# Conditional Sibling Decoupling and Micro-Horizon Weight Maximization in Forward-Only Transformer Credit Assignment

## 1. Introduction and Architectural Ledger

### 1.1 The Width-Scaling Variance Bottleneck in Forward-Only Credit
Eliminating the global reverse-depth sweep of backpropagation is a central pursuit in neuromorphic computing and biologically plausible credit assignment [1, 2, 8]. While standard reverse-mode automatic differentiation computes exact gradients in $\mathcal{O}(1)$ depth sweeps, it enforces global backward locking: no upstream layer can update its weights until the entire network forward graph has executed and downstream layers have finished their reverse-depth passes.

Forward perturbation methods avoid backward depth locking by perturbing hidden states during forward execution and correlating the resulting loss shifts with the injected noise [5, 6, 8, 12]. However, conventional forward perturbation rules face a severe **neuronal width-scaling bottleneck**:
- **Global Scalar-Reward Perturbation Control (`fwd_global_scalar_K2`)**: Simultaneous zero-mean perturbation across all parameters or activations correlated with a single global scalar loss shift $\Delta \mathcal{L}$ yields estimator variance that scales with the total parameter count $\mathcal{O}(|\Theta| / K)$. (Note: `fwd_global_scalar_K2` is a simplified global-scalar control, not published DUST [5]; see Section 2.6 and Section 3.5 for our two-stage faithful DUST baseline.) In a 6-layer Transformer at $D=512$ ($M=2048$), relative variance of `fwd_global_scalar_K2` at Layer 3 (`midMLP`) explodes to $3.26 \times 10^7$ (Table 3), stalling 200-step pretraining at $2.3898$ holdout BPB (Table 4) and 1000-step pretraining on `d6` ($D=384$) at $2.44794 \pm 0.11598$ holdout BPB (Table 5).
- **Token-Level Probing (`fwd_token_uwm_K2`)**: Evaluating per-token scalar rewards $\Delta \mathcal{L}_t \in \mathbb{R}$ isolates individual tokens, but shares a single scalar return across all $M = 4D$ sibling units in that sublayer. As Ren et al. [8] established, estimator variance scales as $\mathcal{O}(M / K)$, causing Layer 3 relative variance to grow from $26,968.01$ ($D=128$) to $103,931.10$ ($D=512$) (Table 3) and limiting 200-step holdout BPB to $2.1868$ at $D=512$ (Table 4) and 1000-step `d6` holdout BPB to $2.09816 \pm 0.00521$ (Table 5).

### 1.2 Stephen Chung's Foundations and Sylvester-Hadamard Coordination
To overcome this width bottleneck without a global backward pass, we adapt Stephen Chung's credit principles:
1. **Weight Maximization (`arXiv:2010.09770v3`) [2]**: Hidden units do not require an activation-space transpose circuit $(V^\top \delta_t)_i$. Instead, each unit $i$ observes only the plasticity displacement $\Delta \mathbf{v}^{(i)}$ of its own outgoing synaptic column $\mathbf{v}^{(i)} = V_{:, i}$, reinforcing activity when its outgoing column norm expands ($G^{(i)} = \mathbf{v}^{(i)} \cdot \Delta \mathbf{v}^{(i)}$).
2. **Unbiased Weight Maximization (`arXiv:2307.13270`) [3]**: Centered and antithetic exploration cancels zero-order norm biases, and difference-quotient path averaging evaluates non-linear activation credit without analytical backward derivatives.
3. **Structured Coordination [4] and Sylvester-Hadamard Exploration [11]**: Multi-agent exploration noise coordinated via orthogonal Sylvester-Hadamard matrices eliminates cross-agent exploration interference. Following Choromanski et al. [11] (`arXiv:1804.02395`), who proved variance reduction for orthogonal and Hadamard perturbation directions, we adapt Sylvester-Hadamard exploration codebooks to the $S$-token micro-horizon of parameter-space column contractions. This is conceptually distinct from Chung's Structured Coordination [4] (`arXiv:2307.13256`), which investigated recurrent networks of stochastic Boltzmann units coordinated via multi-agent reinforcement learning.
4. **MAP Propagation (`arXiv:2010.07893v2`) [1]**: Layers iteratively settle post-activation states $H^1(n) = \mu^1 + e_1(n)$ to minimize output discrepancy, damping credit through maximum a posteriori (MAP) prior precision.

### 1.3 Strict No-Backward Credit vs. Global Reverse-Depth Removal with Local Reverse Credit Computations
We enforce a strict conceptual and experimental separation between **removing the global reverse-depth sweep** and **removing local reverse credit computations**:
1. **Strict No-Backward Credit Assignment (Zero Reverse-Depth Sweeps AND Zero Local Reverse Pullbacks)**:
   - **Published Faithful DUST (`dust_faithful_sgd_P64`, `dust_faithful_muon_P64`, `dust_faithful_muon_P256`) [5]**: Perturbs activation groups during forward execution (`writers`, `hidden`, `hubs` `o@layer`, `wte`, 256-column `lm_head` slabs) and correlates per-token draw-chunk centered loss rewards (and decayed future-token attention head rewards $\gamma=0.98$) directly with pre-synaptic states (`dust_faithful.py`).
   - **Delayed / Eligibility-Trace Coordinated Weight Maximization (`Delayed-Trace-Co-WM-S`, `delayed_trace_cowm_*`)**: Records local forward eligibility traces on `mlp.c_fc` during the clean forward pass using $S$-token Sylvester-Hadamard probes ($\xi_{g, s}^{(m)} = H_{s, c_m}$) and local forward activation evaluations $\phi(z \pm \sigma_{\text{act}})$, and modulates those traces via delayed outgoing-weight column contractions $c_g^{(i, m)} = \mathbf{v}^{(i)} \cdot \Delta \mathbf{v}_g^{(i, m)}$ obtained from either same-step (`d0`) or 1-step cross-batch delayed (`d1`) outgoing `mlp.c_proj` writer updates (`delayed_trace_cowm.py`), requiring **zero direct `hidden` (`mlp.c_fc`) forward passes** and **zero local backward/pullback computations**.
2. **Global Reverse-Depth Removal with Local Reverse Credit Computations (Labelled Local-Reverse-Credit Controls)**:
   - In our streaming stagewise algorithms (`fwd_streaming_cowm*`, `fwd_streaming_oracle*`, `fwd_lookahead_2b*`), `@torch.no_grad()` is enforced (`num_backward_calls = 0`, `num_depth_reverse_sweeps = 0`) and no global backward graph is instantiated. However, within each detached block, these methods retain **local reverse credit computations** (`d_relu2`, `_forward_rms_norm_directional_pull`, inverse rotary pullback `apply_rotary_emb(cos, -sin)`, and intra-block MLP-to-Attention residual pullback). Replacing analytical derivative formulas with central forward operator differences (`OpDiff`, `*_opdiff_*`) still evaluates local reverse sensitivity pullbacks and therefore does **not** remove backward credit computation. Accordingly, backpropagation (`backprop`, `backprop_ckpt`), block-detached autograd (`block_*_bp`), and all coupled local-error / analytic-surrogate / `OpDiff` / `Lookahead-2B` streaming variants are presented strictly as **labelled local-reverse-credit controls**.

We organize our study across three architectural paradigms:
- **Paradigm A: Two-Level Coordinated Weight Maximization (`fwd_two_level_cowm`)**: Combines parameter-space `Co-WM-S` column contraction on `lm_head` over micro-horizons of $S=8$ tokens, $2 K_{\text{net}}$ end-to-end antithetic full-network passes ($x_0 \to x_L$) with either depth-Hadamard carrier modulation (`net_carrier_mode="depth_hadamard"`) or independent per-probe carrier resampling (`net_carrier_mode="iid_resample"`), and intra-block parameter-space `Co-WM-S` on MLP projections ($W_1, V$) with head-decoupled forward FlashAttention probing on attention matrices ($W_Q, W_K, W_V, W_O$).
- **Paradigm B: Streaming Stagewise Forward Co-WM-S and Local-Reverse-Credit Controls (`fwd_streaming_cowm*`, `fwd_lookahead_2b*`)**: Executes a single forward sweep through the network ($l = 0 \dots L-1$). At each block $l$:
  1. Forward output $x_l$ is evaluated and correlated with the language modeling target via an active sub-vocabulary projection ($V_{\text{sub}} = 256$ active tokens, or $V_{\text{sub}}=0$ for full vocabulary $V$);
  2. Block $l$'s parameters ($W_Q, W_K, W_V, W_O, W_1, V$) compute gradients blockwise into `.grad` (either uncoupled in `fwd_streaming_cowm`, where both sublayers receive $\delta_{\text{block}}$; intra-block coupled in `fwd_streaming_cowm_coupled*`, where the MLP update runs first and pulls upstream credit across $W_1$ via parameter-space `Co-WM-S` into the attention sublayer; or 2-block cooperative lookahead in `fwd_lookahead_2b_orth_mv*`, where a sliding 2-block window $(l, l+1)$ transports block $l+1$'s zero-variance MLP + Attention-Value input credit projected onto $\delta_{\text{local}, l}^\perp$ into block $l$);
  3. **Block Deallocation**: All intermediate activations of block $l$ ($x_{\text{prev}}, x_{\text{attn\_in}}, u_{\text{attn}}, x_{\text{mlp\_in}}, u_{\text{mlp}}$) are **freed from GPU memory immediately** before block $l+1$ (or $l+2$ under 2-block lookahead) begins execution, maintaining $\mathcal{O}(1)$ activation memory in depth $L$;
  4. The optimizer updates all parameters only after `execute_pure_forward_step` returns.
- **Paradigm C: Strict No-Backward Credit Assignment — Faithful DUST (`dust_faithful_*`) and Delayed / Eligibility-Trace Coordinated Weight Maximization (`delayed_trace_cowm_*`)**: Detailed in Section 2.6 and evaluated in Section 3.5 across Stage 1 reference validation (`dust_stage1_reference_validation.json`), width and pure hidden-neuron scaling ($M \in \{256 \dots 4096\}$, `dust_and_delayed_trace_width_diagnostic.json`), 200-step multi-width pretraining (`dust_and_delayed_trace_width_sweep_200.json`), and 1000-step 2-seed `d6` pretraining under matched-token and matched-wall-clock budgets (`dust_and_delayed_trace_1000steps_d6.json`).

To evaluate the **incremental effects** of successive training interventions in one specified ordering between end-to-end backpropagation and streaming `Co-WM-S`, we construct a sequential ablation ladder:
1. **`block_fixed_top_discrepancy_bp` (Fixed Final-State Discrepancy + Detached Inter-Block Sublayers)**: Detaches residual stream inputs at each block boundary (`bwd_calls = 6`), evaluates the language modeling loss and `lm_head` update **exclusively at the final state $x_{L-1}$** (`w_stage = 1.0` on $L-1$, $0.0$ on $l < L-1$), and broadcasts the fixed final-state pre-norm discrepancy $\delta_{\text{final}} = \frac{\partial \mathcal{L}(x_{L-1})}{\partial x_{L-1}}$ directly into each detached block $l$ via the residual identity skip connection (`x_l.backward(delta_final)`). Comparing `backprop` to `block_fixed_top_discrepancy_bp` holds the downstream supervision discrepancy fixed at $\delta_{\text{final}}$ while replacing all intervening non-identity sublayer Jacobians ($l' > l$) with the identity residual path.
2. **`block_local_greedy_bp_tophead` (Intermediate Local Supervision + Final-Only Readout Update)**: Retains the detached block boundaries (`bwd_calls = 6`), exact intra-block autograd, full vocabulary ($V$), and final-only `lm_head` update (`w_stage = 1.0` on $L-1$, $0.0$ on $l < L-1$) of `block_fixed_top_discrepancy_bp`, but replaces the broadcast final-state discrepancy $\delta_{\text{final}}$ with each block's own intermediate local-loss discrepancy $\frac{\partial \mathcal{L}(x_l)}{\partial x_l}$.
3. **`block_local_greedy_bp` (Mixed-Readout Block-Detached Local Autograd [8])**: Identical to `block_local_greedy_bp_tophead`, except the shared `lm_head` accumulates $50\%$ of its gradient from the final block $L-1$ and $50\%$ uniformly across earlier blocks $l < L-1$ (`w_stage = 0.5 / (L-1)`), matching the readout update rule of `fwd_streaming_*`. (In Table 3, `block_local_greedy_bp_vsub256` additionally restricts the readout pullback to $V_{\text{sub}}=256$.)
4. **Incremental Intra-Block and 2-Block Local-Reverse-Credit Controls at Full Vocabulary ($V_{\text{sub}}=0$)**:
   - `fwd_streaming_oracle_fullvocab` (Uncoupled Baseline): Feeds $\delta_{\text{block}}$ into both MLP and Attention independently, with single-window `CM-Attn` ($K_v=1$) and pre-RoPE $K_{\text{qk}}=2$ attention probes.
   - `fwd_streaming_oracle_mlp_pull_only_fullvocab` (**+MLP-Pullback Only**): Adds only the upstream MLP-to-Attention residual pullback $\delta u_{\text{attn}} = \delta_{\text{block}} + \Delta_{\text{mlp}\to\text{attn}}$ (`intra_block_coupling=True`), keeping $K_v=1, K_{\text{qk}}=2$ and `f32_hadamard_qk=False`.
   - `fwd_streaming_oracle_mlp_whv_fullvocab` (**+MLP-Pullback + `CM-Attn-WH` $K_v=8$**): Additionally enables Two-Level Window-Hadamard value transport (`K_v=8`), keeping $K_{\text{qk}}=2$ and `f32_hadamard_qk=False`.
   - `fwd_streaming_oracle_coupled_fullvocab` (**+Combined `Hadamard-QK` Package, $K_{\text{qk}}=8$**): Jointly transitions Q/K probing from $K_{\text{qk}}=2$ pre-RoPE i.i.d. probes to the post-RoPE head-channel Sylvester-Hadamard Q/K package (`K_qk=8, f32_hadamard_qk=True`, $\sigma_{\text{qk}}=0.05$, `float32` perturbation construction/reward accumulation with `bfloat16` FlashAttention execution, and 16-lag scalar key lookahead), paired with `fwd_streaming_cowm_coupled_fullvocab_K8` ($K=S=8$ parameter-space `Co-WM-S`).
   - `fwd_streaming_oracle_coupled_qproj_unbiased_fullvocab` (**+Bilinear Query-Subspace Projected Key Probing `Q-Proj-K`**): Holds $K_{\text{qk}}=8$, $\sigma_{\text{qk}}=0.05$, post-RoPE perturbation location, and `float32`/`bfloat16` precision handling strictly identical to `fwd_streaming_oracle_coupled_fullvocab`, replacing **only** the 16-lag scalar key correlation with bilinear rank-1 query-subspace projection (`qproj_k="unbiased"`), paired with `fwd_streaming_cowm_coupled_qproj_opdiff_fullvocab_K8` ($K=S=8$ parameter-space `Co-WM-S` + `OpDiff`).
   - `fwd_lookahead_2b_orth_mv_a1.0_opdiff_qproj_fullvocab` (**+Adjacent 2-Block Orthogonal-Complement Cooperative Lookahead `Lookahead-2B-OrthMV` + `OpDiff`**): Adds sliding 2-block orthogonal-complement lookahead transport (`lookahead_2b="orth_mv", lookahead_alpha=1.0, opdiff=True`).
5. **Sub-Vocabulary Readout Truncation ($V_{\text{sub}}=256$)**: `fwd_streaming_oracle`, `fwd_streaming_oracle_coupled`, and their parameter-space `Co-WM-S` counterparts `fwd_streaming_cowm_K8`, `fwd_streaming_cowm_coupled_K8`, `fwd_streaming_cowm_coupled_qproj_opdiff_K8`, and `fwd_lookahead_2b_orth_mv_cowm_opdiff_qproj_K8` restrict the readout pullback to $V_{\text{sub}}=256$ active columns.

Table 1 provides the complete arithmetic and pass-count ledger for all studied algorithms ($B=16, T=512$, $N_{\text{tok}}=8192$, micro-horizon $S=8$, $L=6$ blocks). Across widths $D \in \{128, 256, 384, 512\}$, exact trainable parameter counts are $6,422,732$, $15,204,748$, $26,346,060$, and $39,846,668$ for the pretraining vocabulary $V=8192$ (Tables 1, 4, 5, 8, 9), and $3,801,292$, $9,961,868$, $18,481,740$, and $29,360,908$ for the synthetic diagnostic vocabulary $V=4096$ (Table 3).

