Hadamard Modal-Decoupled Late Morphogenesis: Overcoming the Cooldown Horizon in Progressive Language Model Growth

1. Introduction: The Late-Stage Model Growth Problem

Progressive model growth—training a smaller parent transformer on early tokens and expanding its depth ($L$) or MLP width ($d_{\text{ff}}$) mid-flight—promises substantial pretraining compute savings [1, 2, 3, 4, 5, 6]. However, the practical utility of progressive growth hinges on when the expansion occurs. Expanding early ($20\%\text{--}44\%$ of the token budget) leaves ample training time in the full-capacity target architecture, where standard depth stacking ($G_{\text{stack}}$ [6]) or symmetry-broken width splitting (Noisy Net2Wider [1], LEMON [5], SPARKLING [10]) easily recover; yet early expansion saves only $9\%\text{--}20\%$ of total floating-point operations (FLOPs). Conversely, late-stage model growth—keeping the trunk small for $50.0\%\text{--}75.1\%$ of the token budget to save $22.3\%\text{--}34.1\%$ of total compute—forces the expanded model to differentiate newly added parameters inside the linear learning-rate cooldown window ($p \in [0.50, 1.00]$).

Under a linear warmdown schedule $\eta(p) = \eta_{\max} \cdot 2(1 - p)$ for $p \in [0.5, 1.0]$ (following constant $\eta(p) = \eta_{\max}$ on $p \in [0, 0.5]$), the remaining cumulative learning-rate budget shrinks quadratically with the remaining token fraction $1 - p_{\text{grow}}$:

$$\Lambda(p_{\text{grow}}) = \int_{p_{\text{grow}}}^{1} \eta(p)\,dp = \eta_{\max}(1 - p_{\text{grow}})^2, \qquad \frac{\Lambda(p_{\text{grow}})}{\Lambda(0.5)} = \frac{(1 - p_{\text{grow}})^2}{(0.5)^2} = 4(1 - p_{\text{grow}})^2.$$

Expanding at $p_{\text{grow}} = 0.659$ (gs=628 of 953 steps) or $p_{\text{grow}} = 0.751$ (gs=716) leaves only $46.5\%$ or $24.8\%$ of the post-peak warmdown learning-rate integral ($\mathbf{15.5\%}$ or $\mathbf{8.3\%}$ of the total run-wide LR integral $\int_0^1 \eta(p)\,dp = 0.75\eta_{\max}$). When we evaluate competitive literature growth operators across this late-stage horizon under the Muon+AdamW optimizer [7, 8, 9], we uncover a sharp Late-Expansion Crossover (Section 3 and Section 5):

1. The Zero-Readout Cooldown Trap ($G_{\text{zero}}$): True identity depth insertion—zeroing the output projection $W_{\text{proj}} = 0$ and setting residual scalars $\lambda_{\text{resid}}=1, \lambda_{x_0}=0$ on budded blocks (identity_zero)—achieves an exact 0.000000-nat forward loss transition in BF16. However, zeroed readouts block backward gradients into the newly inserted internal projections ($W_Q, W_K, W_V, W_{\text{fc}}$) at the first post-expansion step ($t = t_{\text{grow}}$):

$$\left.\frac{\partial \mathcal{L}}{\partial h_{\text{mid}}}\right|_{t = t_{\text{grow}}} = W_{\text{proj}}^\top \frac{\partial \mathcal{L}}{\partial y} = 0,$$

and delay internal feature learning until $W_{\text{proj}}$ accumulates norm under a decaying learning rate. During linear LR warmdown, this delayed warmup causes $G_{\text{zero}}$ ($4\text{L}\to 8\text{L}$) to fall behind our method by $+0.0188$ to $+0.0231$ bits-per-byte (BPB) across $50.0\%\text{--}75.1\%$ expansion timings (1.034054 vs. 1.015243 val BPB at $65.1\%$ tokens). 2. Unattenuated Forward Shock and Differential-Mode Suppression in Stacked Muon ($G_{\text{stack}} + \text{SPARKLING}$): Stacking trained layers ($G_{\text{stack}}$ [6]) with asymmetric optimizer state resetting ($S' = [S, 0]$) and new-parameter learning-rate re-warmup (SPARKLING, Yu et al., 2026 [10]) avoids zero-readout gradient starvation, and at early expansion ($43.9\%$ tokens, gs=418) beats from-scratch baselines (0.983304 vs. 0.987346 Static 12L Iso-FLOPs). Yet when applied during linear LR cooldown ($65.9\%$ and $75.1\%$ tokens), unattenuated layer copying inflicts a $+0.23$ to $+1.12$ nat immediate forward loss shock while re-warmed learning rates amplify transient trajectory drift. Simultaneously, when MLP width doubles ($d_{\text{ff}} = 1024 \to 2048$), standard stacked muon_step_fused normalizes the small differential gradient $g_- = (g_2 - g_1)/\sqrt{2}$ against the dominant common mode $g_+ = (g_1 + g_2)/\sqrt{2}$, preconditions both width halves through a single shared Gram polynomial ($A_{\text{fc}} = X_1^\top X_1 + X_2^\top X_2$ for $W_{\text{fc}}$ and $A_{\text{proj}} = X_1 X_1^\top + X_2 X_2^\top$ for $W_{\text{proj}}$), and constrains the sum of squared update norms across the two halves to $\|U_1\|_F^2 + \|U_2\|_F^2 = \|U\|_F^2$ without granting an aspect-ratio LR boost on the wide readout $W_{\text{proj}} \in \mathbb{R}^{512\times 2048}$ (Theorem 1B), leaving daughter slices poorly differentiated ($\cos(W_{\text{fc}}^{(1)}, W_{\text{fc}}^{(2)}) \in [0.809, 0.933]$). Consequently, at $65.9\%$ and $75.1\%$ expansion, G_stack + SPARKLING falls behind training the final 12-layer model from scratch at matched FLOPs (1.002396 ± 0.001997 vs. 0.998472 ± 0.001947 across 3 seeds at $65.9\%$; 1.008111 vs. 1.002679 at $75.1\%$).

To overcome both late-expansion failure modes, we design Hadamard Modal-Decoupled Late Morphogenesis (HSDM / HSDM-2D), a unified late-growth mechanism combining three synergistic components:

• Whole-Block Dual Straight-Through Residual Homotopy (DSTRH, Theorem 3): Attenuates newly budded transformer blocks in the forward pass by a linear ramp $\beta(t) = \beta_0 + (1 - \beta_0)\min(1, (t - t_{\text{grow}})/W)$ ($\beta_0 = 0, W = 25$) while routing unit-gain direct parameter gradients $\frac{\partial x_{\text{out}}}{\partial \theta_\ell} = \frac{\partial \text{Block}_\ell(\tilde{x}_{\text{sub}})}{\partial \theta_\ell}$ into the budded weights from the very first post-mitosis micro-step and scaling upstream trunk input derivatives by $\frac{\partial \tilde{x}_{\text{sub}}}{\partial x_{\text{sub}}} = \beta(t) I$. This reduces the +0.28-nat forward shock to loss_bf16_abs_diff = 0.000010–0.000287 nats without starving backward gradients during LR cooldown.
• Stiefel Row-Tangent / Isometric Polar-Express Width Splitting (stiefel_tangent & isometric_pe, Theorem 2A) & Hadamard Modal Muon (modal_hadamard, Theorem 2B): Breaks width symmetry at step 0 via either row-orthogonal Stiefel tangent input splitting $W_{\text{fc}}^{\text{new}} = [W_{\text{fc}} + \Delta_{\text{fc}}^\perp; W_{\text{fc}} - \Delta_{\text{fc}}^\perp]$ with symmetric readout $[\frac{1}{2}W_{\text{proj}}, \frac{1}{2}W_{\text{proj}}]$ (stiefel_tangent, which cancels first-order $O(\alpha)$ ReluSquared cross-terms) or exact zero-loss Isometric Polar-Express output splitting $W_{\text{proj}}^{\text{new}} = [\frac{1}{2}(W_{\text{proj}} + \Delta_{\text{proj}}), \frac{1}{2}(W_{\text{proj}} - \Delta_{\text{proj}})]$ (isometric_pe, mlp_fp32_rel_err = 0.000209). In Stage 2, sibling Nesterov gradients are transformed via the $2\times 2$ Hadamard basis $H_2$ (modal_width_muon_step_fused) and $4\times 4$ Kronecker Hadamard basis $H_2 \otimes H_2$ (modal_2d_muon_step_fused) with cosine-decaying modal gain $\gamma(t) = 1 + (\gamma_0 - 1)\cos^2(\frac{\pi}{2}\frac{t - t_{\text{grow}}}{T_{\text{rem}}})$ ($\gamma_0 = 2.0$), so that linearly independent common ($g_+$) and differential ($g_-$) modes are orthogonalized via independent Polar Express Gram matrices, driving $\cos(W_{\text{fc}}^{(1)}, W_{\text{fc}}^{(2)})$ down to $0.714\text{--}0.719$ within 70 steps.
• Post-Mitosis Micro-Batch Subdivision & Momentum Cooldown Synthesis: Subdividing gradient accumulation in Stage 2 (accum 4 -> 2) doubles the count of orthogonalized Muon updates during the remaining LR cooldown window at identical token budget and model FLOPs, while ramping Muon momentum $\mu: 0.95 \to 0.85$ post-mitosis (synthesizing the terminal momentum cooldown insight of cartan_geodesic [13]) further sharpens convergence.

Summary of Contributions (177 Pretraining Runs: ~155 Full-Token 499.65M-Token Runs & ~22 Matched-Compute 328M–477M-Token Static Iso-FLOPs Baselines)

1. Mechanistic Theory of Late Expansion under Muon (Section 3): Building on Yu et al.'s (2026) [10] proof of duplicate gradient symmetry under polynomial Muon (Theorem 1A), we establish the Shared-Gram Differential-Mode Suppression and Norm-Conservation Theorem (Theorem 1B), the Stiefel Row-Tangent / Isometric Polar-Express & Hadamard Modal Muon Theorems (Theorems 2A & 2B) (with explicit non-collinearity conditions), and the Whole-Block Dual Straight-Through Residual Homotopy Theorem (Theorem 3), backed by exact pre/post-expansion FP32 and BF16 forward fidelity certificates (mitosis_fidelity in result.json).
2. Overcoming the Late-Expansion Cooldown Horizon in Coupled 2D Growth (Section 5.1): Across the full timing curve ($43.9\%, 50.0\%, 65.9\%, 75.1\%$ tokens) on $(8\text{L}, r=2) \to (12\text{L}, r=4)$ ($41.94\text{M} \to 71.30\text{M}$ parameters), HSDM-2D strictly dominates both G_stack + SPARKLING and matched-compute Static 12L Iso-FLOPs at every timing, and transfers to reduced scale ($d = 384$) and exact Isometric Polar-Express output splitting (isometric_pe, 0.992815 val / 0.991972 holdout BPB at $65.9\%$ tokens).
3. Clean One-Factor-at-a-Time (OFAT) Causal Ablations, 3-Seed Batch-Matched Static Controls, and Tuned AdamW Comparisons (Section 5.1): Holding fixed batch (accum=4), stiefel_tangent, and gs=628 constant, we isolate the individual causal effect of DSTRH vs. ordinary linear homotopy (lerp) vs. no homotopy (hom=0), and of modal_hadamard vs. standard Muon (dipolar=none); we also evaluate Yu et al.'s exact Eq. (27)–(28) re-warmup schedule (sparkling_eq28), a 3-seed batch-matched Static 12L Iso-FLOPs VarBatch control (s42, s137, s2026) with exact paired $95\%$ $t$-confidence intervals, and tuned AdamW hidden-matrix pretraining (--matrix-opt adamw).
4. 1D Depth Late Growth & Empirical Capacity Horizon (Section 5.2): On pure depth growth $4\text{L}\to 8\text{L}$ at $65.1\%$ tokens (gs=620, $29.1\%$ FLOP savings vs. full Static 8L, 3 seeds), HSDM-w80 (modal_haar Stiefel rotation $\alpha_0=0.45$ over 80 steps + $\beta_0=0.0, W=25$ DSTRH) achieves 1.017490 ± 0.003816 val / 1.016807 ± 0.003905 holdout BPB, beating $G_{\text{stack}}$ (1.019193 ± 0.004076) on all 3 seeds by -0.001703 ± 0.000487 val BPB ($t = -6.05, p = 0.0262$) and beating Static 8L Iso-FLOPs (1.020017 ± 0.002066) on all 3 seeds by -0.002527 val BPB.

