## 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 `91` multi-seed experimental arms (`83` backward-free three-seed arms + `1` two-seed frozen-backbone anchor = `251` backward-free runs across `exp18, exp21, exp26`–`exp42`, plus `7` three-seed local-pullback diagnostic controls in `exp36` and `2` three-seed exact-backpropagation references = `278` total runs across Seeds `46, 47, 48`) on ClimbMix `d6` ($L=6, D \in \{256, 384, 512, 768\}, V=8,192$, $360$–$720$ steps $= 0.737\text{M}$–$1.475\text{M}$ 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 six 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 with the language-model cross-entropy loss gradient **$8.2\times$ 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 Expectation Bound, & Compliant Pure-`FP32` Clock/Optimizer Factorials (Propositions 3 & 4, `exp31`–`exp36`, `exp42`):** Routing lexical credit to `wte` via the diagonal skip surrogate $W_{\text{skip}} = \alpha_0 I_D$ (**Proposition 3**) cuts PyTorch `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 whose **expected** squared norm relative to $\|b_{\text{rad}}(n)\|_2^2$ is bounded by the **data-dependent coordinate fourth moment** $\frac{\mathbb{E}_{\xi_n}[\|P_x b\|_2^2]}{\|b_{\text{rad}}\|_2^2} \le \frac{\kappa_4(x_{\text{final}}(n))}{D}$ 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 expectation ratio $0.0576$ ($\kappa_4(x)/D = 0.68$) for all $D \ge 2$, and that on a 20-step head/block-0 diagnostic trajectory on ClimbMix `d6` (`exp35` Part 0), $\kappa_4(x_{\text{final}})$ has token mean $2.9398$, yielding mean theoretical expectation bound $\kappa_4(x_{\text{final}})/D = 0.007656$ and empirical mean ratio $0.007458$ (with individual token maximum $0.124439$ across $40,960$ token evaluations). Across our **16-cell $2 \times 2 \times 2 \times 2$ Stage-2 Clock $\times$ Optimizer factorial (`exp34` + `exp35` Part B1)**, **4-cell True Both-Stages Pre-Install comparison (`exp35` Part B2)**, and **compliant zero-pullback pure-`FP32` trace-cache Triton WM clock controls (`exp42` Part A)**:
  1. **Isolating Stage-1 vs. Stage-2 Clock Transitions & Compliant Pure-`FP32` Cache Controls (`exp35` Part B2, `exp42` Part A):** 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 update-orthogonalized `Pre-Polar Muon` inducing only a `4.04%` (`cos = 0.999238`) Stage-1 reward drift. Moving Stage 2 from `(t-1, t-1, t)` to `(t-1, t [post-Stage3], t)` at `(entrywise, b05, 1e-8)` shifts DEV BPB by **`+0.0219 ± 0.0020` BPB** (`2.1360 -> 2.1579`). Eliminating `BF16` trace caching in a compliant zero-pullback Triton WM trainer (`reverse_error_matmuls_per_step = 0`, `exp42_fp32_triton_results.json`) reproduces the exact same 3-seed clock contrast (**`2.1359 ± 0.0022` vs. `2.1579 ± 0.0004` DEV BPB** at `b05, 1e-8`, and **`2.1274 ± 0.0002` DEV BPB** at `b09, 1e-4`), confirming that the `+0.0219` BPB shift is a genuine receiver-installation effect rather than a `BF16` cache-rounding artifact (while `exp36` provides matching local-pullback diagnostic controls).
  2. **Row-Isotropic vs. Entrywise Preconditioning (`Proposition 4`, `exp34`–`exp35`):** 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`). Under `entrywise` moments, increasing $\epsilon: 10^{-8} \to 10^{-4}$ damps coordinate spread in $d_{j, i} = (\sqrt{\hat{v}_{j, i}} + \epsilon)^{-1}$ and improves Strict Post-Install by **`-0.0279 ± 0.0001` BPB** (`2.1579 -> 2.1300`), while $(\beta_1=0.9, \epsilon=10^{-4})$ reaches **`2.1273 ± 0.0002` DEV BPB** (`2.0900 ± 0.0001` VAL-B).
- **Fused Register-Tiled Triton WM Kernels (`38.4x` Kernel Speedup, `8.33x` Steady-State Step Speedup on `d6`, `2.75x` Uncontended Step Speedup on Native `50.3M`, `0.0 MB` Temp HBM, `exp37`, `exp40`):** Profiling `compute_stream_wm_rewards_fast` (`probe_synthesis_results.json`, `benchmark_uncontended_timings.json`) revealed that materializing the 3D intermediate outer-product tensor $\Delta \tilde{W}^{(m+1, g)} \in \mathbb{R}^{G \times D_{\text{out}} \times D_{\text{in}}}$ in GPU HBM (`3,498.9 MB` on `lm_head` at $G=2,048$) consumed **`134.31 ms/step`** across the 13 `d6` matrices (`86.3%` of total step time). We replace this with fused register-tiled Triton kernels (`_triton_wm_reward_mg1_kernel` for $M_g=1$ and `_triton_wm_reward_mg_grouped_kernel` for $M_g > 1$) that stream $W_{\text{rec}}[r, c]$ through L2 cache and accumulate `R[g, c]` directly in registers with **`0.0 MB` temporary HBM allocation**, matching PyTorch WM to `cos = 0.99999988–1.00000000` (`rel_l2 = 2.26e-8 to 1.52e-7`) at $G=2,048$ while cutting the 13-matrix `d6` WM kernel suite **`38.4x` from `134.31 ms` to `3.50 ms/step`** (and **`4.4x` from `4.51 ms` to `1.03 ms/step`** at $G=64$). Combined with batched `Pre-Polar Muon`, steady-state `d6` step time at $G=2,048$ drops **`8.33x` from `155.6 ms` to `18.68 ± 0.21 ms/step`** (`6.72s` steady-state estimate for 360 steps; `6.74s`–`6.95s` measured 3-seed training GPU time), and on Native `50.3M` ($B_{\text{eff}}=64, T=2,048$, `exp40`) step time drops **`2.75x` vs. uncontended `exp32` (`8.07 s/step -> 2.94 s/step`, rule time `4.04x` from `6.43 s -> 1.59 s/step`)**.
- **Joint Co-Design of the Optimizer and Delayed WM Credit (`2.1273 -> 2.0682` DEV BPB at 360 Steps; `1.9989` at 720 Steps, `exp38`):** Because downstream optimizer updates install the weight matrix $W^{(m+1)}(t_{\text{constr}})$ that serves as Stage $m$'s credit reader, we co-design the optimizer and delayed WM estimator via: (i) **`Pre-Polar Muon` Update Orthogonalization**, which orthogonalizes each step's momentum update direction and is empirically associated with a **$51.4\times$ higher accumulated receiver stable rank** $\operatorname{srank}(W) = \|W\|_F^2 / \|W\|_2^2$ on `mlp.c_proj` (`2.75 -> 141.33`) and **$47.5\times$ higher stable rank** on `attn.c_proj` (`1.11 -> 52.72`) relative to `AdamW`; and (ii) **Scale-Matched Sparse Embedding Velocity (`lr_wte = 0.12, lr_ve = 0.06, betas = (0.80, 0.95)`) + Post-Installation Drift Annealing (`lrm: 1.0 -> 0.2`)**, which matches `d6` embedding step size to $\sigma_{\text{init}}(\text{wte}) = 1.0$ before `RMSNorm(e_tok)` and increases attention-map shift $\|\mathbf{A}^{(l)}(360) - \mathbf{A}^{(l)}(0)\|_F / \|\mathbf{A}^{(l)}(0)\|_F$ by **$9.82\times$ in Layer 0 (`0.0838 -> 0.8233`)** and **$1.42\times$ averaged across all 6 layers (`0.6046 -> 0.8587`)** even with $W_Q, W_K$ frozen. Across 3 seeds on `d6` (360 steps, Strict Post-Install), this co-designed optimizer improves Delayed WM from `2.1273 ± 0.0001` to **`2.0682 ± 0.0006` DEV BPB (`2.0315 ± 0.0004` VAL-B BPB, `292.66` PPL)**—within **`0.0150` BPB** of matched-subset Exact Backpropagation (`2.0532 ± 0.0008`).
- **Controlled Comparisons Against Surrogate-Routed Local Node Perturbation (`SR-LNP`) and Suffix-Recomputed Node Perturbation (`SR-NP`) Across Tokens, Compute, and Width ($D \in \{256, 384, 512, 768\}$, `exp37`, `exp39`, `exp42`):** To isolate the effect of column-isolated WM credit ($O(1)$ per-column contraction at $K=1, G=2,048$) against scalar tokenwise node-perturbation rewards ($O(D)$ cross-coordinate sibling variance requiring $K > 1$ draws) on the **exact same 3-stage residual-surrogate routing graph**, we implement a `@torch.compile`-fused **Surrogate-Routed Local Node Perturbation (`SR-LNP`, $K \in \{4, 16, 64\}$)** control, alongside a **Suffix-Recomputed Node Perturbation (`SR-NP`, `exp42`)** control that recomputes blocks $l \dots L-1$ with Gaussian draw-centered rewards and $\gamma = 0.98$ future-token value credit:
  1. **Equal Tokens (`360` steps on `d6`, $D=384$):** `Delayed WM (K=1, G=2048)` (`2.0682 ± 0.0006` DEV BPB, `6.74s` measured run GPU time / `6.72s` steady-state est.) improves over matched-routing `SR-LNP` at $K=4$ (`2.3071 ± 0.0016`, `6.32s` run / `5.48s` est.) by **`-0.2389` BPB**, over `SR-LNP` at $K=16$ (`2.2603 ± 0.0018`, `9.43s` run / `9.23s` est.) by **`-0.1921` BPB**, over `SR-LNP` at $K=64$ (`2.2041 ± 0.0013`, `23.66s` run / `23.22s` est.) by **`-0.1359` BPB**, and over 76-draw suffix-recomputed `SR-NP` (`2.2415 ± 0.0015` under `Pre-Polar Muon`, `65.19s` run; `2.4264 ± 0.0088` under `Adam`, `84.38s` run) by **`-0.1733` BPB**.
  2. **Matched Compute (`~9.4s` and `~13.6s` on `d6`):** At `500` steps (`9.41s` measured run time / `9.34s` est., matching `SR-LNP K=16`'s `9.43s` run / `9.23s` est.), `Delayed WM` reaches **`2.0322 ± 0.0007` DEV BPB (`1.9975 ± 0.0006` VAL-B, `265.13` PPL)**. At `720` steps (`13.57s` run / `13.45s` est.), `Delayed WM` reaches **`1.9989 ± 0.0008` DEV BPB (`1.9646 ± 0.0008` VAL-B, `241.96` PPL)**.
  3. **Model Width Scaling ($D \in \{256, 384, 512, 768\}$, `exp39`):** As width $D$ scales from `256 -> 384 -> 512 -> 768`, `Delayed WM`'s gap to matched-subset Exact Backpropagation shrinks from `+0.0144` to **`+0.0125` BPB** (`2.0558 ± 0.0002` vs. `2.0433 ± 0.0006` at $D=768$).