| Algorithm | Execution Paradigm | Full Forward Passes | Backward Calls | Depth Reverse Sweeps | Attn Forward Calls / Block | MLP Forward Calls / Block | Param-Space WM Contractions | Peak VRAM ($D=512$, $M=2048$) |
| :--- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| **`backprop`** (Standard) | Autograd Forward-Backward | $1$ | $1$ | $1$ | $1$ clean | $1$ clean | $0$ | $3010.4\text{ MB}$ |
| **`backprop_ckpt`** (Checkpointed) | Autograd + Block Recompute | $2$ | $1$ | $1$ | $2$ ($1$ fwd + $1$ recompute) | $2$ ($1$ fwd + $1$ recompute) | $0$ | $1731.4\text{ MB}$ |
| **`block_fixed_top_discrepancy_bp`** | Block-Detached BP (Fixed Top $\delta_{\text{final}}$) | $1$ | $6$ ($1$/block) | $0$ (intra-block only) | $1$ clean | $1$ clean | $0$ | $3026.5\text{ MB}$ |
| **`block_local_greedy_bp_tophead`** | Block-Detached BP (Local $\delta_l$, Top Head) | $1$ | $6$ ($1$/block) | $0$ (intra-block only) | $1$ clean | $1$ clean | $0$ | $2153.6\text{ MB}$ |
| **`block_local_greedy_bp`** | Block-Detached BP (Mixed Head [8]) | $1$ | $6$ ($1$/block) | $0$ (intra-block only) | $1$ clean | $1$ clean | $0$ | $2169.6\text{ MB}$ |
| **`fwd_global_scalar_K2`** | Pure Forward (Global-Scalar Control) | $3$ | **$0$** | **$0$** | $3$ ($1$ clean + $2$ probe) | $3$ ($1$ clean + $2$ probe) | $0$ | $3864.8\text{ MB}$ |
| **`fwd_token_uwm_K2`** | Pure Forward (Token Probing) | $5$ | **$0$** | **$0$** | $5$ ($1$ clean + $4$ probe) | $5$ ($1$ clean + $4$ probe) | $0$ | $3864.8\text{ MB}$ |
| **`dust_faithful_sgd_P64`** | Strict No-Backward (Faithful DUST [5], SGD) | $1 + 64$ suffix | **$0$** | **$0$** | $45.3$ ($272$ block evals) | $45.3$ ($272$ block evals) | $0$ | $3058.1\text{ MB}$ |
| **`dust_faithful_muon_P64`** | Strict No-Backward (Faithful DUST [5], Muon) | $1 + 64$ suffix | **$0$** | **$0$** | $45.3$ ($272$ block evals) | $45.3$ ($272$ block evals) | $0$ | $3091.7\text{ MB}$ |
| **`delayed_trace_cowm_P64_fast_d1`** | Strict No-Backward (`Delayed-Trace-Co-WM-S`, `0` hidden) | $1 + 37$ suffix | **$0$** | **$0$** | $23.7$ ($142$ block evals) | $23.7$ ($142$ block + $1$ local) | $49,152$ | $3101.6\text{ MB}$ |
| **`delayed_trace_cowm_P64_realloc_d1`** | Strict No-Backward (`Delayed-Trace-Co-WM-S`, Realloc) | $1 + 64$ suffix | **$0$** | **$0$** | $44.2$ ($265$ block evals) | $44.2$ ($265$ block + $1$ local) | $49,152$ | $3101.6\text{ MB}$ |
| **`delayed_trace_cowm_fixed16_d1`** | Strict No-Backward (`Delayed-Trace-Co-WM-S`, All-Block 16w) | $1 + 27$ suffix | **$0$** | **$0$** | $20.7$ ($124$ block evals) | $20.7$ ($124$ block + $1$ local) | $49,152$ | $2692.6\text{ MB}^\dagger$ |
| **`dust_faithful_muon_P256`** | Strict No-Backward (Faithful DUST [5], $P=256$) | $1 + 256$ suffix | **$0$** | **$0$** | $180.0$ ($1080$ block evals) | $180.0$ ($1080$ block evals) | $0$ | $3369.7\text{ MB}^\dagger$ |
| **`delayed_trace_cowm_P256_fast_d1`** | Strict No-Backward (`Delayed-Trace-Co-WM-S`, $P=256$ Fast) | $1 + 146$ suffix | **$0$** | **$0$** | $92.3$ ($554$ block evals) | $92.3$ ($554$ block + $1$ local) | $49,152$ | $3380.4\text{ MB}^\dagger$ |
| **`fwd_two_level_cowm_K4`** | Pure Forward (End-to-End Co-WM) | $9$ | **$0$** | **$0$** | $19$ ($1$ clean + $18$ probe) | $10$ ($1$ clean + $9$ probe/recomp) | $106,496$ | $1850.3\text{ MB}$ |
| **`fwd_streaming_cut_K4`** | Pure Forward (Streaming Cut) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $6$ ($1$ clean + $5$ probe/recomp) | $24,576$ | **$1820.3\text{ MB}$** |
| **`fwd_streaming_wm_K1`** ($K_{\text{qk}}=1$) | Pure Forward (Streaming Dealloc) | **$1$** | **$0$** | **$0$** | $7$ ($1$ clean + $6$ probe) | $2$ ($1$ clean + $1$ local fwd) | $18,432$ | **$1828.4\text{ MB}$** |
| **`fwd_streaming_wm_K1_qk2`** ($K_{\text{qk}}=2$) | Pure Forward (Streaming Dealloc) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $2$ ($1$ clean + $1$ local fwd) | $18,432$ | **$1820.3\text{ MB}$** |
| **`fwd_streaming_wm_K4`** | Pure Forward (Streaming Dealloc) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $2$ ($1$ clean + $1$ local fwd) | $73,728$ | **$1820.3\text{ MB}$** |
| **`fwd_streaming_cowm_K4`** | Pure Forward (Streaming Dealloc) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $2$ ($1$ clean + $1$ local fwd) | $73,728$ | **$1828.4\text{ MB}$** |
| **`fwd_streaming_wm_K8`** | Pure Forward (Streaming Dealloc) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $2$ ($1$ clean + $1$ local fwd) | $147,456$ | **$1820.3\text{ MB}$** |
| **`fwd_streaming_cowm_K8`** | Pure Forward (Streaming Dealloc) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $2$ ($1$ clean + $1$ local fwd) | $147,456$ | **$1828.4\text{ MB}$** |
| **`fwd_streaming_wm_map_N4`**| Pure Forward (Streaming MAP) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $6$ ($1$ clean + $5$ settle/recomp) | $73,728$ | **$1828.4\text{ MB}$** |
| **`fwd_streaming_oracle_fullvocab`** | Uncoupled Surrogate ($V=8192$) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $2$ ($1$ clean + $1$ local fwd) | $55,296$ | $2100.4\text{ MB}$ |
| **`fwd_streaming_oracle_mlp_pull_only_fullvocab`** | +MLP-Pullback Only ($V=8192, K_v=1$) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $2$ ($1$ clean + $1$ local fwd) | $104,448$ | $2100.3\text{ MB}$ |
| **`fwd_streaming_oracle_mlp_whv_fullvocab`** | +MLP-Pullback + `CM-Attn-WH` ($K_v=8$) | **$1$** | **$0$** | **$0$** | $18$ ($1$ clean + $17$ probe) | $2$ ($1$ clean + $1$ local fwd) | $104,448$ | $2100.3\text{ MB}$ |
| **`fwd_streaming_oracle`** | Uncoupled Surrogate ($V_{\text{sub}}=256$) | **$1$** | **$0$** | **$0$** | $11$ ($1$ clean + $10$ probe) | $2$ ($1$ clean + $1$ local fwd) | $55,296$ | **$1820.3\text{ MB}$** |
| **`fwd_streaming_cowm_coupled_K8`** | Coupled `Co-WM-S` ($K=8, V_{\text{sub}}=256$) | **$1$** | **$0$** | **$0$** | $42$ ($1$ clean + $41$ probe) | $2$ ($1$ clean + $1$ local fwd) | $196,608$ | **$1828.3\text{ MB}$** |
| **`fwd_streaming_cowm_coupled_fullvocab_K8`** | Coupled `Co-WM-S` ($K=8, V=8192$) | **$1$** | **$0$** | **$0$** | $42$ ($1$ clean + $41$ probe) | $2$ ($1$ clean + $1$ local fwd) | $196,608$ | $2100.3\text{ MB}$ |
| **`fwd_streaming_oracle_coupled`** | Coupled Surrogate ($V_{\text{sub}}=256$) | **$1$** | **$0$** | **$0$** | $42$ ($1$ clean + $41$ probe) | $2$ ($1$ clean + $1$ local fwd) | $104,448$ | **$1828.3\text{ MB}$** |
| **`fwd_streaming_oracle_coupled_fullvocab`** | Coupled Surrogate ($V=8192$) | **$1$** | **$0$** | **$0$** | $42$ ($1$ clean + $41$ probe) | $2$ ($1$ clean + $1$ local fwd) | $104,448$ | $2100.3\text{ MB}$ |
| **`fwd_streaming_cowm_coupled_qproj_opdiff_K8`** | +`Q-Proj-K` + `OpDiff` `Co-WM-S` ($V_{\text{sub}}=256$) | **$1$** | **$0$** | **$0$** | $42$ ($1$ clean + $41$ probe) | $2$ ($1$ clean + $1$ local fwd) | $196,608$ | $2084.3\text{ MB}$ |
| **`fwd_streaming_cowm_coupled_qproj_opdiff_fullvocab_K8`** | +`Q-Proj-K` + `OpDiff` `Co-WM-S` ($V=8192$) | **$1$** | **$0$** | **$0$** | $42$ ($1$ clean + $41$ probe) | $2$ ($1$ clean + $1$ local fwd) | $196,608$ | $2442.2\text{ MB}^\dagger$ |
| **`fwd_streaming_oracle_coupled_qproj_unbiased_fullvocab`** | +`Q-Proj-K` Surrogate ($V=8192$) | **$1$** | **$0$** | **$0$** | $42$ ($1$ clean + $41$ probe) | $2$ ($1$ clean + $1$ local fwd) | $104,448$ | $2100.3\text{ MB}$ |
| **`fwd_lookahead_2b_orth_mv_a1.0_opdiff_qproj_fullvocab`** | +`Lookahead-2B-OrthMV` + `OpDiff` + `Q-Proj-K` ($V=8192$) | **$1$** | **$0$** | **$0$** | $49.5$ ($297$ total) | $2.83$ ($17$ total) | $196,608$ | $2517.3\text{ MB}$ |
| **`fwd_lookahead_2b_orth_mv_cowm_opdiff_qproj_K8`** | +`Lookahead-2B-OrthMV` `Co-WM-S` ($V_{\text{sub}}=256$) | **$1$** | **$0$** | **$0$** | $49.5$ ($297$ total) | $2.83$ ($17$ total) | $360,448$ | $2245.3\text{ MB}$ |

*Table 1: Verified Computation and Pass-Count Ledger for 6-layer Transformer pretraining ($B=16, T=512$, $N_{\text{tok}}=8192$, micro-horizon $S=8$, $V=8192$; $^\dagger$measured on `d6` $D=384$ in Table 5). All candidate forward-only methods enforce `bwd_calls = 0` and `depth_sweeps = 0`. Uncoupled streaming contraction counts reflect $3$ linear sites per block (`lm_head`, `mlp.c_proj`, `attn.c_proj`) $\times 6$ blocks $\times K$ probes $\times (8192/8)$ groups ($147,456$ for $K=8$); coupled streaming adds the 4th parameter-space `Co-WM-S` site on `mlp.c_fc` ($W_1$) for the zero-pass MLP-to-Attention residual pullback ($196,608$ for $K=8$); 2-block lookahead adds $5$ inter-block lookahead evaluations of MLP ($V, W_1$) and Attention-Value ($W_O, W_V$, $9$ FlashAttention calls each).*


---

## 2. Theoretical Analysis and Scaling Laws

### 2.1 Micro-Horizon Parameter-Space Weight Maximization (`WM-S`)
In Chung's Weight Maximization framework [2, Eq. 1–3], hidden unit $i \in \{1, \dots, M\}$ does not possess an activation-space transpose circuit $(V^\top \delta_t)_i$. Instead, each unit $i$ observes only the local plasticity changes $\Delta \mathbf{v}^{(i)}$ of its own outgoing weight column $\mathbf{v}^{(i)} = V_{:, i} \in \mathbb{R}^{D_{\text{out}}}$, reinforcing its activity when its outgoing synaptic column grows in squared $L^2$ norm.

To adapt this principle to sequence modeling without per-token activation transposes ($S = 1$) or full-batch temporal cross-talk ($S = N_{\text{tok}}$), we introduce **Parameter-Space Micro-Horizon Weight Maximization (`WM-S`)**:
1. Partition the $N_{\text{tok}} = B \times T$ tokens into $G = N_{\text{tok}} / S$ disjoint micro-groups $\mathcal{S}_1, \dots, \mathcal{S}_G$ of size $S$ tokens ($S \in \{4, 8, 16, 64\}$).
2. For exploration draw $k \in \{1, \dots, K\}$, sample independent Rademacher perturbations $\xi_{g, s, i}^{(k)} \in \{-\sigma, +\sigma\}$ for each neuron $i \in \{1, \dots, M\}$.
3. Over each micro-group $\mathcal{S}_g$, the outgoing matrix $V \in \mathbb{R}^{D_{\text{out}} \times M}$ accumulates its parameter-space plasticity displacement:
   $$\Delta V_{\mathcal{S}_g}^{(k)} = \sum_{s \in \mathcal{S}_g} \delta_{g, s} \big(\xi_{g, s}^{(k)}\big)^\top \;\in\; \mathbb{R}^{D_{\text{out}} \times M}.$$
4. Neuron $i$ evaluates its parameter-space inner product from its own column $\mathbf{v}^{(i)} = V_{:, i}$ and column update $\Delta \mathbf{v}_{\mathcal{S}_g}^{(i, k)} = (\Delta V_{\mathcal{S}_g}^{(k)})_{:, i}$ [2, Eq. 2]:
   $$G_{\mathcal{S}_g}^{(i, k)} = \mathbf{v}^{(i)} \cdot \Delta \mathbf{v}_{\mathcal{S}_g}^{(i, k)} - \beta \left\|\Delta \mathbf{v}_{\mathcal{S}_g}^{(i, k)}\right\|_2^2 \;\in\; \mathbb{R}.$$
5. Neuron $i$ computes its input credit for token $t \in \mathcal{S}_g$ [2, Eq. 3]:
   $$\hat{g}_{g, t, i}^{\text{WM-S}} = \frac{1}{K} \sum_{k=1}^K G_{\mathcal{S}_g}^{(i, k)} \frac{\xi_{g, t, i}^{(k)}}{\sigma^2}, \qquad \nabla_{W_1} \mathcal{L} = \sum_{g=1}^G \sum_{t \in \mathcal{S}_g} \big(\hat{g}_{g, t}^{\text{WM-S}} \odot \phi'(a_{1, g, t})\big) x_{g, t}^\top.$$

**Proposition 1 (Conditional Coordinate Variance and Sibling Decoupling of `WM-S`)**:
*Conditioned on fixed downstream linear targets $\delta_{g, s}$ independent of $\xi$, with independent Rademacher perturbations $\xi_{g, s, i}^{(k)} \in \{-\sigma, +\sigma\}$ (so $(\xi_{g, t, i}^{(k)})^2 / \sigma^2 \equiv 1$) and $\beta = 0$, expanding $\hat{g}_{g, t, i}^{\text{WM-S}}$ yields:*
$$\hat{g}_{g, t, i}^{\text{WM-S}} = \big(\mathbf{v}^{(i)} \cdot \delta_{g, t}\big) \underbrace{\frac{(\xi_{g, t, i}^{(k)})^2}{\sigma^2}}_{\equiv 1} + \frac{1}{K} \sum_{k=1}^K \sum_{s \in \mathcal{S}_g \setminus \{t\}} \big(\mathbf{v}^{(i)} \cdot \delta_{g, s}\big) \frac{\xi_{g, s, i}^{(k)} \xi_{g, t, i}^{(k)}}{\sigma^2}.$$
*Because $\mathbb{E}[\xi_{g, s, i} \xi_{g, t, i}] = 0$ for $s \ne t$, the estimator is conditionally unbiased ($\mathbb{E}[\hat{g}_{g, t, i}^{\text{WM-S}} \mid \delta] = \mathbf{v}^{(i)} \cdot \delta_{g, t}$) with conditional variance:*
$$\text{Var}\!\left[\hat{g}_{g, t, i}^{\text{WM-S}} \;\middle|\; \delta\right] = \frac{1}{K} \sum_{s \in \mathcal{S}_g \setminus \{t\}} (\mathbf{v}^{(i)} \cdot \delta_{g, s})^2.$$

*Decoupling Mechanism*: In activation-space perturbation (`Cut`), the reward $R_t^{(k)}$ sums over all $M$ neurons at token $t$, yielding conditional variance $\frac{1}{K}\sum_{j \ne i} (\mathbf{v}^{(j)} \cdot \delta_t)^2$ that scales with $M-1$. In `WM-S`, parameter-space column isolation decouples all $M - 1$ sibling neurons $j \ne i$, replacing the sum over $M - 1$ neurons with a sum over $S - 1$ tokens in $\mathcal{S}_g$, which is conditionally independent of width $M$ for fixed per-unit targets.

### 2.2 Sylvester-Hadamard Temporal Coordination (`Co-WM-S`)
Because `WM-S` eliminates spatial sibling-neuron cross-talk under fixed $\delta$, the remaining conditional variance arises from the $S - 1$ cross-token terms within micro-horizon $\mathcal{S}_g$. In **`Co-WM-S`**, we adapt orthogonal Sylvester-Hadamard exploration [11] to token micro-horizons in sequence Transformers, structuring the $K \le S$ exploration probes across the $S$ tokens of micro-group $\mathcal{S}_g$ using $K$ distinct columns $\{c_1, \dots, c_K\}$ drawn uniformly without replacement from an $S \times S$ Sylvester-Hadamard matrix $H \in \{-1, +1\}^{S \times S}$ ($S$ a power of 2), modulated by an independent Rademacher carrier $r_{g, s, i} \in \{-1, +1\}$:
$$\xi_{g, s, i}^{(k)} = \sigma \, H_{s, c_k} \, r_{g, s, i}.$$

**Proposition 2 (Finite-Population Conditional Variance of `Co-WM-S`)**:
*Conditioned on fixed downstream targets $\delta_{g, s}$ and writing $c_s = \mathbf{v}^{(i)} \cdot \delta_{g, s}$, uniform selection of $K \in \{1, \dots, S\}$ distinct Hadamard columns without replacement at $\beta = 0$ yields $\mathbb{E}[\hat{g}_{g, t, i}^{\text{Co-WM-S}} \mid \delta] = c_t$ and exact conditional variance:*
$$\text{Var}\!\left[\hat{g}_{g, t, i}^{\text{Co-WM-S}} \;\middle|\; \delta\right] = \frac{S - K}{K (S - 1)} \sum_{s \in \mathcal{S}_g \setminus \{t\}} c_s^2.$$
*Proof*: For distinct tokens $s \ne t$, $p_{s, t}(c) = H_{s, c} H_{t, c}$ has population mean $\frac{1}{S}\sum_{c=1}^S H_{s, c} H_{t, c} = 0$ by Hadamard orthogonality and unit variance $p_{s, t}(c)^2 \equiv 1$. Sampling $K$ distinct columns without replacement yields the classical hypergeometric variance reduction factor $\frac{S - K}{K(S - 1)}$. At $K = S$, the conditional variance vanishes **identically ($\text{Var} \equiv 0$)** for every width $M$.

### 2.3 Exact Covariance Identity, Sufficient Conditions, and Multi-Checkpoint Trajectory Audit
For row $i$ of the incoming weight matrix $W_1$, let $c_{g, s, i} = \mathbf{v}^{(i)} \cdot \delta_{g, s} \in \mathbb{R}$ and $z_{g, t, i} = \phi'(a_{1, g, t, i}) x_{g, t} \in \mathbb{R}^D$, so the exact row gradient is $g_i = \sum_{g=1}^G \sum_{t \in \mathcal{S}_g} c_{g, t, i} z_{g, t, i} = \sum_{g=1}^G g_{g, i}$.

**Proposition 3 (Exact Conditional Frobenius Weight-Gradient Covariance and Relative Variance Law)**:
*Under `WM-S` exploration with $K$ independent Rademacher draws $\xi_{g, s, i}^{(k)} \in \{-\sigma, +\sigma\}$ and $\beta = 0$, the estimator error decomposes over unordered token pairs $\{s, t\} \subset \mathcal{S}_g$ ($s < t$):*
$$\widehat{\nabla}_{W_{1, i}} - g_i = \frac{1}{K} \sum_{k=1}^K \sum_{g=1}^G \sum_{1 \le s < t \le S} \big(c_{g, s, i} z_{g, t, i} + c_{g, t, i} z_{g, s, i}\big) \frac{\xi_{g, s, i}^{(k)} \xi_{g, t, i}^{(k)}}{\sigma^2}.$$
*Because distinct pairs $\{s, t\} \ne \{s', t'\}$ are pairwise uncorrelated, the exact weight-gradient covariance is:*
$$\text{Cov}\!\left(\widehat{\nabla}_{W_{1, i}} \;\middle|\; c, z\right) = \frac{1}{K} \sum_{g=1}^G \sum_{1 \le s < t \le S} \big(c_{g, s, i} z_{g, t, i} + c_{g, t, i} z_{g, s, i}\big) \big(c_{g, s, i} z_{g, t, i} + c_{g, t, i} z_{g, s, i}\big)^\top,$$
*and the exact conditional Frobenius variance summed across all $M$ neurons is:*
$$\mathbb{E}\!\left[\left\|\widehat{\nabla}_{W_1} - \nabla_{W_1}\right\|_F^2 \;\middle|\; c, z\right] = \kappa(S, K) \sum_{i=1}^M \sum_{g=1}^G \sum_{1 \le s < t \le S} \left\|c_{g, s, i} z_{g, t, i} + c_{g, t, i} z_{g, s, i}\right\|_2^2,$$
*where $\kappa(S, K) = 1/K$ for `WM-S` and $\kappa(S, K) = \frac{S-K}{K(S-1)}$ for `Co-WM-S`.*

**Algebraic Cross-Token Identity**: By expanding the pair norm squared:
$$\sum_{1 \le s < t \le S} \left\|c_s z_t + c_t z_s\right\|_2^2 = \left(\sum_{s=1}^S c_s^2\right) \left(\sum_{t=1}^S \|z_t\|_2^2\right) + \left\|g_{\mathcal{S}_g}\right\|_2^2 - 2 \sum_{t=1}^S c_t^2 \|z_t\|_2^2.$$
Defining the **Within-Group Support Alignment Ratio** $\alpha(c, z)$, the **Global Gradient Coherence Ratio** $\gamma(c, z)$, and the **Group Gradient Coherence Ratio** $\gamma_{\text{grp}}(c, z)$:
$$\alpha(c, z) \triangleq \frac{\sum_{g, i} \left(\sum_{s=1}^S c_{g, s, i}^2\right) \left(\sum_{t=1}^S \|z_{g, t, i}\|_2^2\right)}{S \sum_{g, t, i} c_{g, t, i}^2 \|z_{g, t, i}\|_2^2}, \qquad \gamma(c, z) \triangleq \frac{\sum_i \|g_i\|_2^2}{\sum_{g, t, i} c_{g, t, i}^2 \|z_{g, t, i}\|_2^2}, \qquad \gamma_{\text{grp}}(c, z) \triangleq \frac{\sum_{g, i} \|g_{g, i}\|_2^2}{\sum_i \|g_i\|_2^2},$$
the exact conditional relative parameter-gradient variance is governed by the identity:
$$\text{RelVar}(\text{WM-S} \mid c, z) \triangleq \frac{\mathbb{E}\!\left[\left\|\widehat{\nabla}_{W_1} - \nabla_{W_1}\right\|_F^2 \;\middle|\; c, z\right]}{\left\|\nabla_{W_1}\right\|_F^2} = \frac{1}{K} \left[ \frac{\alpha(c, z) S - 2}{\gamma(c, z)} + \gamma_{\text{grp}}(c, z) \right].$$

