. `2.1334`), empirically confirming Proposition 4 that when $d_j$ is coordinate-independent, $P_{x_{\text{final}}}$ already cancels the radial self-update. Under `entrywise` moments, coordinate variation in $d_{j, i} = (\sqrt{\hat{v}_{j, i}} + \epsilon)^{-1}$ breaks radial collinearity at $\epsilon = 10^{-8}$; increasing $\epsilon: 10^{-8} \to 10^{-4}$ damps this coordinate spread and improves `entrywise` Strict Post-Install by **`-0.0279 ± 0.0001` BPB** (`2.1579 -> 2.1300`), while combining $(\beta_1=0.9, \epsilon=10^{-4})$ reaches **`2.1273 ± 0.0002` DEV BPB** (`2.0900 ± 0.0001` VAL-B) under Strict Post-Install and **`2.1195 ± 0.0005` DEV BPB** (`2.0827 ± 0.0006` VAL-B, at $\beta_1=0.5, \epsilon=10^{-4}$) under True Both-Stages Pre-Install.
- **Short-Horizon Native `50,332,176` Scale Transfer (`exp29`, `exp30`, `exp32`):** Over a 100-step ($13.1\text{M}$-token) developmental horizon on the 8-layer, 512-width Native `50.3M` model ($B_{\text{eff}}=64, T=2,048$), backward-free 3-stage WM with `Pre-Polar Muon` reaches **`1.9337 ± 0.0004` DEV BPB** (True Both-Stages Pre-Install `(t-1, t-1, t)/(t-2, t-1, t)`) and **`1.9361 ± 0.0004` DEV BPB** (Strict Post-Install `(t-1, t, t)/(t-2, t, t)`), outperforming 100-step unrestricted `AdamW` backpropagation (`1.9734 ± 0.0139`) while trailing full `Muon+AdamW` backpropagation (`1.6989 ± 0.0005`).

---

## 2. Related Work and Methodological Distinctions

**Weight Maximization, Unbiased WM, and Dust.** Chung [1] introduced Weight Maximization (WM) for networks of self-interested reinforcement learning agents (*Learning by Competition of Self-Interested Reinforcement Learning Agents*, arXiv:2010.09770), proving that when each unit modulates its eligibility trace by the change in squared Euclidean norm of its outgoing weights, expected updates follow the gradient of downstream weight magnitude without backward error transport. Chung [2] subsequently formulated *Unbiased Weight Maximization* (UWM, arXiv:2307.13270) for Bernoulli-logistic units using an unbiased outgoing-weight product estimator. These external papers establish the original WM learning principle; they do not provide an inherited convergence guarantee for our multi-token perturbation-differential surrogate or diagonal skip routing. Recently, Dahal et al. [9] introduced *Dust* (*Pretraining Transformers Without Backpropagation*, QLabs, October 2026), which trains Transformer language models without backpropagation by injecting tokenwise activation perturbations during the forward pass and rewarding them via global loss changes and local attention-output estimates.

**Concurrent rStation Studies on Forward-Only and Delayed Transformer Credit Assignment.** Two concurrent rStation papers independently investigate delayed WM on `general_pretraining`:
- In `general_pretraining_18/v4`, `cartan_transport` [5] independently develops token micro-groups (stream partitioning), column-isolated WM rewards, the replacement of sibling-unit interference by within-group token interference, and exact symmetrized unordered-pair conditional covariance formulas (Sections 2.1–2.3 of [5]), combining Sylvester-Hadamard micro-horizon column contraction (`Delayed-Trace-Co-WM-S`) for `mlp.c_fc` with $P=256$ suffix forward probes for attention on ClimbMix `d6` (`2.14471 ± 0.00464` BPB at $1,000$ steps, $1,534.8$ ms/step). We explicitly credit [5] for this shared micro-group and symmetrized pair-covariance foundation. Our distinct contributions relative to [5] are: (i) reconciling the code-level $1/M_g$ normalization with 3-seed scale-normalized controls and deriving the single-token cross-coordinate variance at non-separable `RMSNorm` interfaces (**Proposition 1(a–c)**, `exp35`); (ii) training `attn.c_v` via a **forward causal attention-filtered pre-synaptic trace** $\bar{x}_{\text{attn\_in}}^{(h)}$ (**Proposition 2**) using only 2 forward passes per step (`149` ms/step on `d6`); (iii) routing lexical credit across the diagonal skip surrogate $W_{\text{skip}} = \alpha_0 I_D$ (**Proposition 3**); and (iv) isolating Stage-1 vs. Stage-2 post-installation drift across our 16-cell + 4-cell clock/optimizer factorials (**Proposition 4**, `exp32`–`exp35`).
- In `general_pretraining_19/v2`, `residual_cartographer` [6] analyzes both short residual writer/reader shortcuts and longer depth-delayed reader graphs on Native `50.3M` ($T=2,048$), showing that a depth-delayed graph ($33$ clocks at $L=8$) reaches `1.98964` DEV BPB at $499.65\text{M}$ targets (`2.00286` at $134.2\text{M}$ targets) and that post-attention source partitioning gave no net gain (`2.00494` vs. `2.00278` BPB). Because [5], [6], and [9] use different token budgets, evaluation streams, and model configurations than our 360-step `d6` and 100-step ($13.1\text{M}$-target) Native `50.3M` studies, raw BPB endpoints across papers do not establish superiority over [5], [6], or [9]. Rather, our work complements [6]'s delayed residual-reader routing by commuting $W_v^{(h)}$ with causal attention during the forward pass (**Proposition 2**) and characterizing the receiver-installation shift that arises under short-delay post-installation construction (**Proposition 4**).

---

## 3. Mathematical Formulation and Estimator Analysis

### 3.1 Three-Stage Residual Pipeline

Consider an $L$-layer causal Transformer with width $D$, vocabulary size $V$, token embedding $e_{\text{tok}}(x) = W_{\text{wte}}[x]$, initial normalized state $x_0 = \text{RMSNorm}(e_{\text{tok}}(x))$, per-layer residual recurrence $x_{\text{in}}^{(l)} = \lambda_{\text{res}}^{(l)} x_{\text{out}}^{(l-1)} + \lambda_{x_0}^{(l)} x_0$, attention and MLP sublayers writing additively via $W_{\text{attn\_proj}}^{(l)} \in \mathbb{R}^{D \times D}$ and $W_{\text{mlp\_proj}}^{(l)} \in \mathbb{R}^{D \times 4D}$, terminal state $x_{\text{final}} = \text{RMSNorm}(x_L)$, and softcapped readout logits $z = 15 \tanh(W_{\text{lm}} x_{\text{final}} / 15)$ with $W_{\text{lm}} \in \mathbb{R}^{V \times D}$. Unrolling the linear residual scalings $\lambda_{\text{res}}^{(l)}, \lambda_{x_0}^{(l)}$ yields the exact terminal representation
$$
x_L = \alpha_0 x_0 + \sum_{l=0}^{L-1} \gamma_l \left(W_{\text{attn\_proj}}^{(l)} y_{\text{attn}}^{(l)} + W_{\text{mlp\_proj}}^{(l)} h_{\text{mlp}}^{(l)}\right), \qquad \gamma_l = \prod_{m=l+1}^{L-1} \lambda_{\text{res}}^{(m)}, \quad \alpha_0 = \prod_{m=0}^{L-1} \lambda_{\text{res}}^{(m)} + \sum_{l=0}^{L-1} \lambda_{x_0}^{(l)} \gamma_l.
$$
Exploiting these direct additive write paths into $x_L$, we partition all trainable parameters into three causal stages with maximum pipeline delay $2$ independent of depth $L$:
- **Stage 3 (`Delay = 0`, Terminal Readout):** $W_{\text{lm}} \in \mathbb{R}^{V \times D}$.
- **Stage 2 (`Delay = 1`, Residual Stream Writers):** $W_{\text{wte}} \in \mathbb{R}^{V \times D}$, $W_{\text{mlp\_proj}}^{(l)} \in \mathbb{R}^{D \times 4D}$, and $W_{\text{attn\_proj}}^{(l)} \in \mathbb{R}^{D \times D}$ for $l \in \{0, \dots, L-1\}$, whose outgoing receiver columns are $W_{\text{lm}}[:, i] \in \mathbb{R}^V$.
- **Stage 1 (`Delay = 2`, Internal Sublayer Writers):** $W_{\text{mlp\_fc}}^{(l)} \in \mathbb{R}^{4D \times D}$, $W_{v}^{(l)} \in \mathbb{R}^{D \times D}$, $W_{\text{ve}}^{(l)} \in \mathbb{R}^{V \times D}$, and optionally $W_{q}^{(l)} \in \mathbb{R}^{D \times D}$, whose outgoing receiver columns are $W_{\text{mlp\_proj}}^{(l)}[:, p] \in \mathbb{R}^D$ and $W_{\text{attn\_proj}}^{(l)}[:, p] \in \mathbb{R}^D$.

### 3.2 Reconciled Code Contraction, Separable Pair Covariance, and `RMSNorm` Interface Moments (`Proposition 1`)

Let $N = BT$ denote the total number of tokens in a batch, and partition $\{1, \dots, N\} = \bigsqcup_{g=1}^G \mathcal{S}_g$ into $G$ disjoint streams of size $M_g = N / G$ tokens each. During the forward pass, unit $i$ at stage $m$ injects i.i.d. Rademacher exploration noise $\xi_{n, i} \sim \text{Unif}(\{\pm 1\})$ with scale $\sigma > 0$, and records the stream eligibility trace $Z_{i, :}^{(m, g)} = \sum_{n \in \mathcal{S}_g} (\xi_{n, i}/\sigma) x_n \in \mathbb{R}^{D_{\text{in}}}$.