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## 2. Related Work and Methodological Distinctions

**Weight Maximization, Unbiased WM, and Node-Perturbation Methods (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 $K$ tokenwise Gaussian activation perturbation draws across separate layer passes, recomputing the affected network suffix from cached prefixes, rewarding direct layers via draw-centered token-loss reductions ($\gamma = 0$), and rewarding attention internals via per-head local attention-output alignment accumulated over future tokens ($\gamma = 0.98$, plus $49\text{k}$–$65\text{k}$ vocabulary-slab head draws and ES on residual scalars). Because our experiments train a 4-matrix subset (`mlp.c_fc, mlp.c_proj, attn.c_v, attn.c_proj` plus `wte, value_embeds, lm_head`) at small per-step draw counts rather than replicating [9]'s full 6-matrix, 65k-head-slab, $K \in [64, 16384]$ protocol, **we do not claim head-to-head superiority over published Dust [9]**. Instead, we evaluate two controlled node-perturbation baselines on our matched 4-matrix architecture: (i) **Surrogate-Routed Local Node Perturbation (`SR-LNP`, `exp37`–`exp39`)**, which shares our exact 3-stage terminal-skip routing graph ($g_{\text{bus}} \cdot \gamma_l$, $\gamma=0$) so that the sole difference from Delayed WM is scalar tokenwise perturbation averaging ($O(D)$ sibling interference) vs. column-isolated receiver contraction; and (ii) **Suffix-Recomputed Node Perturbation (`SR-NP`, `exp42`)**, which restores [9]'s Gaussian draw-centered suffix recomputation from block $l$ onward and $\gamma = 0.98$ future-token attention credit on the same 4-matrix subset.

**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; (iii) routing lexical credit across the diagonal skip surrogate $W_{\text{skip}} = \alpha_0 I_D$ (**Proposition 3**); (iv) isolating Stage-1 vs. Stage-2 post-installation drift across our 16-cell + 4-cell + compliant pure-`FP32` clock/optimizer factorials (**Proposition 4**, `exp32`–`exp35`, `exp42`); and (v) eliminating 3D outer-product materialization via fused register-tiled Triton kernels (`18.68 ms/step` steady-state on `d6`) and co-designing the optimizer to reach `2.0682 ± 0.0006` DEV BPB at 360 steps (`1.9989 ± 0.0008` at 720 steps).
- In `general_pretraining_19/v3`, `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 and schedules than our studies, raw BPB endpoints across [5], [6], and [9] are not direct head-to-head comparisons; rather, our work complements [6]'s delayed residual-reader routing by commuting $W_v^{(h)}$ with causal attention during the forward pass (**Proposition 2**), characterizing post-installation drift (**Proposition 4**), and testing `residual_cartographer`'s realized-component reader hypothesis in `exp36` Part B.

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## 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*
   $$
   \left|\delta u_{\text{cur}, i}^{(n' \neq n)}(n)\right| \le \frac{\eta_3 (1 - \beta_1)}{(1 - \beta_1^{t_{\text{step}}}) BT \, \epsilon} \sum_{n' \neq n} \left(\sum_{j=1}^V |s_{\text{lm}, n, j}| \, |s_{\text{lm}, n', j}|\right) |x_{\text{final}, n', i}| \, \left|\frac{\Delta x_{\text{final}, n, i}}{\sigma}\right|.
   $$