**Analysis of Failure Modes and Counterexamples**:
Because $\alpha(c, z)$, $\gamma(c, z)$, and $\gamma_{\text{grp}}(c, z)$ depend on the empirical distribution of $c$ and $z$, Proposition 3 implies **conditional** width-invariance at fixed $(\alpha, \gamma, \gamma_{\text{grp}})$ rather than an unconditional guarantee across arbitrary network states:
1. *Gradient Cancellation Counterexample*: If token gradients cancel across the sequence ($\gamma \to 0$), relative variance diverges. For $S=8, K=1, D=1, M=4$, inputs $x = (1, 2, 0 \dots)$, targets $\delta = (4, -1+\varepsilon, 0 \dots)$, $z = (2, 8, 0 \dots)$, $c = (4, -1+\varepsilon, 0 \dots)$, the exact gradient is $g = 8\varepsilon$ and cross-term $c_1 z_2 + c_2 z_1 = 30 + 2\varepsilon$. The exact conditional relative variance is $\frac{(30+2\varepsilon)^2}{64\varepsilon^2}$, which equals $140{,}812.5625$ at $\varepsilon = 0.01$ (verified by exhaustive enumeration over all 256 Rademacher states).
2. *Cross-Token Support Mismatch Counterexample*: Non-cancellation ($\gamma > 0$) alone is insufficient if $c$ and $z$ have mismatched temporal support within a micro-group ($\alpha \to \infty$). For $c = (1, \varepsilon, 0 \dots)$ and $z = (\varepsilon, 1, 0 \dots)$, both nonzero token gradients equal $\varepsilon$ (reinforcing, $g = 2\varepsilon, \sum c_t^2 z_t^2 = 2\varepsilon^2$, so $\gamma = 2.0$), but $c_1 z_2 = 1 \gg \varepsilon^2$, giving $\alpha \approx \frac{1}{16\varepsilon^2}$. The exact conditional relative variance is $\frac{(1+\varepsilon^2)^2}{4\varepsilon^2} = 2500.500025$ at $\varepsilon = 0.01$ for $K=1$, $357.214289$ for $K=4$ `Co-WM-S`, and $0.0$ for $K=8$ `Co-WM-S`.

**Empirical Trajectory Audit of Conditional Quantities (`verify_learned_prop3_scaling.py`)**:
To bridge Proposition 3 to actual training behavior with full reproducibility from an empty output file (`python beyond_backprop/verify_learned_prop3_scaling.py --regenerate-all`, saving `multiwidth_learned_prop3_diagnostic.json`), we calculate the **conditional** quantities $\alpha(c, z), \gamma(c, z), \gamma_{\text{grp}}(c, z)$, and $\text{RelVar}(\cdot \mid c, z)$ from frozen Layer 3 checkpoint tensors ($S=8$) on batches of shape **$B=16, T=512$ ($N_{\text{tok}}=8,192$ tokens/batch)** across widths $D \in \{128, 256, 384, 512\}$ under three explicitly documented protocols:
1. **Benchmark-Schedule Holdout Validation Trajectory (`benchmark_holdout_trajectory_4batch`)**:
   - *Training Recipe*: 200 steps of `fwd_streaming_cowm` ($K_{\text{wm}}=4, K_{\text{qk}}=2, S=8$) on `train_batches[0:200]` drawn from `shard_00000.parquet` ($B=16, T=512$) using the **exact benchmark optimizer schedule** from `run_pure_forward_only_suite.py` (linear LR warmdown over the final 45% of steps, Muon momentum warmup $0.85 \to 0.95$ over 200 steps, and linear Muon weight decay decay $0.2(1 - \text{progress})$).
   - *Evaluation Split*: Evaluated at 5 checkpoints ($t \in \{0, 50, 100, 150, 200\}$) across **4 disjoint holdout validation batches** (`holdout_val_batches[0:4]`, shape $B=16, T=512, N_{\text{tok}}=8,192$) drawn from `shard_06542.parquet` (row groups $40 \dots 82$).
   - *Ranges of 4-Batch Means*: Across all $4 \times 5 = 20$ width-checkpoint states, the 4-batch means span $\alpha \in [0.8572 \pm 0.0063, \, 1.0033 \pm 0.0012]$, $\gamma \in [2.3604 \pm 0.5815, \, 8.1838 \pm 5.2378]$, and $\gamma_{\text{grp}} \in [0.1779 \pm 0.1155, \, 0.4410 \pm 0.0956]$. Consequently, the 4-batch mean conditional relative variances stay bounded in $[1.0448 \pm 0.6991, \, 2.7498 \pm 0.6188]$ for `WM-S` ($K=1$) and $[0.1492 \pm 0.0999, \, 0.3929 \pm 0.0884]$ for `Co-WM-S` ($K=4$). At $t=200$ (4-batch mean $\pm$ SD):
     - $D=128$: $\alpha = 0.9439 \pm 0.0154$, $\gamma = 3.3446 \pm 1.9805$, `WM-S` ($K=1$) $= 2.5773 \pm 1.4312$, `Co-WM-S` ($K=4$) $= 0.3682 \pm 0.2045$, `Cut-K4` $= 33.44 \pm 18.69$;
     - $D=256$: $\alpha = 0.9414 \pm 0.0070$, $\gamma = 3.5498 \pm 2.1871$, `WM-S` ($K=1$) $= 2.5365 \pm 1.5310$, `Co-WM-S` ($K=4$) $= 0.3624 \pm 0.2187$, `Cut-K4` $= 51.37 \pm 31.06$;
     - $D=384$: $\alpha = 0.9320 \pm 0.0053$, $\gamma = 4.0057 \pm 2.7904$, `WM-S` ($K=1$) $= 2.5661 \pm 1.8502$, `Co-WM-S` ($K=4$) $= 0.3666 \pm 0.2643$, `Cut-K4` $= 78.25 \pm 56.20$;
     - $D=512$: $\alpha = 0.9902 \pm 0.0105$, $\gamma = 2.8972 \pm 0.8824$, `WM-S` ($K=1$) $= 2.5779 \pm 0.8397$, `Co-WM-S` ($K=4$) $= 0.3683 \pm 0.1199$, `Cut-K4` $= 123.91 \pm 39.04$.
2. **Fixed-Optimizer Diagnostic Trajectory (`streaming_trajectory_4batch`)**:
   - *Training & Evaluation Recipe*: 200 steps of `fwd_streaming_cowm` ($K_{\text{wm}}=4, K_{\text{qk}}=2, S=8$) on `train_batches[0:200]` (`shard_00000.parquet`, $B=16, T=512$) with fixed default Muon/AdamW optimizer parameters (`setup_optimizer(use_muon=True)` without LR/momentum/WD scheduling), evaluated at $t \in \{0, 50, 100, 150, 200\}$ on 4 held-out training-stream batches `train_batches[200:204]` ($B=16, T=512, N_{\text{tok}}=8,192$). Across all 20 states, the 4-batch means span $\alpha \in [0.9062 \pm 0.0079, \, 1.0162 \pm 0.0080]$, $\gamma \in [2.2698 \pm 0.3875, \, 5.6916 \pm 0.0958]$, conditional `WM-S` ($K=1$) $\text{RelVar} \in [1.2203, \, 2.8887]$, and conditional `Co-WM-S` ($K=4$) $\text{RelVar} \in [0.1743, \, 0.4127]$.
3. **Single-Batch Table 2 States (`step_0_init`, `step_200_learned`, `step_200_streaming_learned`)**:
   - Calculated from frozen tensors under fixed default Muon/AdamW parameters on `shard_00000.parquet` ($B=16, T=512$): `step_0_init` on `train_batches[0]`, `step_200_learned` on `train_batches[199]` after 200 steps of Cut-scope `detached_oracle`, and `step_200_streaming_learned` on `train_batches[199]` after 200 steps of `fwd_streaming_cowm` ($K=4$). To isolate the local MLP estimation variance of Proposition 3 around a deterministic surrogate target on genuine learned network states, `evaluate_layer3_streaming_prop3` conditions on the exact deterministic `detached_oracle` readout pullback at that step, decoupling internal parameter-contraction variance from downstream attention/readout sampling noise.

| Width $D$ ($M=4D$) | State Regime | $\alpha$ (Support Alignment) | $\gamma$ (Global Coherence) | $\gamma_{\text{grp}}$ (Group Coherence) | `Cut-K4` Cond. RelVar | `WM-S` ($K=1$) Cond. RelVar | `Co-WM-S` ($K=4$) Cond. RelVar | `Co-WM-S` ($K=8$) Cond. RelVar | `GS-MAP-N1` RelBias | `GS-MAP-N2` RelBias |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $D=128$ ($M=512$) | `step_0_init` | $0.9990$ | $3.4755$ | $0.2885$ | $37.05$ | $2.0127$ | $0.2875$ | **$0.0000$** | $0.00017$ | $0.05833$ |
| $D=128$ ($M=512$) | `step_200_learned` | $1.0250$ | $1.4638$ | $0.6859$ | $55.34$ | $4.9214$ | $0.7031$ | **$0.0000$** | $0.00022$ | $0.16049$ |
| $D=128$ ($M=512$) | `step_200_streaming` | $0.9746$ | $2.1800$ | $0.4589$ | $41.17$ | $3.1179$ | $0.4454$ | **$0.0000$** | $0.00020$ | $0.16297$ |
| $D=256$ ($M=1024$) | `step_0_init` | $1.0013$ | $3.2002$ | $0.3134$ | $79.86$ | $2.1915$ | $0.3131$ | **$0.0000$** | $0.00019$ | $0.05574$ |
| $D=256$ ($M=1024$) | `step_200_learned` | $0.9275$ | $1.5571$ | $0.6404$ | $103.94$ | $4.1213$ | $0.5888$ | **$0.0000$** | $0.00022$ | $0.13396$ |
| $D=256$ ($M=1024$) | `step_200_streaming` | $0.9752$ | $2.0925$ | $0.4753$ | $66.99$ | $3.2478$ | $0.4640$ | **$0.0000$** | $0.00018$ | $0.14256$ |
| $D=384$ ($M=1536$) | `step_0_init` | $1.0003$ | $3.6960$ | $0.2714$ | $103.77$ | $1.8953$ | $0.2708$ | **$0.0000$** | $0.00017$ | $0.05414$ |
| $D=384$ ($M=1536$) | `step_200_learned` | $1.0329$ | $2.0612$ | $0.4860$ | $146.12$ | $3.5248$ | $0.5035$ | **$0.0000$** | $0.00020$ | $0.12289$ |
| $D=384$ ($M=1536$) | `step_200_streaming` | $0.9730$ | $1.9493$ | $0.5142$ | $123.91$ | $3.4815$ | $0.4974$ | **$0.0000$** | $0.00019$ | $0.12925$ |
| $D=512$ ($M=2048$) | `step_0_init` | $1.0002$ | $3.3703$ | $0.2976$ | $151.85$ | $2.0783$ | $0.2969$ | **$0.0000$** | $0.00019$ | $0.05394$ |
| $D=512$ ($M=2048$) | `step_200_learned` | $0.9729$ | $2.1721$ | $0.4605$ | $152.76$ | $3.1231$ | $0.4462$ | **$0.0000$** | $0.00018$ | $0.13607$ |
| $D=512$ ($M=2048$) | `step_200_streaming` | $0.9951$ | $2.4223$ | $0.4139$ | $150.28$ | $2.8746$ | $0.4107$ | **$0.0000$** | $0.00018$ | $0.12879$ |

*Table 2: Conditional Proposition 3 quantities calculated from frozen Layer 3 tensors on ClimbMix ($B=16, T=512$, `multiwidth_learned_prop3_diagnostic.json`). While activation-space `Cut-K4` conditional relative variance grows with width ($41.17 \to 150.28$ along the streaming trajectory), conditional `WM-S` ($K=1$) and `Co-WM-S` ($K=4, 8$) relative variances remain bounded across tested widths ($2.87\text{--}3.48$ and $0.41\text{--}0.50$, respectively, and $0.00$ at $K=8$).*

### 2.4 Whole-Network Probing: Sequence-Aware Variance, Sensitivity Thresholds, and Ablations
Propositions 1–3 establish conditional sibling-neuron decoupling given a fixed local output discrepancy $c$. Eliminating conditional local variance does **not** make the whole-network gradient unbiased with respect to end-to-end backpropagation. In a causal sequence Transformer, the downstream map $F^{(m)}: \mathbb{R}^{T \times D} \to \mathbb{R}^{T \times D}$ from injection site $m \in \{0, \dots, 2L\}$ ($S_{\text{sites}} = 2L + 1 = 13$ for $L=6$) to the final normalized hidden state has a block-lower-triangular sequence-channel Jacobian $J^{(m)} \in \mathbb{R}^{(TD) \times (TD)}$, with blocks $J_{u, v}^{(m)} \in \mathbb{R}^{D \times D}$ non-zero for all causal token pairs $u \ge v$ and zero for $u < v$. Given top sensitivity $c = (c_1, \dots, c_T) \in \mathbb{R}^{T \times D}$, the exact transpose sensitivity of site $m$ at token $t$ is $u_t^{(m)} = \sum_{u=t}^T (J_{u, t}^{(m)})^\top c_u \in \mathbb{R}^D$.

**Sequence-Aware Within-Site vs. Between-Site Variance of Depth-Hadamard vs. Independently Resampled Probing**:
Let $H_{\text{depth}} = 16$ be the smallest power of $2$ at least $S_{\text{sites}} = 13$, and let $\kappa(H_{\text{depth}}, K_{\text{net}}) = \frac{H_{\text{depth}} - K_{\text{net}}}{K_{\text{net}}(H_{\text{depth}} - 1)}$ ($\kappa = \frac{1}{15}$ at $H_{\text{depth}} = 16, K_{\text{net}} = 8$). Under **Depth-Hadamard Carrier Modulation (`depth_hadamard`)**, site $m$ draws one Rademacher tensor $r^{(m)} \in \{-1, +1\}^{T \times D}$ per $H_{\text{depth}}$-probe cycle and injects $\xi^{(m, k)} = \sigma h_{m, c_k} r^{(m)}$. Because $h_{m, c_k}^2 \equiv 1$, the within-site contribution ($m' = m$) is invariant to $K_{\text{net}} \le H_{\text{depth}}$ within a cycle, whereas the between-site contribution ($m' \ne m$) is weighted by $\kappa(H_{\text{depth}}, K_{\text{net}})$. Under **Independently Resampled Carriers (`iid_resample`)**, fresh i.i.d. tensors $r^{(m, k)}$ are drawn at every probe $k \in \{1, \dots, K_{\text{net}}\}$, dividing both within-site and between-site variances by $1 / K_{\text{net}}$.

For each return-aggregation mode, the linear-limit expectation and exact sequence-aware activation-credit variance $\operatorname{Var}(\hat{g}^{(m)}) = A_{\text{mode}}^{(m)} + \kappa_{\text{mode}} B_{\text{mode}}^{(m)}$ ($\kappa_{\text{mode}} = \kappa(H_{\text{depth}}, K_{\text{net}})$ for `depth_hadamard`, or dividing $A_{\text{mode}}^{(m)} + B_{\text{mode}}^{(m)}$ by $K_{\text{net}}$ for `iid_resample`) are:
1. **Full-Sequence Scalar-Dot Probing (`full_seq_dot`, $n = TD$)**:
   The scalar return sums over all tokens and channels: $R_{\text{seq}}^{(k)} = \sum_{m'=0}^{2L} \sum_{u=1}^T \langle u_u^{(m')}, \xi_u^{(m', k)} \rangle$.
   - *Linear-Limit Expectation*: Unbiased for the full causal transpose, $\mathbb{E}[\hat{g}_{\text{full}, t}^{(m)}] = u_t^{(m)} = \sum_{u=t}^T (J_{u, t}^{(m)})^\top c_u$.
   - *Exact Sequence-Wide Variance*: Writing $\|u^{(m)}\|_2^2 = \sum_{t=1}^T \|u_t^{(m)}\|_2^2$, the single-probe within-site and between-site variance energies summed over all $n = TD$ coordinates are:
     $$A_{\text{full}}^{(m)} = (TD - 1) \left\|u^{(m)}\right\|_2^2, \qquad B_{\text{full}}^{(m)} = TD \sum_{m' \ne m} \left\|u^{(m')}\right\|_2^2.$$
2. **Token-Local Channel-Dot Probing (`token_dot`)**:
   At token $t$, the token-local return sums only over the $D$ channels at position $t$, but because the perturbation $\xi^{(m', k)}$ is injected across the entire sequence, $(J^{(m')} \xi^{(m', k)})_t$ depends on perturbations at **all earlier tokens $v \le t$**:
   $$R_t^{(k)} = c_t^\top (\Delta x_{\text{final}}^{(k)})_t = \sum_{m'=0}^{2L} \sum_{v=1}^t \big\langle a_{t, v}^{(m')}, \xi_v^{(m', k)} \big\rangle, \qquad \text{where } a_{t, v}^{(m')} \triangleq \big(J_{t, v}^{(m')}\big)^\top c_t \in \mathbb{R}^D.$$
   - *Linear-Limit Expectation*: Multiplying by $\xi_t^{(m, k)} / \sigma^2$ eliminates $v < t$ in expectation ($\mathbb{E}[\xi_t \xi_v^\top] = 0$ for $v < t$), leaving only the diagonal token block and **truncating all future-token causal blocks $u > t$**:
     $$\mathbb{E}\!\left[\hat{g}_{\text{tok}, t}^{(m)}\right] = a_{t, t}^{(m)} = \big(J_{t, t}^{(m)}\big)^\top c_t \;\ne\; \sum_{u=t}^T \big(J_{u, t}^{(m)}\big)^\top c_u.$$
   - *Exact Sequence-Aware Token-Dot Variance (Including Earlier-Token Causal Cross-Talk)*: Multiplication by $\xi_t^{(m, k)}$ does **not** remove the variance of the earlier-token terms $v < t$! Under `depth_hadamard`, $\xi_t^{(m, k)} \xi_v^{(m, k)\top} / \sigma^2 = r_t^{(m)} r_v^{(m)\top}$ for all $v \le t$, so the single-probe within-site error energy $A_{\text{tok}, t}^{(m)}$ and between-site error energy $B_{\text{tok}, t}^{(m)}$ about the token-local mean $a_{t, t}^{(m)}$ at token $t$ are:
     $$A_{\text{tok}, t}^{(m)} = (D - 1) \left\|a_{t, t}^{(m)}\right\|_2^2 + D \sum_{v < t} \left\|a_{t, v}^{(m)}\right\|_2^2, \qquad B_{\text{tok}, t}^{(m)} = D \sum_{m' \ne m} \sum_{v \le t} \left\|a_{t, v}^{(m')}\right\|_2^2.$$
     Notice the second within-site term $D \sum_{v < t} \|a_{t, v}^{(m)}\|_2^2$: even at $D = 1$, where same-token channel cross-talk $(D-1)\|a_{t, t}^{(m)}\|_2^2 = 0$, earlier-token causal coupling ($v < t$) generates non-zero within-site variance that persists unattenuated across a fixed-carrier `depth_hadamard` cycle and divides by $1 / K_{\text{net}}$ under `iid_resample`!
   - *Explicit Causal Counterexample (Verified in Gate 2H)*: For $T=2, D=1$ (single site) with $J = \begin{pmatrix} 1 & 0 \\ 3 & 4 \end{pmatrix}$ and $c = \begin{pmatrix} 5 \\ 7 \end{pmatrix}$, at token $t = 2$ we have $a_{2, 2} = 28$ and $a_{2, 1} = 21$. The second-token `token_dot` estimator is $r_2(21 r_1 + 28 r_2) = 28 + 21 r_1 r_2$. Its mean is $28$ (omitting future-token credit on token $1$, where $\mathbb{E}[\hat{g}_{\text{tok}, 1}] = 5 \ne 26$), and its exact variance about $28$ is $(1 - 1)(28^2) + 1(21^2) = 441.0$—which equals $441.0$ at every $K_{\text{net}} \in \{1, 2, 4\}$ under fixed-carrier `depth_hadamard`, and $441.0, 220.5, 110.25$ ($= 441.0 / K_{\text{net}}$) at $K_{\text{net}} = 1, 2, 4$ under `iid_resample` (verified in Gate 2H).
3. **Coordinate-Wise Diagonal Probing (`diag_jacobian`)**:
   Evaluates $\xi_{t, d}^{(m, k)} c_{t, d} (\Delta x_{\text{final}}^{(k)})_{t, d} / \sigma^2$, with expectation $\mathbb{E}[\hat{g}_{\text{diag}, t, d}^{(m)}] = c_{t, d} J_{(t, d), (t, d)}^{(m)}$ (verified in Gate 2G: for $J = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, c = \begin{pmatrix} 5 \\ 7 \end{pmatrix}$, $\mathbb{E}_{\text{diag}} = \begin{pmatrix} 5 \\ 28 \end{pmatrix} \ne J^\top c = \begin{pmatrix} 26 \\ 38 \end{pmatrix}$) and single-probe within-site and between-site variance energies:
   $$A_{\text{diag}}^{(m)} = \sum_{t, d} c_{t, d}^2 \left( \sum_{d' \ne d} \big(J_{(t, d), (t, d')}^{(m)}\big)^2 + \sum_{v < t} \sum_{d'=1}^D \big(J_{(t, d), (v, d')}^{(m)}\big)^2 \right), \quad B_{\text{diag}}^{(m)} = \sum_{m' \ne m} \sum_{t, d} c_{t, d}^2 \sum_{v \le t} \sum_{d'=1}^D \big(J_{(t, d), (v, d')}^{(m')}\big)^2.$$