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2. Experimental Setup, FLOPs Accounting, and Evaluation Protocol

2.1 Reference Architecture, Optimizer, and Exact Per-Token FLOPs Accounting

We conduct all experiments on the standardized r/general_pretraining benchmark (ClimbMix corpus tokenized with a GPT-2 BPE tokenizer, vocabulary $V = 8192$, sequence length $T_{\text{seq}} = 2048$, full token budget $B = 499,646,464$ tokens $= 953$ steps at reference batch size $524,288$ tokens/step $= 4$ microbatches of $64 \times 2048$) [11, 12, 13, 14]. The reference Static 8L baseline ($L = 8$, $d = 512$, $n_h = n_{kv} = 4$ heads of dimension $d_h = 128$, MLP expansion ratio $r = d_{\text{ff}}/d = 4$, ReluSquared activation $\phi(u) = (\max(0, u))^2$, repeating 4-layer window pattern SSSL with 3 sliding-window $W_{\text{attn}} = 1024$ layers followed by 1 full-context $W_{\text{attn}} = 2048$ layer, and alternating value embeddings ve on odd layer indices) has $50,332,176$ parameters (at reduced scale $d = 384$, $n_h = n_{kv} = 3$ heads of dimension $d_h = 128$). Two-dimensional hidden matrices in attention and MLP blocks are optimized with fused Muon (muon_step_fused: Nesterov momentum warming from $0.85 \to 0.95$ over 300 steps, 5-step quintic Polar Express orthogonalization, NorMuon per-row/column second-moment normalization $\beta_2 = 0.95$, base matrix learning rate MATRIX_LR = 0.04 with aspect-ratio multiplier $\sqrt{\max(1, m/n)}$, cautious weight decay $0.2(1 - p)$, and linear LR warmdown over the final $50\%$ of token-budget progress $p \in [0.5, 1.0]$); embeddings (EMBEDDING_LR = 0.6), unembeddings (UNEMBEDDING_LR = 0.004), value embeddings, and per-layer residual scalars ($\lambda_{\text{resid}}, \lambda_{x_0}$, SCALAR_LR = 0.5) use AdamW ($\beta = (0.8, 0.95)$). For optimizer-generality controls (--matrix-opt adamw), hidden matrices use AdamW with tuned base learning rate 0.003.

To avoid any ambiguity between matrix-only and attention-inclusive compute accounting, every FLOPs value reported in this paper and in all_runs_summary.csv uses the exact attention-inclusive model FLOP estimator estimate_flops() recorded in result.json:

$$F_{\text{tok}}(L, d, r) = 6\left(N_{\text{params}}(L, d, r) - N_{\text{emb}}(L, d)\right) + 12 \sum_{\ell=0}^{L-1} n_h d_h W_{\text{attn}}(\ell).$$

Note that $F_{\text{tok}}(L, d, r)$ measures forward-and-backward model matrix-multiply and attention FLOPs (scaling with sequence length $B_{\text{tok}}$), excluding $O(d^2)$ per-step optimizer polynomial operations that are independent of batch token count; therefore, we report both cumulative_flops and actual GPU training and wall-clock time (total_training_time, total_seconds) for every experiment. At $d = 512$ ($n_h = 4, d_h = 128$) and $d = 384$ ($n_h = 3, d_h = 128$), the exact parameter counts and per-token FLOPs are:

• d=512, 4L, r=2 ($25,166,088$ params, $25.17\text{M}$): $1.069563 \times 10^8$ FLOPs/tok ($0.4474\times$ Static 8L).
• d=512, 4L, r=4 ($29,360,392$ params, $29.36\text{M}$): $1.321221 \times 10^8$ FLOPs/tok ($0.5526\times$ Static 8L).
• d=512, 8L, r=2 ($41,943,568$ params, $41.94\text{M}$): $1.887468 \times 10^8$ FLOPs/tok ($0.7895\times$ Static 8L; $0.5455\times$ Static 12L).
• d=512, 8L, r=4 (Static 8L, $50,332,176$ params, $50.33\text{M}$): $2.390784 \times 10^8$ FLOPs/tok; 953-step budget $F_{\text{base}, 8\text{L}} = \mathbf{1.194547 \times 10^{17}}$ FLOPs (flops_ratio = 1.0000).
• d=512, 12L, r=4 (Static 12L, $71,303,960$ params, $71.30\text{M}$): $3.460347 \times 10^8$ FLOPs/tok ($1.4474\times$ Static 8L); 953-step budget $F_{\text{base}, 12\text{L}} = \mathbf{1.728950 \times 10^{17}}$ FLOPs (flops_ratio = 1.4474).
• d=512, 16L, r=4 (Static 16L, $92,275,744$ params, $92.28\text{M}$): $4.529910 \times 10^8$ FLOPs/tok ($1.8947\times$ Static 8L); 953-step budget $F_{\text{base}, 16\text{L}} = \mathbf{2.263353 \times 10^{17}}$ FLOPs (flops_ratio = 1.8947).
• d=384 Configurations ($n_h = 3, d_h = 128$): 8L, r=2 has $28,311,952$ params ($1.226857 \times 10^8$ FLOPs/tok); 8L, r=4 (d=384 Static 8L) has $33,030,544$ params ($1.509972 \times 10^8$ FLOPs/tok, $0.6316\times$ d=512 Static 8L); 12L, r=4 (d=384 Static 12L) has $46,400,088$ params ($2.170587 \times 10^8$ FLOPs/tok, $0.9079\times$ d=512 Static 8L).

For every growth schedule expanding at token step $g_1$ (and optional $g_2$), cumulative training compute is integrated micro-step by micro-step as $F_{\text{total}} = \sum_{s} B_{\text{tok}}(s)\,F_{\text{tok}}(L(s), d, r(s))$. When comparing a growth model reaching target architecture $\mathcal{A}_{\text{target}} \in \{8\text{L}, 12\text{L}\}$ at total compute $F_{\text{total}}$, our Matched-Compute Static Baseline (Static Iso-FLOPs) trains $\mathcal{A}_{\text{target}}$ from scratch for $N_{\text{iso}} = \text{round}(F_{\text{total}} / (B_{\text{step}} F_{\text{tok}}(\mathcal{A}_{\text{target}})))$ steps ($328\text{M}\text{--}477\text{M}$ tokens) with its entire learning-rate schedule and $50\%$ linear cooldown compressed to $N_{\text{iso}}$ steps so that it finishes cooldown at exact matched compute $F_{\text{total}}$. Whenever a growth schedule uses Stage-2 micro-batch subdivision (accum 4 -> 2), we also run a 3-Seed Variable-Batch Static Iso-FLOPs Control (VarBatch Static Iso-FLOPs) that switches accum 4 -> 2 at the exact same token fraction, isolating the architectural growth mechanism from batch-size dynamics.

2.2 Dual-Split Evaluation (val_bpb & Untouched holdout_bpb) and BF16 Mitosis Fidelity

Every primary run is evaluated in bfloat16 on two strictly disjoint $20,971,520$-token splits ($20 \times 524,288$-token batches each):

1. Official Validation Split (val_bpb): First $20.97\text{M}$ tokens of val_tokens.bin ($\text{BPB} = \mathcal{L}_{\text{CE}} / 2.80203023$).
2. Untouched Holdout Split (holdout_bpb): Token offsets $[20,971,520 : 41,943,040]$ of val_tokens.bin, never inspected during design or hyperparameter selection.
3. Exact Pre/Post-Expansion Forward Fidelity (mitosis_fidelity): Immediately before and after each expansion step $t_{\text{grow}}$, train_growth.py evaluates the exact bfloat16 cross-entropy loss (loss_bf16_before, loss_bf16_after, loss_bf16_abs_diff) and the relative MLP output discrepancy (mlp_fp32_rel_err, mlp_bf16_rel_err) on a fixed reference batch, storing the certificate inside result.json.

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3. Theory: Why Late Expansion Fails and How HSDM Solves It

3.1 Exact Muon Update Equations, Duplicate Symmetry-Locking, and Shared-Gram Differential-Mode Suppression

In our benchmark (experiments/train_growth.py:495-525), muon_step_fused updates each 2D weight matrix $W_t \in \mathbb{R}^{m \times n}$ via four sequential operations:

1. Nesterov Momentum: $M_t = \mu M_{t-1} + (1 - \mu)G_t$, $\tilde{G}_t = (1 - \mu)G_t + \mu M_t$ ($\mu = 0.95$).
2. 5-Step Quintic Polar Express (polar_express): Normalize $X_0 = \tilde{G}_t / (\|\tilde{G}_t\|_F \cdot 1.02 + 10^{-6})$. For $k = 0, \dots, 4$ with precomputed quintic coefficients $(a_k, b_k, c_k)$:
   • If $m > n$ (tall matrix, e.g., $W_{\text{fc}} \in \mathbb{R}^{d_{\text{ff}}\times d}$): $A_k = X_k^\top X_k \in \mathbb{R}^{n\times n}$, $B_k = b_k A_k + c_k A_k^2$, $X_{k+1} = a_k X_k + X_k B_k$.
   • If $m \le n$ (wide matrix, e.g., $W_{\text{proj}} \in \mathbb{R}^{d\times d_{\text{ff}}}$): $A_k = X_k X_k^\top \in \mathbb{R}^{m\times m}$, $B_k = b_k A_k + c_k A_k^2$, $X_{k+1} = a_k X_k + B_k X_k$.
3. NorMuon Second-Moment Normalization: Maintain exponential moving average $v_t = \beta_2 v_{t-1} + (1 - \beta_2)\,\text{mean}_{\text{axis}}(X_5 \odot X_5)$ (reduced over columns if $m \ge n$, else rows), and rescale $U_t = X_5 \odot s_t$ so that $\|U_t\|_F = \|X_5\|_F$.
4. Cautious Weight Decay & Step: Mask decay where $U_t \odot W_t \ge 0$, and apply $W_{t+1} = W_t (1 - \eta_t \lambda \mathbb{I}[U_t \odot W_t \ge 0]) - \eta_t U_t$, where $\eta_t = \eta_{\text{base}} \max(1, m/n)^{1/2}$.

When an MLP block $y = W_{\text{proj}}\phi(W_{\text{fc}} x)$ is widened from $d_{\text{ff}}$ to $2d_{\text{ff}}$, classical function-preserving width splitting (Net2WiderNet [1], $\alpha = 0$) sets $W_{\text{fc}}^{\text{new}} = [W_{\text{fc}}; W_{\text{fc}}] \in \mathbb{R}^{2d_{\text{ff}}\times d}$ and $W_{\text{proj}}^{\text{new}} = [\frac{1}{2}W_{\text{proj}}, \frac{1}{2}W_{\text{proj}}] \in \mathbb{R}^{d\times 2d_{\text{ff}}}$. As recently proved by Yu et al. (2026) (SPARKLING, Section 4.1, Eqs. 21–23) [10], exact duplicate initialization with duplicated optimizer states remains permanently locked under polynomial Muon:

Theorem 1A (Exact Duplicate Symmetry Preservation — Yu et al., 2026 [10], Section 4.1): Under any deterministic first-order optimizer (SGD, AdamW) or matrix-orthogonalized optimizer (muon_step_fused), initializing $W_{\text{fc}}^{\text{new}} = [W_{\text{fc}}; W_{\text{fc}}]$ and $W_{\text{proj}}^{\text{new}} = [\frac{1}{2}W_{\text{proj}}, \frac{1}{2}W_{\text{proj}}]$ with duplicated optimizer state yields identical activations $h^{(1)}_t \equiv h^{(2)}_t$, identical gradients $G^{(1)}_t \equiv G^{(2)}_t$, and identical updates $U^{(1)}_t \equiv U^{(2)}_t$ for all $t \ge t_{\text{grow}}$. In particular, for any block-duplicated matrix $M = [A; A]$, every odd polynomial step $X_{k+1} = X_k(a_k I + b_k X_k^\top X_k + c_k (X_k^\top X_k)^2)$ and axis-reduced NorMuon normalizer preserves $U^{(1)}_t = U^{(2)}_t$ identically, locking $\cos(W^{(1)}_t, W^{(2)}_t) = 1.0000$ in exact and BF16 arithmetic.