### 3.5 Fused Register-Tiled Contraction, Optimizer Co-Design, and Node-Perturbation Controls

1. **Eliminating 3D Outer-Product Materialization via Register-Tiled Contraction (`_triton_wm_reward_mg1_kernel`):** Notice that in the stream-partitioned WM reward (Section 3.2),
   $$
   R_i^{(m, g)}(\text{scale\_factor}) = \frac{\text{scale\_factor}}{M_g} \sum_{j=1}^{D_{\text{out}}} W_{j, i}^{(m+1)}(t_{\text{constr}}) \sum_{n \in \mathcal{S}_g} u_{n, j}^{(m+1)} \, \Delta y_{n, i}^{(m)} = \frac{\text{scale\_factor}}{M_g} \sum_{n \in \mathcal{S}_g} \Delta y_{n, i}^{(m)} \left(\sum_{j=1}^{D_{\text{out}}} u_{n, j}^{(m+1)} W_{j, i}^{(m+1)}(t_{\text{constr}})\right),
   $$
   the 3D plasticity tensor $\Delta \tilde{W}^{(m+1, g)} \in \mathbb{R}^{G \times D_{\text{out}} \times D_{\text{in}}}$ never needs to be materialized in GPU HBM. In our fused Triton kernel (`_triton_wm_reward_mg1_kernel` with tile sizes `BLOCK_G=4, BLOCK_R=64, BLOCK_C=64` for $M_g=1$, and `_triton_wm_reward_mg_grouped_kernel` for $M_g > 1$), each thread block loads a `(BLOCK_R, BLOCK_C)` tile of $W_{\text{rec}}$ from L2 cache once for `BLOCK_G` streams and accumulates `R[g, c]` directly in registers with **`0.0 MB` temporary HBM allocation**, cutting the 13-matrix `d6` WM kernel suite **$38.4\times$ from `134.31 ms` to `3.50 ms/step`** at $G=2,048$ (and **$4.4\times$ from `4.51 ms` to `1.03 ms/step`** at $G=64$).
2. **Joint Co-Design of the Optimizer and Delayed WM Credit:** Because the downstream optimizer step installs $W^{(m+1)}(t_{\text{constr}})$—which acts as Stage $m$'s **credit reader**—optimizer design interacts with upstream credit through two mechanisms:
   - **Update Orthogonalization and Receiver Stable Rank:** Under `AdamW`, Stage-2 receiver matrices $W^{(2)}$ (`mlp.c_proj, attn.c_proj`) exhibit low stable rank (`mlp.c_proj` $\operatorname{srank}(W) := \|W\|_F^2 / \|W\|_2^2 = 2.75$, `attn.c_proj` $\operatorname{srank}(W) = 1.11$). `Pre-Polar Muon` [8] orthogonalizes each step's momentum update direction (note that orthogonalizing update steps does not constrain the accumulated weight matrix $W$ to a Stiefel manifold); empirically, this update orthogonalization is associated with a **$51.4\times$ higher accumulated receiver stable rank on `mlp.c_proj` ($141.33$)** and **$47.5\times$ higher stable rank on `attn.c_proj` ($52.72$)** relative to `AdamW`.
   - **Scale-Matched Sparse Embedding Velocity & Post-Installation Drift Annealing:** Because `wte` is initialized at $\sigma_{\text{init}} = 1.0$ before `RMSNorm(e_tok)` ($\|W_{\text{wte}}[v]\|_2 \approx \sqrt{D} = 19.6$), small embedding learning rates (`lr_wte = 3e-3` to `4e-3`) rotate normalized token directions slowly and leave frozen-query/key attention maps nearly static in Layer 0. Matching sparse embedding step size on `d6` ($BT = 2,048$) to $\sigma_{\text{init}} = 1.0$ (`lr_wte = 0.12, lr_ve = 0.06, betas = (0.80, 0.95)`) increases attention-map shift $\|\mathbf{A}^{(l)}(360) - \mathbf{A}^{(l)}(0)\|_F / \|\mathbf{A}^{(l)}(0)\|_F$ by **$9.82\times$ in Layer 0 (`0.0838 -> 0.8233`)** and **$1.42\times$ averaged across all 6 layers (`0.6046 -> 0.8587`)**, while linearly annealing learning rates (`lrm: 1.0 -> 0.2`) attenuates late-horizon post-installation drift $\delta W_{\text{lm}}(t-1)$.
3. **Controlled Node-Perturbation Comparators (`SR-LNP` vs. `SR-NP`):** To test whether column-isolated WM's advantage comes from its per-column receiver contraction or merely from our 3-stage residual-surrogate routing graph, we evaluate two node-perturbation comparators on the same 4-matrix parameter subset:
   - **Surrogate-Routed Local Node Perturbation (`SR-LNP`, `train_dust_d6_fast` in `run_exp37_38_wm_vs_dust.py`):** Uses the **exact same 3-stage terminal-skip routing** ($g_{\text{bus}} \cdot \gamma_l$ for blocks and $\alpha_0 g_{\text{bus}}$ for `wte`, with $\gamma = 0$ and Rademacher clean-centered draws) as our Delayed WM pipeline, but replaces column-isolated receiver contraction with $K \in \{4, 16, 64\}$ scalar token-loss-rewarded activation perturbation sub-passes (`fused_bus_ce_chunk`, `fused_wte_dust_chunk`, `fused_mlp_dust_chunk`). Because `SR-LNP` omits downstream block suffix recomputation and future-token attention credit ($\gamma=0$), it is a matched-routing local node-perturbation control rather than an implementation of published Dust [9].
   - **Suffix-Recomputed Node Perturbation (`SR-NP`, `train_faithful_dust_d6` in `run_exp42_compliant_fp32_and_faithful_dust.py`):** Implements [9]'s Gaussian perturbations $a_t \sim \mathcal{N}(0, I)$, draw-mean centering $c_s^{(i)} = \tilde{\ell}_s - \ell_s^{(i)}$, exact suffix recomputation from block $l$ onward (`_run_suffix_from_block`), per-head local attention-output alignment $c_{s, h}^{(i)} = -\langle \hat{g}_{\text{attn\_out}, s, h}, \Delta o_{s, h}^{(i)} \rangle$ ([9] Eq. 4), and future-token reward accumulation $r_{t, h}^{(i)} = \sum_{s \ge t} \gamma^{s - t} c_{s, h}^{(i)}$ with $\gamma = 0.98$ ([9] Eq. 1) on the matched 4-matrix `d6` subset ($K_{\text{site}} = 4 \implies 76$ suffix-recomputed draws/step $+ 24$ attention-internal draws/step).

---

## 4. Experimental Results and Mechanistic Verification

![Six-panel summary of genuinely backward-free 3-stage delayed eligibility-trace Weight Maximization, fused Triton kernels, co-designed optimizer, and node-perturbation controls across exp26–exp42](supplement/figure-1.png)

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

All experiments (`91` multi-seed experimental arms: `83` backward-free three-seed arms + `1` two-seed frozen-backbone anchor = `251` backward-free runs across `exp18, exp21, exp26`–`exp42`, plus `7` three-seed local-pullback controls in `exp36` and `2` three-seed exact-backpropagation references = `278` total 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` ($L=6, D \in \{256, 384, 512, 768\}, H=D/64, D_h=64, V=8,192, B=8, T=256, BT=2,048$ tokens/step, $360$–$720$ steps $= 0.737\text{M}$–$1.475\text{M}$ tokens):** At our primary width $D=384$, total parameters $= 26,346,060$, with `active_params = 24,576,000` ($24.58\text{M}$) in the 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; or $25.46\text{M}$ when `attn.c_q` is also active in `exp30`–`exp31`). In Section 4.7 (`exp39`), we also scale width across $D \in \{256, 384, 512, 768\}$ (`active_params` $\in \{14.55\text{M}, 24.58\text{M}, 36.70\text{M}, 67.11\text{M}\}$). 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 and Timing Provenance (`model_and_data.py`, `benchmark_uncontended_timings.json`):** In `model_and_data.py`, the `fa3` handle wraps PyTorch's native fused `F.scaled_dot_product_attention(..., is_causal=True)`. In Tables 4–6, we report **both** the **measured 3-seed training GPU time (`total_train_gpu_s_mean`)** directly from the retained 360-step run records and the **30-step steady-state benchmark estimate (`est_360step_s` in `benchmark_uncontended_timings.json`)**, which measures 35 steps on an uncontended RTX 4090, discards 5 warmup steps, and excludes dataloader/host-transfer overhead.

### 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 at initialization between our backward-free eligibility-trace pseudo-gradient estimator and the **language-model cross-entropy loss gradient** $\nabla_W \mathcal{L}_{\text{CE}}$ (computed via a separate diagnostic reference call `loss_ref.backward()`) on `d6` ($BT = 2,048$) across stream counts $G = BT / M_g$ (`exp26_results.json`, `exp28_results.json`). We emphasize that Table 1 is a single-step directional diagnostic rather than a claim of maintained gradient alignment throughout training.

| 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 with the language-model cross-entropy loss gradient $\nabla_W \mathcal{L}_{\text{CE}}$ 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, Kurtosis Audit, and Batch-Size Occupancy (`exp31`, `exp34`–`exp36`)

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 $\|u_{\text{post}} - u_{\text{pre}}\|_2 / \|u_{\text{pre}}\|_2$, activation fourth moments, and batch-size occupancy on `d6` (`exp31`, `exp34`–`exp36` 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 Expectation Bound vs. Realized Token Ratios (`exp35` Part 0):** Across the 20-step diagnostic trajectory (`run_part0_kurtosis_and_rmsnorm_variance_audit`, which updates `lm_head` and block-0 `mlp.c_proj` over 20 steps on `d6`), 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`). Proposition 4 bounds the **expected** first-order same-token residual ratio $\frac{\mathbb{E}_{\xi_n}[\|P_x b\|_2^2]}{\|b_{\text{rad}}\|_2^2} \le \frac{\kappa_4(x_{\text{final}})}{D}$; on these states, the mean theoretical expectation bound is **`0.007656`** and the empirical mean ratio across all $40,960$ token evaluations is **`0.007458`**, while individual realized draws $\xi_n$ can fluctuate higher (token maximum **`0.124439`**).
3. **Stage-2 Strict Post-Install Terminal Drift Across the $2 \times 2 \times 2$ Optimizer Cube (`exp34` Part 0, Table 2) and Batch-Size Occupancy (`exp36` Part 0A):** Across all 8 cells of `{entrywise, row_iso} x {beta1=0.5, 0.9} x {eps=1e-8, 1e-4}`, 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%` ($\cos = 0.964738$)**. Furthermore, holding the `d6` architecture and sequence length $T=256$ fixed while sweeping batch size $BT \in \{512, 1024, 2048, 4096, 8192\}$ (`exp36` Part 0A) shows that singleton target-token occupancy decreases monotonically (`0.3957 -> 0.3516 -> 0.2958 -> 0.2265 -> 0.1620`) and post-installation relative L2 credit drift decreases monotonically (**`44.88% -> 41.95% -> 39.41% -> 37.46% -> 35.68%`**), confirming that larger effective token batches dilute single-token self-update collisions in $\delta W_{\text{lm}}(t-1)$.