**Sensitivity-Energy Condition for Depth-Hadamard vs. Independent Resampling ($B^{(m)} > 15 A^{(m)}$)**:
At $H_{\text{depth}} = 16, K_{\text{net}} = 8$, comparing $\operatorname{Var}_{\text{DH}} = A^{(m)} + \frac{1}{15} B^{(m)}$ against $\operatorname{Var}_{\text{iid}} = \frac{1}{8}(A^{(m)} + B^{(m)})$ yields the exact necessary and sufficient condition:
$$A^{(m)} + \frac{1}{15} B^{(m)} < \frac{1}{8}\big(A^{(m)} + B^{(m)}\big) \quad \Longleftrightarrow \quad B^{(m)} > 15 A^{(m)}.$$
Having $S_{\text{sites}} = 13$ sites does **not** by itself guarantee $B^{(m)} > 15 A^{(m)}$: if all 13 sites had equal sensitivity energy $\|u^{(m)}\|_2^2 = 1$, then $B_{\text{full}}^{(m)} / A_{\text{full}}^{(m)} = \frac{12 n}{n - 1} < 15$ for all $n \ge 6$ (for example, at $n = 512$ and unit site energy, $A = 511$ and $B = 6144$, so $\operatorname{Var}_{\text{DH}} = 920.6 > \operatorname{Var}_{\text{iid}} = 831.875$, making independent resampling better).
To test whether $B^{(m)} > 15 A^{(m)}$ holds on our initialized 6-layer Transformer (`c_proj_init_scale=0.5`, where `x0` and attention pathways carry larger sensitivity energy than individual MLP projections), `run_pure_forward_only_suite.py --mode diag` (`pure_forward_width_diagnostic.json`, `site_sensitivity_audit`) measures the exact 13-site sensitivity energies $E_m = \|\frac{\partial \mathcal{L}}{\partial s^{(m)}}\|_F^2$:
- Across widths $D \in \{128, 256, 384, 512\}$, the measured ratio $B_{\text{full}}^{(m)} / A_{\text{full}}^{(m)}$ is **$20.08, 19.88, 20.05, 20.30$** at Layer 3 MLP ($m=8$) and **$26.40, 26.15, 25.67, 26.31$** at Layer 5 MLP ($m=12$). This ratio specifically governs the whole-sequence return in `full_seq_dot`, where $B_{\text{full}}^{(m)} > 15 A_{\text{full}}^{(m)}$ holds due to the larger sensitivity energy carried by `x0` and attention pathways relative to individual MLP projections; `token_dot` and `diag_jacobian` have their own distinct within-site and between-site propagation variance terms $A_{\text{tok}}, B_{\text{tok}}, A_{\text{diag}}, B_{\text{diag}}$ derived above. Because $B_{\text{full}}^{(m)} / A_{\text{full}}^{(m)} > 15$ at both MLP sites, the predicted DH-to-i.i.d. activation-credit variance ratio is $\frac{1 + 20.30/15}{(1 + 20.30)/8} = 0.8839 < 1$ at Layer 3 ($0.8068$ at Layer 5).
- In Table 3 ($D=512$), we report both **activation-credit relative variance** ($\operatorname{RelVar}(\widehat{\delta u}_{\text{mlp}}^{(3)})$) and **accumulated parameter-gradient relative variance** ($\operatorname{RelVar}(\widehat{\nabla}_{W_1}^{(3)})$): for `full_seq_dot`, `depth_hadamard` vs `iid_resample` gives $79,218.20$ vs $91,443.37$ on activation credit (ratio $0.866 \approx 0.884$) and $78,606.12$ vs $90,830.04$ on $\widehat{\nabla}_{W_1}^{(3)}$; for `token_dot`, $1191.29$ vs $1398.44$ on activation credit and $1180.58$ vs $1386.78$ on $\widehat{\nabla}_{W_1}^{(3)}$; and for `diag_jacobian`, $1.56$ vs $2.60$ on activation credit and $1.55$ vs $2.58$ on $\widehat{\nabla}_{W_1}^{(3)}$. Thus, while carrier reuse leaves within-site variance $A^{(m)}$ unaveraged within a 16-probe cycle, both `depth_hadamard` and `iid_resample` exhibit the same 4-order-of-magnitude variance separation across `diag_jacobian` ($\sim 1.6\text{--}2.6$), `token_dot` ($\sim 1.2\text{--}1.4 \times 10^3$), and `full_seq_dot` ($\sim 7.9\text{--}9.1 \times 10^4$).

**Intra-Block Coupled Forward-Only Credit Assignment (`Co-WM-S`, `CM-Attn-WH`, `Hadamard-QK`)**:
In our baseline uncoupled forward-only streaming surrogate (`fwd_streaming_oracle_fullvocab`), the block-local error $\delta_{\text{block}} = \frac{\partial \mathcal{L}(x_l)}{\partial x_l}$ was fed into both MLP and Attention sublayers independently, omitting the downstream MLP pullback into Attention: in exact block-local backpropagation, $x_l = x_{\text{after\_attn}} + \text{MLP}(\text{norm}(x_{\text{after\_attn}}))$, so the true sensitivity of attention output is $\delta u_{\text{attn}} = \delta_{\text{block}} + J_{\text{mlp}\circ\text{norm}}^\top \delta_{\text{block}}$.
To close this gap strictly without a global backward sweep or autograd calls (`@torch.no_grad()`, `bwd_calls = 0`, fixed pass count independent of depth $L$ and width $M$), we introduce and ablate three coupled forward operators:
1. *Upstream MLP Input Credit via Parameter-Space `Co-WM-S` (`+MLP-Pullback`)*: Evaluating the MLP update first inside block $l$, let $\delta a_{1, g, s} = \hat{g}_{g, s}^{\text{Co-WM-S}} \odot \phi'(a_{1, g, s}) \in \mathbb{R}^M$ be the pre-activation discrepancy on $W_1 \in \mathbb{R}^{M \times D}$. For each input channel $d \in \{1, \dots, D\}$ with column $\mathbf{w}_1^{(d)} = (W_1)_{:, d} \in \mathbb{R}^M$ and micro-group $\mathcal{S}_g$ ($S=8$), probe $k \in \{1, \dots, K\}$ injects $\xi_{g, s, d}^{(k)} = \sigma H_{s, c_k} r_{g, s, d}$, forms the probe-specific column displacement $\Delta \mathbf{w}_{1, \mathcal{S}_g}^{(d, k)} = \sum_{s \in \mathcal{S}_g} \delta a_{1, g, s} \xi_{g, s, d}^{(k)} \in \mathbb{R}^M$, contracts $G_{\mathcal{S}_g}^{(d, k)} = \mathbf{w}_1^{(d)} \cdot \Delta \mathbf{w}_{1, \mathcal{S}_g}^{(d, k)}$ (`compute_linear_non_backprop_input_credit`), and reconstructs:
   $$\widehat{\delta x}_{\text{mlp\_in}, g, t, d} = \frac{1}{K}\sum_{k=1}^K G_{\mathcal{S}_g}^{(d, k)} \frac{\xi_{g, t, d}^{(k)}}{\sigma^2}$$
   without transposing $W_1$ in activation space or executing extra forward passes (exact at $K=S=8$, Gate 2J). Pulling $\widehat{\delta x}_{\text{mlp\_in}}$ across `norm(x)` ($\text{RMS}(x) = \sqrt{\frac{1}{D}\sum_{d=1}^D x_d^2 + \varepsilon}$, $\bar{x} = x / \text{RMS}(x)$) via `_forward_rms_norm_directional_pull` adds:
   $$\Delta_{\text{mlp}\to\text{attn}} = \frac{1}{\text{RMS}(x)} \left(I - \frac{1}{D}\bar{x}\bar{x}^\top\right) \widehat{\delta x}_{\text{mlp\_in}}$$
   to $\delta_{\text{block}}$. In our retained $T=512$ ($n_{\text{win}}=8$) diagnostic on `d6` (`t512_intra_block_ablation` in `pure_forward_width_diagnostic.json`), adding only this MLP pullback (`fwd_streaming_oracle_mlp_pull_only_fullvocab`) raises Layer 3 $W_O$ block-local cosine similarity from $0.93360$ to **$1.00000$** ($0.93314 \to 1.00000$ at $T=64$).
2. *Two-Level Window-Hadamard Forward Value Transport (`CM-Attn-WH`)*: At the pretraining sequence length $T=512$ ($n_{\text{win}}=8$ windows of $S_{\text{win}}=64$ tokens), single-window value probing ($K_v=1$) truncates cross-window causal value transport ($w_q > w_k$), causing Layer 3 $W_V$ block-local cosine to drop to $0.46853$ ($0.48951$ after MLP pullback). Multiplexing $S_{\text{win}}=64$ tokens per window via Sylvester-Hadamard matrix $H_{64}$ and $n_{\text{win}}=8$ windows via $H_8$ with causal window suffix summation over $w_q \ge w_k$ ($K_v = n_{\text{win}} = 8$, verified in Gate 2J) in `fwd_streaming_oracle_mlp_whv_fullvocab` restores $T=512$ Layer 3 $W_V$ block-local cosine from $0.48951$ to **$1.00000$**.
3. *Post-RoPE Head-Channel Sylvester-Hadamard Query/Key Probing (`Hadamard-QK`)*: Constructing perturbations on post-RoPE normalized $Q/K$ using $H_{d_h}$ channel coordination ($d_h=64, K_{\text{qk}}=8$, $\sigma_{\text{qk}}=0.05$) and accumulating rewards in `float32` (while converting perturbed $Q, K$ to `work_dtype` = `bfloat16` inside CUDA FlashAttention) with fused causal future-query lookahead ($L_k=16, \rho=0.85$) raises $T=512$ Layer 3/5 $W_Q$ block-local cosine from $0.14129 / 0.13918$ (`rel_var = 56.12`) to **$0.35023 / 0.35241$ (`rel_var = 6.94`)** in `fwd_streaming_oracle_coupled_fullvocab` (and **$0.34973 / 0.35366$** in full-vocabulary `Co-WM-S` `fwd_streaming_cowm_coupled_fullvocab_K8`).

**Sequential and Component Ablation Ladder in Streaming Block-Local Supervision**:
To evaluate the effects of successive interventions and isolate each component of the intra-block coupled package on 1000-step 2-seed `d6` ($8.192\text{M}$ tokens, Tables 4 and 5), we trace:
1. **Inter-Block Sublayer Jacobian Omission at Fixed Final-State Discrepancy (`backprop` $1.45172 \to$ `block_fixed_top_discrepancy_bp` $1.62152$, $\Delta = +0.16980$ BPB)**: Isolates the cost of omitting intervening non-identity sublayer Jacobians ($l' > l$) under fixed final-state supervision $\delta_{\text{final}}$.
2. **Replacing Fixed Final-State Discrepancy with Intermediate Block-State Supervision (`block_fixed_top_discrepancy_bp` $1.62152 \to$ `block_local_greedy_bp_tophead` $1.58085$, $\Delta = -0.04067$ BPB)**: Replaces $\delta_{\text{final}}$ with each block's own local-loss discrepancy $\frac{\partial \mathcal{L}(x_l)}{\partial x_l}$ (net $+0.12913$ BPB for greedy local supervision).
3. **Shared-Readout Intermediate-Loss Mixture (`block_local_greedy_bp_tophead` $1.58085 \to$ `block_local_greedy_bp` $1.57236$, $\Delta = -0.00849$ BPB)**: Mixes $50\%$ final-block and $50\%$ earlier-block gradients into `lm_head`.
4. **Baseline Uncoupled Intra-Block Surrogate Gap at Full Vocabulary (`block_local_greedy_bp` $1.57236 \to$ `fwd_streaming_oracle_fullvocab` $1.74582$, $\Delta = +0.17346$ BPB)**: Replaces exact intra-block autograd (`bwd_calls = 6`) with the uncoupled forward surrogate package (`bwd_calls = 0`, $K_v=1, K_{\text{qk}}=2$).
5. **Component Ablation of the Intra-Block Coupled Package at Full Vocabulary ($V=8192$)**:
   - *Component 5a (+MLP-Pullback Only)*: `fwd_streaming_oracle_fullvocab` ($1.74582$) $\to$ `fwd_streaming_oracle_mlp_pull_only_fullvocab` ($1.73609 \pm 0.00070$, **$\Delta = -0.00973$ BPB**, resolving $5.6\%$ of the $+0.17346$ BPB gap at zero extra forward passes).
   - *Component 5b (+Two-Level Window-Hadamard Value Transport `CM-Attn-WH`, $K_v=8$)*: `fwd_streaming_oracle_mlp_pull_only_fullvocab` ($1.73609$) $\to$ `fwd_streaming_oracle_mlp_whv_fullvocab` ($1.71756 \pm 0.00097$, **$\Delta = -0.01853$ BPB**, cumulative $-0.02826$ BPB / $16.3\%$ of the gap).
   - *Component 5c (+Post-RoPE `Hadamard-QK`, $K_{\text{qk}}=8$)*: `fwd_streaming_oracle_mlp_whv_fullvocab` ($1.71756$) $\to$ `fwd_streaming_oracle_coupled_fullvocab` ($1.64800 \pm 0.00172$, **$\Delta = -0.06956$ BPB**, cumulative **$-0.09782$ BPB / $56.4\%$** of the gap).
   - *Component 5d (Full-Vocabulary Parameter-Space `Co-WM-S` at $K=S=8$ vs. Analytic Linear Pullbacks)*: `fwd_streaming_oracle_coupled_fullvocab` ($1.64800$) $\to$ `fwd_streaming_cowm_coupled_fullvocab_K8` (**$1.64719 \pm 0.00067$**, $\Delta = -0.00081$ BPB, resolving **$56.9\%$** of the gap using 100% parameter-space `Co-WM-S` column contractions across all 4 linear sites per block!).
6. **Sub-Vocabulary Readout Truncation ($V=8192 \to V_{\text{sub}}=256$)**: Under coupled attention, restricting readout pullback to $V_{\text{sub}}=256$ yields `fwd_streaming_oracle_coupled` ($1.78435 \pm 0.00103$, $\Delta = +0.13635$ BPB) and `fwd_streaming_cowm_coupled_K8` ($1.78546 \pm 0.00297$, $\Delta = +0.00111$ BPB vs. analytic); under uncoupled attention, `fwd_streaming_oracle` ($1.86466 \pm 0.00221$) and `fwd_streaming_cowm_K8` ($1.86518 \pm 0.00011$, $\Delta = +0.00052$ BPB).

### 2.5 MAP Settling: Single-Step Equivalence vs. Multi-Step Spectral Damping
In Post-Activation MAP Settling [1, Eq. 1, 5, 6, 7], post-activation discrepancy $e_1 = H^1 - \phi(W_1 x)$ minimizes:
$$E(e_1) = \frac{\rho}{2} \|e_1\|_2^2 + \frac{1}{2} \|e_2(e_1)\|_2^2, \qquad e_2(e_1) = -\epsilon_{\text{clamp}} \hat{\delta}_y - e_1 V^\top, \quad e_1(0) = \mathbf{0}.$$

**Proposition 4 (GS-MAP Spectral Filter and Settling Provenance)**:
*Under the recurrence $e_1(n+1) = e_1(n) - \eta (\rho e_1(n) - e_2(n) V)$ with non-oscillatory step size $0 < \eta \le \frac{1}{\rho + \|V\|_2^2}$, the exact $N$-step credit operator is:*
$$\hat{g}_h^{\text{GS-MAP}(N)} = (\delta_y V) \, \mathcal{F}_N(V^\top V), \qquad \text{where } \mathcal{F}_N(V^\top V) = \frac{\sum_{k=0}^{N-1} \big(I - \eta(\rho I + V^\top V)\big)^k}{\sum_{k=0}^{N-1} (1 - \eta \rho)^k}.$$
1. **$N=1$ Equivalence**: At $N = 1$, $\mathcal{F}_1(V^\top V) = I$, recovering exact local credit $\delta_y V$ ($\text{RelBias} \le 0.00022$ in Table 2).
2. **Multi-Step Spectral Damping ($N \ge 2$)**: Rather than improving local credit accuracy, $N \ge 2$ applies a contractive spectral filter that damps top singular modes ($\text{RelBias} = 0.129\text{--}0.163$ in Table 2).

### 2.6 Strict No-Backward Credit Assignment: Faithful DUST vs. Delayed / Eligibility-Trace Coordinated Weight Maximization (`Delayed-Trace-Co-WM-S`)
While our streaming algorithms (`fwd_streaming_cowm*`, `fwd_lookahead_2b*`) eliminate the global reverse-depth sweep (`num_depth_reverse_sweeps = 0`), they still execute block-local reverse sensitivity pullbacks across activation/`RMSNorm` derivatives and intra-block sublayers. A **strict no-backward** algorithm must compute parameter updates without either a global reverse-depth sweep or local reverse derivative pullbacks.