While Theorem 1A (due to Yu et al. [10]) explains why $\alpha = 0$ is rank-locked, Theorem 1B identifies the coupled-Gram preconditioner and norm-conservation bottlenecks that persist in stacked muon_step_fused even after symmetry is broken by LEMON [5], Noisy Net2Wider [1], or SPARKLING [10]:

Theorem 1B (Shared-Gram Differential-Mode Suppression and Total Frobenius Energy Conservation in Stacked Muon): *Let $g_1, g_2$ denote the two Nesterov-blended width-half gradients of a widened MLP matrix ($W_{\text{fc}} = [W_{\text{fc}}^{(1)}; W_{\text{fc}}^{(2)}] \in \mathbb{R}^{2d_{\text{ff}}\times d}$ with $2d_{\text{ff}} > d$, or $W_{\text{proj}} = [W_{\text{proj}}^{(1)}, W_{\text{proj}}^{(2)}] \in \mathbb{R}^{d\times 2d_{\text{ff}}}$ with $d < 2d_{\text{ff}}$), and define the orthonormal $2\times 2$ Hadamard modes $g_+ = \frac{1}{\sqrt{2}}(g_1 + g_2)$ (common mode) and $g_- = \frac{1}{\sqrt{2}}(g_2 - g_1)$ (differential mode). When standard muon_step_fused is applied to the stacked matrix $g$:

1. $O(\epsilon)$ Differential-Mode Attenuation under Shared-Gram Preconditioning: Shortly after symmetry-broken mitosis, when the differential gradient is small relative to the common mode ($\|g_-\|_F = \epsilon \|g_+\|_F$ with $\epsilon \ll 1$), stacked muon_step_fused normalizes $X_0 = g / (\sqrt{\|g_+\|_F^2 + \|g_-\|_F^2}\cdot 1.02)$ jointly. Thus the differential component enters Polar Express at amplitude $\|X_{0,-}\|_F = O(\epsilon / \sqrt{1+\epsilon^2})$ and is right- or left-multiplied at every iteration by the shared Gram polynomial $B_k(A_k)$ where $A_k = X_{k,+}^\top X_{k,+} + X_{k,-}^\top X_{k,-} = X_{k,+}^\top X_{k,+} + O(\epsilon^2)$ (for $W_{\text{fc}}$) or $A_k = X_{k,+} X_{k,+}^\top + X_{k,-} X_{k,-}^\top$ (for $W_{\text{proj}}$). Rather than orthogonalizing $g_-$ along its own singular directions, stacked Muon preconditions $g_-$ using the singular spectrum of the dominant common mode $g_+$.
2. Total Squared Norm Conservation and Asymmetric $W_{\text{fc}}$ vs. $W_{\text{proj}}$ LR Scaling: Because the non-singleton dimension of the shared Gram matrix $A_k \in \mathbb{R}^{d \times d}$ remains $d = \min(m, n)$ when $d_{\text{ff}}$ doubles from $1024 \to 2048$, the total Polar Express output squared Frobenius norm satisfies the exact trace conservation identity

$$\|U^{(1)}\|_F^2 + \|U^{(2)}\|_F^2 = \|U\|_F^2 = \|X_5\|_F^2 \approx c_{\text{PE}}^2 d \quad (\approx 525.2\text{ at }d = 512\text{ vs. }522.0\text{ in Stage 1}).$$

Under exact sibling symmetry ($X_1 = X_2$) or orthogonal equivalence ($X_2 = Q X_1$), both halves receive equal halves $\|U^{(1)}\|_F^2 = \|U^{(2)}\|_F^2 = \frac{1}{2}\|U\|_F^2$ (while under asymmetric momentum reset $S' = [S, 0]$ where $X_1 \neq X_2$, the shared-Gram polynomial distributes the fixed total energy $\|U\|_F^2$ unequally between the two halves). Crucially, for the tall matrix $W_{\text{fc}} \in \mathbb{R}^{2048\times 512}$, the aspect-ratio LR factor $\eta_{\text{fc}} = \eta_{\text{base}}\sqrt{\max(1, 2048/512)} = 2\eta_{\text{base}}$ increases by $\sqrt{2}$ relative to Stage 1 ($\sqrt{1024/512}\eta_{\text{base}} = \sqrt{2}\eta_{\text{base}}$), doubling total squared step energy $\|\eta_{\text{fc}} U_{\text{fc}}\|_F^2$. Conversely, for the wide readout matrix $W_{\text{proj}} \in \mathbb{R}^{512\times 2048}$, $\max(1, 512/2048) = 1$ in both Stage 1 and Stage 2, so $\eta_{\text{proj}} = \eta_{\text{base}}$ receives no $\sqrt{2}$ increase and the average per-half squared post-LR update norm $\frac{1}{2}\|\eta_{\text{proj}} U_{\text{proj}}\|_F^2$ is halved ($0.105$ vs. $0.209$ in Stage 1).

Empirically, in our diagnostic logs (diagnostics.jsonl), even when SPARKLING resets half of the momentum buffer and applies LR re-warmup, standard stacked Muon leaves the two MLP halves at $\cos(W_{\text{fc}}^{(1)}, W_{\text{fc}}^{(2)}) = 0.809$ (gs=418) to $0.933$ (gs=628).

3.2 Theorems 2A & 2B: Width-Splitting Operators (dipolar_pe, isometric_pe & stiefel_tangent) and Hadamard Modal Muon (modal_hadamard)

To break width symmetry cleanly at mitosis and eliminate differential-mode suppression during late-stage cooldown, our implementation in experiments/train_growth.py provides two families of width-initialization operators (Theorem 2A) paired with $2\times 2$ (H_2) and $4\times 4$ (H_2 \otimes H_2) Hadamard Modal Muon (Theorem 2B):

Theorem 2A (Exact Isometric Readout Dipolar Split vs. Stiefel Row-Tangent Input Split): Consider an MLP block $y = W_{\text{proj}}\phi(W_{\text{fc}}x)$ with quadratic ReluSquared activation $\phi(u) = (\max(0, u))^2$ widened from $d_{\text{ff}}$ to $2d_{\text{ff}}$:

1. Exact Function-Preserving Readout Dipolar Split (dipolar_pe & isometric_pe, train_growth.py:1244-1265): Duplicates $W_{\text{fc}}^{\text{new}} = [W_{\text{fc}}; W_{\text{fc}}]$ and splits the readout asymmetrically as $W_{\text{proj}}^{\text{new}} = [\frac{1}{2}(W_{\text{proj}} + \Delta_{\text{proj}}), \frac{1}{2}(W_{\text{proj}} - \Delta_{\text{proj}})]$, where $\Delta_{\text{proj}} = \alpha\,U_{\text{proj}} \odot (\|W_{\text{proj}}\|_{\text{col}} / \|U_{\text{proj}}\|_{\text{col}})$ (dipolar_pe) or $\Delta_{\text{proj}} = \alpha\,(\|W_{\text{proj}}\|_F / \|U_{\text{proj}}\|_F) U_{\text{proj}}$ (isometric_pe) and $U_{\text{proj}} = \text{PolarExpress}(Z)$. Because $W_{\text{proj}}^{(1)} + W_{\text{proj}}^{(2)} = W_{\text{proj}}$ and $\phi(W_{\text{fc}}^{(1)}x) = \phi(W_{\text{fc}}^{(2)}x)$, the forward map is preserved with zero error in exact arithmetic (mlp_fp32_rel_err = 0.000000–0.000209, loss_bf16_abs_diff = 0.000047–0.000079 nats) while injecting a non-collinear step-1 backward gradient dipole $G_{\text{fc}}^{(1)} - G_{\text{fc}}^{(2)} = (\Delta_{\text{proj}}^\top \frac{\partial\mathcal{L}}{\partial y} \odot \phi'(W_{\text{fc}}x))x^\top \ne 0$.
2. Stiefel Row-Tangent Input Split (stiefel_tangent, train_growth.py:1178-1206): Perturbs the incoming weights along the row-orthogonal Stiefel tangent space of $W_{\text{fc}}$ while splitting the readout symmetrically:

$$W_{\text{fc}}^{\text{new}} = \begin{bmatrix} W_{\text{fc}} + \Delta_{\text{fc}}^\perp \\ W_{\text{fc}} - \Delta_{\text{fc}}^\perp \end{bmatrix} \in \mathbb{R}^{2d_{\text{ff}}\times d}, \qquad W_{\text{proj}}^{\text{new}} = \begin{bmatrix} \frac{1}{2}W_{\text{proj}}, & \frac{1}{2}W_{\text{proj}} \end{bmatrix} \in \mathbb{R}^{d\times 2d_{\text{ff}}},$$

where $U_{\text{fc}} = \text{PolarExpress}(Z)$ is projected row-wise orthogonal to each row $w_{\text{fc}, i}$ of $W_{\text{fc}}$:

$$u_{\text{fc}, i}^\perp = u_{\text{fc}, i} - \frac{\langle u_{\text{fc}, i}, w_{\text{fc}, i}\rangle}{\|w_{\text{fc}, i}\|_2^2} w_{\text{fc}, i}, \qquad \Delta_{\text{fc}, i}^\perp = \frac{\alpha}{2}\frac{\|w_{\text{fc}, i}\|_2}{\|u_{\text{fc}, i}^\perp\|_2} u_{\text{fc}, i}^\perp.$$

By row-orthogonality $\langle \Delta_{\text{fc}, i}^\perp, w_{\text{fc}, i}\rangle = 0$, every daughter pair has exact row-norm parity $\|w_{\text{fc}, i} + \Delta_{\text{fc}, i}^\perp\|_2 = \|w_{\text{fc}, i} - \Delta_{\text{fc}, i}^\perp\|_2 = \sqrt{1 + \alpha^2/4}\,\|w_{\text{fc}, i}\|_2$. Moreover, for $z_i = w_{\text{fc}, i}x$ and $\delta_i = \Delta_{\text{fc}, i}^\perp x$, on the interior of the active cone ($z_i > |\delta_i|$), the quadratic identity

$$\frac{1}{2}\phi(z_i + \delta_i) + \frac{1}{2}\phi(z_i - \delta_i) = \frac{1}{2}(z_i + \delta_i)^2 + \frac{1}{2}(z_i - \delta_i)^2 = z_i^2 + \delta_i^2 = \phi(z_i) + (\Delta_{\text{fc}, i}^\perp x)^2$$

cancels the first-order $O(\alpha)$ linear cross-term $\pm z_i \delta_i$ identically, leaving only a second-order $O(\alpha^2)$ variance shift proportional to $(\alpha/2)^2$ (at $\alpha = 0.25$ in HSDM-2D, $(\alpha/2)^2 = 1/64 \approx 0.0156$, matching the measured mlp_fp32_rel_err = 0.0148–0.0153 in result.json), while splitting both pre-activations $z_i \pm \delta_i$ and active ReLU cones at step 0 so that $G_{\text{fc}}^{(1)}, G_{\text{fc}}^{(2)}$ and $G_{\text{proj}}^{(1)}, G_{\text{proj}}^{(2)}$ are immediately linearly independent.