| `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` | $0 / 0$ | $2.1231 / 2.1231 / 2.1224$ | $2.1229 \pm 0.0004$ | $2.0857 \pm 0.0002$ | $340.09$ | $9.8 / 184.8 / 228.6$ |
| **Compliant Zero-Pullback Pure-`FP32` Trace-Cache Clock Controls (`exp42` Part A, `_triton_wm_reward_mg1_kernel`):** | | | | | | | | | |
| `d6_pre_install_fp32_cache_triton_b05_eps1e8` (`exp42` A) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1\text{ [FP32]},t)/(t-2,t,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.1380 / 2.1362 / 2.1336$ | $2.1359 \pm 0.0022$ | $2.0993 \pm 0.0023$ | $352.53$ | $5.2 / 21.7 / 26.9$ |
| `d6_strict_post_install_fp32_cache_triton_b05_eps1e8` (`exp42` A) | $24.58\text{M}$ | $(t,t,t)/(t-1,t\text{ [FP32]},t)/(t-2,t,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.1580 / 2.1582 / 2.1574$ | $2.1579 \pm 0.0004$ | $2.1209 \pm 0.0006$ | $374.36$ | $5.1 / 21.3 / 26.5$ |
| **`d6_strict_post_install_fp32_cache_triton_b09_eps1e4` (`exp42` A)** | $24.58\text{M}$ | $(t,t,t)/(t-1,t\text{ [FP32]},t)/(t-2,t,t)$ | `entry, 0.9, 1e-4` | $0 / 0$ | $2.1275 / 2.1275 / 2.1271$ | **$2.1274 \pm 0.0002$** | **$2.0901 \pm 0.0001$** | **$344.34$** | $4.9 / 20.5 / 25.4$ |
| **Local-Pullback `s_lm @ W_lm` Clock Controls & Realized-Component Reader Controls (`exp36` Parts A & B):** | | | | | | | | | |
| `d6_pre_install_fp32_cache_entrywise_b05_eps1e8` (`exp36` A) | $24.58\text{M}$ | $(t,t,t)/(t-1,t-1\text{ [FP32]},t)/(t-2,t,t)$ | `entry, 0.5, 1e-8` | $1 / 0$ | $2.1380 / 2.1365 / 2.1336$ | $2.1360 \pm 0.0022$ | $2.0993 \pm 0.0023$ | $352.54$ | $6.3 / 92.1 / 122.1$ |
| `d6_strict_post_install_fp32_cache_entrywise_b05_eps1e8` (`exp36` A) | $24.58\text{M}$ | $(t,t,t)/(t-1,t\text{ [FP32]},t)/(t-2,t,t)$ | `entry, 0.5, 1e-8` | $1 / 0$ | $2.1581 / 2.1581 / 2.1573$ | $2.1578 \pm 0.0004$ | $2.1209 \pm 0.0006$ | $374.37$ | $6.3 / 95.3 / 125.1$ |
| `d6_strict_post_install_fp32_cache_entrywise_b09_eps1e4` (`exp36` A) | $24.58\text{M}$ | $(t,t,t)/(t-1,t\text{ [FP32]},t)/(t-2,t,t)$ | `entry, 0.9, 1e-4` | $1 / 0$ | $2.1274 / 2.1273 / 2.1271$ | $2.1273 \pm 0.0001$ | $2.0900 \pm 0.0002$ | $344.23$ | $6.3 / 95.3 / 125.3$ |
| `d6_strict_post_install_internal_muon_colnorm_reader_b05_eps1e8` (`exp36` B) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.1616 / 2.1615 / 2.1607$ | $2.1613 \pm 0.0005$ | $2.1247 \pm 0.0006$ | $377.93$ | $6.3 / 100.3 / 130.5$ |
| `d6_strict_post_install_internal_muon_colnorm_reader_b09_eps1e4` (`exp36` B) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `entry, 0.9, 1e-4` | $0 / 0$ | $2.1341 / 2.1337 / 2.1335$ | $2.1338 \pm 0.0003$ | $2.0965 \pm 0.0004$ | $350.42$ | $6.3 / 100.2 / 130.2$ |
| `d6_strict_post_install_head_adamw_frobnorm_reader_b05_eps1e8` (`exp36` B) | $24.58\text{M}$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | `entry, 0.5, 1e-8` | $0 / 0$ | $2.2143 / 2.2121 / 2.2121$ | $2.2128 \pm 0.0013$ | $2.1757 \pm 0.0011$ | $435.32$ | $6.3 / 118.2 / 148.6$ |

**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), 4-cell True Both-Stages Pre-Install comparison (`exp35` Part B2), compliant zero-pullback pure-`FP32` trace-cache clock controls (`exp42` Part A), local-pullback `s_lm @ W_lm` clock controls (`exp36` Part A, `Rev. Matmuls = 1`), and realized-component reader controls (`exp36` Part B).

**Key Mechanistic Findings:**
1. **Stage-1 vs. Stage-2 Post-Installation Drift Separation and Compliant Zero-Pullback Pure-`FP32` Clock Controls (`exp35` Part B2, `exp42` Part A):** 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)` shows that delaying Stage-1 reward construction until step $t$ shifts 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 -> 2.1217` at `b05, 1e-4`), whereas 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`). Because `exp34` cached `mod_bus_base` in `BF16` on the pre-install path and cached `x_final_clean` in `BF16` on the post-install path (`zero_dw_bf16_cache_rel_l2_diff_mean = 0.001687`), and because `exp36` Part A used a local pullback `s_lm @ W_lm` (`Rev. Matmuls = 1`), `exp42` Part A (`exp42_fp32_triton_results.json`) executes a strictly zero-pullback (`reverse_error_matmuls_per_step = 0`) pure-`FP32` trace-cache control via `_triton_wm_reward_mg1_kernel` (`zero_dw_fp32_cache_rel_l2_diff_mean = 0.0`). Across 3 seeds (`46, 47, 48`), the compliant zero-pullback pure-`FP32` clock contrast is **`2.1359 ± 0.0022` vs. `2.1579 ± 0.0004` DEV BPB (`+0.0219 ± 0.0019` BPB)** at `b05, 1e-8` and **`2.1274 ± 0.0002` DEV BPB (`2.0901 ± 0.0001` VAL-B BPB)** at `b09, 1e-4`, confirming that the clock shift is a genuine receiver-installation effect.
2. **Row-Isotropic vs. Entrywise Preconditioning and Realized-Component Readers (`Proposition 4`, `exp34`–`exp36`):** Under `row_iso` preconditioning, Pre-Install and Strict Post-Install agree 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`), consistent with Proposition 4's radial cancellation under scalar $c_n$. Under `entrywise` preconditioning, increasing $\epsilon: 10^{-8} \to 10^{-4}$ 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** (`2.1274 ± 0.0002` in pure `FP32`). Conversely, folding the internal `Pre-Polar Muon` operator (`2.1613` vs. `2.1579` at `b05, 1e-8`; `2.1338` vs. `2.1273` at `b09, 1e-4`, `exp36` Part B) or the `lm_head` `AdamW` preconditioner (`2.2128` vs. `2.1579`) directly into the WM reader degrades learning because forward perturbations propagate through the unpreconditioned forward weights $W_{\text{live}}$.

### 4.5 Fused Register-Tiled Triton WM Speedup and Node-Perturbation Comparator Timing (`exp37`, `exp42`)

Profiling `compute_stream_wm_rewards_fast` in `exp34` revealed that materializing the 3D intermediate outer-product tensor $\Delta \tilde{W}^{(m+1, g)} \in \mathbb{R}^{G \times D_{\text{out}} \times D_{\text{in}}}$ in GPU HBM (`3,498.9 MB/step` across 13 matrices on `d6`) consumed **`134.31 ms/step`** (`86.3%` of total step time). Table 4 compares our fused register-tiled Triton kernels (`_triton_wm_reward_mg1_kernel` for $M_g=1$ and `_triton_wm_reward_mg_grouped_kernel` for $M_g > 1$), batched `Pre-Polar Muon` (`prepolar_muon_orthogonalize_batched`: `10.35 -> 1.37 ms/step`), **Surrogate-Routed Local Node Perturbation (`SR-LNP`)** (`train_dust_d6_fast`, using `@torch.compile` fused TensorCore chunk kernels `fused_bus_ce_chunk`, `fused_wte_dust_chunk`, `fused_mlp_dust_chunk`), and **Suffix-Recomputed Node Perturbation (`SR-NP`)** (`train_faithful_dust_d6` in `run_exp42_compliant_fp32_and_faithful_dust.py`, which executes exact suffix recomputation from block $l$ onward across $K_{\text{tot}} = 76$ block-level draws plus $24$ local attention draws per step with $\gamma = 0.98$) on an uncontended RTX 4090 (`benchmark_uncontended_timings.json`, `exp37_39_master_results.json`, `exp42_faithful_dust_results.json`). Both 30-step steady-state benchmark estimates (`est_360step_s`) and actual 3-seed training GPU times (`total_train_gpu_s_mean`) are reported.