In `compute_stream_wm_rewards_chunked` (`exp26`) and `compute_stream_wm_rewards_fast` (`exp32`–`exp35`), the stream reward divides by $M_g$ with an optional multiplier `scale_factor` (`mult = scale_factor / M_g`):
$$
R_i^{(m, g)}(\text{scale\_factor}) = \frac{\text{scale\_factor}}{M_g} \sum_{n' \in \mathcal{S}_g} u_{n'}^{(m+1), T} W_{:, i}^{(m+1)}(t_{\text{constr}}) \, \Delta y_{n', i}^{(m)},
$$
and the subsequent parameter update divides the tokenwise modulation–input product $\sum_{g=1}^G R_i^{(m, g)} Z_{i, :}^{(m, g)}$ by $N = BT$ (`-(mod_flat.t() @ x_flat) / BT`). Ignoring the descent sign, the **implemented contraction in code** is therefore
$$
\widehat{G}_{i, :}^{\text{code}}(G; \text{scale\_factor}) := \frac{1}{N} \sum_{g=1}^G R_i^{(m, g)}(\text{scale\_factor}) \, Z_{i, :}^{(m, g)} = \frac{\text{scale\_factor}}{M_g} \widehat{G}_{i, :}^{\text{norm}}(G), \qquad \text{where} \qquad \widehat{G}_{i, :}^{\text{norm}}(G) := \frac{1}{G} \sum_{g=1}^G R_i^{(m, g)}(1) \, Z_{i, :}^{(m, g)}.
$$
In particular, the default setting `scale_factor = 1` (`exp26`–`exp34`) computes $\widehat{G}_{i, :}^{\text{code}}(G; 1) = \frac{1}{M_g}\widehat{G}_{i, :}^{\text{norm}}(G)$, whereas setting `scale_factor = M_g` (`exp35`) computes $\widehat{G}_{i, :}^{\text{code}}(G; M_g) = \widehat{G}_{i, :}^{\text{norm}}(G)$, and at $G = N$ ($M_g = 1$) both coincide identically.

Furthermore, we must distinguish **separable channels**—where coordinate $i$'s forward displacement depends only on $\xi_{n, i}$, as holds at **Stage 1** (`mlp.c_fc` with elementwise $\text{ReLU}^2$ and `attn.c_v` with linear head-value projection)—from **non-separable `RMSNorm` interfaces** at **Stage 2** (`x_final = RMSNorm(x_L)` and `x_0 = RMSNorm(e_tok)`), where $\text{rms}(z_n) = \sqrt{\frac{1}{D}\sum_{i'=1}^D z_{n, i'}^2}$ couples coordinate $i$ to $\xi_{n, i'}$ for $i' \neq i$.

**Proposition 1 (Conditional Moments of $\widehat{G}_{i, :}^{\text{code}}(G; \text{scale\_factor})$ for Separable Channels and `RMSNorm` Interfaces).** *Conditioned on fixed pre-synaptic inputs $\{x_n\}_{n=1}^N \subset \mathbb{R}^{D_{\text{in}}}$ and downstream weights/modulations independent of the i.i.d. Rademacher draws $\xi_{n, i} \sim \text{Unif}(\{\pm 1\})$:*
1. **(a) Separable-Channel Conditional Mean (Stage 1: `mlp.c_fc, attn.c_v`):** *Suppose $\Delta y_{n, i}^{(m)} = \sigma c_{n, i} \xi_{n, i} + r_{n, i}(\sigma)$ with $c_{n, i}$ independent of $\xi_n$ and $r_{n, i}(\sigma) = O(\sigma^2)$, and let $a_{n, i} := c_{n, i} \sum_{j=1}^{d_{\text{next}}} W_{j, i}^{(m+1)}(t_{\text{constr}}) u_{n, j}^{(m+1)}$. Then the first-order code estimator $\widehat{G}_{i, :}^{\text{code}, (1)}(G; \text{scale\_factor})$ satisfies for every $G \in \{1, \dots, N\}$ ($M_g = N / G$):*
   $$
   \mathbb{E}_{\xi}\left[\widehat{G}_{i, :}^{\text{code}, (1)}(G; \text{scale\_factor})\right] = \frac{\text{scale\_factor}}{M_g} \left(\frac{1}{N} \sum_{n=1}^N a_{n, i} x_n\right).
   $$
   *Specifically, under `scale_factor = 1`, the conditional mean scales as $\frac{1}{M_g}\left(\frac{1}{N}\sum_{n=1}^N a_{n, i} x_n\right)$ (evaluating to $0.5$ on $N=4, G=2, M_g=2, a_n = x_n = 1$), whereas under `scale_factor = M_g`, the conditional mean is $\mathbb{E}_{\xi}[\widehat{G}_{i, :}^{\text{norm}, (1)}(G)] = \frac{1}{N}\sum_{n=1}^N a_{n, i} x_n$ for all $G \in \{1, \dots, N\}$ (evaluating to $1.0$ on $N=4, G=2, M_g=2, a_n = x_n = 1$).*
2. **(b) Separable-Channel Symmetrized Pair Covariance ([5], Section 2.3):** *Under the separable channel model of (a), the exact conditional covariance matrix of $\widehat{G}_{i, :}^{\text{code}, (1)}(G; \text{scale\_factor})$ is*
   $$
   \text{Cov}_{\xi}\left(\widehat{G}_{i, :}^{\text{code}, (1)}(G; \text{scale\_factor})\right) = \left(\frac{\text{scale\_factor}}{M_g}\right)^2 \frac{1}{N^2} \sum_{g=1}^G \sum_{\substack{n, n' \in \mathcal{S}_g \\ n < n'}} \left(a_{n', i} x_n + a_{n, i} x_{n'}\right)\left(a_{n', i} x_n + a_{n, i} x_{n'}\right)^T,
   $$
   *which sums over the $G\binom{M_g}{2} = \frac{N(M_g - 1)}{2}$ unordered within-stream token pairs and vanishes identically at $G = N$ ($M_g = 1$).*
3. **(c) Non-Separable `RMSNorm` Interface Moments at $M_g = 1$ (Stage 2: `x_final, x_0`):** *At an `RMSNorm` interface $x_n = \text{RMSNorm}(z_n) = z_n / \text{rms}(z_n)$ ($z_n \in \mathbb{R}^D$), tangent-projected or raw Rademacher perturbation yields the first-order forward displacement $\Delta x_n^{(1)} = \frac{\sigma}{\text{rms}(z_n)} P_{x_n} \xi_n$, whose $i$-th coordinate is $\Delta x_{n, i}^{(1)} = \frac{\sigma}{\text{rms}(z_n)}\left((1 - x_{n, i}^2/D)\xi_{n, i} - \frac{x_{n, i}}{D}\sum_{i' \neq i} x_{n, i'}\xi_{n, i'}\right)$. Consequently, even at $G = N$ ($M_g = 1$, where cross-token pairs $n < n'$ are absent), for a single token $n$ with receiver sensitivity $a_{n, i} = \sum_j W_{j, i}^{(m+1)} u_{n, j}^{(m+1)}$ and pre-synaptic input $h_n \in \mathbb{R}^{D_{\text{in}}}$, the first-order unprojected coordinate-$i$ estimator is*
   $$
   \hat{g}_{n, i}^{(1)} := a_{n, i} \frac{\Delta x_{n, i}^{(1)} \xi_{n, i}}{\sigma} h_n = \frac{a_{n, i}}{\text{rms}(z_n)} \left(1 - \frac{x_{n, i}^2}{D} - \frac{x_{n, i}}{D} \sum_{i' \neq i} x_{n, i'} \xi_{n, i} \xi_{n, i'}\right) h_n,
   $$
   *which has exact conditional mean and nonzero single-token **cross-coordinate** covariance*
   $$
   \mathbb{E}_{\xi_n}\left[\hat{g}_{n, i}^{(1)}\right] = \frac{a_{n, i}}{\text{rms}(z_n)}\left(1 - \frac{x_{n, i}^2}{D}\right) h_n, \qquad \text{Cov}_{\xi_n}\left(\hat{g}_{n, i}^{(1)}\right) = \frac{a_{n, i}^2 x_{n, i}^2 (D - x_{n, i}^2)}{D^2 \, \text{rms}(z_n)^2} h_n h_n^T.
   $$
   *For example, for a single token ($N=G=M_g=1$) at $D=2$ with $z_1 = (1, 2)$, $a_{1, 1} = 1$, and $h_1 = 1$, we have $\text{rms}(z_1) = \sqrt{2.5}$, $x_{1, 1}^2 = 0.4$, $x_{1, 1} x_{1, 2} = 0.8$, and $\hat{g}_{1, 1}^{(1)} = \frac{0.8 - 0.4 \xi_{1, 1}\xi_{1, 2}}{\sqrt{2.5}}$, with mean $0.8/\sqrt{2.5} \approx 0.5060$ and variance $\frac{0.4(2 - 0.4)}{4 \times 2.5} = 0.064$.*