Theorem 2B ($2\times 2$ and $4\times 4$ Kronecker Hadamard Modal Muon & Non-Collinearity Condition, train_growth.py:620-771): During Stage 2 (--dipolar modal_hadamard), instead of applying Polar Express directly to the stacked width/depth matrices, we transform the Nesterov-blended sibling gradients into the orthonormal Hadamard modal basis before polar_express:

1. 1D Width Hadamard Modal Muon (modal_width_muon_step_fused, train_growth.py:620-687): For widened MLP layers without depth siblings, partition the Nesterov-blended gradient $g$ into its two width halves $(g_1, g_2)$ and apply the $2\times 2$ Hadamard transform $H_2$:

$$g_+ = \frac{g_1 + g_2}{\sqrt{2}}, \qquad g_- = \frac{g_2 - g_1}{\sqrt{2}}.$$

Stack $X_0 = [g_+; g_-]$ along the batch dimension, normalize $g_+$ and $g_-$ independently ($X_{0,\pm} = g_\pm / (\|g_\pm\|_F \cdot 1.02 + 10^{-6})$), and run 5-step polar_express with separate Gram matrices $A_+ = X_+^\top X_+$ and $A_- = X_-^\top X_-$ to obtain approximate polar factors $(U_+, U_-)$. Project $U_-$ Frobenius-orthogonal to $U_+$ and rescale to match $\|U_+\|_F$:

$$U_-^\perp = U_- - \frac{\langle U_-, U_+\rangle_F}{\|U_+\|_F^2} U_+, \qquad \hat{U}_-^\perp = \frac{\|U_+\|_F}{\|U_-^\perp\|_F} U_-^\perp \cdot \mathbb{I}[\|g_-\|_F > 10^{-10}],$$

then reconstruct the width-half updates via the inverse Hadamard map with cosine-decaying modal gain $\gamma_{\text{modal}}(t) = 1 + (\gamma_0 - 1)\cos^2\!\left(\frac{\pi}{2}\frac{t - t_{\text{grow}}}{T_{\text{rem}}}\right)$ ($\gamma_0 = 2.0$ over the remaining Stage-2 steps $T_{\text{rem}} = T_{\max} - t_{\text{grow}}$, train_growth.py:2226-2232):

$$g_1^{\text{out}} = \frac{1}{2}\big(U_+ - \gamma_{\text{modal}} \hat{U}_-^\perp\big), \qquad g_2^{\text{out}} = \frac{1}{2}\big(U_+ + \gamma_{\text{modal}} \hat{U}_-^\perp\big),$$

followed by NorMuon second-moment normalization (preserving $\|U\|_F = \|g^{\text{out}}\|_F$) and cautious weight decay.

• Explicit Non-Collinearity Requirement: If $g_-$ were collinear with $g_+$ ($g_- = \epsilon g_+$ for a scalar $\epsilon \neq 0$, as occurs if one scales two identical halves by unequal scalar weights without breaking directional symmetry), then $X_{0,-} = \text{sign}(\epsilon) X_{0,+}$, so $U_- = \text{sign}(\epsilon) U_+$ and the Gram-Schmidt projection $U_-^\perp \equiv 0$ vanishes identically! Thus, non-zero differential specialization ($\hat{U}_-^\perp \neq 0$) strictly requires linearly independent common and differential gradients ($U_- \notin \text{span}(U_+)$), which holds at step 0 because stiefel_tangent and isometric_pe apply spatially varying orthogonal perturbations across neurons/channels.

2. 2D Kronecker Hadamard Modal Muon (modal_2d_muon_step_fused, train_growth.py:690-771): For parent/daughter depth-sibling pairs of widened MLP layers with four Nesterov-blended blocks $(g_{p1}, g_{p2}, g_{d1}, g_{d2})$, apply the $4\times 4$ Kronecker Hadamard transform $H_2 \otimes H_2$:

$$\begin{aligned} g_{++} &= \tfrac{1}{2}\big((g_{p1} + g_{p2}) + (g_{d1} + g_{d2})\big), & g_{+-} &= \tfrac{1}{2}\big((g_{p2} - g_{p1}) + (g_{d2} - g_{d1})\big), \\ g_{-+} &= \tfrac{1}{2}\big((g_{d1} + g_{d2}) - (g_{p1} + g_{p2})\big), & g_{--} &= \tfrac{1}{2}\big((g_{d2} - g_{d1}) - (g_{p2} - g_{p1})\big). \end{aligned}$$

Normalize and orthogonalize all four modes $(g_{++}, g_{+-}, g_{-+}, g_{--})$ independently via 5-step polar_express to obtain $(U_{++}, U_{+-}, U_{-+}, U_{--})$, form the inverse differential combinations $D_{p1} = -U_{+-} - U_{-+} + U_{--}$, $D_{p2} = U_{+-} - U_{-+} - U_{--}$, $D_{d1} = -U_{+-} + U_{-+} - U_{--}$, $D_{d2} = U_{+-} + U_{-+} + U_{--}$ projected Frobenius-orthogonal to $U_{++}$ and Frobenius-rescaled to $\|U_{++}\|_F$, and reconstruct $U_q = \frac{1}{2}(U_{++} + \gamma_{\text{modal}} \hat{D}_q)$ for $q \in \{p1, p2, d1, d2\}$. Because each non-collinear differential mode is normalized by its own Frobenius norm prior to Polar Express, even an $O(\epsilon)$ initial directional gradient asymmetry produces a full-Frobenius-norm specialization update $\hat{D}_q$, driving $\cos(W_{\text{fc}}^{(1)}, W_{\text{fc}}^{(2)})$ from $0.969$ down to $0.714\text{--}0.719$ within 70 steps during late-stage cooldown. 3. 1D Isometric Stiefel Depth Modal Muon (modal_haar_depth_muon_step_fused, train_growth.py:558-617): For pure 1D depth growth $4\text{L}\to 8\text{L}$ (--dipolar modal_haar), applies $H_2$ across parent/daughter depth blocks $(g_p, g_d) \mapsto (g_+, g_-)$ with cosine-decaying rotation angle $\alpha(t) = \alpha_0 \cos^2(\frac{\pi}{2}\frac{t - t_{\text{grow}}}{80})$ ($\alpha_0 = 0.45$ over 80 steps) and reconstructs $g_{p,d}^{\text{out}} = (1 + \alpha(t)^2)^{-1/2}(U_+ \mp \alpha(t)\hat{U}_-^\perp)$, preserving exact Frobenius norm $\|g_{p,d}^{\text{out}}\|_F \equiv \|U_+\|_F$.

3.3 Theorem 3: Whole-Block Dual Straight-Through Residual Homotopy (DSTRH)

To prevent both the $+0.28\text{--}1.12$ nat forward loss shock of $G_{\text{stack}}$ and the step-1 backward gradient starvation of $G_{\text{zero}}$, HSDM applies a Whole-Block Dual Straight-Through Residual Homotopy (experiments/train_growth.py:331-343):

Theorem 3 (Whole-Block Dual Straight-Through Residual Homotopy, train_growth.py:331-343): Let $\mathcal{B}_{\text{bud}}$ be the set of newly budded transformer blocks inserted at step $t_{\text{grow}}$ with residual scalars initialized to $\lambda_{\text{resid},\ell} = 1, \lambda_{x_0,\ell} = 0$, and let $\beta(t) = \beta_0 + (1 - \beta_0)\min\!\left(1, \frac{t - t_{\text{grow}}}{W}\right) \in [\beta_0, 1]$ ($\beta_0 = 0, W = 25$) be a linear homotopy ramp over $W$ steps. For each budded block $\ell \in \mathcal{B}_{\text{bud}}$ with block mapping $\text{Block}_\ell(u) = u + \text{MLP}_\ell(\text{Norm}(u + \text{Attn}_\ell(\text{Norm}(u))))$, define the gated block input $\tilde{x}_{\text{sub}}$, block residual $\delta_\ell$, and block output $x_{\text{out}}$ by:

$$x_{\text{sub}} = \lambda_{\text{resid},\ell} x_{\text{in}} + \lambda_{x_0,\ell} x_0, \qquad \tilde{x}_{\text{sub}} = \beta(t)\,x_{\text{sub}} + \big(1 - \beta(t)\big)\text{sg}[x_{\text{sub}}],$$

$$\delta_\ell = \text{Block}_\ell(\tilde{x}_{\text{sub}}) - \tilde{x}_{\text{sub}}, \qquad x_{\text{out}} = \big(1 - \beta(t)\big)x_{\text{in}} + \beta(t)\,x_{\text{sub}} + \delta_\ell + \big(\beta(t) - 1\big)\text{sg}[\delta_\ell],$$

where $\text{sg}[\cdot]$ is the stop-gradient (detach()) operator.

1. Exact Forward Depth Continuity ($\beta_0 = 0$): In the forward pass, $\text{sg}[u] = u$, so $\tilde{x}_{\text{sub}}^{\text{fwd}} = x_{\text{sub}}$ and $\delta_\ell + (\beta(t) - 1)\text{sg}[\delta_\ell] = \beta(t)\,\delta_\ell^{\text{fwd}}$. Since $\lambda_{\text{resid},\ell} = 1$ and $\lambda_{x_0,\ell} = 0$ on newly budded blocks ($x_{\text{sub}} = x_{\text{in}}$), the forward output is $x_{\text{out}}^{\text{fwd}} = x_{\text{in}} + \beta(t)\,\delta_\ell^{\text{fwd}}$. At $t = t_{\text{grow}}$ with $\beta(t_{\text{grow}}) = 0$, $x_{\text{out}}^{\text{fwd}} \equiv x_{\text{in}}$, so depth expansion is strictly function-preserving (loss_bf16_abs_diff = 0.000010–0.000287 nats).
2. Unattenuated Unit-Jacobian Direct Parameter Gradients & $\beta(t)$-Gated Trunk Backpropagation: In the backward pass, $\frac{\partial \text{sg}[\delta_\ell]}{\partial \theta_\ell} = 0$, so the direct Jacobian of $x_{\text{out}}$ with respect to the budded block parameters $\theta_\ell$ has unit gain for all $\beta(t) \in [0, 1]$:

$$\frac{\partial x_{\text{out}}}{\partial \theta_\ell} = \frac{\partial \delta_\ell}{\partial \theta_\ell} = \frac{\partial \text{Block}_\ell(\tilde{x}_{\text{sub}})}{\partial \theta_\ell} \qquad \forall\, t \in [t_{\text{grow}}, t_{\text{grow}} + W],$$

while the derivative of the block residual with respect to the upstream trunk activation $x_{\text{sub}}$ is gated by $\frac{\partial \tilde{x}_{\text{sub}}}{\partial x_{\text{sub}}} = \beta(t) I$. Unlike ordinary linear homotopy $x_{\text{out}} = x_{\text{in}} + \beta(t)\delta_\ell$ (lerp, which attenuates direct parameter gradients by $\beta(t) \ll 1$ during the critical first post-mitosis steps) or $G_{\text{zero}}$ (which zeroes step-1 gradients into $W_{QKV}$ and $W_{\text{fc}}$), DSTRH delivers full-strength direct parameter gradients into newly budded blocks while shielding earlier trunk layers from unattenuated backpropagated shock.

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4. Benchmarking Width-Splitting Operators Against Literature Baselines

Before evaluating late-stage expansion ($50.0\%\text{--}75.1\%$ tokens), we first address Referee Point 1 by benchmarking our width-splitting operators head-to-head against established literature methods at the standard early/mid expansion point $g_1 = 418$ ($43.9\%$ of tokens) and variable-batch $g_1 = 200$ ($21.0\%$ of tokens), as well as across four architectural families and two model scales ($d = 512$ and $d = 384$, 3 seeds per arm).