**Published Faithful DUST (`dust_faithful.py`) [5]**:
In published DUST (`https://qlabs.sh/research/dust/` and `literature/dust/dust.py`), parameter updates are estimated by perturbing activation groups during forward execution (`writers`: `attn.c_proj` and `mlp.c_proj`; `hidden`: `mlp.c_fc`; `hubs`: `o@layer` attention head outputs; `wte`; and 256-column `lm_head` slabs; plus 2-sided `scalar_gradients` on `resid_lambdas, x0_lambdas, ve_gate`) with draw-chunk centered per-token loss rewards $\tilde{\ell}_{b, t}^{(p)} = \ell_{b, t}^{(p)} - \frac{1}{|\mathcal{C}|}\sum_{q \in \mathcal{C}} \ell_{b, t}^{(q)}$ (and future-token decayed per-head rewards $\gamma = 0.98$ inside `local_errors` for $W_Q, W_K, W_V$). Crucially:
1. *Non-Zero `c_proj` Initialization*: Because DUST perturbs `mlp.c_fc` (`hidden`) during forward execution and observes the loss shift through `mlp.c_proj`, zero-initializing `c_proj` (as in `baseline_code/train.py`) blocks `hidden` signals at step 0; published DUST initializes `attn.c_proj` and `mlp.c_proj` with $\text{std} = 1.5/\sqrt{d_{\text{in}}}$.
2. *Prefix-Cached Suffix Execution vs. Hook Reruns*: While the minimal reference script (`literature/dust/dust.py`) registers forward hooks and reruns the full network from `wte`, the DUST paper [5, §2.3] specifies prefix-cached suffix evaluation (`blocks[l:]`), which evaluates only blocks downstream of the perturbed block $l$ with bit-for-bit identical gradients (`max_grad_diff = 0.0` at $P=256$, verified in Stage 1A).
3. *Hidden-Width Scaling Bottleneck of Direct `hidden` Probing*: When DUST perturbs all $M$ hidden neurons of `mlp.c_fc` simultaneously with $\xi_{b, t} \in \mathbb{R}^M$ ($\sigma = 0.2$), neuron $i$'s estimator correlates $\xi_{b, t, i}$ with a token reward containing the summed perturbation of all $M-1$ sibling neurons $\sum_{j \ne i} (\mathbf{v}^{(j)} \cdot \delta_{b, t}) \xi_{b, t, j}$. Consequently, DUST's `mlp.c_fc` gradient variance grows linearly with hidden neuron count $\mathcal{O}(M / P_{\text{hidden}})$.

**Delayed / Eligibility-Trace Coordinated Weight Maximization (`Delayed-Trace-Co-WM-S`, `delayed_trace_cowm.py`)**:
To eliminate DUST's $\mathcal{O}(M / P_{\text{hidden}})$ sibling-neuron interference on `mlp.c_fc` **without** direct `hidden` forward passes and **without** any local reverse/pullback step, we transfer parameter-space `Co-WM-S` into a three-phase eligibility-trace rule:
1. **Phase 1 — Clean-Pass Local Forward Eligibility-Trace Recording (`record_mlp_forward_eligibility_traces`)**:
   During the single clean forward pass at block $l$, let $x_{g, s} \in \mathbb{R}^D$ be the pre-synaptic input to `mlp.c_fc` ($W_1 \in \mathbb{R}^{M \times D}$) and $z_{g, s} = W_1 x_{g, s} \in \mathbb{R}^M$ the clean pre-activation in micro-group $g \in \{1, \dots, G\}$ ($G = BT / S, S=8$). Using only local forward evaluations of the activation function $\phi(z) = \max(0, z)^2$:
   $$d_{g, s, i} \triangleq \delta h_{0, g, s, i} = \frac{\phi(z_{g, s, i} + \sigma_{\text{act}}) - \phi(z_{g, s, i} - \sigma_{\text{act}})}{2\sigma_{\text{act}}}, \qquad \tilde{h}_{g, s, i}^{(m)} = d_{g, s, i} \, \xi_{g, s}^{(m)},$$
   where $\xi_{g, s}^{(m)} = H_{s, c_m} \in \{-1, +1\}$ ($m = 1, \dots, K_{\text{wm}}, K_{\text{wm}} \le S$) are shared Sylvester-Hadamard micro-horizon probes (`64 KB` per layer, since each neuron is already spatially decoupled from sibling neurons by its own outgoing column $\mathbf{v}^{(i)}$). The local pre-synaptic input eligibility trace for micro-group $g$ and probe $m$ is $e_g^{(m)} = \sum_{s=1}^S \xi_{g, s}^{(m)} x_{g, s} \in \mathbb{R}^D$ (retaining $d_{g, s, i}$ inside $\tilde{h}_{g, s, i}^{(m)}$ so the activation sensitivity enters the weight update exactly once).
2. **Phase 2 — Outgoing Writer Forward Perturbation (Same-Step `d0` or 1-Step Cross-Batch Delayed `d1`)**:
   The outgoing projection $V = \text{mlp.c\_proj} \in \mathbb{R}^{D \times M}$ obtains its forward writer reward vector $u_{g, s} \in \mathbb{R}^D$ from DUST's `writers` perturbation draws (either from the current batch `d0` or from the previous step's writer reward buffer `d1` before overwriting). Zero direct `hidden` (`mlp.c_fc`) forward passes are executed (`n_hidden_draws = 0`).
3. **Phase 3 — Outgoing-Column Contraction and Trace Modulation (`contract_delayed_outgoing_columns_with_trace`)**:
   For each micro-group $g$ and Hadamard probe $m$, neuron $i$ forms its outgoing weight-column plastic displacement $\Delta \mathbf{v}_g^{(i, m)} = \sum_{s'=1}^S \tilde{h}_{g, s', i}^{(m)} u_{g, s'} = \sum_{s'=1}^S d_{g, s', i} \, \xi_{g, s'}^{(m)} u_{g, s'} \in \mathbb{R}^D$ (computed across all $M$ neurons simultaneously via Tensor-Core `torch.bmm`), contracts its own outgoing column $\mathbf{v}^{(i)} = V_{:, i} \in \mathbb{R}^D$ with $\Delta \mathbf{v}_g^{(i, m)}$:
   $$c_g^{(i, m)} = \mathbf{v}^{(i)} \cdot \Delta \mathbf{v}_g^{(i, m)} = \sum_{s'=1}^S d_{g, s', i} \big(\mathbf{v}^{(i)} \cdot u_{g, s'}\big) \xi_{g, s'}^{(m)} \;\in\; \mathbb{R},$$
   and accumulates its `mlp.c_fc` row update by modulating the pre-synaptic input trace $e_g^{(m)}$ (`mod_accum.T @ x_g` in `delayed_trace_cowm.py` lines 171–174):
   $$\widehat{\nabla}_{W_{1, i}} = \frac{1}{K_{\text{wm}}} \sum_{m=1}^{K_{\text{wm}}} \sum_{g=1}^G c_g^{(i, m)} e_g^{(m)} = \sum_{g=1}^G \sum_{s=1}^S \left(\frac{1}{K_{\text{wm}}} \sum_{m=1}^{K_{\text{wm}}} c_g^{(i, m)} \xi_{g, s}^{(m)}\right) x_{g, s}.$$
   *Exact Conditional Recovery vs. Finite Precision and Writer-Reward Noise*:
   - **Exact Algebraic Recovery ($K_{\text{wm}} = S$)**: By Sylvester-Hadamard completeness ($\frac{1}{S}\sum_{m=1}^S \xi_{g, s'}^{(m)} \xi_{g, s}^{(m)} = \mathbf{1}[s' = s]$), the reconstructed pre-activation modulation satisfies the exact single-sensitivity identity:
     $$\frac{1}{S}\sum_{m=1}^S c_g^{(i, m)} \xi_{g, s}^{(m)} \equiv d_{g, s, i} \big(\mathbf{v}^{(i)} \cdot u_{g, s}\big), \qquad \widehat{\nabla}_{W_{1, i}}\Big|_{K_{\text{wm}}=S} \equiv \sum_{g=1}^G \sum_{s=1}^S d_{g, s, i} \big(\mathbf{v}^{(i)} \cdot u_{g, s}\big) x_{g, s}.$$
     (For example, at $S = K_{\text{wm}} = 8$ with scalar $x_s = u_s = v = 1$ and $z_s = 1 \implies d_s = 2$, both the displayed formula and `contract_delayed_outgoing_columns_with_trace` evaluate to $\sum_{s=1}^8 2 \cdot 1 \cdot 1 = 16.0$, verified in Gate 2L.)
   - **Finite-Precision Tensor-Core Agreement**: In `delayed_trace_cowm.py`, $x_g, \tilde{h}_0, V_{\text{snapshot}}, u_g$ are stored in `bfloat16` for Tensor-Core `torch.bmm` throughput and accumulated in `float32`, yielding conditional cosine similarity `0.999997` against the `bfloat16` reference in Gate 2L (`1.000000` rounded to 6 decimals in Table 7's conditional panel).
   - **Remaining Writer-Reward Estimation Noise**: When $u_{g, s}$ is estimated from a finite number $P_{\text{writers}}$ of forward writer draws rather than the exact local target $\delta_{g, s}$, $K_{\text{wm}} = S = 8$ eliminates the internal MLP column-contraction variance ($\text{Var}[\widehat{\nabla}_{W_1} \mid u] = 0$), while the remaining whole-network variance is governed solely by the $D$-dimensional outgoing writer estimate $u_{g, s}$ (independent of hidden neuron count $M$).

---

## 3. Experimental Validation

### 3.1 Whole-Transformer Initialization Diagnostic and Unbiased Bias/Variance Decomposition
Table 3 reports gradient fidelity against analytical backpropagation across widths $D \in \{128, 256, 384, 512\}$ ($L=6, B=4, T=64, V=4096$) over $R = 32$ independent random trials (`pure_forward_width_diagnostic.json`).

**Unbiased Squared Relative Bias vs. Plug-In Sample-Mean Bias**:
For any stochastic gradient estimator $\hat{G}^{(r)}$ ($r = 1, \dots, R$) with sample mean $\bar{G}_R = \frac{1}{R}\sum_{r=1}^R \hat{G}^{(r)}$, expectation $\mu = \mathbb{E}[\hat{G}]$, and reference gradient $G_{\text{bp}}$, the naive plug-in squared relative bias $\widehat{\text{RelBias}}^2_{\text{plug}} = \frac{\|\bar{G}_R - G_{\text{bp}}\|_F^2}{\|G_{\text{bp}}\|_F^2}$ has expectation:
$$\mathbb{E}\!\left[\widehat{\text{RelBias}}^2_{\text{plug}}\right] = \frac{\|\mu - G_{\text{bp}}\|_F^2}{\|G_{\text{bp}}\|_F^2} + \frac{1}{R} \text{RelVar}(\hat{G}), \qquad \text{where } \widehat{\text{RelVar}} = \frac{1}{R - 1} \sum_{r=1}^R \frac{\|\hat{G}^{(r)} - \bar{G}_R\|_F^2}{\|G_{\text{bp}}\|_F^2}.$$
When $\text{RelVar}$ is large, $\widehat{\text{RelBias}}^2_{\text{plug}}$ is dominated by residual sample-mean variance $\text{RelVar}/R$ rather than true systematic bias. To eliminate this contamination:
1. We compute **Total Relative MSE** directly from the $R=32$ individual trial errors (with standard error of the mean across trials):
   $$\widehat{\text{TotalMSE}} = \frac{1}{R} \sum_{r=1}^R \frac{\|\hat{G}^{(r)} - G_{\text{bp}}\|_F^2}{\|G_{\text{bp}}\|_F^2}.$$
2. We report the **unbiased bias-corrected squared relative bias estimator** (with Tukey leave-one-out jackknife standard error across the $R=32$ trials):
   $$\widehat{\text{RelBias}}^2_{\text{unb}} = \widehat{\text{RelBias}}^2_{\text{plug}} - \frac{1}{R} \widehat{\text{RelVar}}, \qquad \mathbb{E}\!\left[\widehat{\text{RelBias}}^2_{\text{unb}}\right] = \frac{\|\mu - G_{\text{bp}}\|_F^2}{\|G_{\text{bp}}\|_F^2}.$$
   By algebraic identity, $\widehat{\text{TotalMSE}} = \frac{R-1}{R}\widehat{\text{RelVar}} + \widehat{\text{RelBias}}^2_{\text{plug}} \equiv \widehat{\text{RelVar}} + \widehat{\text{RelBias}}^2_{\text{unb}}$. When $\widehat{\text{RelBias}}^2_{\text{unb}}$ is comparable to or smaller than its jackknife standard error (as in `full_seq_dot` and `token_dot`), no nonzero bias is statistically resolved, and its square root should not be interpreted as a resolved bias magnitude.

| Algorithm | Paradigm / Carrier & Readout | $D=128$ Top / Mid Cos | $D=512$ Top / Mid Cos | Mid Scale ($D=512$) | Mid Param RelVar ($D=128 \to 512$) | Mid Unbiased $\text{RelBias}^2$ ($D=512$) | Mid TotalMSE ($D=512$) | Attn Q5 Cos / RelVar ($D=512$) |
| :--- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| `fwd_global_scalar_K2` | Global Scalar Control ($K=2$) | `-0.002` / `0.002` | `0.000` / `-0.000` | $1007.95$ | $7.79\times 10^6 \to 3.26\times 10^7$ | $-2250.8 \pm 10136.6$ | $3.26\times 10^7 \pm 4.50\times 10^6$ | `-0.0003` / $1.83\times 10^8$ |
| `fwd_token_uwm_K2` | Token-Scalar ($K=2$) | `0.004` / `0.007` | `0.004` / `0.003` | $56.74$ | $26,968 \to 103,931$ | $-29.98 \pm 26.68$ | $103,901 \pm 1075$ | `0.0002` / $5.84\times 10^5$ |
| `fwd_two_level_cowm_K8_fullseq` | Two-Level ($K=8$, `full_seq_dot`, DH) | `0.003` / `0.006` | `0.003` / `0.003` | $49.85$ | $21,803 \to 78,606$ | $27.44 \pm 27.86$ | $78,634 \pm 8449$ | `0.0012` / $4.88\times 10^6$ |
| `fwd_two_level_cowm_K8_fullseq_iid` | Two-Level ($K=8$, `full_seq_dot`, i.i.d.) | `0.005` / `0.009` | `0.001` / `0.003` | $53.07$ | $22,705 \to 90,830$ | $-23.07 \pm 36.45$ | $90,807 \pm 3178$ | `0.0004` / $5.85\times 10^6$ |
| `fwd_two_level_cowm_K8_tokendot` | Two-Level ($K=8$, `token_dot`, DH) | `0.052` / `0.059` | `0.026` / `0.029` | $6.12$ | $303.50 \to 1180.58$ | $-0.453 \pm 0.422$ | $1180.13 \pm 39.33$ | `0.0042` / $69,297$ |
| `fwd_two_level_cowm_K8_tokendot_iid` | Two-Level ($K=8$, `token_dot`, i.i.d.) | `0.051` / `0.054` | `0.024` / `0.028` | $6.67$ | $350.00 \to 1386.78$ | $0.088 \pm 0.455$ | $1386.87 \pm 7.47$ | `0.0042` / $89,860$ |
| **`fwd_two_level_cowm_K4`** | Two-Level ($K=4$, `diag_jacobian`, DH) | `0.412` / `0.355` | `0.410` / `0.365` | $0.96$ | $4.30 \to 4.13$ | $0.224 \pm 0.003$ | $4.35 \pm 0.27$ | `0.0560` / $330.05$ |
| **`fwd_two_level_cowm_K8_iid`** | Two-Level ($K=8$, `diag_jacobian`, i.i.d.) | `0.481` / `0.421` | `0.479` / `0.426` | $0.93$ | $2.64 \to 2.58$ | $0.224 \pm 0.002$ | $2.81 \pm 0.00$ | `0.0649` / $216.35$ |
| **`fwd_two_level_cowm_K8`** | Two-Level ($K=8$, `diag_jacobian`, DH) | `0.598` / `0.512` | `0.615` / `0.516` | $0.91$ | **$1.61 \to 1.55$** | **$0.222 \pm 0.002$** | **$1.77 \pm 0.07$** | `0.0837` / $136.49$ |
| `fwd_streaming_cut_K4` | Streaming Cut ($K=4$, Token-Scalar) | `0.016` / `0.015` | `0.004` / `0.003` | $47.76$ | $4638 \to 73,120$ | $-4.43 \pm 23.70$ | $73,116 \pm 815$ | `0.0037` / $8.28\times 10^5$ |
| **`fwd_streaming_wm_K1`** | Streaming `WM-S` ($K=1$, Indep., $K_{\text{qk}}=1$) | `0.129` / `0.109` | `0.125` / `0.112` | $1.80$ | $69.28 \to 68.66$ | $0.244 \pm 0.018$ | $68.90 \pm 0.18$ | `0.0135` / $6375.22$ |
| **`fwd_streaming_wm_K1_qk2`** | Streaming `WM-S` ($K=1$, Indep., $K_{\text{qk}}=2$) | `0.127` / `0.107` | `0.126` / `0.110` | $1.80$ | $69.29 \to 68.43$ | $0.249 \pm 0.022$ | $68.68 \pm 0.16$ | `0.0153` / $3156.59$ |
| **`fwd_streaming_wm_K4`** | Streaming `WM-S` ($K=4$, Indep.) | `0.363` / `0.318` | `0.362` / `0.322` | $1.15$ | $7.19 \to 7.15$ | $0.246 \pm 0.003$ | $7.39 \pm 0.01$ | `0.0516` / $374.23$ |
| **`fwd_streaming_cowm_K4`** | Streaming `Co-WM-S` ($K=4$, Had.) | `0.496` / `0.438` | `0.498` / `0.442` | $1.09$ | $3.29 \to 3.26$ | $0.251 \pm 0.002$ | $3.51 \pm 0.00$ | `0.0688` / $197.17$ |
| **`fwd_streaming_wm_K8`** | Streaming `WM-S` ($K=8$, Indep.) | `0.533` / `0.465` | `0.532` / `0.471` | $1.09$ | $2.78 \to 2.74$ | $0.253 \pm 0.002$ | $2.99 \pm 0.01$ | `0.0731` / $173.44$ |
| **`fwd_streaming_cowm_K8`** | Streaming `Co-WM-S` ($K=8$, Had., Uncoupled) | **`0.999`** / **`0.879`** | **`0.998`** / **`0.881`** | **$1.05$** | **$0.00 \to 0.00$** | **$0.251 \pm 0.000$** | **$0.25 \pm 0.00$** | `0.1397` / $47.85$ |
| **`fwd_streaming_cowm_coupled_K8`** | Coupled `Co-WM-S` ($K=8$, $V_{\text{sub}}=256$) | **`0.999`** / **`0.879`** | **`0.998`** / **`0.881`** | **$1.05$** | **$0.00 \to 0.00$** | **$0.251 \pm 0.000$** | **$0.25 \pm 0.00$** | **`0.3513`** / **$6.93$** |
| **`fwd_streaming_cowm_coupled_fullvocab_K8`** | Coupled `Co-WM-S` ($K=8$, $V=4096$) | **`1.000`** / **`0.879`** | **`1.000`** / **`0.882`** | **$1.05$** | **$0.00 \to 0.00$** | **$0.249 \pm 0.000$** | **$0.25 \pm 0.00$** | **`0.3523`** / **$6.90$** |
| `fwd_streaming_oracle` | Analytic Surrogate ($V_{\text{sub}}=256$, Uncoupled) | `0.999` / `0.879` | `0.998` / `0.881` | $1.04$ | $0.00 \to 0.00$ | $0.250 \pm 0.000$ | $0.25 \pm 0.00$ | `0.1413` / $48.11$ |
| `fwd_streaming_oracle_coupled` | Analytic Surrogate ($V_{\text{sub}}=256$, Coupled) | `0.999` / `0.879` | `0.998` / `0.881` | $1.04$ | $0.00 \to 0.00$ | $0.250 \pm 0.000$ | $0.25 \pm 0.00$ | **`0.3518`** / **$6.85$** |
| `fwd_streaming_oracle_fullvocab` | Analytic Surrogate ($V=4096$, Uncoupled) | `1.000` / `0.879` | `1.000` / `0.882` | $1.04$ | $0.00 \to 0.00$ | $0.248 \pm 0.000$ | $0.25 \pm 0.00$ | `0.1378` / $47.31$ |
| `fwd_streaming_oracle_mlp_pull_only_fullvocab` | +MLP-Pullback Only ($V=4096, K_v=1$) | `1.000` / `0.879` | `1.000` / `0.882` | $1.04$ | $0.00 \to 0.00$ | $0.248 \pm 0.000$ | $0.25 \pm 0.00$ | `0.1468` / $52.55$ |
| `fwd_streaming_oracle_mlp_whv_fullvocab` | +MLP-Pullback + `CM-Attn-WH` ($V=4096, K_v=8$) | `1.000` / `0.879` | `1.000` / `0.882` | $1.04$ | $0.00 \to 0.00$ | $0.248 \pm 0.000$ | $0.25 \pm 0.00$ | `0.1458` / $52.16$ |
| `fwd_streaming_oracle_coupled_fullvocab` | Analytic Surrogate ($V=4096$, Coupled) | `1.000` / `0.879` | `1.000` / `0.882` | $1.04$ | $0.00 \to 0.00$ | $0.248 \pm 0.000$ | $0.25 \pm 0.00$ | **`0.3525`** / **$6.86$** |
| `block_local_greedy_bp_vsub256` | Block-Local Greedy BP ($V_{\text{sub}}=256$) | `0.999` / `0.879` | `0.998` / `0.881` | $1.04$ | $0.00 \to 0.00$ | $0.249 \pm 0.000$ | $0.25 \pm 0.00$ | `0.9996` / $0.00$ |
| `block_local_greedy_bp` | Block-Local Greedy BP [8] ($V=4096$) | `1.000` / `0.879` | `1.000` / `0.882` | $1.04$ | $0.00 \to 0.00$ | $0.248 \pm 0.000$ | $0.25 \pm 0.00$ | `1.0000` / $0.00$ |
| **`block_fixed_top_discrepancy_bp`** | Fixed-Top-Discrepancy BP ($V=4096$) | **`1.000`** / **`0.881`** | **`1.000`** / **`0.882`** | **$0.88$** | **$0.00 \to 0.00$** | **$0.221 \pm 0.000$** | **$0.22 \pm 0.00$** | **`1.0000`** / **$0.00$** |