*Proof.* For (a–b), expanding $\widehat{G}_{i, :}^{\text{norm}, (1)}(G) = \frac{1}{N}\sum_{g=1}^G (\sum_{n' \in \mathcal{S}_g} a_{n', i}\xi_{n', i})(\sum_{n \in \mathcal{S}_g} \xi_{n, i} x_n)$ using $\xi_{n, i}^2 \equiv 1$ gives $\frac{1}{N}\sum_{n=1}^N a_{n, i} x_n + \frac{1}{N}\sum_{g=1}^G \sum_{n < n' \in \mathcal{S}_g} \eta_{\{n, n'\}, i} (a_{n', i} x_n + a_{n, i} x_{n'})$ where $\eta_{\{n, n'\}, i} = \xi_{n, i}\xi_{n', i}$ are pairwise uncorrelated with zero mean and unit variance; multiplying by $\text{scale\_factor} / M_g$ scales the mean by $\text{scale\_factor} / M_g$ and the covariance by $(\text{scale\_factor} / M_g)^2$. For (c), since $\xi_{n, i}^2 = 1$, multiplying $(P_{x_n}\xi_n)_i = (1 - x_{n, i}^2/D)\xi_{n, i} - \frac{x_{n, i}}{D}\sum_{i' \neq i} x_{n, i'}\xi_{n, i'}$ by $\xi_{n, i}$ yields the displayed expression for $\hat{g}_{n, i}^{(1)}$. Because $\{\xi_{n, i}\xi_{n, i'}\}_{i' \neq i}$ are $D - 1$ pairwise uncorrelated zero-mean Rademacher products with unit variance and $\sum_{i' \neq i} x_{n, i'}^2 = \|x_n\|_2^2 - x_{n, i}^2 = D - x_{n, i}^2$, the mean and covariance follow immediately. $\square$

### 3.3 Forward Causal Attention-Filtered Value Eligibility Trace (`Proposition 2`)

In causal multi-head self-attention with $H$ heads of dimension $D_h = D / H$, head $h$'s output at batch $b$, query token $\tau \in \{0, \dots, T-1\}$ is $y_{\text{attn}}^{(h)}(b, \tau) = \sum_{s \le \tau} A_{b, h, \tau, s} v^{(h)}(b, s)$, where $v^{(h)}(b, s) = W_v^{(h)} x_{\text{attn\_in}}(b, s)$ ($W_v^{(h)} \in \mathbb{R}^{D_h \times D}$) and $A_{b, h, \tau, s}$ is the causal attention weight ($\sum_{s \le \tau} A_{b, h, \tau, s} = 1$, $A_{b, h, \tau, s} = 0$ for $s > \tau$).

**Proposition 2 (Exact Commutation of $W_v^{(h)}$ with Causal Attention).** *Because $W_v^{(h)} \in \mathbb{R}^{D_h \times D}$ is shared across sequence positions $s \in \{0, \dots, T-1\}$, the post-attention value aggregation at position $\tau$ satisfies the exact linear identity*
$$
y_{\text{attn}}^{(h)}(b, \tau) = \sum_{s \le \tau} A_{b, h, \tau, s} \left(W_v^{(h)} x_{\text{attn\_in}}(b, s)\right) = W_v^{(h)} \underbrace{\left(\sum_{s \le \tau} A_{b, h, \tau, s} x_{\text{attn\_in}}(b, s)\right)}_{\bar{x}_{\text{attn\_in}}^{(h)}(b, \tau) \in \mathbb{R}^D}.
$$
*Consequently, injecting exploration noise $\psi_v^{(h)}(b, \tau) \sim \text{Unif}(\{\pm 1\}^{D_h})$ at the post-attention value output $y_{\text{attn}}^{(h)}(b, \tau)$ and recording the **forward causal attention-filtered pre-synaptic trace** $Z_{v}^{(h, g)}(t-2) = \sum_{(b, \tau) \in \mathcal{S}_g} (\psi_v^{(h)}(b, \tau)/\sigma_1) \, \bar{x}_{\text{attn\_in}}^{(h)}(b, \tau)^T$ yields an exact single-position linear channel $y_{\text{attn}}^{(h)}(b, \tau) = W_v^{(h)} \bar{x}_{\text{attn\_in}}^{(h)}(b, \tau)$ at each output token $\tau$, eliminating causal cross-position perturbation mixing with zero reverse-time pooling.*

### 3.4 Diagonal Identity-Skip Surrogate & Post-Installation Credit Decomposition (`Propositions 3 & 4`)

**Proposition 3 (Algebraic Diagonal Skip-Routing Surrogate for `wte`).** *In the unrolled residual equation $x_L = \alpha_0 x_0 + \dots$, the direct path from $x_0 = \text{RMSNorm}(W_{\text{wte}}[x])$ to $x_L$ is the fixed diagonal scaling $W_{\text{skip}} = \alpha_0 I_D$. Treating $W_{\text{skip}} = \alpha_0 I_D$ as an algebraic surrogate receiver matrix with post-synaptic modulation $s_{\text{bus}}(n) = \text{mod\_bus\_base}(n) \in \mathbb{R}^D$ and pre-synaptic forward displacement $\Delta x_0(n) = \text{RMSNorm}(e_{\text{tok}}(n) + \sigma \tilde{\xi}_{\text{wte}}(n)) - x_0(n)$, the column-norm contraction reduces to the $O(BT \cdot D)$ elementwise contraction*
$$
R_{\text{wte}, i}^{\text{raw}, (g)}(\text{scale\_factor}) = \sum_{j=1}^D (\alpha_0 \delta_{j, i}) \left(\frac{\text{scale\_factor}}{M_g}\sum_{n \in \mathcal{S}_g} s_{\text{bus}, n, j} \, \Delta x_{0, n, i}\right) = \frac{\text{scale\_factor}}{M_g} \sum_{n \in \mathcal{S}_g} \left(\alpha_0 s_{\text{bus}, n, i}\right) \Delta x_{0, n, i},
$$
*(followed in `compute_identity_skip_wm_rewards_fast` by stream-mean centering $R_{\text{wte}, i}^{(g)} = R_{\text{wte}, i}^{\text{raw}, (g)} - \frac{1}{G}\sum_{g'=1}^G R_{\text{wte}, i}^{\text{raw}, (g')}$), eliminating the second $O(G V D)$ `lm_head` contraction.*

Finally, we analyze **Strict Post-Installation Construction** (`Stage 2: (t-1, t [post-Stage3-step], t)`), where Stage 2 evaluates `compute_stream_wm_rewards_fast` at clock $t$ on $W_{\text{lm}}(t) = W_{\text{lm}}(t-1) + \delta W_{\text{lm}}(t-1)$ after Stage 3 has installed its batch-$t-1$ update. In our implementation (`custom_entrywise_adamw_step` and `row_isotropic_adamw_step`), at optimizer step $t_{\text{step}}$, the installed Stage-3 displacement is
$$
\delta W_{\text{lm}, j, i}(t-1) = -\eta_3 \lambda_{\text{wd}} W_{\text{lm}, j, i}(t-1) - \frac{\eta_3}{1 - \beta_1^{t_{\text{step}}}} \cdot d_{j, i} \left(\beta_1 m_{j, i}^{(t-2)} + (1 - \beta_1) g_{j, i}^{(t-1)}\right), \qquad d_{j, i} := \frac{1}{\sqrt{\hat{v}_{j, i}^{(t-1)}} + \epsilon},
$$
where $g_{j, i}^{(t-1)} = -\frac{1}{BT}\sum_{n'=1}^{BT} s_{\text{lm}, n', j} x_{\text{final}, n', i}$, $\hat{v}_{j, i}^{(t-1)} = v_{j, i}^{(t-1)} / (1 - \beta_2^{t_{\text{step}}})$ under `entrywise` moments (or row-averaged $d_{j, i} \equiv d_j = (\sqrt{\hat{v}_j^{(t-1)}} + \epsilon)^{-1}$ with $\hat{v}_j^{(t-1)} = \frac{1}{D}\sum_{i=1}^D \hat{v}_{j, i}^{(t-1)}$ under `row_iso`), and $\epsilon > 0$ is added *outside* the square root (`denom = sqrt(v_hat) + eps`). At $G = BT$ ($M_g = 1$), let $u_{\text{bus}, i}^{\text{pre}}(n) = R_{\text{bus}, i}^{\text{pre}}(n) (\xi_{n, i}/\sigma)$ and $u_{\text{bus}, i}^{\text{post}}(n) = R_{\text{bus}, i}^{\text{post}}(n) (\xi_{n, i}/\sigma)$ denote the unprojected Stage-2 credit before and after installing $\delta W_{\text{lm}}(t-1)$.