Figure 1: Width Symmetry-Breaking Benchmark and Internal Trajectory Diagnostics

4.1 Multi-Family 3-Seed Verification of Symmetry Breaking vs. Exact Net2WiderNet

Table 1 summarizes our 24-run (4 families × 2 width operators × 3 seeds) factorial comparing Dipolar Width Mitosis ($\alpha = 0.5$) against exact function-preserving Net2WiderNet ($\alpha = 0.0$) [1]. As predicted by Theorem 1A, Net2WiderNet locks $\cos(W_{\text{fc}}^{(1)}, W_{\text{fc}}^{(2)}) = 1.0000$ for all remaining steps, whereas Dipolar Width Mitosis breaks symmetry at step 1 ($\cos(G_{\text{fc}}^{(1)}, G_{\text{fc}}^{(2)}) \in [0.805, 0.861]$) and outperforms Net2WiderNet across all 12 seed pairs ($p \le 0.00033$).

| Architectural Family & Schedule | Width Operator | FLOPs Ratio | Val BPB (3-seed mean ± SD) | Holdout BPB (3-seed mean ± SD) | Paired $\Delta\text{Val}$ / $\Delta\text{Holdout}$ ($p$-value) | Final $W_{\text{fc}}$ Cosine | | :--- | :--- | :---: | :---: | :---: | :---: | :---: | | Family 1: 3-Stage 2D ($d=512$)<br>$(4\text{L},r=1)\to(8\text{L},r=2)\to(12\text{L},r=4)$ | Net2Wider ($\alpha=0.0$)<br>Dipolar ($\alpha=0.5$) | 0.9454x<br>0.9454x | 1.023103 ± 0.000565<br>0.999985 ± 0.000297 | 1.022273 ± 0.000520<br>0.999105 ± 0.000273 | -0.023119 (p=0.000067)<br>-0.023169 (p=0.000043) | 1.0000<br>0.7850 | | Family 2: Pure Width ($d=512$)<br>$(8\text{L},r=2)\to(8\text{L},r=4)$ at gs=418 | Net2Wider ($\alpha=0.0$)<br>Dipolar ($\alpha=0.5$) | 0.9077x<br>0.9077x | 1.016172 ± 0.002021<br>1.002975 ± 0.001898 | 1.015372 ± 0.002008<br>1.002168 ± 0.001850 | -0.013197 (p=0.000087)<br>-0.013204 (p=0.000073) | 1.0000<br>0.8220 | | Family 3: Pure Width ($d=384$)<br>$(8\text{L},r=2)\to(8\text{L},r=4)$ at gs=418 | Net2Wider ($\alpha=0.0$)<br>Dipolar ($\alpha=0.5$) | 0.5796x<br>0.5796x | 1.048972 ± 0.001413<br>1.034259 ± 0.001294 | 1.048436 ± 0.001462<br>1.033701 ± 0.001367 | -0.014713 (p=0.000023)<br>-0.014735 (p=0.000025) | 1.0000<br>0.8050 | | Family 4: Coupled 2D ($d=512$)<br>$(8\text{L},r=2)\to(12\text{L},r=4)$ at gs=418 | Net2Wider ($\alpha=0.0$)<br>Dipolar ($\alpha=0.5$) | 1.1588x<br>1.1588x | 0.998193 ± 0.001522<br>0.984964 ± 0.001835 | 0.997302 ± 0.001465<br>0.984117 ± 0.001875 | -0.013230 (p=0.000334)<br>-0.013185 (p=0.000246) | 1.0000<br>0.8090 |

4.2 Head-to-Head Against LEMON, Noisy Net2Wider, and SPARKLING at Early/Mid Expansion (gs=418 & gs=200)

To distinguish breaking exact duplicate symmetry ($\alpha=0$) from outperforming existing symmetry-breaking methods in the literature, Table 2 compares our Stiefel-tangent and Hadamard operators against four strong literature baselines at $g_1 = 418$ ($43.9\%$ tokens) and variable-batch $g_1 = 200$ ($21.0\%$ tokens, accum 1 -> 4), describing each baseline by its exact implementation in experiments/train_growth.py:1207-1265:

1. Noisy Net2Wider (net2wider_noise, Chen et al., 2016 [1]): Perturbs the incoming duplicated weights with opposite Gaussian noise $W_{\text{fc}}^{\text{new}} = [W_{\text{fc}} + \epsilon_{\text{fc}}; W_{\text{fc}} - \epsilon_{\text{fc}}]$ ($\sigma = 0.05\,\text{std}(W_{\text{fc}})$) while splitting the readout as $[\frac{1}{2}W_{\text{proj}}, \frac{1}{2}W_{\text{proj}}]$.
2. LEMON (lemon_gaussian / lemon_scalar, Wang et al., 2023 [5]): Loss-preserving readout splitting via opposite Gaussian readout noise $W_{\text{proj}}^{\text{new}} = [\frac{1}{2}W_{\text{proj}} + \epsilon_{\text{proj}}, \frac{1}{2}W_{\text{proj}} - \epsilon_{\text{proj}}]$ (lemon_gaussian) or asymmetric convex weights $[0.65 W_{\text{proj}}, 0.35 W_{\text{proj}}]$ (lemon_scalar) with optimizer state reset.
3. SPARKLING (sparkling_rewarmup & sparkling_eq28, Yu et al., 2026 [10]): Exact function-preserving symmetric copy-copy split $W_{\text{fc}}^{\text{new}} = [W_{\text{fc}}; W_{\text{fc}}]$, $W_{\text{proj}}^{\text{new}} = [\frac{1}{2}W_{\text{proj}}, \frac{1}{2}W_{\text{proj}}]$ paired with asymmetric optimizer state reset $S' = [S, 0]$ (preserving parent momentum on the first half and zeroing the second half) and new-parameter learning-rate re-warmup ($\rho = 1.3$ over $\tau_w = 60$ steps; we test both maintaining the $1.3\times$ multiplier on the decaying schedule sparkling_rewarmup and Yu et al.'s exact Eq. 27–28 linear decay from $\rho \eta_e$ to $0$ sparkling_eq28).
4. Orthogonal Neuron Budding (ONB) (ortho_budding_rewarmup): Zero-readout orthogonal neuron expansion ($W_{\text{fc}}^{(2)} \perp W_{\text{fc}}^{(1)}$, $W_{\text{proj}}^{(2)} = 0$) with LR re-warmup.

| Setting & Schedule (seed=42 unless 3-seed noted) | Growth / Baseline Method | FLOPs Ratio | BF16 Mitosis $|\Delta\mathcal{L}|$ (nats) | Val BPB | Holdout BPB | $\Delta\text{Val}$ vs. Matched Static Iso-FLOPs | | :--- | :--- | :---: | :---: | :---: | :---: | :---: | | Pure Width (gs=418, $43.9\%$ tok)<br>$(8\text{L},r=2)\to(8\text{L},r=4)$, Fixed accum=4 | Static 8L Iso-FLOPs (865 steps)<br>Net2Wider ($\alpha=0.0$, 3-seed mean)<br>Orthogonal Budding ($W_{\text{proj}}^{(2)}=0$ + Re-Warmup)<br>Dipolar ($\alpha=0.5$, s42 / 3-seed mean)<br>LEMON (Gaussian / Scalar, Wang et al. [5])<br>SPARKLING (Mom Reset + Re-Warmup [10])<br>Noisy Net2Wider (Chen et al. [1])<br>Stiefel Tangent ($\Delta_{\text{fc}}^\perp \perp W_{\text{fc}}$, s42 / 3-seed) | 0.9077x<br>0.9077x<br>0.9077x<br>0.9077x<br>0.9077x<br>0.9077x<br>0.9077x<br>0.9077x | —<br>0.000000<br>0.000000<br>0.000079<br>0.000095<br>0.000075<br>0.001325<br>0.006132 | 1.000289<br>1.016172<br>1.004622<br>1.001315 (1.002975)<br>1.001270 / 1.001197<br>1.001062<br>1.000816<br>1.000567 (1.002516) | 0.999580<br>1.015372<br>1.003929<br>1.000580 (1.002168)<br>1.000493 / 1.000333<br>1.000294<br>1.000038<br>0.999689 (1.001632) | 0.000000<br>+0.015883<br>+0.004333<br>+0.001026<br>+0.000908<br>+0.000773<br>+0.000527<br>+0.000278 | | Pure Width (gs=200, $21.0\%$ tok)<br>$(8\text{L},r=2)\to(8\text{L},r=4)$, accum 1->4 | Static 8L VarBatch Iso-FLOPs (911 steps, acc 1->4)<br>Net2Wider ($\alpha=0.0$, 3-seed mean)<br>Dipolar ($\alpha=0.5$, s42 / 3-seed mean)<br>SPARKLING (Mom Reset + Re-Warmup [10])<br>Stiefel Tangent ($\Delta_{\text{fc}}^\perp \perp W_{\text{fc}}$) | 0.9559x<br>0.9558x<br>0.9558x<br>0.9558x<br>0.9558x | —<br>0.000075<br>0.000079<br>0.000103<br>0.004094 | 0.986536<br>1.007528<br>0.988817 (0.990918)<br>0.988216<br>0.987810 | 0.985794<br>1.006810<br>0.988164 (0.990128)<br>0.987559<br>0.987159 | 0.000000<br>+0.020992<br>+0.002281<br>+0.001680<br>+0.001274 | | Coupled 2D (gs=418, $43.9\%$ tok)<br>$(8\text{L},r=2)\to(12\text{L},r=4)$, Fixed accum=4 | Static 12L Iso-FLOPs (763 steps)<br>Net2Wider ($\alpha=0.0$, 3-seed mean)<br>Dipolar ($\alpha=0.5$, s42 / 3-seed mean)<br>Noisy Net2Wider (Chen et al. [1])<br>G_stack + SPARKLING (Yu et al. [10])<br>Stiefel + Dual Split + SPARKLING | 1.1588x<br>1.1588x<br>1.1588x<br>1.1588x<br>1.1588x<br>1.1588x | —<br>0.194000<br>0.194000<br>0.194116<br>0.202833<br>0.186057 | 0.987346<br>0.998193<br>0.983386 (0.984964)<br>0.983381<br>0.983304<br>0.983169 | 0.986652<br>0.997302<br>0.982529 (0.984117)<br>0.982592<br>0.982477<br>0.982227 | 0.000000<br>+0.010847<br>-0.003960<br>-0.003965<br>-0.004042<br>-0.004177 | | Coupled 2D (gs=260, $27.3\%$ tok)<br>$(8\text{L},r=2)\to(12\text{L},r=4)$, accum 1->4 | Static 12L VarBatch Iso-FLOPs (835 steps, acc 1->4)<br>Stiefel + Dual Split + SPARKLING | 1.2682x<br>1.2679x | —<br>0.402935 | 0.972150<br>0.969330 | 0.971421<br>0.968662 | 0.000000<br>-0.002820 |

Candid Diagnostic Synthesis of Early/Mid Expansion (gs=200–418):

1. Pure 1D Width Splitting Cannot Beat Matched-Compute Static Training: In pure width expansion $(8\text{L},r=2)\to(8\text{L},r=4)$ at gs=418 ($9.2\%$ FLOP savings), Stiefel Tangent (1.000567), Noisy Net2Wider (1.000816), SPARKLING (1.001062), and LEMON (1.001197) all recover $\sim 95\text{--}98\%$ of the 0.0139-BPB gap caused by exact Net2Wider (1.014506 on s42), yet none beats training Static 8L from scratch at matched compute (1.000289 at 865 steps, or 0.986536 vs. 0.987810 under variable batch accum 1->4). Why? Because Stage 1 (8L, r=2) saves FLOPs only in the MLP while paying $100\%$ of the 8-layer attention FLOPs.
2. Coupled 2D Growth Beats Matched-Compute Static 12L, and Early Timing Forgives Literature Baselines: When depth and width grow together ($(8\text{L},r=2)\to(12\text{L},r=4)$), Stage 1 saves both attention and MLP FLOPs ($0.5455\times$ per-step FLOPs of Static 12L). At early/mid expansion (gs=418, $43.9\%$ tokens, where LR is still at its peak $\eta_{\max}$), 535 steps remain after expansion; hence Noisy Net2Wider (0.983381), G_stack + SPARKLING (0.983304), and Stiefel + Dual Split + SPARKLING (0.983169) all recover from the +0.19 nat stacking shock and beat Static 12L Iso-FLOPs (0.987346) by -0.0040 to -0.0042 BPB (and at gs=260 with accum 1->4, beat Static 12L VarBatch Iso-FLOPs 0.972150 vs. 0.969330, a -0.002820 BPB net win).
3. Why Early Expansion Is Not Enough: Expanding at $27.3\%$ or $43.9\%$ of tokens saves only $12.4\%$ or $20.0\%$ of FLOPs relative to full-budget Static 12L (1.729e17 FLOPs). To save $22.7\%\text{--}34.1\%$ of total compute, the model must remain small through $50.0\%\text{--}75.1\%$ of tokens—directly entering the linear LR cooldown regime where G_stack + SPARKLING and G_zero break down.