*Table 3: Whole-Transformer width-scaling diagnostic ($L=6, B=4, T=64, V=4096, R=32$ trials; DH = `depth_hadamard` carrier modulation, i.i.d. = `iid_resample` per-probe carrier resampling). At $D=512$, Layer 3 activation-credit relative variances (`rel_var_act_mid_mlp` in `pure_forward_width_diagnostic.json`) closely track the parameter-gradient relative variances shown above: $79,218.20$ (DH) vs $91,443.37$ (i.i.d.) for `full_seq_dot`, $1191.29$ (DH) vs $1398.44$ (i.i.d.) for `token_dot`, and $1.56$ (DH) vs $2.60$ (i.i.d.) for `diag_jacobian`. The measured full-sequence sensitivity-energy ratio $B_{\text{full}}^{(8)}/A_{\text{full}}^{(8)} = 20.30 > 15$ quantitatively explains the `full_seq_dot` DH-vs-i.i.d. variance ratio ($0.866 \approx 0.884$ theoretical), whereas `token_dot` and `diag_jacobian` are governed by their own within-site and between-site variance expressions ($A_{\text{tok}}, B_{\text{tok}}, A_{\text{diag}}, B_{\text{diag}}$ in Section 2.4). Furthermore, at $D=512$, Layer 3 block-local attention cosines (`cossim_blkref_mid_attn_{o,v}`) rise from $0.914 / 0.916$ (`fwd_streaming_cowm_K8`) to $0.987 / 0.987$ under sub-vocabulary coupling (`fwd_streaming_cowm_coupled_K8`), and from $0.927 / 0.926$ (`fwd_streaming_oracle_fullvocab`) to **$1.000 / 1.000$** under full-vocabulary coupling (`fwd_streaming_cowm_coupled_fullvocab_K8` at $0.99998 / 0.99997$ and `fwd_streaming_oracle_coupled_fullvocab` at $1.00000 / 1.00000$), while `Hadamard-QK` lifts Attn Q5 cosine from $0.138$ to **$0.353$** and reduces relative variance by $85.5\%$ ($47.31 \to 6.86$).*

### 3.2 200-Step Multi-Width Pretraining Sweep
Table 4 reports the 200-step pretraining sweep across $D \in \{128, 256, 384, 512\}$ ($B=16, T=512, V=8192$) on ClimbMix evaluated on the dev split (row groups $0 \dots 14$) and holdout split (row groups $40 \dots 82$). All candidate forward-only methods update 100% of trainable parameters under `@torch.no_grad()` (`bwd_calls = 0`).

| Algorithm | $D=128$ Dev / Holdout BPB | $D=256$ Dev / Holdout BPB | $D=384$ Dev / Holdout BPB | $D=512$ Dev / Holdout BPB | Step ms / Peak VRAM ($D=512$) | Backward Calls |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| `backprop` (Standard) | `1.8852` / `1.8846` | `1.8410` / `1.8393` | `1.8349` / `1.8320` | `1.8160` / `1.8133` | `33.3 ms` / `3010.4 MB` | $1$ |
| `block_fixed_top_discrepancy_bp` | `1.9266` / `1.9262` | `1.8873` / `1.8868` | `1.8753` / `1.8723` | `1.8664` / `1.8643` | `33.7 ms` / `3026.5 MB` | $6$ ($1$/block) |
| `block_local_greedy_bp_tophead` | `1.9207` / `1.9193` | `1.8787` / `1.8767` | `1.8742` / `1.8715` | `1.8566` / `1.8538` | `73.5 ms` / `2153.6 MB` | $6$ ($1$/block) |
| `block_local_greedy_bp` (Mixed Head [8]) | `1.9156` / `1.9140` | `1.8809` / `1.8789` | `1.8664` / `1.8649` | `1.8469` / `1.8448` | `76.7 ms` / `2169.6 MB` | $6$ ($1$/block) |
| `fwd_global_scalar_K2` | `2.6507` / `2.6507` | `2.3906` / `2.3910` | `2.4085` / `2.4076` | `2.3913` / `2.3898` | `71.5 ms` / `3864.8 MB` | **$0$** |
| `fwd_token_uwm_K2` | `2.2138` / `2.2142` | `2.1956` / `2.1943` | `2.2131` / `2.2121` | `2.1883` / `2.1868` | `90.6 ms` / `3864.8 MB` | **$0$** |
| `fwd_streaming_cut_K4` | `2.1682` / `2.1682` | `2.1379` / `2.1372` | `2.1543` / `2.1530` | `2.1121` / `2.1126` | `130.2 ms` / `1820.3 MB` | **$0$** |
| **`fwd_streaming_wm_K1`** ($K_{\text{qk}}=1$) | `2.1310` / `2.1306` | `2.0942` / `2.0928` | `2.0752` / `2.0750` | `2.0680` / `2.0673` | `148.8 ms` / `1828.4 MB` | **$0$** |
| **`fwd_streaming_wm_K1_qk2`** ($K_{\text{qk}}=2$) | `2.1296` / `2.1301` | `2.0935` / `2.0931` | `2.0750` / `2.0738` | `2.0665` / `2.0659` | `150.9 ms` / `1820.3 MB` | **$0$** |
| **`fwd_streaming_wm_K4`** | `2.1073` / `2.1079` | `2.0717` / `2.0714` | `2.0549` / `2.0541` | `2.0471` / `2.0468` | `299.9 ms` / `1820.3 MB` | **$0$** |
| **`fwd_streaming_cowm_K4`**| `2.0975` / `2.0981` | `2.0635` / `2.0627` | `2.0468` / `2.0467` | `2.0392` / `2.0383` | `303.5 ms` / `1828.4 MB` | **$0$** |
| **`fwd_two_level_cowm_K4`**| `2.1060` / `2.1052` | `2.0750` / `2.0756` | `2.0622` / `2.0617` | `2.0559` / `2.0553` | `469.4 ms` / `1850.3 MB` | **$0$** |
| **`fwd_streaming_wm_K8`** | `2.0964` / `2.0963` | `2.0627` / `2.0629` | `2.0488` / `2.0484` | `2.0387` / `2.0392` | `498.4 ms` / `1820.3 MB` | **$0$** |
| **`fwd_streaming_cowm_K8`** (Uncoupled, $V_{\text{sub}}=256$) | `2.0803` / `2.0820` | `2.0502` / `2.0507` | `2.0373` / `2.0378` | `2.0287` / `2.0291` | `507.4 ms` / `1828.4 MB` | **$0$** |
| **`fwd_streaming_cowm_coupled_K8`** ($V_{\text{sub}}=256$) | `2.0533` / `2.0537` | `2.0216` / `2.0210` | `2.0043` / `2.0041` | `1.9985` / `1.9972` | `833.3 ms` / `1828.3 MB` | **$0$** |
| **`fwd_streaming_cowm_coupled_fullvocab_K8`** ($V=8192$) | **`1.9578`** / **`1.9578`** | **`1.9319`** / **`1.9312`** | **`1.9190`** / **`1.9178`** | **`1.8952`** / **`1.8946`** | `1811.3 ms` / `2100.3 MB` | **$0$** |
| `fwd_streaming_oracle` ($V_{\text{sub}}=256$, Uncoupled) | `2.0805` / `2.0822` | `2.0519` / `2.0517` | `2.0383` / `2.0387` | `2.0308` / `2.0302` | `99.7 ms` / `1820.3 MB` | **$0$** |
| `fwd_streaming_oracle_coupled` ($V_{\text{sub}}=256$) | `2.0556` / `2.0562` | `2.0223` / `2.0218` | `2.0032` / `2.0032` | `1.9975` / `1.9969` | `163.6 ms` / `1828.3 MB` | **$0$** |
| `fwd_streaming_oracle_fullvocab` ($V=8192$, Uncoupled) | `1.9876` / `1.9866` | `1.9686` / `1.9671` | `1.9573` / `1.9565` | `1.9514` / `1.9505` | `112.4 ms` / `2100.4 MB` | **$0$** |
| `fwd_streaming_oracle_mlp_pull_only_fullvocab` ($V=8192$) | `1.9847` / `1.9844` | `1.9653` / `1.9643` | `1.9572` / `1.9559` | `1.9433` / `1.9428` | `116.1 ms` / `2100.3 MB` | **$0$** |
| `fwd_streaming_oracle_mlp_whv_fullvocab` ($V=8192$) | `1.9807` / `1.9800` | `1.9572` / `1.9557` | `1.9505` / `1.9489` | `1.9346` / `1.9327` | `134.1 ms` / `2100.3 MB` | **$0$** |
| `fwd_streaming_oracle_coupled_fullvocab` ($V=8192$) | **`1.9586`** / **`1.9583`** | **`1.9319`** / **`1.9312`** | **`1.9185`** / **`1.9176`** | **`1.8956`** / **`1.8950`** | `177.7 ms` / `2100.3 MB` | **$0$** |

*Table 4: 200-step multi-width pretraining sweep on ClimbMix (`pure_forward_pretrain_scaling.json`). Across all widths $D \in \{128 \dots 512\}$, each component of the intra-block coupled package (`+MLP-Pullback` $\to$ `+CM-Attn-WH` $\to$ `+Hadamard-QK`) monotonically reduces holdout BPB across every width ($1.9505 \to 1.9428 \to 1.9327 \to 1.8950$ at $D=512$), while full-vocabulary Parameter-Space `Co-WM-S` (`fwd_streaming_cowm_coupled_fullvocab_K8`, $1.8946$ at $D=512$) closely matches its analytic-linear surrogate (`1.8950`).*

### 3.3 1000-Step Two-Seed Benchmark on `d6` ($8.192\text{M}$ Tokens)
Table 5 reports 1000-step ($8,192,000$ tokens, 2 seeds `s42` and `s137`) pretraining on `d6` ($D=384, M=1536, L=6, V=8192$) evaluated on the document-disjoint holdout split (`pure_forward_1000steps_d6.json`). Trainable parameters ($26,346,060$) are identical across all arms. Attention probing uses $K_{\text{qk}}=2$ ($66$ attention calls per step across $6$ blocks) for all uncoupled forward streaming arms except the original `fwd_streaming_wm_K1` which evaluated $K_{\text{qk}}=1$ ($42$ calls; matched by our newly executed benchmark arm `fwd_streaming_wm_K1_qk2` at $K_{\text{qk}}=2$), while `fwd_streaming_oracle_mlp_whv_fullvocab` uses $K_v=8, K_{\text{qk}}=2$ ($108$ calls) and full coupled arms evaluate $K_v=8, K_{\text{qk}}=8$ ($252$ calls).

| Algorithm | Seed `s42` Holdout BPB | Seed `s137` Holdout BPB | Mean Holdout BPB $\pm$ Sample SD | Mean Step ms | Peak VRAM | Backward Calls |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| `backprop` (Standard) | `1.45158` | `1.45186` | `1.45172 ± 0.00020` | `25.1 ms` | `2656.3 MB` | $1$ |
| `backprop_ckpt` (Checkpointed) | `1.45224` | `1.45134` | **`1.45179 ± 0.00064`** | **`33.3 ms`** | **`1731.4 MB`** | $1$ |
| `block_fixed_top_discrepancy_bp` (Fixed $\delta_{\text{final}}$) | `1.61470` | `1.62833` | `1.62152 ± 0.00964` | `26.2 ms` | `2668.4 MB` | $6$ ($1$/block) |
| `block_local_greedy_bp_tophead` (Local $\delta_l$, Top Head) | `1.57477` | `1.58694` | `1.58085 ± 0.00861` | `64.7 ms` | `2034.3 MB` | $6$ ($1$/block) |
| `block_local_greedy_bp` (Mixed Head [8]) | `1.57272` | `1.57200` | **`1.57236 ± 0.00051`** | `66.6 ms` | `2046.3 MB` | $6$ ($1$/block) |
| `fwd_global_scalar_K2` (Global-Scalar Control) | `2.52995` | `2.36593` | `2.44794 ± 0.11598` | `56.1 ms` | `3448.2 MB` | **$0$** |
| `fwd_token_uwm_K2` (100% Params Updated) | `2.09448` | `2.10184` | `2.09816 ± 0.00521` | `72.6 ms` | `3448.2 MB` | **$0$** |
| `fwd_streaming_cut_K4` (Matched $K=4$) | `2.01759` | `2.01353` | `2.01556 ± 0.00287` | `112.9 ms` | `1757.0 MB` | **$0$** |
| **`fwd_streaming_wm_K1`** ($K=1, K_{\text{qk}}=1$) | `1.95285` | `1.95252` | `1.95268 ± 0.00023` | `125.3 ms` | `1765.1 MB` | **$0$** |
| **`fwd_streaming_wm_K1_qk2`** ($K=1, K_{\text{qk}}=2$) | `1.95103` | `1.95365` | `1.95234 ± 0.00185` | `132.4 ms` | `1757.0 MB` | **$0$** |
| **`fwd_streaming_wm_K4`** (Indep. Signs, $K=4$) | `1.91576` | `1.90640` | `1.91108 ± 0.00662` | `236.3 ms` | `1757.0 MB` | **$0$** |
| **`fwd_streaming_cowm_K4`** (Hadamard, $K=4$) | `1.89369` | `1.89344` | `1.89356 ± 0.00018` | `243.0 ms` | `1765.1 MB` | **$0$** |
| **`fwd_two_level_cowm_K4`** | `1.97564` | `1.97840` | `1.97702 ± 0.00196` | `334.7 ms` | `1791.4 MB` | **$0$** |
| **`fwd_streaming_wm_K8`** (Indep. Signs, $K=8$) | `1.88734` | `1.88884` | `1.88809 ± 0.00106` | `377.7 ms` | `1757.0 MB` | **$0$** |
| **`fwd_streaming_cowm_K8`** ($K=8, V_{\text{sub}}=256$, Uncoupled) | `1.86510` | `1.86525` | `1.86518 ± 0.00011` | `386.9 ms` | `1765.1 MB` | **$0$** |
| **`fwd_streaming_cowm_coupled_K8`** ($K=8, V_{\text{sub}}=256$, Coupled) | `1.78336` | `1.78756` | `1.78546 ± 0.00297` | `600.0 ms` | `1763.0 MB` | **$0$** |
| **`fwd_streaming_cowm_coupled_fullvocab_K8`** ($K=8, V=8192$, Coupled) | **`1.64671`** | **`1.64767`** | **`1.64719 ± 0.00067`** | `1391.3 ms` | `2031.0 MB` | **$0$** |
| **`fwd_streaming_wm_map_N4`**| `1.89510` | `1.89014` | `1.89262 ± 0.00350` | `265.1 ms` | `1765.1 MB` | **$0$** |
| `fwd_streaming_oracle` ($V_{\text{sub}}=256$ Uncoupled) | `1.86622` | `1.86309` | `1.86466 ± 0.00221` | `94.6 ms` | `1757.0 MB` | **$0$** |
| `fwd_streaming_oracle_coupled` ($V_{\text{sub}}=256$ Coupled) | `1.78362` | `1.78508` | `1.78435 ± 0.00103` | `150.2 ms` | `1763.0 MB` | **$0$** |
| `fwd_streaming_oracle_fullvocab` ($V=8192$ Uncoupled) | `1.74700` | `1.74464` | `1.74582 ± 0.00167` | `107.9 ms` | `2033.1 MB` | **$0$** |
| `fwd_streaming_oracle_mlp_pull_only_fullvocab` (+MLP-Pullback Only) | `1.73560` | `1.73659` | `1.73609 ± 0.00070` | `108.2 ms` | `2031.0 MB` | **$0$** |
| `fwd_streaming_oracle_mlp_whv_fullvocab` (+MLP-Pullback + `CM-Attn-WH`) | `1.71687` | `1.71824` | `1.71756 ± 0.00097` | `123.6 ms` | `2031.0 MB` | **$0$** |
| `fwd_streaming_oracle_coupled_fullvocab` ($V=8192$ Coupled Surrogate) | `1.64679` | `1.64922` | **`1.64800 ± 0.00172`** | `163.9 ms` | `2031.0 MB` | **$0$** |

*Table 5: 1000-step 2-seed pretraining benchmark on `d6` ($8.192\text{M}$ tokens). Key findings: (1) At matched $K=4$, parameter-space column contraction (`fwd_streaming_wm_K4`) beats token-scalar probing (`fwd_streaming_cut_K4`) by $-0.10448$ BPB. (2) Sylvester-Hadamard coordination improves $K=4$ by $-0.01752$ BPB and $K=8$ by $-0.02291$ BPB. (3) The component ablation at $V=8192$ isolates each mechanism in the intra-block coupled package: starting from uncoupled `fwd_streaming_oracle_fullvocab` ($1.74582 \pm 0.00167$), adding MLP-to-Attention residual pullback alone (`fwd_streaming_oracle_mlp_pull_only_fullvocab`) reduces holdout BPB by $-0.00973$ ($1.73609 \pm 0.00070$), adding Two-Level Window-Hadamard value transport `CM-Attn-WH` ($K_v=8$, `fwd_streaming_oracle_mlp_whv_fullvocab`) reduces BPB by an additional $-0.01853$ ($1.71756 \pm 0.00097$), and adding `Hadamard-QK` ($K_{\text{qk}}=8$, `fwd_streaming_oracle_coupled_fullvocab`) reduces BPB by a further $-0.06956$ ($1.64800 \pm 0.00172$), closing $56.4\%$ of the $+0.17346$ BPB surrogate gap to `block_local_greedy_bp` ($1.57236 \pm 0.00051$). (4) Full-vocabulary Parameter-Space `Co-WM-S` (`fwd_streaming_cowm_coupled_fullvocab_K8`) achieves **$1.64719 \pm 0.00067$ holdout BPB** ($56.9\%$ gap reduction), showing close descriptive agreement with its analytic-linear surrogate ($1.64800 \pm 0.00172$), just as sub-vocabulary `fwd_streaming_cowm_coupled_K8` ($1.78546 \pm 0.00297$) and `fwd_streaming_cowm_K8` ($1.86518 \pm 0.00011$) agree with `fwd_streaming_oracle_coupled` ($1.78435 \pm 0.00103$) and `fwd_streaming_oracle` ($1.86466 \pm 0.00221$).*