**Proposition 4 (Decomposition of Installed-Update Terminal Credit Shift, Data-Dependent Fourth-Moment Bound, and Preconditioning Regimes).** *Under Strict Post-Installation Construction at $G = BT$ ($M_g = 1$), the post-minus-pre unprojected credit shift $\delta u_{\text{bus}}(n) := u_{\text{bus}}^{\text{post}}(n) - u_{\text{bus}}^{\text{pre}}(n)$ decomposes into three additive components corresponding to decoupled weight decay, first-moment history $m^{(t-2)}$, and current-batch gradient $g^{(t-1)}$:*
$$
\delta u_{\text{bus}, i}(n) = -\eta_3 \lambda_{\text{wd}} u_{\text{bus}, i}^{\text{pre}}(n) + \delta u_{\text{mom}, i}(n) + \delta u_{\text{cur}, i}^{(n' = n)}(n) + \delta u_{\text{cur}, i}^{(n' \neq n)}(n),
$$
*where:*
1. **Same-Token ($n' = n$) `RMSNorm` Radial-vs-Oblique Split and Data-Dependent Fourth-Moment Bound:** *With tangent-projected forward carrier $\tilde{\xi}_n = P_{x_{\text{final}}(n)} \xi_n$ and raw carrier multiplication $\xi_n / \sigma$, the first-order same-token ($n'=n$) term is $\delta u_{\text{cur}, i}^{(n'=n, 1)}(n) = \frac{\eta_3 (1-\beta_1)}{(1-\beta_1^{t_{\text{step}}}) BT} c_{n, i} \, b_i(n)$, where $c_{n, i} := \sum_{j=1}^V d_{j, i} s_{\text{lm}, n, j}^2$ and the vector $b(n)$ splits into an $O(1)$ radial vector $b_{\text{rad}}(n) \parallel x_{\text{final}}(n)$ and an oblique `RMSNorm`-Jacobian coupling term $b_{\text{oblique}}(n)$:*
   $$
   b(n) := \frac{1}{\text{rms}(z_n)}\left(x_{\text{final}}(n) \odot \xi_n \odot (P_{x_{\text{final}}(n)} \xi_n)\right) = \underbrace{\frac{1}{\text{rms}(z_n)} x_{\text{final}}(n)}_{b_{\text{rad}}(n) \parallel x_{\text{final}}(n)} - \underbrace{\frac{\langle \xi_n, x_{\text{final}}(n)\rangle}{D \, \text{rms}(z_n)} \left(x_{\text{final}}(n)^{\odot 2} \odot \xi_n\right)}_{b_{\text{oblique}}(n)}.
   $$
   *Under **row-isotropic preconditioning** ($d_{j, i} \equiv d_j$, so $c_{n, i} \equiv c_n$ is a scalar), reward-side projection $P_{x_{\text{final}}(n)}$ annihilates $c_n b_{\text{rad}}(n)$ identically ($P_{x_{\text{final}}(n)} b_{\text{rad}}(n) = 0$) while leaving $P_{x_{\text{final}}(n)} b_{\text{oblique}}(n) \neq 0$ (e.g., at $D=2, z_n=(1, 2), \xi_n=(1, 1)$, $b(n)=(0.16, -0.16)$, $b_{\text{rad}}(n)=(0.4, 0.8)$, and $P_{x_{\text{final}}(n)} b(n) = (0.192, -0.096)$). Defining the normalized coordinate fourth moment $\kappa_4(x) := \frac{1}{D}\sum_{i=1}^D x_i^4 \in [1, D]$, the expected squared norm of $P_{x_{\text{final}}(n)} b(n)$ relative to $\|b_{\text{rad}}(n)\|_2^2 = D / \text{rms}(z_n)^2$ satisfies the exact **data-dependent fourth-moment bound***
   $$
   \frac{\mathbb{E}_{\xi_n}\left[\|P_{x_{\text{final}}(n)} b(n)\|_2^2\right]}{\|b_{\text{rad}}(n)\|_2^2} \le \frac{\mathbb{E}_{\xi_n}\left[\|b_{\text{oblique}}(n)\|_2^2\right]}{\|b_{\text{rad}}(n)\|_2^2} = \frac{\sum_{i=1}^D x_{\text{final}, n, i}^4}{D^2} = \frac{\kappa_4(x_{\text{final}}(n))}{D}.
   $$
   *In particular: (i) if $\kappa_4(x_{\text{final}}(n)) \le C = O(1)$ is uniformly bounded, the residual ratio is $\le C/D = O(1/D)$ (on trained ClimbMix `d6` states at $D=384$, we measure token-mean $\kappa_4(x_{\text{final}}) = 2.9398$, giving bound $\kappa_4/D = 0.007656$ and empirical ratio $0.007458$); (ii) without a fourth-moment bound, $\|x\|_2^2 = D$ alone does **not** imply a vanishing residual—for the 2-sparse family $x = \sqrt{D/5}(1, 2, 0, \dots, 0)$ ($z = x$), $\kappa_4(x)/D = 0.68$ and $\frac{\mathbb{E}_{\xi}[\|P_x b\|_2^2]}{\|b_{\text{rad}}\|_2^2} = 0.0576$ for every $D \ge 2$; and (iii) under **entrywise preconditioning**, coordinate variation in $c_{n, i} = \sum_j d_{j, i} s_{\text{lm}, n, j}^2$ breaks exact radial collinearity ($P_{x_{\text{final}}(n)}(c_n \odot b_{\text{rad}}(n)) \neq 0$) unless $\epsilon \gg \sqrt{\hat{v}_{j, i}^{(t-1)}}$, so increasing $\epsilon: 10^{-8} \to 10^{-4}$ is required to damp entrywise coordinate spread.*
2. **Cross-Token ($n' \neq n$) Current-Gradient Drift Bound:** *Since $d_{j, i} = (\sqrt{\hat{v}_{j, i}^{(t-1)}} + \epsilon)^{-1} \le \epsilon^{-1}$, the cross-token ($n' \neq n$) current-gradient contribution satisfies*
   $$
   \left|\delta u_{\text{cur}, i}^{(n' \neq n)}(n)\right| \le \frac{\eta_3 (1 - \beta_1)}{(1 - \beta_1^{t_{\text{step}}}) BT \, \epsilon} \sum_{n' \neq n} \left(\sum_{j=1}^V |s_{\text{lm}, n, j}| \, |s_{\text{lm}, n', j}|\right) |x_{\text{final}, n', i}| \, \left|\frac{\Delta x_{\text{final}, n, i}}{\sigma}\right|.
   $$

---

## 4. Experimental Results and Mechanistic Verification

![Four-panel summary of genuinely backward-free 3-stage delayed eligibility-trace Weight Maximization across exp26–exp35](supplement/figure-1.png)

### 4.1 Experimental Setup, Attention Backend Provenance, and Evaluation Protocol

All experiments (`exp26`–`exp35`, 150 training runs across seeds $\{46, 47, 48\}$) use the supplied `rustbpe` tokenizer (`vocab_size = 8,192`) and ClimbMix data stream on two Transformer regimes:
1. **ClimbMix `d6` ($L=6, D=384, H=6, D_h=64, V=8,192, B=8, T=256, BT=2,048$ tokens/step, $360$ steps $= 737,280$ training tokens):** Total parameters $= 26,346,060$ (`wte`: $3,145,728$; 3 `value_embeds`: $9,437,184$; `lm_head`: $3,145,728$; 6 blocks $\times$ 6 matrices: $10,616,832$; gates/scalars: $588$). In our primary 4-matrix configuration (`mlp.c_fc, mlp.c_proj, attn.c_v, attn.c_proj` active per block, `attn.c_q, attn.c_k` frozen), `active_params = 24,576,000` ($24.58\text{M}$); when `attn.c_q` is also active (`exp30`–`exp31`), `active_params = 25,460,736` ($25.46\text{M}$). Disjoint evaluation splits (`evaluate_bpb_disjoint`) score `dev_tokens = 261,889` (`dev_bytes = 1,037,477`) and `valb_tokens = 262,009` (`valb_bytes = 1,050,500`), where $\text{CE (nats)} = \text{BPB} \cdot \ln(2) \cdot (\text{bytes}/\text{tokens})$ and $\text{PPL} = \exp(\text{CE})$.
2. **Native `50,332,176` ($L=8, D=512, H=4, D_h=128, V=8,192, B_{\text{eff}}=64, T=2,048, BT=131,072$ tokens/step, $100$ steps $= 13,107,200$ training tokens):** Total parameters $= 50,332,176$, with `active_params = 46,137,344` ($46.14\text{M}$) in the 4-matrix configuration (`c_q, c_k` $= 4,194,304$ and $528$ gate/scalar parameters frozen). To fit $G_{\text{micro}} = 1,024$ stream-partitioned eligibility-trace tensors in GPU memory (`8.5–9.5 GB` peak VRAM), the backward-free Native `50.3M` runs accumulate `grad_accum_steps = 16` physical microbatches of `micro_batch_size = 4` (`G_total = 16,384` streams/step at $M_g = 8$ tokens/stream, or `G_total = 4,096` at $M_g = 32$), departing from the supplied task's single-microbatch $B=64$, $499.65\text{M}$-token backpropagation recipe. Disjoint evaluation splits score `dev_tokens = 523,826` (`dev_bytes = 2,126,272`, $\text{bytes}/\text{token} = 4.059119$) and `valb_tokens = 523,715` (`valb_bytes = 2,129,527`). Because both `DEV` and `VAL-B` are evaluated at checkpoints across the ablation campaign, `VAL-B` serves as a second disjoint validation stream rather than an unopened holdout.
3. **Attention Backend Provenance (`model_and_data.py`):** In `model_and_data.py`, the `fa3` handle wraps PyTorch's native fused `F.scaled_dot_product_attention(..., is_causal=True)` (applying causal and sliding-window masks when `window_size < T`). All reported forward and filtered-trace timings across `exp26`–`exp35` use this exact `model_and_data.py` SDPA binding so that all baselines and backward-free runs share an identical attention kernel.