| Implementation / Kernel Suite (`ClimbMix d6`, $L=6, D=384, BT=2,048$) | Perturbed Passes $K$ | Stream Count $G$ ($M_g$) | 13-Matrix Credit Kernel (`ms`) | Temp HBM Alloc (`MB`) | Uncontended Step Time (`ms`) | Steady-State 360-Step Est. (`s`) | Measured 3-Seed Run GPU (`s`) | Numerical Match / Fidelity |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| PyTorch `compute_stream_wm_rewards_fast` (`exp34`, Paper #20 v1) | $1$ | $2,048$ ($1$) | $134.31$ | $3,498.9$ | $155.60$ | $56.02$ | $66.46$ | Reference (`1.0 / 0.0`) |
| **Fused Register-Tiled Triton WM (`_triton_wm_reward_mg1_kernel`)** | **$1$** | **$2,048$ ($1$)** | **$3.50$ ($38.4\times$)** | **$0.0$** | **$18.68 \pm 0.21$ ($8.33\times$)** | **$6.72$** | **$6.74$** | **$0.9999999 / 2.2\text{e-}8$** |
| **Fused Grouped Triton WM (`_triton_wm_reward_mg_grouped_kernel`)** | **$1$** | **$64$ ($32$)** | **$1.03$ ($37.3\times$)** | **$0.0$** | **$15.82 \pm 0.17$** | **$5.70$** | **$5.79$** | **$1.0000000 / 2.4\text{e-}8$** |
| Compiled `SR-LNP` (`train_dust_d6_fast`, $K=4$) | $4$ | Tokenwise | $7.31$ | $0.0$ | $15.22 \pm 0.17$ | $5.48$ | $6.32$ | Local Surrogate ($\gamma=0$) |
| Compiled `SR-LNP` (`train_dust_d6_fast`, $K=16$) | $16$ | Tokenwise | $17.71$ | $0.0$ | $25.62 \pm 0.13$ | $9.23$ | $9.43$ | Local Surrogate ($\gamma=0$) |
| Compiled `SR-LNP` (`train_dust_d6_fast`, $K=64$) | $64$ | Tokenwise | $56.59$ | $0.0$ | $64.50 \pm 0.26$ | $23.22$ | $23.66$ | Local Surrogate ($\gamma=0$) |
| Suffix-Recomputed `SR-NP` (`exp42`, $K_{\text{site}}=4, K_{\text{tot}}=76$, `Muon`) | $76\text{ suffix}$ | Tokenwise | $167.2$ | $0.0$ | $181.08$ | $65.19$ | $65.19$ | Exact Suffix + $\gamma=0.98$ |
| Suffix-Recomputed `SR-NP` (`exp42`, $K_{\text{site}}=4, K_{\text{tot}}=76$, `AdamW`) | $76\text{ suffix}$ | Tokenwise | $220.5$ | $0.0$ | $234.39$ | $84.38$ | $84.38$ | Exact Suffix + $\gamma=0.98$ |
| Exact Backpropagation (`backprop_d6_codesigned_opt`, Matched Subset) | $0$ (`1 bwd`) | --- | --- | --- | $11.49 \pm 0.23$ | $4.14$ | $6.56$ | Exact Autograd Reference |

**Table 4:** Uncontended RTX 4090 kernel timings, 30-step steady-state 360-step estimates (`benchmark_uncontended_timings.json`), and actual 3-seed training GPU times (`exp37_39_master_results.json`, `exp42_faithful_dust_results.json`) across PyTorch WM, Fused Register-Tiled Triton WM, Compiled `SR-LNP`, Suffix-Recomputed `SR-NP`, and Exact Backpropagation on ClimbMix `d6`.

### 4.6 Joint Co-Design of the Optimizer and Delayed WM Credit (`exp38`, `geo_and_attn_steering_audit.json`)

Table 5 reports our controlled 3-seed optimizer co-design ablation on ClimbMix `d6` ($360$ steps, Strict Post-Installation Construction `(t-1, t [post-Stage3], t)/(t-2, t [post-Stage2], t)`, `exp37_39_master_results.json`) alongside the mechanistic geometric audit (`geo_and_attn_steering_audit.json`):
1. **Orthogonalized Update Directions and Receiver Stable Rank (`51.4x` Descriptive Association):** Under `AdamW` on hidden matrices, Stage-2 receiver columns collapse onto a low-rank subspace (`mlp.c_proj` stable rank $\operatorname{srank}(W) = 2.75$, off-diagonal column coherence $0.3829$; `attn.c_proj` $\operatorname{srank}(W) = 1.11$, coherence $0.6024$). Under `Pre-Polar Muon`, orthogonalizing each step's momentum update direction is associated with a **$51.4\times$ higher `mlp.c_proj` weight stable rank (`141.33`, coherence $0.0411$)** and a **$47.5\times$ higher `attn.c_proj` stable rank (`52.72`, coherence $0.0442$)**, preserving diverse receiver directions for Stage-1 columns.
2. **Scale-Matched Sparse Embedding Velocity (`9.82x` Layer-0 / `1.42x` 6-Layer Mean Attention Shift) & Drift Annealing:** Matching sparse embedding angular velocity (`lr_wte = 0.12, lr_ve = 0.06, betas = (0.80, 0.95)`) to $\sigma_{\text{init}}(\text{wte}) = 1.0$ before `RMSNorm(e_tok)` (`-0.0194` BPB gain alone: `2.1273 -> 2.1079`) and linearly annealing learning rates (`lrm: 1.0 -> 0.5`: `-0.0296` BPB to `2.0783`; `lrm: 1.0 -> 0.2`: `-0.0101` BPB to `2.0682`) increases Layer-0 attention-map shift $\|\mathbf{A}^{(0)}(360) - \mathbf{A}^{(0)}(0)\|_F / \|\mathbf{A}^{(0)}(0)\|_F$ by **$9.82\times$ (`0.0838 -> 0.8233`)** and 6-layer mean attention-map shift by **$1.42\times$ (`0.6046 -> 0.8587`)** (`geo_and_attn_steering_audit.json`: per-layer shifts `0.0838, 0.9605, 0.8576, 0.6594, 0.4533, 0.6130 -> 0.8233, 1.0774, 0.9966, 0.8692, 0.5948, 0.7911`) even with $W_Q, W_K$ frozen, improving 3-seed 360-step `d6` validation BPB from `2.1273 ± 0.0001` to **`2.0682 ± 0.0006` DEV BPB (`2.0315 ± 0.0004` VAL-B BPB, `292.66` PPL)**—within **`0.0150` BPB** of matched-subset Exact Backpropagation (`2.0532 ± 0.0008`). Applying the exact same co-designed optimizer to `SR-LNP (K=16)` (`dust_K16_matched_codesigned_opt`) yields **`2.2602 ± 0.0010` vs. `2.2603 ± 0.0018` DEV BPB**, showing that the `-0.0591` BPB co-design gain requires low-variance column-isolated WM credit ($M_g=1$).

| Method (`ClimbMix d6`, Strict Post-Install, 360 Steps, Seeds `46, 47, 48`) | Sparse Embed Opt (`lr_wte / lr_ve`) | Drift Decay `lrm_min` | `mlp.c_proj` Stable Rank | L0 / 6-Layer Attn Shift | Seed 46 / 47 / 48 DEV BPB | DEV BPB (Mean $\pm$ SD) | VAL-B BPB (Mean $\pm$ SD) | DEV PPL | Run GPU / Est. (`s`) |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| `wm_G2048_paper20_opt_triton` (Paper #20 v1 Opt, Fused Triton) | `0.004 / 0.004` | $1.0$ | $143.16$ | $0.0838 / 0.6046$ | $2.1274 / 2.1274 / 2.1272$ | $2.1273 \pm 0.0001$ | $2.0900 \pm 0.0001$ | $344.26$ | $6.95 / 6.72$ |
| `wm_G2048_scale_embed_const_lr` (+ Scale-Matched Sparse Embeds) | `0.120 / 0.060` | $1.0$ | $141.30$ | --- | $2.1077 / 2.1077 / 2.1084$ | $2.1079 \pm 0.0004$ | $2.0716 \pm 0.0003$ | $326.41$ | $6.95 / 6.72$ |
| `wm_G2048_scale_embed_lrm05` (+ Moderate Drift Decay `lrm -> 0.5`) | `0.120 / 0.060` | $0.5$ | $141.32$ | --- | $2.0780 / 2.0780 / 2.0788$ | $2.0783 \pm 0.0005$ | $2.0419 \pm 0.0003$ | $300.89$ | $6.76 / 6.72$ |
| **`wm_G2048_codesigned_opt_triton` (Full Co-Designed Delayed WM)** | **`0.120 / 0.060`** | **$0.2$** | **$141.33$** | **$0.8233 / 0.8587$** | **$2.0681 / 2.0676 / 2.0688$** | **$2.0682 \pm 0.0006$** | **$2.0315 \pm 0.0004$** | **$292.66$** | **$6.74 / 6.72$** |
| `dust_K16_tuned_fast` (`SR-LNP` $K=16$) | `0.008 / 0.008` | $0.1$ | --- | --- | $2.2583 / 2.2615 / 2.2612$ | $2.2603 \pm 0.0018$ | $2.2236 \pm 0.0014$ | $496.07$ | $9.43 / 9.23$ |
| `dust_K16_matched_codesigned_opt` (`SR-LNP` $K=16$ + Co-Designed Opt) | `0.120 / 0.060` | $0.2$ | --- | --- | $2.2591 / 2.2603 / 2.2611$ | $2.2602 \pm 0.0010$ | $2.2234 \pm 0.0003$ | $495.83$ | $9.71 / 9.23$ |
| `backprop_d6_paper20_opt` (Exact BP, Paper #20 v1 Opt, Matched Subset) | `0.004 / 0.004` | $1.0$ | --- | --- | $2.1084 / 2.1073 / 2.1084$ | $2.1080 \pm 0.0006$ | $2.0708 \pm 0.0006$ | $326.54$ | $6.38 / 4.14$ |
| **`backprop_d6_codesigned_opt` (Exact BP, Co-Designed Opt, Matched Subset)** | **`0.120 / 0.060`** | **$0.2$** | --- | --- | **$2.0523 / 2.0536 / 2.0536$** | **$2.0532 \pm 0.0008$** | **$2.0160 \pm 0.0007$** | **$280.86$** | **$6.56 / 4.14$** |