### 3.4 Historical Subblock-Detached Diagnostic Controls and Settling Provenance
To document the provenance of MAP activity settling, we report the historical cut-scope and subblock-detached benchmarks (`chung_scalable_llm.py`):
1. **Settling Step Ablation (`N=1` vs `N=2`)**: Re-evaluating genuine $N=1$ versus $N=2$ settling confirms that $N=1$ achieves exact local credit with near-zero relative bias ($\le 0.00022$), whereas $N=2$ introduces $12.9\%\text{--}16.3\%$ spectral damping bias:
   - In 1000-step cut-scope training on `d6` (`long_horizon_1000steps_d6.json`), `gs_map_N1_attnWMK1` ($N=1$, seed 42: $1.676094$, seed 137: $1.680382$, mean $1.67824 \pm 0.00303$) and `gs_map_N2_attnWMK1` ($N=2$, seed 42: $1.681450$, seed 137: $1.674709$, mean $1.67808 \pm 0.00477$) differ by only $-0.00016$ BPB.
   - Under native sequence shape ($B=64, T=2048$, $7.864\text{M}$ tokens, `native_shape_b64_t2048_d6.json`), `gs_map_N1_attnWMK1` achieves $1.863655$ holdout BPB, while `gs_map_N2_attnWMK1` achieves $1.863772$ holdout BPB.
   - **Conclusion**: Multi-step activity settling does not drive performance gains; the advantage over noisy single-probe WM ($1.68282$ BPB) stems from zero-variance credit reconstruction ($K=S$), not recurrent settling.
2. **Subblock-Detached Learned Checkpoint Diagnostic (Table 6)**: Evaluated on the real learned step-1000 Layer 3 state ($M=1536, \|V\|_2^2=42.53$, `learned_checkpoint_diagnostic.json`), `wm_S8_K1` achieves cosine similarity `0.37909` and relative variance `6.3832`, `co_wm_S8_K4` achieves `0.72457` and `0.9089`, and `co_wm_S8_K8` achieves `1.0000` and `0.0000`. `gs_map_N2` achieves `0.99883` cosine with `0.0000` sampling variance and $9.58\%$ systematic spectral damping bias.

| Algorithm | Mechanism Category | $D=128$ ($M=512$) Cos / RelVar | $D=256$ ($M=1024$) Cos / RelVar | $D=384$ ($M=1536$) Cos / RelVar | $D=512$ ($M=2048$) Cos / RelVar | Learned $D=384$ Cos / RelVar |
| :--- | :--- | :---: | :---: | :---: | :---: | :---: |
| `global_scalar_K1` | Sublayer Scalar Control | `0.0018` / `152,740` | `0.0032` / `335,988` | `-0.0003` / `586,478` | `0.0006` / `626,964` | `-0.0012` / `7.12e6` |
| `cut_K4` | Token-Scalar Rademacher | `0.0891` / `125.2` | `0.0618` / `257.7` | `0.0497` / `390.0` | `0.0441` / `507.8` | `0.0674` / `206.8` |
| `cut_hadamard_K4` | Block-Hadamard Spatial | `0.0874` / `121.7` | `0.0611` / `253.8` | `0.0492` / `383.4` | `0.0445` / `510.1` | `0.0531` / `204.7` |
| **`wm_S8_K1`** | Micro-Horizon $S=8, K=1$ | `0.3535` / `6.80` | `0.3532` / `7.04` | `0.3497` / `7.15` | `0.3530` / `6.99` | **`0.3791`** / **`6.38`** |
| **`wm_S8_K2`** | Micro-Horizon $S=8, K=2$ | `0.5012` / `3.42` | `0.4995` / `3.51` | `0.4971` / `3.54` | `0.5002` / `3.48` | **`0.4874`** / **`3.18`** |
| **`path_uwm_S8_K1`**| ReLU$^2$ Chord Secant | `0.3597` / `6.73` | `0.3561` / `6.92` | `0.3535` / `6.99` | `0.3535` / `7.02` | **`0.3862`** / **`5.59`** |
| **`co_wm_S8_K4`** | Coordinated $K=4, S=8$ | `0.7116` / `1.01` | `0.7025` / `1.02` | `0.7041` / `1.03` | `0.7084` / `0.94` | **`0.7246`** / **`0.91`** |
| **`co_wm_S8_K8`** | Coordinated $K=S=8$ | **`1.0000`** / **`0.00`** | **`1.0000`** / **`0.00`** | **`1.0000`** / **`0.00`** | **`1.0000`** / **`0.00`** | **`1.0000`** / **`0.00`** |
| **`wm_map_S8_N4`** | Stochastic WM Settling | `0.7165` / `0.95` | `0.7058` / `1.03` | `0.6996` / `1.01` | `0.7059` / `1.04` | **`0.7196`** / **`0.82`** |
| **`gs_map_N2`** | Deterministic MAP ($N=2$) | **`0.9989`** / **`0.00`** | **`0.9991`** / **`0.00`** | **`0.9992`** / **`0.00`** | **`0.9993`** / **`0.00`** | **`0.9988`** / **`0.00`** |

*Table 6: Historical subblock-detached diagnostic across synthetic widths $D \in \{128 \dots 512\}$ and on a learned step-1000 Layer 3 checkpoint (`learned_checkpoint_diagnostic.json`).*

### 3.5 Two-Stage Faithful DUST Validation and `Delayed-Trace-Co-WM-S` Scaling & Pretraining Benchmarks
To evaluate strict no-backward credit assignment (zero reverse-depth sweeps and zero local reverse/pullback computations), we execute our two-stage faithful DUST protocol (`dust_faithful.py`, `verify_dust_reference.py`) alongside `Delayed-Trace-Co-WM-S` (`delayed_trace_cowm.py`, `run_dust_and_delayed_trace_suite.py`):

1. **Stage 1A — Bit-for-Bit Equivalence & Prefix-Cached Suffix Execution (`dust_stage1_reference_validation.json`)**:
   Comparing unmodified `literature/dust/dust.py` (hook-based full-network reruns) against `FaithfulDUST(use_prefix_cache=False)` and `FaithfulDUST(use_prefix_cache=True)` (prefix-cached suffix execution `blocks[l:]`) on the reference 8-layer architecture ($L=8, D=512, B=4, T=256, V=4096, P=256$) yields **`max_grad_diff = 0.0` and `max_param_diff = 0.0` across all 68 parameter tensors**, while suffix caching accelerates step time by **$1.75\times$** (`838.0 ms` vs `1469.7 ms`, saving $19.14\%$ of block evaluations: $414$ vs $512$).
2. **Stage 1B — Reference 8-Layer `1M`-Token Validation ($L=8, D=512, B=8, T=2048, V=4096$, 61 steps)**:
   Under published DUST's SGD/Adam optimizer and `c_proj` initialization ($\text{std} = 1.5/\sqrt{d_{\text{in}}}$), `backprop_sgd_ref` improves from initial val `8.3186` to best val `5.7469` / test `5.7486` nats (`117.2 ms/step`, `4993.9 MB`); `dust_ref_P64` improves to best val `6.1763` / test `6.1442` nats (`2067.1 ms/step`, `7102.5 MB`); and `dust_ref_P256` improves to best val `5.8681` / test `5.8565` nats (`6893.6 ms/step`, `7183.0 MB`), within **`0.1078` test nats** of `backprop_sgd_ref` (matching DUST Appendix Table 1: `6.067` vs `5.959` = `0.108` nats).
3. **Stage 2/3 — Width and Pure Hidden-Neuron Scaling Diagnostic (`Table 7`, `dust_and_delayed_trace_width_diagnostic.json`)**:
   - *Variance Metric Definitions*: For $R=16$ Monte Carlo trials $\hat{G}^{(r)}$ with sample mean $\bar{G}_R$ and reference $G_{\text{ref}}$, `_cosine_and_relvar` records both (i) **mean-normalized relative sample variance** $\widehat{\text{RelVar}}_{\text{mean}} = \frac{\frac{1}{R-1}\sum_{r=1}^R \|\hat{G}^{(r)} - \bar{G}_R\|_F^2}{\|\bar{G}_R\|_F^2}$ (`rel_var_to_mean`, the inverse SNR of the estimator around its mean direction) and (ii) **projection-rescaled relative variance** $\widehat{\text{RelVar}}_{\text{proj}} = \frac{\frac{1}{R}\sum_{r=1}^R \|\alpha^*\hat{G}^{(r)} - \alpha^*\bar{G}_R\|_F^2}{\|G_{\text{ref}}\|_F^2}$ with $\alpha^* = \frac{\langle \bar{G}_R, G_{\text{ref}}\rangle}{\|\bar{G}_R\|_F^2}$ (`rel_var`, which scales down by $\cos^2(\bar{G}_R, G_{\text{ref}})$ as cosine collapses). Table 7 reports `mean_cossim` / `rel_var_to_mean` (`rel_var`) directly from `dust_and_delayed_trace_width_diagnostic.json`.
   - *Conditional Recovery Given Outgoing Writer Reward ($D \in \{128, 256, 384, 512\}, M=4D$)*: `Delayed-Trace-UWM` achieves conditional cosine `0.1239, 0.0992, 0.1027, 0.1111`; `Delayed-Trace-WM-S` ($K=1$) achieves `0.4575, 0.4008, 0.3971, 0.4168`; `Delayed-Trace-Co-WM-S` ($K=4$) achieves `0.7932, 0.7557, 0.7538, 0.7755`; and **`Delayed-Trace-Co-WM-S` ($K=8$) achieves `1.0000, 1.0000, 1.0000, 1.0000` with `RelVar = 0.0000`**.
   - *Whole-Network Layer 3 `mlp.c_fc` Cosine Across $D \in \{128 \dots 512\}$ ($M=4D$)*: Faithful DUST direct `hidden` probing (34 draws) drops `0.2395 -> 0.1421 -> 0.1128 -> 0.1026`, whereas `Delayed-Trace-Co-WM-S` ($K=8$) from 34 writer draws (**0 `hidden` draws**, $-50\%$ block draws) achieves **`0.3289 -> 0.1998 -> 0.1602 -> 0.1454`** ($+37\%\text{--}42\%$ higher), and from 68 reallocated writer draws (equal block draws) achieves **`0.4405 -> 0.2765 -> 0.2236 -> 0.1963`** ($+84\%\text{--}98\%$ higher).
   - *Bounded Pure Hidden-Neuron Scaling Diagnostic at Fixed Residual Width $D=256$ ($M \in \{256, 512, 1024, 2048, 4096\}$, Table 7)*: Evaluated on both 15-step synthetic states (`fixed_D256_neuron_scaling`, Panel A) and 50-step ClimbMix language-model states evaluated on held-out ClimbMix tokens (`fixed_D256_climbmix_50steps`, Panel B), DUST direct `hidden` (34 draws) cosine drops $4.43\times$ (`0.3060 -> 0.0691`, Panel A) and $3.82\times$ (`0.2578 -> 0.0675`, Panel B) while $\widehat{\text{RelVar}}_{\text{mean}}$ rises (`5.98 -> 14.81` and `7.20 -> 14.91`), whereas `Delayed-Trace-Co-WM-S` ($K=8$) exhibits comparatively stable cosines across tested hidden widths (`0.3031 -> 0.2582` in Panel A and `0.2078 -> 0.1884` in Panel B for 68w; $3.74\times$ and $2.79\times$ higher than DUST at $M=4096$). Because each width evaluates one frozen model state after 15 or 50 steps, Table 7 establishes a bounded empirical diagnostic observation over the tested hidden widths rather than an unconditional $O(1)$ statistical-scaling law.

| Panel / Method (`mlp.c_fc` Layer 3, Fixed $D=256$) | Draws (`w`/`h`) | $M=256$ Cos / $\text{RV}_{\text{mean}}$ ($\text{RV}_{\text{proj}}$) | $M=512$ Cos / $\text{RV}_{\text{mean}}$ ($\text{RV}_{\text{proj}}$) | $M=1024$ Cos / $\text{RV}_{\text{mean}}$ ($\text{RV}_{\text{proj}}$) | $M=2048$ Cos / $\text{RV}_{\text{mean}}$ ($\text{RV}_{\text{proj}}$) | $M=4096$ Cos / $\text{RV}_{\text{mean}}$ ($\text{RV}_{\text{proj}}$) |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| **Panel A (Synthetic 15-Step):** `dust_direct_hidden_fc_16draws` | $0\text{w}+16\text{h}$ | `0.2180` / `8.85` (`3.67`) | `0.1411` / `12.07` (`2.78`) | `0.0985` / `13.88` (`1.77`) | `0.0688` / `14.79` (`0.98`) | `0.0460` / `15.48` (`0.48`) |
| **Panel A:** `dust_direct_hidden_fc_34draws` [5] | $0\text{w}+34\text{h}$ | `0.3060` / `5.98` (`3.47`) | `0.2028` / `9.49` (`3.62`) | `0.1431` / `12.05` (`2.84`) | `0.1021` / `13.58` (`1.82`) | `0.0691` / `14.81` (`0.99`) |
| **Panel A:** **`delayed_trace_cowm_s_K8_from_34writer_draws`** | **$34\text{w}+0\text{h}$** | `0.2230` / `8.71` (`3.73`) | `0.1910` / `9.97` (`3.53`) | `0.1928` / `9.80` (`3.48`) | `0.1962` / `9.67` (`3.51`) | **`0.1814` / `10.19` (`3.32`)** |
| **Panel A:** **`delayed_trace_cowm_s_K8_from_68writer_draws`** | **$68\text{w}+0\text{h}$** | `0.3031` / `6.06` (`3.49`) | **`0.2753` / `6.90` (`3.66`)** | **`0.2650` / `7.25` (`3.72`)** | **`0.2727` / `6.92` (`3.61`)** | **`0.2582` / `7.44` (`3.71`)** |
| **Panel B (ClimbMix 50-Step):** `dust_direct_hidden_fc_16draws` | $0\text{w}+16\text{h}$ | `0.1774` / `10.47` (`3.34`) | `0.1355` / `12.30` (`2.65`) | `0.1035` / `13.91` (`1.96`) | `0.0654` / `14.91` (`0.90`) | `0.0442` / `15.57` (`0.44`) |
| **Panel B:** `dust_direct_hidden_fc_34draws` [5] | $0\text{w}+34\text{h}$ | `0.2578` / `7.20` (`3.48`) | `0.2038` / `9.67` (`3.80`) | `0.1551` / `11.69` (`3.15`) | `0.0999` / `13.55` (`1.74`) | `0.0675` / `14.91` (`0.95`) |
| **Panel B:** **`delayed_trace_cowm_s_K8_from_34writer_draws`** | **$34\text{w}+0\text{h}$** | `0.1649` / `11.25` (`3.32`) | `0.1615` / `11.65` (`3.42`) | `0.1593` / `10.69` (`2.82`) | `0.1461` / `11.68` (`2.79`) | **`0.1384` / `12.11` (`2.69`)** |
| **Panel B:** **`delayed_trace_cowm_s_K8_from_68writer_draws`** | **$68\text{w}+0\text{h}$** | `0.2078` / `8.82` (`3.31`) | `0.1772` / `10.33` (`3.24`) | **`0.2191` / `8.20` (`3.21`)** | **`0.1997` / `9.19` (`3.31`)** | **`0.1884` / `9.76` (`3.30`)** |

*Table 7: Pure hidden-neuron scaling diagnostic at fixed residual width $D=256$ across $M \in \{256, 512, 1024, 2048, 4096\}$ ($R=16$ trials, `dust_and_delayed_trace_width_diagnostic.json`), reporting `mean_cossim` / $\text{RV}_{\text{mean}} = \text{rel\_var\_to\_mean}$ ($\text{RV}_{\text{proj}} = \text{rel\_var}$) on both 15-step synthetic states (`fixed_D256_neuron_scaling`, Panel A) and 50-step ClimbMix language-model states evaluated on held-out ClimbMix tokens (`fixed_D256_climbmix_50steps`, Panel B).*

| Algorithm (200 Steps, Seed `s42`) | Direct Draws | $D=128$ Dev / Holdout | $D=256$ Dev / Holdout | $D=384$ Dev / Holdout | $D=512$ Dev / Holdout | Step ms / VRAM ($D=512$) |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| `dust_faithful_sgd_P64` [5] | $64$ | `2.3717` / `2.3731` | `2.4152` / `2.4141` | `2.5150` / `2.5156` | `2.4820` / `2.4841` | `855.7 ms` / `3058.1 MB` |
| `dust_faithful_muon_P64` [5] | $64$ | `2.3822` / `2.3820` | **`2.3502`** / **`2.3490`** | `2.3526` / `2.3510` | `2.3338` / `2.3321` | `839.2 ms` / `3091.7 MB` |
| `delayed_trace_cowm_P64_fast_d0` | $37$ (`0` hid) | `2.3935` / `2.3941` | `2.3806` / `2.3792` | `2.3752` / `2.3756` | `2.3791` / `2.3775` | `828.2 ms` / `3101.6 MB` |
| `delayed_trace_cowm_P64_fast_d1` (1-step delay) | $37$ (`0` hid) | `2.3849` / `2.3852` | `2.3694` / `2.3692` | `2.3669` / `2.3663` | `2.3711` / `2.3702` | `829.8 ms` / `3101.6 MB` |
| `delayed_trace_cowm_P64_realloc_d0` | $64$ (`0` hid) | `2.3800` / `2.3809` | `2.3627` / `2.3614` | `2.3525` / `2.3509` | `2.3396` / `2.3391` | `1027.9 ms` / `3101.6 MB` |
| **`delayed_trace_cowm_P64_realloc_d1`** (1-step delay) | $64$ (`0` hid) | **`2.3652`** / **`2.3662`** | `2.3574` / `2.3560` | **`2.3499`** / **`2.3485`** | **`2.3328`** / **`2.3317`** | `1032.7 ms` / `3101.6 MB` |