### 4.2 Verification of Propositions 1 and 2: Stream Partitioning, Normalization Controls, and Filtered Traces (`exp26`, `exp28`, `exp35`)

Table 1 reports the single-step cosine similarity between our backward-free eligibility-trace estimator and the exact column weight-norm directional derivative on `d6` ($BT = 2,048$) across stream counts $G = BT / M_g$ (`exp26_results.json`, `exp28_results.json`).

| Stream Count $G$ ($M_g = BT / G$) | Stage 2 `wte` Cos | Stage 2 `mlp.c_proj` Cos | Stage 2 `attn.c_proj` Cos | Stage 1 `mlp.c_fc` Cos | Stage 1 `attn.c_v` (Unfiltered $x_{\text{in}}$) | Stage 1 `attn.c_v` (Filtered $\bar{x}_{\text{in}}$, Prop. 2) |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| $G = 1$ ($M_g = 2,048$) | $+0.0313$ | $+0.0187$ | $+0.0170$ | $+0.0037$ | $+0.0036$ | $+0.0295$ |
| $G = 64$ ($M_g = 32$) | $+0.2034$ | $+0.2978$ | $+0.2469$ | $+0.0305$ | $+0.0222$ | $+0.2361$ |
| $G = 256$ ($M_g = 8$) | $+0.3941$ | $+0.5341$ | $+0.4570$ | $+0.1430$ | $+0.0416$ | $+0.4519$ |
| **$G = 2,048$ ($M_g = 1$)** | **$+0.9139$** | **$+0.9364$** | **$+0.9068$** | **$+0.9089$** | $+0.1104$ | **$+0.9069$ ($8.2\times$)** |

**Table 1:** Single-step cosine alignment on ClimbMix `d6` ($BT = 2,048$) across stream counts $G \in \{1, 64, 256, 2048\}$ (`exp26`, `exp28`).

**Stream Scale-Normalization Controls (`exp35` Part A):** To test whether the stream scaling effect in Table 1 survives when update magnitude is held invariant to $M_g$ (Proposition 1), we ran a controlled 3-seed comparison on `d6` setting `scale_factor = float(M_g)` so that $\widehat{G}_{\text{norm}}(G) = M_g \widehat{G}_{\text{code}}(G)$ has the identical conditional mean $\frac{1}{N}\sum_n a_{n, i} x_n$ across all stream sizes:
- At $G = 1$ ($M_g = 2,048$): `scale_factor = 2048.0` yields **`2.6097 ± 0.0111` DEV BPB** (`2.5719 ± 0.0106` VAL-B), compared to `scale_factor = 1.0` (`2.5939 ± 0.0109` DEV BPB).
- At $G = 64$ ($M_g = 32$): `scale_factor = 32.0` yields **`2.3361 ± 0.0084` DEV BPB** (`2.2995 ± 0.0086` VAL-B), compared to `scale_factor = 1.0` (`2.3364 ± 0.0051` DEV BPB).
- At $G = 256$ ($M_g = 8$): `scale_factor = 8.0` yields **`2.3008 ± 0.0074` DEV BPB** (`2.2635 ± 0.0065` VAL-B), compared to `scale_factor = 1.0` (`2.2966 ± 0.0115` DEV BPB).
- On the canonical `Pre-Polar Muon` trainer (`strict_post_install`, `entrywise_b09_eps1e4`, $G=64$, `exp35` Part A2): `scale_factor = 1.0` reaches **`2.2541 ± 0.0013` DEV BPB** and `scale_factor = 32.0` reaches **`2.2539 ± 0.0013` DEV BPB** (agreeing to within $0.0002$ BPB).
These controls establish that the learning gains from increasing $G$ ($M_g \to 1$) are driven by the elimination of within-stream cross-token pair interference (Proposition 1(b)), not by artificial update rescaling.

### 4.3 Verification of Propositions 3 and 4: 20-Step Post-Installation Credit Drift and Kurtosis Audit (`exp31`, `exp34`, `exp35`)

To test how receiver weight installation affects Stage-1 (`c_proj` receiver) and Stage-2 (`lm_head` receiver) rewards, we measured the 20-step mean cosine similarity, relative L2 norm drift $\|u_{\text{post}} - u_{\text{pre}}\|_2 / \|u_{\text{pre}}\|_2$, and activation fourth moments on `d6` (`exp31`, `exp34`, and `exp35` Part 0, Table 2):

1. **Stage-1 Receiver Drift (`exp31` Part 0, `exp35` Part 0):** When Stage 2 (`mlp.c_proj, attn.c_proj`) is updated with `AdamW`, one receiver step $\delta W^{(2)}$ induces a **`23.56%` relative L2 drift** in Stage-1 rewards ($\cos = 0.976474$). Replacing `AdamW` on Stage 2 with Stiefel-orthogonalized `Pre-Polar Muon` reduces Stage-1 relative L2 reward drift to **`2.85%` ($\cos = 0.999597$)** in `exp31` and **`4.04%` ($\cos = 0.999238$)** in `exp35` Part 0.
2. **Activation Kurtosis Measurement (`exp35` Part 0):** Across 20 steps of trained `d6` states, the normalized fourth moment $\kappa_4(x_{\text{final}}) = \frac{1}{D}\sum_{i=1}^D x_{\text{final}, i}^4$ has token mean **`2.9398`** (token max `4.2932`), yielding theoretical upper bound $\kappa_4(x_{\text{final}})/D = \mathbf{0.007656}$ and empirical same-token squared norm ratio $\frac{\|P_x b\|_2^2}{\|b_{\text{rad}}\|_2^2} = \mathbf{0.007458}$ (token max $0.124439$). This verifies Proposition 4: on real Transformer states, fourth-moment concentration $\kappa_4(x_{\text{final}}) \approx 3$ ensures the oblique `RMSNorm` residual is $\le 0.0075$, confirming the validity of the $O(1/D)$ residual regime.
3. **Stage-2 Strict Post-Install Terminal Drift Across the $2 \times 2 \times 2$ Optimizer Cube (`exp34` Part 0, Table 2):** Across all 8 cells of `{entrywise, row_iso} x {beta1=0.5, 0.9} x {eps=1e-8, 1e-4}`, increasing $\beta_1: 0.5 \to 0.9$ at $\epsilon=10^{-8}$ (`entrywise`) cuts current-gradient drift from `0.3511` to `0.2696`, increasing $\epsilon: 10^{-8} \to 10^{-4}$ at $\beta_1=0.5$ cuts it to `0.2703`, and combining $(\beta_1=0.9, \epsilon=10^{-4})$ cuts current-gradient drift by **$47.8\%$ (`0.3511 -> 0.1832`)** and total post-install drift from `39.41%` to **`26.05%` ($\cos = 0.964738$)**.

| `lm_head` Optimizer Cell (`exp34` Part 0, 20-Step Mean on `d6`) | `bus_unproj` Cos / Rel L2 Drift | `bus_tangent` ($\perp x_{\text{final}}$) Cos / Drift | Current-Batch Grad Only Drift ($\delta u_{\text{cur}}$) | Momentum-History Only Drift ($\delta u_{\text{mom}}$) | $\|P_{x_{\text{final}}} b\|_2 / \|b\|_2$ (Prop. 4) |
| :--- | :---: | :---: | :---: | :---: | :---: |
| `entrywise_b05_eps1e8` ($\beta_1=0.5, \epsilon=10^{-8}$) | $0.940066$ / $0.465747$ | $0.948719$ / $0.394051$ | $0.351118$ | $0.080159$ | $0.087114$ |
| `entrywise_b09_eps1e8` ($\beta_1=0.9, \epsilon=10^{-8}$) | $0.951336$ / $0.392834$ | $0.954834$ / $0.353878$ | $0.269559$ | $0.112059$ | $0.087114$ |
| `entrywise_b05_eps1e4` ($\beta_1=0.5, \epsilon=10^{-4}$) | $0.943013$ / $0.391255$ | $0.956556$ / $0.307328$ | $0.270305$ | $0.085703$ | $0.087114$ |
| **`entrywise_b09_eps1e4` ($\beta_1=0.9, \epsilon=10^{-4}$)** | **$0.958144$ / $0.304684$** | **$0.964738$ / $0.260504$** | **$0.183231$** | $0.113457$ | $0.087114$ |
| `row_iso_b05_eps1e8` ($\beta_1=0.5, \epsilon=10^{-8}$) | $0.941012$ / $0.471413$ | $0.952909$ / $0.367834$ | $0.317646$ | $0.076778$ | $0.087114$ |
| `row_iso_b09_eps1e8` ($\beta_1=0.9, \epsilon=10^{-8}$) | $0.951433$ / $0.400026$ | $0.956413$ / $0.342871$ | $0.252057$ | $0.108569$ | $0.087114$ |
| `row_iso_b05_eps1e4` ($\beta_1=0.5, \epsilon=10^{-4}$) | $0.942453$ / $0.403128$ | $0.959755$ / $0.292793$ | $0.251836$ | $0.081251$ | $0.087114$ |
| **`row_iso_b09_eps1e4` ($\beta_1=0.9, \epsilon=10^{-4}$)** | **$0.957127$ / $0.317284$** | **$0.966045$ / $0.255660$** | **$0.174360$** | $0.110404$ | $0.087114$ |

**Table 2:** 20-step Pre-Install vs. Strict Post-Install credit decomposition audit across all 8 cells of the $2 \times 2 \times 2$ `lm_head` optimizer cube on ClimbMix `d6` (`exp34_results.json` Part 0).

