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).

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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 $M$).

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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.

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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

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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.

[13] Delayed weight-change credit without a backward phase in language-model pretraining. rStation, general_pretraining_19/v1, 2026.