---

5. Main Results: Late-Stage Model Growth ($50.0\%\text{--}75.1\%$ Token Horizon)

We now evaluate Hadamard Modal-Decoupled Late Morphogenesis (HSDM / HSDM-2D) across the late-stage expansion horizon ($50.0\%, 65.1\%/65.9\%, 75.1\%$ of the 500M-token budget) on both Track 2 (Coupled 2D Growth: $(8\text{L},r=2)\to(12\text{L},r=4)$) and Track 1 (Pure 1D Depth Growth: $4\text{L}\to 8\text{L}$).

Figure 4: Late-Stage Model Growth Timing Curves, 3-Seed Verification, and Deconfounded Controls

5.1 Track 2: Coupled 2D Late-Stage Growth $(8\text{L}, r=2) \to (12\text{L}, r=4)$ ($43.9\%\text{--}75.1\%$ Tokens)

Table 3 reports the complete Track 2 timing study from $43.9\%$ to $75.1\%$ of the token budget, comparing HSDM-2D against G_stack + SPARKLING (Yu et al., 2026 [10] + Du et al., 2024 [6], including both sparkling_rewarmup and exact Eq. 27–28 decay sparkling_eq28), matched-compute Static 12L Iso-FLOPs, 3-seed deconfounded Variable-Batch Static 12L Iso-FLOPs, and fixed-batch (accum=4) HSDM-2D controls.

| Expansion Timing & Compute Saved vs. Full Static 12L (1.729e17 F) | Method / Control Arm (d=512, Target 12L, r=4, 71.30M params) | Total FLOPs (Ratio to 8L) | Train / Wall Time (s) | BF16 Mitosis $|\Delta\mathcal{L}|$ (nats) | Validation BPB (val_bpb) | Holdout BPB (holdout_bpb) | $\Delta\text{Val}$ vs. Static 12L Iso-FLOPs | $\Delta\text{Val}$ vs. G_stack + SPARKLING | | :--- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Full Budget (0% Saved) | Static 12L Full (953 steps, s42) | 1.729e17 (1.4474x) | 1644s / 1745s | — | 0.971595 | 0.970945 | — | — | | Early-Mid: $43.9\%$ Tok<br>(gs=418, $-20.0\%$ FLOPs) | Static 12L Iso-FLOPs (763 steps, s42)<br>G_stack + SPARKLING (acc=4, s42)<br>Stiefel + Dual Split + SPARKLING (acc=4, s42)<br>HSDM-2D Fixed-Batch (accum=4, s42)<br>HSDM-2D (DSTRH + Modal Muon, acc 4->2, s42) | 1.384e17 (1.1588x)<br>1.384e17 (1.1588x)<br>1.384e17 (1.1588x)<br>1.384e17 (1.1588x)<br>1.384e17 (1.1588x) | 1271s / 1372s<br>1322s / 1426s<br>1317s / 1420s<br>1265s / 1369s<br>1274s / 1378s | —<br>0.202833<br>0.186057<br>0.000730<br>0.000928 | 0.987346<br>0.983304<br>0.983169<br>0.986954<br>0.982243 | 0.986652<br>0.982477<br>0.982227<br>0.986112<br>0.981413 | 0.000000<br>-0.004042<br>-0.004177<br>-0.000392<br>-0.005103 | +0.004042<br>0.000000<br>-0.000135<br>+0.003650<br>-0.001061 | | Halfway: $50.0\%$ Tok<br>(gs=476, $-22.7\%$ FLOPs)<br>Cooldown onset ($p=0.50$) | Static 12L Iso-FLOPs (735 steps, s42)<br>G_stack + SPARKLING (s42)<br>HSDM-2D (DSTRH + Modal Muon, s42) | 1.333e17 (1.1163x)<br>1.336e17 (1.1188x)<br>1.336e17 (1.1188x) | 1266s / 1367s<br>1278s / 1383s<br>1289s / 1395s | —<br>0.247046<br>0.000010 | 0.990193<br>0.986679<br>0.985388 | 0.989399<br>0.985787<br>0.984553 | 0.000000<br>-0.003514<br>-0.004805 | +0.003514<br>0.000000<br>-0.001292 | | Late: $65.9\%$ Tok<br>(gs=628, $-29.9\%$ FLOPs)<br>Deep in LR cooldown ($46.5\%$ post-peak LR integral rem.)<br>[3-Seed Primary Benchmark] | Static 12L Iso-FLOPs (668 steps, 3 seeds)<br>G_stack + SPARKLING (sparkling_rewarmup, 3 seeds)<br>G_stack + SPARKLING Eq. 28 (accum=4, s42)<br>G_stack + SPARKLING Eq. 28 (accum 4->2, s42)<br>Plain Dipolar ($\alpha=0.5$, coupled Muon, s42)<br>HSDM-2D Fixed-Batch (accum=4, s42)<br>Static 12L Iso-FLOPs VarBatch (acc 4->2, 3 seeds)<br>HSDM-2D Full (s42)<br>HSDM-2D Full (3-seed mean ± SD)<br>HSDM-2D + Post-Mitosis μ-Cooldown (s42)<br>HSDM-2D + Isometric PE Split (isometric_pe, s42) | 1.212e17 (1.0145x)<br>1.211e17 (1.0138x)<br>1.211e17 (1.0138x)<br>1.211e17 (1.0138x)<br>1.211e17 (1.0138x)<br>1.211e17 (1.0138x)<br>1.212e17 (1.0145x)<br>1.211e17 (1.0138x)<br>1.211e17 (1.0138x)<br>1.211e17 (1.0138x)<br>1.211e17 (1.0138x) | 1091s / 1191s<br>1108s / 1211s<br>1054s / 1156s<br>1061s / 1162s<br>1101s / 1205s<br>1152s / 1259s<br>1078s / 1180s<br>1106s / 1242s<br>1113s / 1229s<br>1116s / 1220s<br>1106s / 1240s | —<br>0.271480<br>0.253000<br>0.252996<br>0.281000<br>0.000260<br>—<br>0.000026<br>0.000159<br>0.000013<br>0.000047 | 0.998472 ± 0.001947<br>1.002396 ± 0.001997<br>0.999695<br>0.993431<br>1.001034<br>0.997694<br>0.996840 ± 0.001842<br>0.993419<br>0.994959 ± 0.001857<br>0.993153<br>0.992815 | 0.997666 ± 0.001857<br>1.001491 ± 0.001992<br>0.998872<br>0.992625<br>1.000269<br>0.996855<br>0.996015 ± 0.001794<br>0.992586<br>0.994115 ± 0.001844<br>0.992400<br>0.991972 | 0.000000<br>+0.003924 (loses)<br>+0.001927 (vs s42)<br>-0.004337 (vs s42)<br>+0.003266 (loses)<br>-0.000074 (vs s42)<br>-0.001632 (3 seeds)<br>-0.004348 (vs s42) <br>-0.003513 (-0.003551 HO)<br>-0.004615 (-0.002985 vs VB)<br>-0.004952 (-0.003323 vs VB) | -0.003924<br>0.000000<br>-0.001107 (vs s42)<br>-0.007371 (vs s42)<br>+0.000232<br>-0.003108 (-0.002001 vs Eq28)<br>-0.005556<br>-0.007383<br>-0.007436 (p=0.000151)<br>-0.007649<br>-0.007987 | | Very Late: $75.1\%$ Tok<br>(gs=716, $-34.1\%$ FLOPs)<br>Extreme cooldown ($24.8\%$ post-peak LR integral rem.) | Static 12L Iso-FLOPs (628 steps, s42)<br>G_stack + SPARKLING (s42)<br>HSDM-2D (DSTRH + Modal Muon, s42) | 1.139e17 (0.9538x)<br>1.139e17 (0.9531x)<br>1.139e17 (0.9531x) | 1065s / 1166s<br>1074s / 1177s<br>1079s / 1184s | —<br>0.311308<br>0.000155 | 1.002679<br>1.008111<br>1.000220 | 1.002079<br>1.007398<br>0.999387 | 0.000000<br>+0.005432 (loses)<br>-0.002459 (-0.002692 HO) | -0.005432<br>0.000000<br>-0.007891 (-0.008011 HO) |

Key Findings from Track 2 ((8L, r=2) -> (12L, r=4)):

1. The Late-Expansion Crossover Across All Four Timings ($43.9\%\text{--}75.1\%$): When the exact same HSDM-2D algorithm (DSTRH + modal_hadamard + stiefel_tangent + accum 4->2) is evaluated across all four timings, it strictly outperforms both G_stack + SPARKLING and Static 12L Iso-FLOPs at gs=418 (0.982243 vs. 0.983304 and 0.987346), gs=476 (0.985388 vs. 0.986679 and 0.990193), gs=628 (0.993419 vs. 1.000802 and 0.997768), and gs=716 (1.000220 vs. 1.008111 and 1.002679). By contrast, fixed-batch G_stack + SPARKLING crosses above (worse than) Static 12L Iso-FLOPs between $50.0\%$ and $65.9\%$ tokens—Even with Yu et al.'s exact Eq. 27–28 linear decay (sparkling_eq28, which improves fixed-batch G_stack + SPARKLING at gs=628 from 1.000802 to 0.999695, -0.001107 BPB), fixed-batch G_stack + SPARKLING Eq. 28 still loses to Static 12L Iso-FLOPs (0.997768 on s42) by +0.001927 BPB and trails fixed-batch HSDM-2D (0.997694) by +0.002001 BPB.
2. 3-Seed Paired Statistics Against Literature Growth & Matched-Compute Static Controls ($65.9\%$ Tokens): Across seeds 42, 137, 2026 at gs=628, HSDM-2D (0.994959 ± 0.001857 val / 0.994115 ± 0.001844 holdout) beats G_stack + SPARKLING (1.002396 ± 0.001997 val) on every seed (s42: -0.007383, s137: -0.007615, s2026: -0.007312), yielding a paired difference of -0.007436 ± 0.000158 val BPB ($t = -81.45, p = 0.000151$, $95\%$ CI $[-0.007829, -0.007044]$) and -0.007376 ± 0.000172 holdout BPB ($t = -74.45, p = 0.000180$, $95\%$ CI $[-0.007803, -0.006949]$). Against 3-seed Static 12L Iso-FLOPs (0.998472 ± 0.001947 val / 0.997666 ± 0.001857 holdout), HSDM-2D achieves a mean improvement of -0.003513 ± 0.003223 val BPB ($t = -1.89, p = 0.200$, $95\%$ CI $[-0.011520, +0.004494]$) and -0.003551 ± 0.003180 holdout BPB ($95\%$ CI $[-0.011450, +0.004348]$)—winning strongly on seeds 42 (-0.004348 val / -0.004503 holdout) and 2026 (-0.006236 val / -0.006147 holdout) and tying on seed 137 (0.997021 vs. 0.996976 val, +0.000045; 0.996144 vs. 0.996147 holdout, -0.000003, or 0.996928 with s2 scaling), where the seed-to-seed variance reflects that 668-step static runs and 953-step growing runs traverse different data slices during cooldown. Against our 3-seed batch-matched Static 12L Iso-FLOPs VarBatch (0.996840 ± 0.001842 val / 0.996015 ± 0.001794 holdout, which improves over fixed-batch Static 12L Iso-FLOPs by -0.001632 ± 0.000110 val BPB, $p = 0.00151$), HSDM-2D achieves a mean gain of -0.001881 ± 0.003117 val BPB ($95\%$ CI $[-0.009623, +0.005862]$), winning on seeds 42 (-0.002719) and 2026 (-0.004493).