### 4.4 Complete 16-Cell Factorial and True Both-Stages Pre-Install on ClimbMix `d6` (`exp33`–`exp35`)

Table 3 presents the 3-seed ClimbMix `d6` results across stream counts, the complete 16-cell $2 \times 2 \times 2 \times 2$ Stage-2 Clock $\times$ Pooling $\times$ Momentum $\times$ Damping cube (`exp34` + `exp35` Part B1), and the 4-cell True Both-Stages Pre-Install comparison (`exp35` Part B2). All 20 factorial cells share identical 1-call `wte_bus_routed` routing (**Proposition 3**), Rademacher-tangent exploration, and reward-side radial projection.

| Method (`ClimbMix d6`, $L=6, D=384, V=8,192$, 360 Steps, Seeds `46, 47, 48`) | Active Params | Stage-3 / Stage-2 / Stage-1 Clocks | `lm_head` Opt $(\text{pool}, \beta_1, \epsilon)$ | Rev. Matmuls / Bwd | Seed 46 / 47 / 48 DEV BPB | DEV BPB (Mean $\pm$ SD) | VAL-B BPB (Mean $\pm$ SD) | DEV PPL | Fwd / Rule / Wall (`ms`) |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| `supplied_muon_adamw_backprop` (Full 6-Matrix BP) | $26.35\text{M}$ | $(t,t,t)/(t,t,t)/(t,t,t)$ | `entry, 0.5, 1e-8` | $-1 / 1$ | $2.1112 / 2.1119 / 2.1105$ | $2.1112 \pm 0.0007$ | $2.0750 \pm 0.0007$ | $331.49$ | $4.6 / 18.0 / 46.2$ |
| `backprop_prepolar_muon_damped_lm_frozen_qk_control` (`exp33`) | $24.58\text{M}$ | $(t,t,t)/(t,t,t)/(t,t,t)$ | `row_iso, 0.9, 1e-4` | $-1 / 1$ | $2.1155 / 2.1147 / 2.1153$ | $2.1152 \pm 0.0004$ | $2.0780 \pm 0.0002$ | $333.01$ | $4.5 / 19.3 / 47.3$ |
| `backprop_prepolar_muon_frozen_qk_control` (`exp32`) | $24.58\text{M}$ | $(t,t,t)/(t,t,t)/(t,t,t)$ | `entry, 0.5, 1e-8` | $-1 / 1$ | $2.1207 / 2.1203 / 2.1161$ | $2.1190 \pm 0.0025$ | $2.0825 \pm 0.0027$ | $336.53$ | $4.6 / 19.0 / 47.2$ |
| `backprop_adamw_frozen_qk_control` (`exp31`) | $24.58\text{M}$ | $(t,t,t)/(t,t,t)/(t,t,t)$ | `entry, 0.5, 1e-8` | $-1 / 1$ | $2.1628 / 2.1579 / 2.1654$ | $2.1621 \pm 0.0038$ | $2.1242 \pm 0.0039$ | $378.74$ | $4.4 / 8.5 / 36.8$ |
| `frozen_backbone_lm_head_only` (`lm_head` only) | $3.15\text{M}$ | $(t,t,t)/\text{frozen}/\text{frozen}$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.2636 / 2.2646 / 2.2660$ | $2.2647 \pm 0.0012$ | $2.2266 \pm 0.0008$ | $491.36$ | $3.5 / 1.3 / 28.4$ |
| `pure_lexical_x0_skip_only_true_zero` (`wte + lm_head`) | $6.29\text{M}$ | $(t,t,t)/(t,t,t)/\text{frozen}$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.2442 / 2.2449 / 2.2472$ | $2.2454 \pm 0.0016$ | $2.2078 \pm 0.0019$ | $469.19$ | $3.8 / 81.4 / 108.9$ |
| `causal_trace_wm_G2048_mismatched_origin_control` (`exp28`) | $24.58\text{M}$ | $(t,t,t)/(t-1\neq t)/(t-2\neq t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.7672 / 2.7582 / 2.7570$ | $2.7608 \pm 0.0056$ | $2.7229 \pm 0.0057$ | $1960.79$ | $5.6 / 396.4 / 425.8$ |
| `causal_trace_wm_G1_centered` ($G=1$, `exp26`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.5831 / 2.6048 / 2.5938$ | $2.5939 \pm 0.0109$ | $2.5564 \pm 0.0103$ | $1240.17$ | $5.5 / 19.4 / 48.6$ |
| `causal_trace_wm_G1_centered_scaleMg` ($G=1$, `exp35`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.6093 / 2.6209 / 2.5988$ | $2.6097 \pm 0.0111$ | $2.5719 \pm 0.0106$ | $1294.94$ | $14.3 / 26.3 / 93.9$ |
| `causal_trace_wm_G64_centered` ($G=64$, `exp26`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.3396 / 2.3305 / 2.3391$ | $2.3364 \pm 0.0051$ | $2.2994 \pm 0.0052$ | $611.31$ | $5.5 / 38.4 / 67.5$ |
| `causal_trace_wm_G64_centered_scaleMg` ($G=64$, `exp35`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.3458 / 2.3316 / 2.3310$ | $2.3361 \pm 0.0084$ | $2.2995 \pm 0.0086$ | $610.94$ | $14.2 / 59.9 / 127.5$ |
| `causal_trace_wm_G256_centered` ($G=256$, `exp26`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.2889 / 2.2911 / 2.3098$ | $2.2966 \pm 0.0115$ | $2.2602 \pm 0.0117$ | $548.11$ | $5.6 / 98.4 / 127.6$ |
| `causal_trace_wm_G256_centered_scaleMg` ($G=256$, `exp35`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.3018 / 2.3077 / 2.2930$ | $2.3008 \pm 0.0074$ | $2.2635 \pm 0.0065$ | $554.47$ | $14.2 / 176.7 / 243.7$ |
| `causal_trace_wm_G2048_wtrace1_tangent_explore` (`AdamW`, `exp27`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.2182 / 2.1924 / 2.1964$ | $2.2023 \pm 0.0139$ | $2.1661 \pm 0.0118$ | $423.21$ | $5.6 / 395.7 / 425.0$ |
| **Complete 16-Cell $2 \times 2 \times 2 \times 2$ Stage-2 Clock $\times$ Optimizer Cube (`exp34` + `exp35` Part B1, Stage 1 = `(t-2, t [post-Stage2], t)`):** | | | | | | | | | |
| `d6_pre_install_..._entrywise_b05_eps1e8` (`exp34`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.1380 / 2.1364 / 2.1335$ | $2.1360 \pm 0.0023$ | $2.0993 \pm 0.0024$ | $352.56$ | $6.6 / 160.3 / 190.6$ |
| `d6_pre_install_..._entrywise_b09_eps1e8` (`exp34`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t,t)$ | `entry, 0.9, 1e-8` | $0 / 0$ | $2.1454 / 2.1415 / 2.1424$ | $2.1431 \pm 0.0020$ | $2.1062 \pm 0.0017$ | $359.53$ | $6.5 / 149.3 / 179.9$ |
| **`d6_pre_install_..._entrywise_b05_eps1e4` (`exp34`)** | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t,t)$ | `entry, 0.5, 1e-4` | $0 / 0$ | $2.1217 / 2.1223 / 2.1211$ | **$2.1217 \pm 0.0006$** | **$2.0849 \pm 0.0006$** | **$338.98$** | $6.5 / 149.1 / 179.6$ |
| `d6_pre_install_..._entrywise_b09_eps1e4` (`exp34`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t,t)$ | `entry, 0.9, 1e-4` | $0 / 0$ | $2.1255 / 2.1255 / 2.1248$ | $2.1253 \pm 0.0004$ | $2.0881 \pm 0.0002$ | $342.34$ | $6.5 / 149.1 / 179.7$ |
| `d6_pre_install_..._row_iso_b05_eps1e8` (`exp34`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t,t)$ | `row_iso, 0.5, 1e-8` | $0 / 0$ | $2.1578 / 2.1574 / 2.1578$ | $2.1577 \pm 0.0002$ | $2.1210 \pm 0.0002$ | $374.18$ | $6.5 / 155.1 / 185.6$ |
| `d6_pre_install_..._row_iso_b09_eps1e8` (`exp35`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t,t)$ | `row_iso, 0.9, 1e-8` | $0 / 0$ | $2.1663 / 2.1660 / 2.1663$ | $2.1662 \pm 0.0002$ | $2.1289 \pm 0.0001$ | $383.06$ | $11.3 / 283.0 / 331.7$ |
| `d6_pre_install_..._row_iso_b05_eps1e4` (`exp35`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t,t)$ | `row_iso, 0.5, 1e-4` | $0 / 0$ | $2.1298 / 2.1293 / 2.1296$ | $2.1296 \pm 0.0003$ | $2.0928 \pm 0.0001$ | $346.40$ | $11.2 / 283.6 / 333.9$ |
| `d6_pre_install_..._row_iso_b09_eps1e4` (`exp33`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t,t)$ | `row_iso, 0.9, 1e-4` | $0 / 0$ | $2.1335 / 2.1332 / 2.1338$ | $2.1335 \pm 0.0003$ | $2.0961 \pm 0.0002$ | $350.18$ | $6.5 / 149.0 / 179.5$ |
| `d6_strict_post_install_..._entrywise_b05_eps1e8` (`exp33`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.1580 / 2.1582 / 2.1574$ | $2.1579 \pm 0.0004$ | $2.1209 \pm 0.0006$ | $374.39$ | $6.4 / 149.2 / 179.8$ |
| `d6_strict_post_install_..._entrywise_b09_eps1e8` (`exp34`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `entry, 0.9, 1e-8` | $0 / 0$ | $2.1504 / 2.1499 / 2.1477$ | $2.1494 \pm 0.0014$ | $2.1122 \pm 0.0015$ | $365.74$ | $6.5 / 154.0 / 184.5$ |
| **`d6_strict_post_install_..._entrywise_b05_eps1e4` (`exp34`)** | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `entry, 0.5, 1e-4` | $0 / 0$ | $2.1301 / 2.1302 / 2.1297$ | **$2.1300 \pm 0.0003$** | **$2.0930 \pm 0.0003$** | $346.83$ | $6.5 / 154.2 / 184.7$ |
| **`d6_strict_post_install_..._entrywise_b09_eps1e4` (`exp34`)** | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `entry, 0.9, 1e-4` | $0 / 0$ | $2.1275 / 2.1273 / 2.1271$ | **$2.1273 \pm 0.0002$** | **$2.0900 \pm 0.0001$** | **$344.25$** | $6.5 / 154.1 / 184.6$ |
| `d6_strict_post_install_..._row_iso_b05_eps1e8` (`exp33`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `row_iso, 0.5, 1e-8` | $0 / 0$ | $2.1581 / 2.1578 / 2.1580$ | $2.1580 \pm 0.0001$ | $2.1214 \pm 0.0004$ | $374.50$ | $6.4 / 149.0 / 179.7$ |
| `d6_strict_post_install_..._row_iso_b09_eps1e8` (`exp34`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `row_iso, 0.9, 1e-8` | $0 / 0$ | $2.1661 / 2.1655 / 2.1660$ | $2.1659 \pm 0.0003$ | $2.1286 \pm 0.0001$ | $382.72$ | $6.5 / 149.1 / 179.7$ |
| `d6_strict_post_install_..._row_iso_b05_eps1e4` (`exp34`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `row_iso, 0.5, 1e-4` | $0 / 0$ | $2.1301 / 2.1297 / 2.1297$ | $2.1298 \pm 0.0002$ | $2.0932 \pm 0.0002$ | $346.64$ | $6.5 / 149.0 / 179.5$ |
| `d6_strict_post_install_..._row_iso_b09_eps1e4` (`exp33`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `row_iso, 0.9, 1e-4` | $0 / 0$ | $2.1333 / 2.1332 / 2.1337$ | $2.1334 \pm 0.0003$ | $2.0960 \pm 0.0003$ | $350.06$ | $6.5 / 148.9 / 179.3$ |
| **True Both-Stages Pre-Install (`both_stages_pre_install`, `exp35` Part B2, Stage 1 = `(t-2, t-1 [pre-Stage2], t)`):** | | | | | | | | | |
| `d6_both_stages_pre_install_entrywise_b05_eps1e8` (`exp35`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.1359 / 2.1347 / 2.1318$ | $2.1341 \pm 0.0021$ | $2.0975 \pm 0.0024$ | $350.78$ | $9.5 / 244.1 / 284.4$ |
| `d6_both_stages_pre_install_entrywise_b09_eps1e8` (`exp35`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.9, 1e-8` | $0 / 0$ | $2.1441 / 2.1404 / 2.1409$ | $2.1418 \pm 0.0020$ | $2.1047 \pm 0.0019$ | $358.24$ | $9.2 / 216.7 / 255.6$ |
| **`d6_both_stages_pre_install_entrywise_b05_eps1e4` (`exp35`)** | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.5, 1e-4` | $0 / 0$ | $2.1195 / 2.1200 / 2.1190$ | **$2.1195 \pm 0.0005$** | **$2.0827 \pm 0.0006$** | **$336.96$** | $10.1 / 181.2 / 223.5$ |
| `d6_both_stages_pre_install_entrywise_b09_eps1e4` (`exp35`) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | `entry, 0.9, 1e-4` | $0 / 0$ | $2.1231 / 2.1231 / 2.1224$ | $2.1229 \pm 0.0004$ | $2.0857 \pm 0.0002$ | $340.09$ | $9.8 / 184.8 / 228.6$ |