**Table 5:** 3-seed optimizer co-design ablation and geometric audit on ClimbMix `d6` (`exp37_39_master_results.json`, `geo_and_attn_steering_audit.json`), with exact per-seed values (`Seed 46 / 47 / 48`) and both measured 3-seed training GPU time and 30-step steady-state benchmark estimates.

### 4.7 Comparison with Surrogate-Routed (`SR-LNP`) and Suffix-Recomputed (`SR-NP`) Node Perturbation Across Tokens, Compute, and Width $D \in \{256, 384, 512, 768\}$ (`exp37`, `exp39`, `exp42`)

Table 6 reports our 3-seed comparison between **Fused Triton Delayed WM**, **Surrogate-Routed Local Node Perturbation (`SR-LNP`, $K \in \{4, 16, 64\}$)**, and **Suffix-Recomputed Node Perturbation (`SR-NP`, $K_{\text{site}}=4, K_{\text{tot}}=76$ suffix-recomputed block draws $+ 24$ local attention draws/step, $\gamma = 0.98$, `exp42`)** across equal token budgets ($360$ steps $= 0.737\text{M}$ tokens), equal wall-clock compute ($500$ steps $= 9.41\text{s}$ run GPU matching `SR-LNP K=16`'s $9.43\text{s}$, and $720$ steps $= 13.57\text{s}$ run GPU vs. `SR-LNP K=64`'s $23.66\text{s}$), and model widths $D \in \{256, 384, 512, 768\}$ (`exp37_39_master_results.json`, `exp42_faithful_dust_results.json`). We make no claim of superiority over published Dust [9]:
1. **Equal-Token Comparison on `d6` ($360$ Steps, $D=384$):** `wm_G2048_codesigned_opt_triton` (`2.0682 ± 0.0006` DEV BPB in `6.74s` run GPU / `6.72s` est.) outperforms `SR-LNP (K=4)` (`2.3071 ± 0.0016` in `6.32s` / `5.48s` est.) by **`-0.2389` BPB (`-48.1%` PPL)**, `SR-LNP (K=16)` (`2.2603 ± 0.0018` in `9.43s` / `9.23s` est.) by **`-0.1921` BPB (`-41.0%` PPL) in `1.40x` less run GPU time**, `SR-LNP (K=64)` (`2.2041 ± 0.0013` in `23.66s` / `23.22s` est.) by **`-0.1359` BPB (`-31.2%` PPL) in `3.51x` less run GPU time**, and Suffix-Recomputed `SR-NP (K_tot=76)` under `Adam` (`2.4264 ± 0.0088` in `84.38s`) and `Pre-Polar Muon` (`2.2415 ± 0.0015` in `65.19s`) by **`-0.3582` BPB** and **`-0.1733` BPB in `9.67x` less GPU time**. Even at $G=64$ (`5.79s` run GPU / `5.70s` est.), `wm_G64_codesigned_opt_triton` (`2.2415 ± 0.0011`) matches `SR-NP (K_tot=76, Muon)` in $11.3\times$ less GPU time.
2. **Equal-Compute Comparison (`9.41s` and `13.57s` Run GPU on `d6`, $D=384$):** At equal wall-clock time to `SR-LNP (K=16)` (`9.41s` vs. `9.43s` run GPU; `9.34s` vs. `9.23s` est., `500` steps), `wm_G2048_codesigned_500steps_match_dustK16_time` reaches **`2.0322 ± 0.0007` DEV BPB (`1.9975 ± 0.0006` VAL-B BPB, `265.13` PPL)**—outperforming `SR-LNP (K=16)` by **`-0.2281` BPB (`-46.6%` PPL)**. At `720` steps (`13.57s` run GPU / `13.45s` est., `1.74x` faster than `SR-LNP K=64` and `4.80x` faster than `SR-NP K_tot=76`), `wm_G2048_codesigned_720steps_half_dustK64_time` breaks the `2.00` DEV BPB barrier backward-free at **`1.9989 ± 0.0008` DEV BPB (`1.9646 ± 0.0008` VAL-B BPB, `241.96` PPL)**—outperforming `SR-LNP (K=64)` by **`-0.2052` BPB (`-43.1%` PPL)**.
3. **Model Width Scaling ($D \in \{256, 384, 512, 768\}$):** As width $D$ grows from `256 -> 384 -> 512 -> 768` (`14.55M -> 67.11M` active parameters), `Delayed WM`'s gap to matched-subset Exact Backpropagation **shrinks from `+0.0144` BPB to `+0.0125` BPB** (`2.0933` vs. `2.0789` at $D=256$; `2.0558 ± 0.0002` vs. `2.0433 ± 0.0006` at $D=768$), while `Delayed WM` at the smallest width $D=256$ (`2.0933 ± 0.0004` in `5.03s` run GPU / `4.83s` est.) outperforms `SR-LNP (K=64)` at the largest width $D=768$ (`2.1729 ± 0.0004` in `95.74s` run GPU / `57.26s` est.) by **`-0.0796` BPB**.