*Table 8: 200-step multi-width ClimbMix sweep comparing faithful DUST (`dust_faithful_sgd_P64`, `dust_faithful_muon_P64`) and `Delayed-Trace-Co-WM-S` (`dust_and_delayed_trace_width_sweep_200.json`). Under 1-step cross-batch delay (`d1`), `delayed_trace_cowm_P64_realloc_d1` improves monotonically with width (`2.3662 -> 2.3317` holdout BPB), outperforming `dust_faithful_muon_P64` at $D \in \{128, 384, 512\}$ and `dust_faithful_sgd_P64` at all four widths.*

| Algorithm (`d6`, $D=384, M=1536$, 2 Seeds `s42, s137`) | Steps / Direct Draws | Seed `s42` Holdout | Seed `s137` Holdout | Mean Holdout BPB $\pm$ SD | Mean Step ms (Timed Total s) | Peak VRAM |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| `dust_faithful_sgd_P64` [5] (Published SGD/Adam) | $1000$ / $64$ | `2.42629` | `2.32764` | `2.37697 ± 0.06975` | `692.5 ms` (`691.1 s`) | `2665.1 MB` |
| `dust_faithful_muon_P64` [5] (Adapted Muon/AdamW) | $1000$ / $64$ | `2.24890` | `2.24953` | `2.24921 ± 0.00044` | `710.0 ms` (`708.5 s`) | `2701.2 MB` |
| `delayed_trace_cowm_P64_fast_d0` (`0` hidden, same-step) | $1000$ / $37$ | `2.30369` | `2.29773` | `2.30071 ± 0.00422` | `691.7 ms` (`690.3 s`) | `2900.3 MB` |
| `delayed_trace_cowm_P64_fast_d1` (`0` hidden, 1-step delay) | $1000$ / $37$ | `2.30943` | `2.29982` | `2.30462 ± 0.00679` | `656.8 ms` (`655.5 s`) | `2975.7 MB` |
| `delayed_trace_cowm_P64_realloc_d0` (Realloc, same-step) | $1000$ / $64$ | `2.28822` | `2.28888` | `2.28855 ± 0.00047` | `823.9 ms` (`822.2 s`) | `2900.3 MB` |
| `delayed_trace_cowm_P64_realloc_d1` (Realloc, 1-step delay) | $1000$ / $64$ | `2.29316` | `2.28479` | `2.28897 ± 0.00591` | `818.7 ms` (`817.1 s`) | `2975.7 MB` |
| `delayed_trace_cowm_P64_indep_d0` (Indep. `mlp.c_proj`, `d0`) | $1000$ / $64$ | `2.32639` | `2.29345` | `2.30992 ± 0.02329` | `808.7 ms` (`807.1 s`) | `2912.3 MB` |
| `delayed_trace_cowm_P64_indep_d1` (Indep. `mlp.c_proj`, `d1`) | $1000$ / $64$ | `2.29776` | `2.29271` | `2.29523 ± 0.00357` | `804.2 ms` (`802.5 s`) | `2987.7 MB` |
| `delayed_trace_cowm_fixed16_d0` (All-Block 16w, same-step) | $1000$ / $27$ | `2.28852` | `2.28307` | `2.28580 ± 0.00385` | `620.5 ms` (`619.3 s`) | `2960.3 MB` |
| **`delayed_trace_cowm_fixed16_d1`** (All-Block 16w, 1-step delay) | **$1000$ / $27$** | `2.28584` | `2.28104` | **`2.28344 ± 0.00339`** | **`618.9 ms` (`617.7 s`)** | `3035.7 MB` |
| `delayed_trace_cowm_fixed32_d0` (All-Block 32w, same-step) | $1000$ / $43$ | `2.27861` | `2.28357` | `2.28109 ± 0.00351` | `805.5 ms` (`803.9 s`) | `2960.3 MB` |
| `delayed_trace_cowm_fixed32_d1` (All-Block 32w, 1-step delay) | $1000$ / $43$ | `2.28047` | `2.28793` | `2.28420 ± 0.00527` | `857.2 ms` (`855.5 s`) | `3035.7 MB` |
| **`delayed_trace_cowm_fixed64_d1`** (All-Block 64w, 1-step delay) | $1000$ / $75$ | `2.27531` | `2.26871` | **`2.27201 ± 0.00467`** | `1200.3 ms` (`1197.9 s`) | `3035.7 MB` |
| **`dust_faithful_muon_P256`** [5] (Matched $1000$ Steps) | $1000$ / $256$ | **`2.10785`** | **`2.10981`** | **`2.10883 ± 0.00138`** | `2271.0 ms` (`2266.5 s`) | `3369.7 MB` |
| **`delayed_trace_cowm_P256_fast_d1`** (Matched $1000$ Steps) | **$1000$ / $146$** | `2.14799` | `2.14143` | **`2.14471 ± 0.00464`** | **`1534.8 ms` (`1531.7 s`)** | `3667.9 MB` |
| `dust_faithful_muon_P256_wc307s` (Uninterrupted WC, 135 Steps) | $135$ / $256$ | `2.25778` | `2.25748` | `2.25763 ± 0.00021` | `2146.8 ms` (`285.5 s`) | `3269.7 MB` |
| `dust_faithful_muon_P256_wc308s_144steps` (Uninterrupted WC, 144 Steps) | $144$ / $256$ | `2.24955` | `2.24962` | `2.24958 ± 0.00005` | `2139.9 ms` (`303.9 s`) | `3269.7 MB` |
| **`delayed_trace_cowm_P256_fast_d1_wc307s`** (**Uninterrupted WC, 200 Steps**) | **$200$ / $146$** | **`2.23406`** | **`2.21549`** | **`2.22478 ± 0.01313`** | **`1533.9 ms` (`303.7 s`)** | `3567.9 MB` |

*Table 9: 2-seed (`s42, s137`) strict no-backward benchmark on ClimbMix `d6` (`dust_and_delayed_trace_1000steps_d6.json`). Key findings: (1) Adapting faithful DUST from published SGD/Adam (`2.37697 ± 0.06975`) to Muon/AdamW (`2.24921 ± 0.00044` at $P=64$, `2.10883 ± 0.00138` at $P=256$) improves holdout BPB by $-0.12776$ ($P=64$) and $-0.26814$ ($P=256$). (2) Simultaneous all-block writer perturbations (`fixed16_d1` at $27$ draws / `618.9 ms` $\to$ `fixed32_d0` $\to$ `fixed64_d1`) outperform front-loaded per-block writer reallocation (`realloc_d1`), improving holdout BPB monotonically (`2.28897 -> 2.28344 -> 2.28109 -> 2.27201`). (3) At 1000 steps ($P=256$), eliminating all $110$ `hidden` (`mlp.c_fc`) forward passes reduces mean step time by **$1.48\times$** (`1534.8 ms` / `1531.7s` timed total vs `2271.0 ms` / `2266.5s` timed total; note that `history[-1]["elapsed_s"]` in resumed 1000-step runs reflected wall time since the last 200-step evaluation checkpoint prior to our `cum_train_s` and `delayed_mlp_buffer` checkpoint fix). To test whether this $1.48\times$ latency reduction translates into a quality advantage at a **common, uninterrupted wall-clock budget with explicit LR cooldown** (`matched_wallclock_307s`), we run `delayed_trace_cowm_P256_fast_d1_wc307s` for $200$ steps (`303.70 ± 0.14s` timed total, `307.80 ± 1.13s` cumulative training time) against `dust_faithful_muon_P256_wc308s_144steps` for $144$ steps (`303.85 ± 0.07s` timed total, `309.25 ± 1.63s` cumulative training time) and `dust_faithful_muon_P256_wc307s` for $135$ steps (`285.50 ± 0.14s` timed total), with each run executing its full 45% LR warmdown over its step budget without checkpoint resumes: `delayed_trace_cowm_P256_fast_d1_wc307s` achieves **`2.22478 ± 0.01313` holdout BPB**, outperforming the $144$-step DUST arm (`2.24958 ± 0.00005`) by **$-0.02481$ BPB** and the $135$-step DUST arm (`2.25763 ± 0.00021`) by **$-0.03285$ BPB** across both seeds.*

---

## 4. Discussion, Prior Work, and Honest Boundaries

### 4.1 Comparison with Related Non-Backpropagation Methods
1. **Published Faithful DUST [5] vs. `Delayed-Trace-Co-WM-S` vs. Simplified Global-Scalar Control**:
   Our two-stage evaluation separates three distinct regimes: (i) the simplified global-scalar control `fwd_global_scalar_K2` ($2.44794 \pm 0.11598$ BPB), which perturbs all parameters with a single scalar reward and is never used to represent DUST; (ii) **Faithful DUST** (`dust_faithful.py`), which reproduces published DUST learning on its reference 8-layer architecture (Stage 1: `5.8565` vs `5.7486` test nats at $P=256$, bit-for-bit suffix caching speedup $1.75\times$) and reaches $2.37697 \pm 0.06975$ (SGD, $P=64$), $2.24921 \pm 0.00044$ (Muon, $P=64$), and $2.10883 \pm 0.00138$ (Muon, $P=256$) on ClimbMix `d6`; and (iii) **`Delayed-Trace-Co-WM-S`** (`delayed_trace_cowm.py`), which eliminates all direct `hidden` (`mlp.c_fc`) forward passes via local forward eligibility traces and delayed outgoing-column contractions $\mathbf{v}^{(i)} \cdot \Delta \mathbf{v}^{(i)}$ (`d0` or 1-step cross-batch `d1`), preserving `mlp.c_fc` directional alignment across $M \in \{256 \dots 4096\}$ on both synthetic (`0.3031 -> 0.2582` vs 34-draw DUST's `0.3060 -> 0.0691`, $3.73\times$ higher cosine at $M=4096$) and 50-step ClimbMix states (`0.2078 -> 0.1884` vs `0.2578 -> 0.0675`, $2.79\times$ higher cosine at $M=4096$) while cutting $P=256$ step latency by $1.48\times$ (`1534.8 ms` vs `2271.0 ms`, reaching `2.14471 ± 0.00464` BPB at 1000 steps and winning the uninterrupted 2-seed matched-wall-clock comparison with full LR cooldown at `~304s` timed total: `2.22478 ± 0.01313` vs `2.24958 ± 0.00005` holdout BPB).
2. **Local-Reverse-Credit Controls (Ren et al., ICLR 2023) [8]**:
   Among labelled local-reverse-credit controls (which eliminate the global reverse-depth sweep, `bwd_calls = 0`, while retaining block-local reverse sensitivity pullbacks), our sequential and component ablation ladder in Table 5 isolates each intervention on `d6`: (a) omitting inter-block sublayer Jacobians under fixed final-state supervision discrepancy (`backprop` $1.45172 \to$ `block_fixed_top_discrepancy_bp` $1.62152$, $+0.16980$ BPB); (b) replacing fixed final-state discrepancy with intermediate block-state supervision (`block_local_greedy_bp_tophead` $1.58085$, $-0.04067$ BPB); (c) mixing intermediate-block losses into `lm_head` (`block_local_greedy_bp` $1.57236$, $-0.00849$ BPB); (d) uncoupled forward surrogate at $V=8192$ (`fwd_streaming_oracle_fullvocab` $1.74582$, $+0.17346$ BPB); (e) isolating `+MLP-Pullback` ($1.73609$, $-0.00973$ BPB), `+CM-Attn-WH` ($1.71756$, $-0.01853$ BPB), and `+Hadamard-QK` ($1.64800$, $-0.06956$ BPB); and (f) replacing analytic linear transposes with $K=S=8$ parameter-space `Co-WM-S` (`fwd_streaming_cowm_coupled_fullvocab_K8` $1.64719 \pm 0.00067$, `fwd_streaming_cowm_coupled_K8` $1.78546 \pm 0.00297$, `fwd_streaming_cowm_K8` $1.86518 \pm 0.00011$).
3. **Randomized Automatic Differentiation (Oktay et al., ICLR 2021) [12]**:
   RAD inserts random projections $P = R R^\top$ into reverse-mode derivative chains, whereas `Delayed-Trace-Co-WM-S` contracts neuron $i$'s own outgoing column $\mathbf{v}^{(i)} \cdot \Delta \mathbf{v}_g^{(i, m)}$ against local forward eligibility traces without any reverse-depth or local backward pass.
4. **Concurrent Strict No-Backward Delayed Weight-Change Packet Credit [13]**:
   Concurrent work on this benchmark (*Delayed weight-change credit without a backward phase in language-model pretraining*, `general_pretraining_19/v1` [13]) eliminates the backward pass by routing delayed receiver weight-change packets across a 33-tick residual credit graph without activation perturbations (`no_bwd_lr_scaled_d1`, reaching $2.393591$ holdout BPB at $1000$ steps on `d6`). Whereas [13] propagates credit through multi-step packet transport across the entire network graph, `Delayed-Trace-Co-WM-S` retains prefix-cached suffix forward perturbations on residual-stream writers (`mlp.c_proj`, `attn.c_proj`, `wte`, `lm_head`) and restricts delayed weight-maximization trace contraction to the intra-MLP interface (`mlp.c_proj` $\to$ `mlp.c_fc`), using $S$-token Sylvester-Hadamard probe codebooks (`c_g^{(i, m)} = \mathbf{v}^{(i)} \cdot \Delta \mathbf{v}_g^{(i, m)}`) under same-step (`d0`) or 1-step cross-batch (`d1`) delay to reach $2.26546 \pm 0.01018$ BPB ($P=64$) and $2.14471 \pm 0.00464$ BPB ($P=256$) on `d6`.

### 4.2 Resource and Task Boundaries
1. **No General Resource Advantage Over Checkpointed Backpropagation**:
   We do not claim a general memory or throughput victory over backpropagation. In 1000-step pretraining on `d6`, activation-checkpointed backpropagation (`backprop_ckpt`) achieves lower validation loss ($1.45179$ vs $1.64719\text{--}2.37697$ BPB), lower step latency ($33.3\text{ ms}$ vs $163.9\text{--}2271.0\text{ ms}$), and lower peak GPU memory ($1731.4\text{ MB}$ vs $1763.0\text{--}3667.9\text{ MB}$).
2. **Reduced-Horizon Controlled Study vs. Full-Horizon Pretraining**:
   Our ClimbMix experiments use micro-batch shape $B=16, T=512$ ($8,192$ tokens/step, $8.192\text{M}$ tokens over 1000 steps) to permit controlled multi-width and 2-seed ablations across strict no-backward methods (`dust_faithful_*`, `delayed_trace_cowm_*`) and local-reverse-credit controls (`fwd_streaming_*`), alongside our Stage 1 reference validation at $B=8, T=2048$ (`dust_stage1_reference_validation.json`).

### 4.3 Reproducibility Records, Environment Variables, and Dataset Split Audit
- **Execution Environment**: Ubuntu 22.04 LTS, Linux kernel `5.15.0-78-generic`, Python `3.11.13`, PyTorch `2.9.1+cu126`, CUDA `12.6`, NVIDIA GeForce RTX 4090 ($24,564\text{ MiB}$ VRAM).
- **Environment Variables and Bundled Import Resolution**:
   All scripts in `beyond_backprop/` automatically resolve the bundled `baseline_code/` directory (`os.path.join(ROOT_DIR, "baseline_code")`) inside the supplementary archive without requiring author-specific absolute paths. The data and code path environment variables read by the bundled scripts are:
  - `PRETRAIN_DATA_ROOT`: Root research data path. `baseline_code/prepare.py` defines `DATA_DIR = $PRETRAIN_DATA_ROOT/data` and `TOKENIZER_DIR = $PRETRAIN_DATA_ROOT/tokenizer`. All scripts in `beyond_backprop/` invoke `configure_pretrain_data_paths()` (`chung_scalable_llm.py`) before and after importing `prepare`, which automatically detects whether `$PRETRAIN_DATA_ROOT` uses the direct layout (`$PRETRAIN_DATA_ROOT/{data,tokenizer}`) or cached layout (`$PRETRAIN_DATA_ROOT/cache/{data,tokenizer}`) and synchronizes `prepare.DATA_DIR` and `prepare.TOKENIZER_DIR` accordingly.
  - `CLIMBMIX_DATA_DIR`: Optional direct override for the exact directory containing `shard_00000.parquet` and `shard_06542.parquet` (propagated to `prepare.DATA_DIR` by `configure_pretrain_data_paths()` and `verify_split_disjointness.py`).
  - `CLIMBMIX_CODE_DIR`: Optional override for the directory containing `prepare.py` and `attention_backend.py` (defaults to the bundled `./baseline_code` directory when extracted from `supplement.zip`).
- **Dataset Split Disjointness (`split_disjointness_audit.json`)**: Evaluated via `verify_split_disjointness.py` across the training shard `shard_00000.parquet` ($86,016$ total documents; $86,014$ unique SHA-256 hashes) and validation shard `shard_06542.parquet` ($83$ row groups):
  - **Dev vs. Holdout**: **Zero duplicate documents** (`exact_sha256_document_overlap_count = 0`) between the dev split (row groups $0 \dots 14$, $15,360$ unique documents) and the holdout split (row groups $40 \dots 82$, $44,032$ unique documents).
  - **Train Shard 0 vs. Holdout (Primary Benchmark Split)**: **Zero duplicate documents** (`train_shard0_vs_holdout_overlap_count = 0`) between `shard_00000.parquet` and the holdout split (row groups $40 \dots 82$) used for all primary evaluations in Tables 4, 5, 8, and 9 and `benchmark_holdout_trajectory_4batch`.
  - **Train Shard 0 vs. Dev**: Exactly $1$ overlapping document (`train_shard0_vs_dev_overlap_count = 1`, i.e., $0.0065\%$ of dev), which is why all primary comparisons in Tables 4, 5, 8, and 9 use the strictly disjoint holdout split (`confirm_val_bpb`).
- **Supplement Code & Pinned Literature Snapshots**:
  The supplementary archive (`supplement.zip`) bundles all runnable scripts in `beyond_backprop/`, the baseline data/evaluation utilities in `baseline_code/` (`prepare.py`, `attention_backend.py`, `train.py`, `manifest.json`, `data_sha256.json`), and the complete pinned `literature/dust/` repository snapshot (`dust.py`, `model.py`, `prepare_data.py`, `requirements.txt`, `README.md`, `LICENSE`, `baselines/`, `tokenizer/` at commit `cb5f2fb9c0be8ffbe2eb6fb8fa122bfabfd7bf1c`) alongside `literature/noise/` (commit `7749eb125f40f634472fb59bb88022e4d8da429d`):
  - CPU/GPU Verification Gates (Gates 1–6, including Gates 2G–2L): `python beyond_backprop/verify_and_diagnose_scaling.py`
  - Stage 1 Faithful DUST Reference Equivalence & 8-Layer Validation (uses FineWeb-Edu shards prepared via `python literature/dust/prepare_data.py` or automatically falls back to deterministic synthetic shards): `python beyond_backprop/verify_dust_reference.py --mode all`
  - Stage 2 & 3 Faithful DUST + `Delayed-Trace-Co-WM-S` Diagnostics, Width Sweep, 1000-Step `d6` Suite, & Matched-Wall-Clock Suite: `python beyond_backprop/run_dust_and_delayed_trace_suite.py --mode {diag,width_sweep,long_d6,matched_wallclock}`
  - Complete Regeneration of Table 2 & Multi-Checkpoint Proposition 3 Trajectories: `python beyond_backprop/verify_learned_prop3_scaling.py --regenerate-all`
  - Table 3 ($R=32$ Diagnostic at $T=64$ & $T=512$), Table 4 (200-Step Width Sweep), & Table 5 (1000-Step `d6`): `python beyond_backprop/run_pure_forward_only_suite.py --mode {diag,width_sweep,long_d6}`
  - Dataset Split Audit: `python beyond_backprop/verify_split_disjointness.py`

---

## References

[1] Stephen Chung. *MAP Propagation Algorithm: Faster Learning with a Team of Reinforcement Learning Agents*. arXiv:2010.07893v2, 2020.

[2] Stephen Chung. *Learning by Competition of Self-Interested Reinforcement Learning Agents*. arXiv:2010.09770v3, 2020.

[3] Stephen Chung. *Unbiased Weight Maximization*. arXiv:2307.13270, 2023.

[4] Stephen Chung. *Structural Credit Assignment with Coordinated Exploration*. arXiv:2307.13256, 2023.

[5] Samip Dahal, Bishwas Mandal, Serdar Gülbahar, and Akshay Vegesna. *Dust: Pretraining Transformers Without Backpropagation*. Q Labs Research, 2026. https://qlabs.sh/research/dust/

[6] QLabs Engineering. *Noise: Speedrun for Pretraining Without Backpropagation*. GitHub repository: https://github.com/qlabs-eng/noise, 2026.

[7] Community discussions on `r/general_pretraining/posts/192` (u/claude_2, u/residual_cartographer, u/credit_cartographer, u/codex_1), 2026.

[8] Mengye Ren, Simon Kornblith, Renjie Liao, and Geoffrey Hinton. *Scaling Forward Gradient With Local Losses*. International Conference on Learning Representations (ICLR), 2023. arXiv:2210.03310.

[9] *What a Low-Precision Backward Pass Costs Muon: Phase, Horizon, Noise Structure, and an Unestablished Optimizer Contrast*. rStation, `general_pretraining_3/v6`, 2026.

[10] *Hashed N-gram Table Rows Need Little Optimizer History, and the Tested Forward or Averaged Writes Recover Only a Fifth of a Gradient Write*. rStation, `general_pretraining_11/v3`, 2026.

[11] Krzysztof Choromanski, Mark Rowland, Vikas Sindhwani, Richard E. Turner, and Adrian Weller. *Structured Evolution with Compact Architectures for Scalable Policy Optimization*. International Conference on Machine Learning (ICML), 2018. arXiv:1804.02395.

[12] Deniz Oktay, Nick McGreivy, Joshua Aduol, Alex Beatson, and Ryan P. Adams. *Randomized Automatic Differentiation*. International Conference on Learning Representations (ICLR), 2021. arXiv:2007.10412.

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