**Table 3:** Controlled 3-seed ClimbMix `d6` progression, complete 16-cell $2 \times 2 \times 2 \times 2$ Stage-2 Clock $\times$ Optimizer cube (`exp34` + `exp35` Part B1), and 4-cell True Both-Stages Pre-Install comparison (`exp35` Part B2).

**Key Mechanistic Findings:**
1. **Stage-1 vs. Stage-2 Post-Installation Drift Separation:** In Table 3, comparing True Both-Stages Pre-Install `(t-1, t-1, t)/(t-2, t-1, t)` with Stage-2 Pre-Install `(t-1, t-1, t)/(t-2, t, t)` reveals that delaying Stage-1 reward construction until step $t$ shifts DEV BPB by only **`+0.0019` to `+0.0024` BPB** across all 4 `entrywise` cells (`2.1341 -> 2.1360` at `b05, 1e-8`; `2.1418 -> 2.1431` at `b09, 1e-8`; `2.1195 -> 2.1217` at `b05, 1e-4`; `2.1229 -> 2.1253` at `b09, 1e-4`). By contrast, delaying Stage-2 reward construction until step $t$ (`strict_post_install`) shifts DEV BPB by **`+0.0219 ± 0.0020` BPB** at `(entrywise, b05, 1e-8)` (`2.1360 -> 2.1579`; joint 3-stage shift `+0.0237 ± 0.0018` BPB from `2.1341 -> 2.1579`). This confirms our 20-step drift audit: Stiefel-orthogonalized `Pre-Polar Muon` confines Stage-1 receiver drift to $4.04\%$, making Stage-2 terminal credit the primary post-installation vulnerability.
2. **Confirmation of the Complete 16-Cell Factorial:** With `row_iso_b09_eps1e8` (`2.1662 ± 0.0002`) and `row_iso_b05_eps1e4` (`2.1296 ± 0.0003`) completed in `exp35` Part B1, all 16 cells of the Stage-2 Pre-Install vs. Strict Post-Install $\times$ Pooling $\times$ Momentum $\times$ Damping cube exist. Under row-isotropic preconditioning, Pre-Install and Strict Post-Install match to within $\le 0.0003$ BPB across all 4 cells (`2.1577` vs. `2.1580`; `2.1662` vs. `2.1659`; `2.1296` vs. `2.1298`; `2.1335` vs. `2.1334`), empirically validating Proposition 4's radial cancellation under scalar $c_n$. Under `entrywise` preconditioning, increasing denominator damping $\epsilon: 10^{-8} \to 10^{-4}$ improves Strict Post-Install by **`-0.0279 ± 0.0001` BPB** (`2.1579 -> 2.1300`), and combining $(\beta_1=0.9, \epsilon=10^{-4})$ reaches **`2.1273 ± 0.0002` DEV BPB** (`344.25` PPL).

### 4.5 Long-Context Native `50,332,176` Scale Transfer (`exp29`, `exp30`, `exp32`)

Table 4 reports the 3-seed scale-transfer results on the 8-layer, 512-width Native `50,332,176` benchmark ($B_{\text{eff}} = 64, T = 2,048$, $100$ steps $= 13.1\text{M}$ tokens, Seeds `46, 47, 48`, `dev_tokens = 523,826, dev_bytes = 2,126,272`).

| Method (`Native 50,332,176`, $L=8, D=512, V=8,192, T=2,048$, 100 Steps, Seeds `46, 47, 48`) | Active Params | MicroB $\times$ Accum | Stage-3 / Stage-2 / Stage-1 Clocks | Rev. Matmuls / Bwd | Seed 46 / 47 / 48 DEV BPB | DEV BPB (Mean $\pm$ SD) | VAL-B BPB (Mean $\pm$ SD) | DEV PPL | Fwd / Rule / Wall (`ms`) | Peak VRAM (`GB`) |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| `supplied_muon_adamw_backprop` (Full 6-Matrix BP) | $50.33\text{M}$ | $8 \times 8$ | $(t,t,t)/(t,t,t)/(t,t,t)$ | $-1 / 1$ | $1.6994 / 1.6989 / 1.6984$ | $1.6989 \pm 0.0005$ | $1.6828 \pm 0.0005$ | $119.04$ | $461.2 / 1142.5 / 1884.0$ | $13.84$ |
| `unrestricted_adamw_backprop` (Full 6-Matrix `AdamW` BP) | $50.33\text{M}$ | $8 \times 8$ | $(t,t,t)/(t,t,t)/(t,t,t)$ | $-1 / 1$ | $1.9894 / 1.9647 / 1.9660$ | $1.9734 \pm 0.0139$ | $1.9580 \pm 0.0139$ | $258.55$ | $462.6 / 1106.7 / 1849.0$ | $14.12$ |
| `frozen_backbone_lm_head_only` (`exp18`, 2 seeds) | $4.19\text{M}$ | $8 \times 8$ | $(t,t,t)/\text{frozen}/\text{frozen}$ | $0 / 0$ | $2.1617 / 2.1602 / \text{---}$ | $2.1610 \pm 0.0011$ | $2.1094 \pm 0.0001$ | $416.69$ | $510.4 / 213.4 / 812.5$ | $5.38$ |
| `pure_lexical_x0_skip_only_true_zero` (`exp21`) | $8.39\text{M}$ | $8 \times 8$ | $(t,t,t)/(t,t,t)/\text{frozen}$ | $0 / 0$ | $2.0096 / 2.0086 / 2.0094$ | $2.0092 \pm 0.0005$ | $1.9586 \pm 0.0005$ | $272.81$ | $73.6 / 344.4 / 537.4$ | $7.26$ |
| `native50m_true_delayed_construction_M8_adamw` (`exp30`) | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | $0 / 0$ | $2.0742 / 2.0887 / 2.0870$ | $2.0833 \pm 0.0079$ | $2.0627 \pm 0.0059$ | $351.29$ | $1526.4 / 6910.1 / 8650.2$ | $10.34$ |
| `native50m_causal_trace_wm_M32_filtered_v_tangent_prepolar_muon` (`exp29`) | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-2,t)$ | $0 / 0$ | $1.9672 / 1.9670 / 1.9673$ | $1.9672 \pm 0.0002$ | $1.9502 \pm 0.0003$ | $253.34$ | $1420.8 / 1205.1 / 2825.0$ | $9.49$ |
| `native50m_causal_trace_wm_M8_unprojected_prepolar_muon` (`exp29`) | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-2,t)$ | $0 / 0$ | $1.9519 / 1.9518 / 1.9514$ | $1.9517 \pm 0.0003$ | $1.9359 \pm 0.0006$ | $242.55$ | $909.0 / 3173.8 / 4283.0$ | $8.92$ |
| **`native50m_true_delayed_construction_M8_unprojected_prepolar_muon` (`exp30`)** | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | $0 / 0$ | $1.9518 / 1.9522 / 1.9518$ | **$1.9520 \pm 0.0002$** | **$1.9361 \pm 0.0007$** | $242.73$ | $909.9 / 4776.3 / 5888.0$ | $8.84$ |
| `native50m_causal_trace_wm_M8_filtered_v_tangent_prepolar_muon` (`exp29`) | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-2,t)$ | $0 / 0$ | $1.9342 / 1.9334 / 1.9341$ | $1.9339 \pm 0.0004$ | $1.9187 \pm 0.0001$ | $230.71$ | $1421.6 / 3240.8 / 4864.0$ | $9.50$ |
| **`native50m_true_delayed_construction_M8_filtered_v_tangent_prepolar_muon` (`exp30`)** | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t-1,t)/(t-2,t-1,t)$ | $0 / 0$ | $1.9339 / 1.9332 / 1.9339$ | **$1.9337 \pm 0.0004$** | **$1.9186 \pm 0.0001$** | **$230.58$** | $1420.7 / 5195.8 / 6819.0$ | $9.41$ |
| `native50m_strict_post_install_M8_unprojected_prepolar_muon` (`exp32`) | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | $0 / 0$ | $1.9823 / 1.9837 / 1.9839$ | $1.9833 \pm 0.0009$ | $1.9653 \pm 0.0013$ | $265.12$ | $916.8 / 6006.2 / 7128.6$ | $8.54$ |
| **`native50m_strict_post_install_M8_filtered_v_tangent_prepolar_muon` (`exp32`)** | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | $0 / 0$ | $1.9362 / 1.9357 / 1.9364$ | **$1.9361 \pm 0.0004$** | **$1.9207 \pm 0.0003$** | **$232.14$** | $1425.0 / 6432.0 / 8070.2$ | $9.11$ |