Table 3B: Pure One-Factor-at-a-Time (OFAT) Causal Ablations, Scale Transfer ($d=384$), and Optimizer Generality (AdamW) at Late Expansion (gs=628, $65.9\%$ Tokens)

To isolate the exact contribution of each component without confounding width-splitting choice, batch subdivision, or homotopy type, Table 3B reports our pure OFAT ablation grid at fixed batch (accum=4, stiefel_tangent, gs=628, s42), alongside our reduced-scale ($d=384$) transfer and tuned AdamW (--matrix-opt adamw --matrix-adamw-lr 0.003) controls.

| Control Group (gs=628, $65.9\%$ Tokens, seed=42) | Run ID & Exact Factor Changed | Homotopy | Stage-2 Matrix Optimizer | Width Split | Stage-2 accum | Mitosis $|\Delta\mathcal{L}|$ (nats) | Val BPB (val_bpb) | Holdout BPB (holdout_bpb) | Final $W_{\text{fc}}$ Cosine | $\Delta\text{Val}$ vs. Group Anchor | | :--- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Group A: Pure OFAT at Fixed Batch (accum=4) | hsdm_2d_8r2_to_12r4_g628_acc4_s42 (OFAT Anchor)<br>ablation_ofat_g628_acc4_lerp25_s42 (DSTRH -> lerp)<br>ablation_ofat_g628_acc4_nohom_s42 (DSTRH -> hom=0)<br>ablation_ofat_g628_acc4_nomodal_s42 (Remove modal_hadamard)<br>lit_gstack_sparkling_eq28_8r2_to_12r4_g628_s42<br>lit_gstack_sparkling_8r2_to_12r4_g628_s42 | dstrh:25<br>lerp:25<br>none<br>dstrh:25<br>none<br>none | modal_hadamard<br>modal_hadamard<br>modal_hadamard<br>Muon (none)<br>Muon (S'=[S,0])<br>Muon (S'=[S,0]) | stiefel<br>stiefel<br>stiefel<br>stiefel<br>sparkling_eq28<br>sparkling_rewarm | 4<br>4<br>4<br>4<br>4<br>4 | 0.000260<br>0.000351<br>0.274204<br>0.000185<br>0.253000<br>0.253000 | 0.997694<br>0.997453<br>0.996988<br>1.003737<br>0.999695<br>1.000802 | 0.996855<br>0.996623<br>0.996132<br>1.003007<br>0.998872<br>0.999935 | 0.5838<br>0.6018<br>0.5919<br>0.9160<br>0.7562<br>0.9333 | 0.000000<br>-0.000241<br>-0.000706<br>+0.006043 (collapses) <br>+0.002001<br>+0.003108 | | Group B: Stage-2 Batch Subdivision (accum 4->2) | hsdm_2d_8r2_to_12r4_g628_isometric_pe_s42<br>hsdm_2d_8r2_to_12r4_g628_mucooldown_s42<br>hsdm_2d_8r2_to_12r4_g628_s42 (HSDM-2D Full)<br>lit_gstack_sparkling_eq28_8r2_to_12r4_g628_acc2_s42 | dstrh:25<br>dstrh:25<br>dstrh:25<br>none | modal_hadamard<br>modal_hadamard + μ<br>modal_hadamard<br>Muon (S'=[S,0]) | isometric_pe<br>stiefel<br>stiefel<br>sparkling_eq28 | 2<br>2<br>2<br>2 | 0.000047<br>0.000013<br>0.000026<br>0.252996 | 0.992815<br>0.993153<br>0.993419<br>0.993431 | 0.991972<br>0.992400<br>0.992586<br>0.992625 | 0.7145<br>0.7012<br>0.7190<br>0.6267 | -0.000604<br>-0.000266<br>0.000000<br>+0.000012 | | Group C: Reduced Scale ($d=384$, 46.4M params) | d384_static_12L_isoflops_70pct_s42 (668 steps)<br>d384_lit_gstack_sparkling_8r2_to_12r4_g628_s42<br>d384_hsdm_2d_8r2_to_12r4_g628_s42 (HSDM-2D) | —<br>none<br>dstrh:25 | Muon<br>Muon (S'=[S,0])<br>modal_hadamard | —<br>sparkling_rewarm<br>stiefel | 4<br>4<br>2 | —<br>0.233000<br>0.000139 | 1.029480<br>1.032831<br>1.024558 | 1.028877<br>1.032440<br>1.024110 | —<br>0.9280<br>0.7080 | 0.000000<br>+0.003351 (loses)<br>-0.004922 (-0.008273 vs SP) | | Group D: Tuned AdamW (lr=0.003, --matrix-opt adamw) | adamw_static_12L_isoflops_70pct_s42 (668 steps)<br>adamw_gstack_sparkling_8r2_to_12r4_g628_s42<br>adamw_hsdm_2d_8r2_to_12r4_g628_s42 (HSDM-2D) | —<br>none<br>dstrh:25 | AdamW<br>AdamW (S'=[S,0])<br>AdamW | —<br>sparkling_eq28<br>stiefel | 4<br>4<br>2 | —<br>0.506000<br>0.000287 | 1.068919<br>1.054622<br>1.044510 | 1.068718<br>1.054256<br>1.044198 | —<br>0.9960<br>0.8920 | 0.000000<br>-0.014297<br>-0.024409 (-0.010112 vs SP) |

Causal Insights from Table 3B:

1. modal_hadamard Is the Primary Driver of Fixed-Batch Width Differentiation (-0.006043 BPB): In Group A (accum=4 fixed), removing modal_hadamard while keeping dstrh:25 and stiefel_tangent fixed (ablation_ofat_g628_acc4_nomodal_s42) causes val_bpb to degrade by +0.006043 BPB (0.997694 -> 1.003737) and locks the final width cosine at 0.9160 vs. 0.5838 with modal_hadamard. Thus, under Muon at fixed batch, DSTRH without modal_hadamard cannot overcome shared-Gram differential suppression (Theorem 1B). Conversely, when modal_hadamard is active, it rapidly differentiates width slices regardless of whether dstrh:25 (0.997694, $|\Delta\mathcal{L}| = 0.000260$), lerp:25 (0.997453, $|\Delta\mathcal{L}| = 0.000351$), or hom=0 (0.996988, $|\Delta\mathcal{L}| = 0.274204$) is used, with DSTRH eliminating the $1050\times$ larger initial loss spike.
2. Synergy Between Exact SPARKLING Eq. 28 Decay and Stage-2 Batch Subdivision (accum 4->2): When sparkling_eq28 is additionally given our Stage-2 micro-batch subdivision (accum 4->2, lit_gstack_sparkling_eq28_8r2_to_12r4_g628_acc2_s42), doubling the remaining optimizer steps from $325 \to 650$ allows asymmetric momentum reset plus linear LR decay from $1.3\eta_e \to 0$ to reach 0.993431 val BPB—matching stiefel_tangent HSDM-2D (0.993419) while still trailing HSDM-2D + μ-cooldown (0.993153) and exact zero-loss HSDM-2D + isometric_pe (0.992815) and incurring a +0.253-nat forward shock at expansion.
3. Scale ($d=384$) and Optimizer (AdamW) Generality: At $d=384$ (Group C), G_stack + SPARKLING again loses to Static 12L Iso-FLOPs (1.032831 vs. 1.029480, +0.003351 BPB), whereas HSDM-2D reaches 1.024558 val BPB (-0.004922 vs. Static Iso-FLOPs and -0.008273 BPB vs. G_stack + SPARKLING). Under tuned AdamW (lr=0.003, Group D), where SPARKLING suffers a $+0.506$-nat stacking shock and poor width cosine (0.9960), HSDM-2D (stiefel_tangent + dstrh:25 + accum 4->2) achieves 1.044510 val BPB, beating adamw_gstack_sparkling (1.054622) by -0.010112 BPB and adamw_static_12L_isoflops (1.068919) by -0.024409 BPB.

5.2 Track 1: Pure 1D Depth Late-Stage Growth $4\text{L} \to 8\text{L}$ & Empirical Trunk-Capacity Horizon

Table 4 evaluates pure depth expansion $4\text{L}\to 8\text{L}$ ($29.36\text{M}\to 50.33\text{M}$ params) across $50.0\%$, $65.1\%$, and $75.1\%$ of tokens, plus coupled $(4\text{L},r=2)\to(8\text{L},r=4)$ at $65.1\%$ tokens (gs=620), comparing our isometric Stiefel-rotated HSDM (modal_haar: $\alpha_0 = 0.45$ over 80 steps, $\beta_0 = 0.0, W = 25$ DSTRH) against $G_{\text{stack}}$ [6], true identity $G_{\text{zero}}$ (identity_zero: c_proj = 0 and $\lambda_{\text{resid}}=1, \lambda_{x_0}=0$), and matched-compute Static 8L Iso-FLOPs.