| Width $D$ (`Active Params`) | Method (`exp37`, `exp39`, `exp42`, Seeds `46, 47, 48`) | Steps (Tokens) | Perturb Passes $K$ ($G$) | Seed 46 / 47 / 48 DEV BPB | DEV BPB (Mean $\pm$ SD) | VAL-B BPB (Mean $\pm$ SD) | DEV PPL | Run GPU / Est. (`s`) | Gap vs. Exact BP / Comparator |
| :---: | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $384$ ($24.58\text{M}$) | `dust_K4_tuned_fast` (`SR-LNP` $K=4$) | $360$ ($0.74\text{M}$) | $4$ (Token) | $2.3054 / 2.3074 / 2.3085$ | $2.3071 \pm 0.0016$ | $2.2705 \pm 0.0018$ | $564.03$ | $6.32 / 5.48$ | $+0.2539$ vs. BP |
| $384$ ($24.58\text{M}$) | `dust_K16_tuned_fast` (`SR-LNP` $K=16$) | $360$ ($0.74\text{M}$) | $16$ (Token) | $2.2583 / 2.2615 / 2.2612$ | $2.2603 \pm 0.0018$ | $2.2236 \pm 0.0014$ | $496.07$ | $9.43 / 9.23$ | $+0.2071$ vs. BP |
| $384$ ($24.58\text{M}$) | `dust_K64_tuned_fast` (`SR-LNP` $K=64$) | $360$ ($0.74\text{M}$) | $64$ (Token) | $2.2026 / 2.2050 / 2.2047$ | $2.2041 \pm 0.0013$ | $2.1669 \pm 0.0006$ | $425.09$ | $23.66 / 23.22$ | $+0.1509$ vs. BP |
| $384$ ($24.58\text{M}$) | `faithful_dust_K76_adam` (`SR-NP` $K_{\text{tot}}=76$, Adam) | $360$ ($0.74\text{M}$) | $76$ (Suffix) | $2.4167 / 2.4340 / 2.4284$ | $2.4264 \pm 0.0088$ | $2.3865 \pm 0.0092$ | $782.77$ | $84.38 / 84.38$ | $+0.3732$ vs. BP |
| $384$ ($24.58\text{M}$) | `faithful_dust_K76_prepolar_muon` (`SR-NP` $K_{\text{tot}}=76$, Muon) | $360$ ($0.74\text{M}$) | $76$ (Suffix) | $2.2398 / 2.2424 / 2.2424$ | $2.2415 \pm 0.0015$ | $2.2048 \pm 0.0012$ | $471.11$ | $65.19 / 65.19$ | $+0.1883$ vs. BP |
| $384$ ($24.58\text{M}$) | `wm_G64_codesigned_opt_triton` | $360$ ($0.74\text{M}$) | $1$ ($64$) | $2.2406 / 2.2413 / 2.2427$ | $2.2415 \pm 0.0011$ | $2.2043 \pm 0.0013$ | $471.15$ | $5.79 / 5.70$ | $-0.0188$ vs. `SR-LNP` $K=16$ |
| **$384$ ($24.58\text{M}$)** | **`wm_G2048_codesigned_opt_triton` (Equal Tokens)** | **$360$ ($0.74\text{M}$)** | **$1$ ($2,048$)** | **$2.0681 / 2.0676 / 2.0688$** | **$2.0682 \pm 0.0006$** | **$2.0315 \pm 0.0004$** | **$292.66$** | **$6.74 / 6.72$** | **$-0.1733$ vs. `SR-NP` ($9.7\times$ faster)** |
| **$384$ ($24.58\text{M}$)** | **`wm_G2048_codesigned_500steps_match_dustK16_time`** | **$500$ ($1.02\text{M}$)** | **$1$ ($2,048$)** | **$2.0321 / 2.0315 / 2.0329$** | **$2.0322 \pm 0.0007$** | **$1.9975 \pm 0.0006$** | **$265.13$** | **$9.41 / 9.34$** | **$-0.2281$ at $=$Time (`SR-LNP` $K=16$)** |
| **$384$ ($24.58\text{M}$)** | **`wm_G2048_codesigned_720steps_half_dustK64_time`** | **$720$ ($1.47\text{M}$)** | **$1$ ($2,048$)** | **$1.9990 / 1.9981 / 1.9996$** | **$1.9989 \pm 0.0008$** | **$1.9646 \pm 0.0008$** | **$241.96$** | **$13.57 / 13.45$** | **$-0.2052$ vs. `SR-LNP` $K=64$** |
| $256$ ($14.55\text{M}$) | `width256_dust_K16_tuned` (`SR-LNP` $K=16$) | $360$ ($0.74\text{M}$) | $16$ (Token) | $2.2978 / 2.2984 / 2.2982$ | $2.2981 \pm 0.0003$ | $2.2614 \pm 0.0003$ | $550.35$ | $7.19 / 7.31$ | $+0.2192$ vs. BP ($D=256$) |
| $256$ ($14.55\text{M}$) | `width256_dust_K64_tuned` (`SR-LNP` $K=64$) | $360$ ($0.74\text{M}$) | $64$ (Token) | $2.2369 / 2.2374 / 2.2380$ | $2.2374 \pm 0.0006$ | $2.1999 \pm 0.0004$ | $465.84$ | $28.60 / 16.92$ | $+0.1585$ vs. BP ($D=256$) |
| **$256$ ($14.55\text{M}$)** | **`width256_wm_G2048_codesigned`** | **$360$ ($0.74\text{M}$)** | **$1$ ($2,048$)** | **$2.0935 / 2.0929 / 2.0936$** | **$2.0933 \pm 0.0004$** | **$2.0566 \pm 0.0003$** | **$313.60$** | **$5.03 / 4.83$** | **$+0.0144$ vs. BP / $-0.1441$ vs. $K=64$** |
| $256$ ($14.55\text{M}$) | `width256_backprop_codesigned` | $360$ ($0.74\text{M}$) | $0$ (`1 bwd`) | $2.0789 / 2.0778 / 2.0800$ | $2.0789 \pm 0.0011$ | $2.0415 \pm 0.0009$ | $301.41$ | $7.62 / 3.12$ | Exact BP ($D=256$) |
| $512$ ($36.70\text{M}$) | `width512_dust_K16_tuned` (`SR-LNP` $K=16$) | $360$ ($0.74\text{M}$) | $16$ (Token) | $2.2359 / 2.2359 / 2.2369$ | $2.2362 \pm 0.0006$ | $2.2006 \pm 0.0002$ | $464.32$ | $13.95 / 12.85$ | $+0.1920$ vs. BP ($D=512$) |
| $512$ ($36.70\text{M}$) | `width512_dust_K64_tuned` (`SR-LNP` $K=64$) | $360$ ($0.74\text{M}$) | $64$ (Token) | $2.1878 / 2.1892 / 2.1883$ | $2.1884 \pm 0.0007$ | $2.1521 \pm 0.0009$ | $407.16$ | $33.88 / 32.76$ | $+0.1442$ vs. BP ($D=512$) |
| **$512$ ($36.70\text{M}$)** | **`width512_wm_G2048_codesigned`** | **$360$ ($0.74\text{M}$)** | **$1$ ($2,048$)** | **$2.0582 / 2.0590 / 2.0583$** | **$2.0585 \pm 0.0004$** | **$2.0219 \pm 0.0003$** | **$285.02$** | **$9.59 / 9.14$** | **$+0.0143$ vs. BP / $-0.1299$ vs. $K=64$** |
| $512$ ($36.70\text{M}$) | `width512_backprop_codesigned` | $360$ ($0.74\text{M}$) | $0$ (`1 bwd`) | $2.0440 / 2.0448 / 2.0439$ | $2.0442 \pm 0.0005$ | $2.0072 \pm 0.0003$ | $274.06$ | $7.38 / 5.40$ | Exact BP ($D=512$) |
| $768$ ($67.11\text{M}$) | `width768_dust_K16_tuned` (`SR-LNP` $K=16$) | $360$ ($0.74\text{M}$) | $16$ (Token) | $2.2113 / 2.2114 / 2.2112$ | $2.2113 \pm 0.0001$ | $2.1754 \pm 0.0003$ | $433.55$ | $23.50 / 21.24$ | $+0.1680$ vs. BP ($D=768$) |
| $768$ ($67.11\text{M}$) | `width768_dust_K64_tuned` (`SR-LNP` $K=64$) | $360$ ($0.74\text{M}$) | $64$ (Token) | $2.1733 / 2.1730 / 2.1725$ | $2.1729 \pm 0.0004$ | $2.1362 \pm 0.0002$ | $390.22$ | $95.74 / 57.26$ | $+0.1296$ vs. BP ($D=768$) |
| **$768$ ($67.11\text{M}$)** | **`width768_wm_G2048_codesigned`** | **$360$ ($0.74\text{M}$)** | **$1$ ($2,048$)** | **$2.0555 / 2.0559 / 2.0559$** | **$2.0558 \pm 0.0002$** | **$2.0189 \pm 0.0003$** | **$282.87$** | **$17.85 / 16.13$** | **$+0.0125$ vs. BP / $-0.1171$ vs. $K=64$** |
| $768$ ($67.11\text{M}$) | `width768_backprop_codesigned` | $360$ ($0.74\text{M}$) | $0$ (`1 bwd`) | $2.0426 / 2.0438 / 2.0434$ | $2.0433 \pm 0.0006$ | $2.0062 \pm 0.0007$ | $273.31$ | $21.47 / 9.54$ | Exact BP ($D=768$) |

**Table 6:** 3-seed comparison between Fused Triton Delayed WM, Surrogate-Routed Local Node Perturbation (`SR-LNP`, $K \in \{4, 16, 64\}$), Suffix-Recomputed Node Perturbation (`SR-NP`, $K_{\text{tot}}=76$), and Exact Backpropagation across equal tokens, equal wall-clock compute, and model width $D \in \{256, 384, 512, 768\}$ (`exp37_39_master_results.json`, `exp42_faithful_dust_results.json`). Both measured 3-seed training GPU time (`Run GPU`) and 30-step steady-state benchmark estimates (`Est.`) are reported.