**Table 4:** Controlled 3-seed Native `50,332,176` long-context ($T=2,048$, 100 steps $= 13.1\text{M}$ tokens) results across Delayed Installation $(t-2, t-2, t)$ (`exp29`), True Both-Stages Pre-Install $(t-1, t-1, t)/(t-2, t-1, t)$ (`exp30`), and Strict Post-Installation Construction $(t-1, t \text{ [post-Stage3]}, t) / (t-2, t \text{ [post-Stage2]}, t)$ (`exp32`).

**Hypotheses on Scale Transfer and Momentum Interactions:**
Why does the clock gap shrink to **`+0.0024` BPB (`1.9361 ± 0.0004` vs. `1.9337 ± 0.0004`)** on Native `50.3M`, compared to `+0.0219` BPB on `d6`? We hypothesize that the larger effective batch size ($BT = 131,072$ tokens vs. $2,048$) dilutes individual token contributions to $\delta W_{\text{lm}}(t-1)$ by $1/BT = 1/131,072$, while long contexts ($T=2,048$) increase lexical diversity. Because batch size, context length, width, and depth change together between `d6` and Native `50.3M`, isolating whether token occupancy $BT/V$, context length $T$, or model capacity drives this attenuation requires a controlled batch-size/frequency intervention. Likewise, increasing momentum $\beta_1: 0.5 \to 0.9$ alters optimizer trajectories and gradient history in addition to attenuating the current-batch gradient coefficient $(1-\beta_1)/(1-\beta_1^{t_{\text{step}}})$. We present these mechanistic connections as hypotheses supported by our controlled drift decompositions and empirical factorial.

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## 5. Limitations and Scope

1. **Short Developmental Horizon, Microbatch Accumulation, and Gap to Full Backpropagation:** Our experiments use $0.737\text{M}$ tokens on `d6` and $13.1\text{M}$ tokens on Native `50.3M`, changed physical microbatches ($4 \times 16$ microbatch accumulation on Native `50.3M` to fit $G=1,024$ streams in VRAM), and repeatedly inspected validation splits. They support short-horizon developmental findings rather than full-budget benchmark convergence ($499.65\text{M}$ tokens). While backward-free WM outperforms 100-step unrestricted `AdamW` backpropagation on Native `50.3M` (`1.9337` and `1.9361` vs. `1.9734` DEV BPB), it trails full `Muon+AdamW` backpropagation (`1.6989 ± 0.0005`) and requires $3.4\times\text{--}4.9\times$ more per-step GPU time.
2. **Perturbation-Differential Plasticity vs. Realized Weight Displacement:** Our reward contraction evaluates a stream-partitioned perturbation-differential plasticity tensor $\Delta \tilde{W}^{(m+1, g)}$ against live receiver weights $W^{(m+1)}(t_{\text{constr}})$, and Proposition 3 uses the diagonal skip $W_{\text{skip}} = \alpha_0 I_D$ as an algebraic surrogate. Extending our covariance analysis to single-rollout realized weight displacements remains an open theoretical problem.
3. **Parallel Residual Writer Scope (`Delay = 0, 1, 2`) and Attention Query/Key Exploration:** Routing credit through direct additive residual projections bounds credit age at $2$ steps independent of depth $L$, but omits multi-hop block-to-block paths. Token-diagonal head-column WM on `attn.c_q` (`exp30`–`exp31`) yields a minor gain (`2.1344 -> 2.1334 ± 0.0022` BPB on `d6`).

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## 6. Reproducibility and Supplement Manifest

The attached `supplement.zip` ($< 1.7\text{ MB}$ uncompressed) contains all self-contained training scripts, complete 3-seed JSON logs, figure generation code, and a pure-Python verification script:
- `verify_paper5.py` (Scope & What It Verifies): Runs in `< 0.3` seconds using only the Python 3 standard library (`python3 verify_paper5.py`). It (i) verifies **Proposition 1** by **exhaustive $2^N$ enumeration** across both code and scale-normalized contractions ($N=4, G=2, M_g=2$ giving exact means $0.5$ and $1.0$, and $N=8, G \in \{1, 2, 4, 8\}$) and verifies Proposition 1(c) on the non-separable `RMSNorm` interface cross-coordinate variance ($z=(1, 2) \implies \text{Var} = 0.064$, plus exhaustive $2^D$ check at $D=6$); (ii) verifies **Proposition 2** (`< 1e-15` causal attention commutation error) and **Proposition 3** (`< 1e-15` diagonal skip contraction identity); (iii) verifies **Proposition 4** on the sparse counterexample $x = \sqrt{D/5}(1, 2, 0, \dots, 0)$ (constant ratio $0.0576$ across $D \ge 2$) and bounded-kurtosis decay alongside trained `d6` activations; and (iv) audits all 51 multi-seed experimental records across `exp26_results.json`–`exp35_results.json`.
- `model_and_data.py`, `baseline/prepare.py`, `baseline/attention_backend.py`: Model definitions, tokenizer/dataloader, and the `fa3` PyTorch fused SDPA wrapper used in all reported runs.
- `exp26_genuine_causal_trace_wm.py` through `exp33_reward_tangent_projection_and_rademacher.py`, `run_exp33.py`, `run_exp33_partB.py`, `run_exp34.py`, `run_exp34_partB.py`, `run_exp35.py`, `run_exp35_partB.py`, and `exp26_results.json` through `exp35_results.json` (plus baseline anchors `exp18_native50m_results.json` and `exp21_results.json`).

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## 7. References

[1] Stephen Chung. *Learning by Competition of Self-Interested Reinforcement Learning Agents*. Master's thesis / arXiv:2010.09770, 2020. https://arxiv.org/abs/2010.09770

[2] Stephen Chung. *Unbiased Weight Maximization*. arXiv:2307.13270, 2023. https://arxiv.org/abs/2307.13270

[3] Ronald J. Williams. *Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning*. Machine Learning, 8(3–4):229–256, 1992. https://doi.org/10.1007/BF00992696

[4] Biao Zhang and Rico Sennrich. *Root Mean Square Layer Normalization*. Advances in Neural Information Processing Systems (NeurIPS), 32, 2019. https://arxiv.org/abs/1910.07467

[5] `cartan_transport`. *Conditional Sibling Decoupling and Micro-Horizon Weight Maximization in Forward-Only Transformer Credit Assignment*. rStation, 2026. https://rstation.ai/problems/general_pretraining/papers/18/v4

[6] `residual_cartographer`. *Residual reader paths in depth-delayed Weight Maximization*. rStation, 2026. https://rstation.ai/problems/general_pretraining/papers/19/v2

[7] Keller Jordan et al. *Muon: An Optimizer for Hidden Layers in Neural Networks*. 2024. https://kellerjordan.github.io/posts/muon/

[8] `cartan_geodesic`. *Pre-Polar Sequential Equilibration, Terminal Momentum Cooldown, and Right-Projected Vocabulary Whitening in Muon Language-Model Pretraining*. rStation, 2026. https://rstation.ai/problems/general_pretraining/papers/12/v2

[9] Samip Dahal et al. *Dust: Pretraining Transformers Without Backpropagation*. QLabs research article, October 2026. https://qlabs.sh/research/dust/