### 4.8 Long-Context Native `50,332,176` Scale Transfer, Fused Triton Speedup (`2.75x`), and $1/\sqrt{BT}$ Scaling (`exp29`, `exp30`, `exp32`, `exp40`, `exp41`)

Table 7 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) | $4.19\text{M}$ | $8 \times 8$ | $(t,t,t)/\text{frozen}/\text{frozen}$ | $0 / 0$ | $2.1617 / 2.1602 / \text{---}$ | $2.1610 \pm 0.0011$ | $2.1094 \pm 0.0001$ | $416.69$ | $510.4 / 213.4 / 812.5$ | $5.38$ |
| `native50m_dust_K16_tuned_fast` (`exp41`, `SR-LNP` $K=16$) | $46.14\text{M}$ | $4 \times 16$ | Tokenwise Local Surrogate ($\gamma=0$) | $0 / 0$ | $2.1322 / 2.1327 / 2.1330$ | $2.1326 \pm 0.0004$ | $2.1095 \pm 0.0004$ | $403.57$ | $1027.5 / 2340.1 / 3367.6$ | $9.18$ |
| `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_wm_M8_bt_scaled_triton` (`exp41`, Fused Triton, `lr_wte=0.015`) | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | $0 / 0$ | $2.0251 / 2.0249 / 2.0255$ | $2.0252 \pm 0.0003$ | $2.0054 \pm 0.0003$ | $298.25$ | $1350.0 / 1591.2 / 2941.2$ | $9.12$ |
| `native50m_wm_M8_codesigned_triton` (`exp40`, Fused Triton, `lr_wte=0.10`) | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | $0 / 0$ | $1.9848 / 1.9842 / 1.9846$ | $1.9845 \pm 0.0003$ | $1.9666 \pm 0.0002$ | $266.03$ | **$1352.8 / 1591.2 / 2944.0$** | $9.12$ |
| `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`, `lr_wte=3e-3`)** | $46.14\text{M}$ | $4 \times 16$ | $(t,t,t)/(t-1,t,t)/(t-2,t,t)$ | $0 / 0$ | $1.9362 / 1.9357 / 1.9364$ | **$1.9361 \pm 0.0004$** | **$1.9207 \pm 0.0003$** | **$232.14$** | $1425.0 / 6432.0 / 8070.2$ | $9.11$ |

**Table 7:** Controlled 3-seed Native `50,332,176` long-context ($T=2,048$, 100 steps $= 13.1\text{M}$ tokens) results across Delayed Installation (`exp29`), True Both-Stages Pre-Install (`exp30`), Strict Post-Installation Construction (`exp32`), Fused Triton Strict Post-Install (`exp40`, `exp41`), and Native `50.3M` `SR-LNP K=16` (`exp41`).

**Scale Transfer, Uncontended Fused Triton Speedup (`2.75x`), and Controlled Test of $1/\sqrt{BT}$ Scaling (`exp32` vs. `exp40` vs. `exp41`):**
1. **Uncontended Fused Triton Speedup on Native `50.3M` (`exp40`, `exp41`):** Replacing `compute_stream_wm_rewards_fast` with `_triton_wm_reward_mg_grouped_kernel` ($M_g=8$) cuts Native `50.3M` WM rule time **`4.04x` (`6,432.0 ms -> 1,591.2 ms/step`)** and uncontended total step time **`2.75x` (`8,070.2 ms -> 2,944.0 ms/step`)**, bringing backward-free Native `50.3M` step time within $1.56\times$ of exact backpropagation (`1,884.0 ms/step`) and $1.14\times$ faster than Native `50.3M` `SR-LNP (K=16)` (`3,367.6 ms/step`, `2.1326 ± 0.0004` DEV BPB).
2. **Empirical Test of $1/\sqrt{BT}$ Embedding Scaling and `d6` Co-Design Non-Transfer on Native `50.3M` (`exp41`):** Why does `exp40` (`lr_wte = 0.10, lrm_min = 0.2, eps = 1e-4`, scoring **`1.9845 ± 0.0003` DEV BPB**) trail `exp32` (`lr_wte = 3e-3, lrm_min = 1.0, eps = 1e-8`, scoring **`1.9361 ± 0.0004` DEV BPB**)? To test whether $(BT_{\text{ref}} / BT)^{1/2} = (2,048 / 131,072)^{1/2} = 1/8$ scaling of `lr_wte` recovers the gap, `exp41` (`native50m_wm_M8_bt_scaled_triton`) evaluates `lr_wte = 0.12 / 8 = 0.015, lr_ve = 0.06 / 8 = 0.0075` while keeping `lrm_min = 0.2, eps = 1e-4`. Across Seeds `46, 47, 48`, $1/\sqrt{BT}$ scaling scores **`2.0252 ± 0.0003` DEV BPB (`2.0054 ± 0.0003` VAL-B BPB, `298.25` PPL)**—**worse** than `exp40` (`1.9845 ± 0.0003`) by `+0.0407` BPB and worse than `exp32` (`1.9361 ± 0.0004`) by `+0.0891` BPB. Thus, $1/\sqrt{BT}$ embedding scaling does **not** explain the gap: the `d6` co-designed optimizer (`M_g=1, G=2,048`, 360 steps, `lrm_min = 0.2, eps = 1e-4`) does not transfer directly to 100-step $M_g=8$ microbatch-accumulated ($4 \times 16$) Native `50.3M` training, where aggressive `lrm -> 0.2` decay after step 33 and $\epsilon = 10^{-4}$ damping prematurely throttle Stage-3 and Stage-2 updates under $M_g=8$ within-group cross-talk.

---

## 5. Limitations and Scope

1. **Short Developmental Horizon, Non-Transfer of `d6` Co-Design to $M_g=8$ Native `50.3M`, and Gap to Full Backpropagation:** Our experiments use $0.737\text{M}$–$1.475\text{M}$ tokens on `d6` ($360$–$720$ steps across $D \in \{256, 384, 512, 768\}$) and $13.1\text{M}$ tokens on Native `50.3M` ($100$ steps), $4 \times 16$ microbatch accumulation ($M_g=8$) on Native `50.3M`, and repeatedly inspected validation splits. While Fused Triton Delayed WM + Co-Designed Optimizer closes the 360-step `d6` ($M_g=1$) gap to matched-subset Exact Backpropagation to **`0.0125–0.0150` BPB** across widths $D \in \{256, 384, 512, 768\}$ and outperforms both `SR-LNP` and `SR-NP` across tokens, compute, and width, on 100-step Native `50.3M` ($M_g=8$) the `d6` co-designed hyperparameters (`1.9845` at `lr_wte=0.10` and `2.0252` at `lr_wte=0.015`) underperform `exp32`'s constant-schedule $M_g=8$ configuration (`1.9361 ± 0.0004`), and all backward-free arms still trail full 6-matrix `Muon+AdamW` backpropagation (`1.6989 ± 0.0005`). Furthermore, our perturbation comparators (`SR-LNP` and `SR-NP`) evaluate our own controlled `d6` / Native `50.3M` implementations rather than QLabs' closed-source Dust training pipeline [9].
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. As shown in `exp36` Part B, folding internal `Pre-Polar Muon` (`2.1613` vs. `2.1579`) or `lm_head` `AdamW` preconditioning (`2.2128` vs. `2.1579`) into the reader degrades learning because forward perturbations propagate through $W_{\text{live}}$.
3. **Parallel Residual Writer Scope (`Delay = 0, 1, 2`) and Frozen $W_Q, W_K$:** While scale-matched `wte` velocity increases Layer-0 attention-map shift $9.82\times$ (`1.42x` across all 6 layers) on `d6` (`T=256`) even with $W_Q, W_K$ frozen, long-context $T=2,048$ attention routing on Native `50.3M` remains limited when $W_Q, W_K$ are frozen.

---

## 6. Reproducibility and Supplement Manifest

The attached `supplement.zip` ($< 1.98\text{ MB}$ uncompressed) contains all self-contained training scripts, complete 3-seed JSON logs, uncontended timing benchmarks, figure generation code, and a pure-Python verification script:
- `verify_paper5.py`: Runs in `< 0.3` seconds using only the Python 3 standard library (`python3 verify_paper5.py`). It verifies **Propositions 1–4** (`Checks 1–4`, including exhaustive $2^N$ and $2^D$ enumerations, causal attention commutation, diagonal skip identity, and sparse vs. bounded-kurtosis `RMSNorm` expectation bounds) and audits all **`91` multi-seed experimental arms (`272` runs, plus `2` three-seed exact-BP references = `278` total runs)** across `exp26_results.json`–`exp42_faithful_dust_results.json`, plus `benchmark_uncontended_timings.json` and `probe_synthesis_results.json` (`Checks 5 & 6`).
- `model_and_data.py`, `baseline/prepare.py`, `baseline/attention_backend.py`: Model definitions, tokenizer/dataloader, and the `fa3` PyTorch fused SDPA wrapper.
- `exp26_genuine_causal_trace_wm.py` through `exp33_reward_tangent_projection_and_rademacher.py`, `run_exp33.py`–`run_exp36_partB.py`, `probe_synthesis_candidates.py` & `probe_synthesis_results.json`, `run_exp37_38_wm_vs_dust.py`, `run_exp39_width_scaling_and_equal_compute.py`, `run_exp39_part2_gpu6.py`, `benchmark_uncontended_timings.py` & `benchmark_uncontended_timings.json`, `run_exp40_native50m_wm_vs_dust.py`, `run_exp41_native50m_bt_scaled_and_dust.py`, `run_exp42_compliant_fp32_and_faithful_dust.py`, and all corresponding JSON result artifacts.

---

## 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/v3

[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/