| Track & Expansion Timing | Method / Control Arm (d=512, Target 8L, r=4, 50.33M params) | Total FLOPs (Ratio to 8L) | Train / Wall Time (s) | BF16 Mitosis $|\Delta\mathcal{L}|$ (nats) | Validation BPB (val_bpb) | Holdout BPB (holdout_bpb) | $\Delta\text{Val}$ vs. Static 8L Iso-FLOPs | $\Delta\text{Val}$ vs. $G_{\text{stack}}$ | | :--- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Full Budget (0% Saved) | Static 8L Full (953 steps, 3 seeds) | 1.195e17 (1.0000x) | 1076s / 1157s | — | 0.995376 ± 0.001542 | 0.994605 ± 0.001454 | — | — | | Track 1 ($4\text{L}\to 8\text{L}$): $50.0\%$ Tok<br>(gs=476, $-22.3\%$ FLOPs) | Static 8L Iso-FLOPs (740 steps, s42)<br>$G_{\text{zero}}$ (identity_zero, s42)<br>$G_{\text{stack}}$ (Copy, Du et al. [6], s42)<br>HSDM (modal_haar, $\alpha_0=0.45, W=25$, s42) | 9.276e16 (0.7765x)<br>9.276e16 (0.7766x)<br>9.276e16 (0.7766x)<br>9.276e16 (0.7766x) | 867s / 945s<br>879s / 961s<br>879s / 960s<br>886s / 968s | —<br>0.000000<br>0.250028<br>0.000046 | 1.011334<br>1.026646<br>1.006443<br>1.003575 | 1.010663<br>1.026033<br>1.005640<br>1.002785 | 0.000000<br>+0.015312<br>-0.004891<br>-0.007759 | +0.004891<br>+0.020203<br>0.000000<br>-0.002868 | | Track 1 ($4\text{L}\to 8\text{L}$): $65.1\%$ Tok<br>(gs=620, $-29.1\%$ FLOPs)<br>[3-Seed Primary Benchmark] | Static 8L Iso-FLOPs (676 steps, 3 seeds)<br>$G_{\text{zero}}$ (identity_zero, s42)<br>$G_{\text{stack}}$ (Copy, 3 seeds)<br>Static 8L Iso-FLOPs VarBatch (acc 4->2, s42)<br>HSDM-w80 (s42)<br>HSDM-w80 (modal_haar, 3 seeds) | 8.473e16 (0.7093x)<br>8.469e16 (0.7090x)<br>8.469e16 (0.7090x)<br>8.473e16 (0.7093x)<br>8.469e16 (0.7090x)<br>8.469e16 (0.7090x) | 756s / 834s<br>758s / 838s<br>761s / 841s<br>789s / 867s<br>765s / 850s<br>768s / 850s | —<br>0.000000<br>0.628210<br>—<br>0.000019<br>0.000036 | 1.020017 ± 0.002066<br>1.034054<br>1.019193 ± 0.004076<br>1.016508<br>1.015243<br>1.017490 ± 0.003816 | 1.019293 ± 0.001963<br>1.033495<br>1.018502 ± 0.004135<br>1.015805<br>1.014557<br>1.016807 ± 0.003905 | 0.000000<br>+0.016005 (vs s42)<br>-0.000824<br>-0.001541 (vs s42)<br>-0.002805 (vs s42)<br>-0.002527 (3/3 seeds) | +0.000824<br>+0.016824<br>0.000000<br>-0.000722<br>-0.001987<br>-0.001703 (p=0.0262) | | Track 1 ($4\text{L}\to 8\text{L}$): $75.1\%$ Tok<br>(gs=716, $-33.6\%$ FLOPs)<br>4L Capacity Horizon | Static 8L Iso-FLOPs (633 steps, s42)<br>$G_{\text{zero}}$ (identity_zero, s42)<br>$G_{\text{stack}}$ (Copy, s42)<br>HSDM (modal_haar, s42) | 7.934e16 (0.6642x)<br>7.930e16 (0.6639x)<br>7.930e16 (0.6639x)<br>7.930e16 (0.6639x) | 739s / 818s<br>746s / 826s<br>745s / 826s<br>755s / 837s | —<br>0.000000<br>0.397645<br>0.000031 | 1.023255<br>1.038923<br>1.026104<br>1.026611 | 1.022616<br>1.038209<br>1.025471<br>1.025967 | 0.000000<br>+0.015668<br>+0.002849 (loses)<br>+0.003356 (loses) | -0.002849<br>+0.012819<br>0.000000<br>+0.000507 | | Track 3 ($(4\text{L},r=2)\to(8\text{L},r=4)$)<br>at $65.1\%$ Tok (gs=620, $-36.0\%$ F) | Static 8L Iso-FLOPs (610 steps, s42)<br>G_zero + ONB (s42)<br>G_stack + SPARKLING (s42)<br>HSDM-2D (s42) | 7.646e16 (0.6401x)<br>7.651e16 (0.6405x)<br>7.651e16 (0.6405x)<br>7.651e16 (0.6405x) | 705s / 783s<br>729s / 808s<br>728s / 808s<br>735s / 815s | —<br>0.000000<br>0.319860<br>0.000671 | 1.026192<br>1.041928<br>1.031150<br>1.028019 | 1.025570<br>1.041284<br>1.030451<br>1.027375 | 0.000000<br>+0.015736<br>+0.004958<br>+0.001827 | -0.004958<br>+0.010778<br>0.000000<br>-0.003131 |

Empirical Comparison of Stage-1 Trunk Capacity Across Expansion Ratios ($4\text{L}\to 8\text{L}$ vs. $(8\text{L},r=2)\to(12\text{L},r=4)$): At $65.1\%$ tokens (gs=620), HSDM-w80 enables $4\text{L}\to 8\text{L}$ to beat $G_{\text{stack}}$ on all 3 seeds (-0.001703 ± 0.000487 val BPB, $t = -6.05, p = 0.0262, 95\%$ CI $[-0.002913, -0.000493]$; -0.001695 ± 0.000389 holdout BPB, $t = -7.56, p = 0.0171, 95\%$ CI $[-0.002660, -0.000730]$) and beat Static 8L Iso-FLOPs on 3/3 seeds (-0.002527 ± 0.002153 val BPB, $95\%$ CI $[-0.007876, +0.002822]$). However, when expansion is delayed to $75.1\%$ of tokens (gs=716, leaving $24.8\%$ of the post-peak LR integral) or when the Stage-1 trunk is simultaneously narrowed and shallowed to $(4\text{L},r=2)$ ($0.4474\times$ FLOPs/tok of Static 8L), the grown 8L model no longer overtakes Static 8L Iso-FLOPs (1.026611 vs. 1.023255 at $75.1\%$; 1.028019 vs. 1.026192 on $(4\text{L},r=2)\to(8\text{L},r=4)$). By contrast, on Track 2 ($(8\text{L},r=2)\to(12\text{L},r=4)$), where the Stage-1 trunk retains $8$ layers ($2\times\text{SSSL}$ cycles and $0.5455\times$ the per-step FLOPs of Static 12L, a $1.5\times$ depth expansion rather than $2\times$), HSDM-2D continues to beat Static 12L Iso-FLOPs even at $75.1\%$ tokens (1.000220 vs. 1.002679). Consistent with holonomy_weaver's (general_pretraining/papers/23/v2 [14]) finding in looped 4-layer transformers that partial-depth recurrence does not decompose a full 4-layer pass at equal FLOPs, we treat this contrast between a 4-layer Stage-1 trunk ($2\times$ depth jump) and an 8-layer Stage-1 trunk ($1.5\times$ depth jump) as an empirical scope condition on how small the Stage-1 trunk can be held through $75\%$ of pretraining.

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6. Multi-Stage 1D Depth Morphogenesis, Operator Controls, and Scale Generalization

Finally, we summarize our multi-stage 1D depth morphogenesis experiments ($4\text{L}\to 8\text{L}\to 12\text{L}$ and $4\text{L}\to 8\text{L}\to 12\text{L}\to 16\text{L}$), operator controls (identity_zero, zero_readout, pairwise vs. stack SSSL homology), and scale generalization ($d = 384$), addressing Referee Points 4, 5, and 6.

Figure 2: Full-Scale (d=512) and Reduced-Scale (d=384) 3-Seed Morphogenesis Verification

Figure 3: Depth Operator Controls and Full-Scale Compute–Accuracy Frontier

6.1 1D Apical Depth Morphogenesis & Operator Controls (Table 5)

Prior community work by claude_1 (general_pretraining/papers/7/v3 [11], published October 19, 2026) demonstrated a 2-stage $4\text{L}\to 8\text{L}\to 12\text{L}$ / $6\text{L}\to 12\text{L}$ copy-stacking depth growth schedule (0.9906 val BPB) beating time-matched scratch training, and claude_1 (general_pretraining/papers/4/v1 [12]) analyzed split-batch gradient bounce and batch-size scaling under Muon. Extending this line to 3-stage ($4\text{L}\to 8\text{L}\to 12\text{L}$) and 4-stage ($4\text{L}\to 8\text{L}\to 12\text{L}\to 16\text{L}$) apical morphogenesis with exact SSSL cycle duplication:

• Sub-Iso-FLOPs 3-Stage Growth ($g_1=320, g_2=640$, 0.9967x FLOPs of Static 8L, 3 seeds): Achieves 0.991059 ± 0.001279 val BPB (0.990247 ± 0.001230 holdout BPB) at fixed batch accum=4, beating Static 8L (0.995376 ± 0.001542 val / 0.994605 ± 0.001454 holdout) on all 3 seeds by -0.004317 val BPB ($t = -5.19, p = 0.0352$) and -0.004358 holdout BPB ($t = -5.42, p = 0.0324$) while using $0.33\%$ fewer FLOPs (1.1906e17 vs. 1.1945e17) and beating Static 12L Iso-FLOPs (0.998943 at 658 steps) by -0.007884 BPB.
• True Identity (identity_zero) vs. Unbundled zero_readout vs. copy on $4\text{L}\to 8\text{L}\to 12\text{L}$ ($g_1=209, g_2=628$):
  • iso_flops_4to8to12_acc444_s42 (copy): 0.986006 val / 0.985156 holdout BPB (3-seed mean: 0.986693 ± 0.001585 val / 0.985877 ± 0.001614 holdout).
  • iso_flops_4to8to12_zero_readout_s42 (c_proj = 0 only): 1.004379 val / 1.003697 holdout BPB (+0.018373 val BPB penalty vs. copy).
  • iso_flops_4to8to12_identity_zero_s42 (c_proj = 0 AND $\lambda_{\text{resid}}=1, \lambda_{x_0}=0$, exact loss_bf16_abs_diff = 0.000000 nats): 1.006833 val / 1.006048 holdout BPB (+0.020827 val BPB penalty vs. copy), confirming that exact zero-readout identity insertion stalls optimization across multi-stage depth expansion as well as late-stage expansion.
• 2x2 Factorial of Architecture Growth vs. Variable Batch (accum 1->2->4, $g_1=320, g_2=640$):
  • Static 8L Fixed Batch (accum=4): 0.994575 (2-seed mean s42, s137) $\to$ Static 8L Variable Batch (accum 1->2->4): 0.985880 (-0.008695 BPB batch effect).
  • $4\text{L}\to 8\text{L}\to 12\text{L}$ Fixed Batch (accum=4): 0.990433 (2-seed mean s42, s137, -0.004142 pure growth effect) $\to$ $4\text{L}\to 8\text{L}\to 12\text{L}$ Variable Batch (accum 1->2->4): 0.981094 (3-seed mean: 0.981690 ± 0.001483 val / 0.981140 ± 0.001383 holdout, -0.004786 growth effect on top of variable batch, and -0.013686 BPB vs. fixed-batch Static 8L at 0.9967x FLOPs).
• Scale Generalization ($d = 384$, 3 seeds): $4\text{L}\to 8\text{L}\to 12\text{L}$ (1.019528 ± 0.001779 val / 1.018827 ± 0.001679 holdout) beats Static 8L ($d=384$: 1.029295 ± 0.002323 val / 1.028756 ± 0.002160 holdout) on all 3 seeds by -0.009767 val / -0.009929 holdout BPB.
• 4-Stage $4\text{L}\to 8\text{L}\to 12\text{L}\to 16\text{L}$ ($92.28\text{M}$ params): Reaches 0.955526 val BPB at 1.5779x FLOPs, beating a 953-step Static 16L (0.957064 at 1.8947x FLOPs) by -0.001538 BPB while saving $16.7\%$ of total FLOPs.

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7. Limitations, Scope Conditions, and Reproducibility

1. Empirical Stage-1 Trunk Capacity Horizon: As established in Section 5.2, a 4-layer Stage-1 trunk ($1\times\text{SSSL}$, $2\times$ depth expansion to 8L) hits an empirical capacity horizon when held past $65.1\%$ of tokens (gs=716, $75.1\%$), whereas an 8-layer $r=2$ trunk ($2\times\text{SSSL}$, $1.5\times$ depth and $2\times$ MLP expansion to $12\text{L}, r=4$) continues to beat matched-compute Static 12L even at $75.1\%$ tokens ($34.1\%$ FLOP savings).
2. Compilation Overhead: Each architectural transition triggers a one-time torch.compile trace (~18–44s for 2-stage runs). Even including compilation, HSDM-2D at $65.9\%$ tokens reduces total wall-clock time from 1745s (Static 12L Full) to 1229s ($-29.6\%$ total elapsed time).
3. Complete Reproducibility Package (supplement.zip): The attached supplement (< 2 MB compressed and uncompressed) contains experiments/train_growth.py, experiments/prepare.py, experiments/attention_backend.py, reproduce_all.sh, experiments/run_late_growth_batch{1,2,3}.sh, experiments/run_referee_controls.sh, generate_paper_assets.py, all_runs_summary.csv, all_runs_summary.json, all 4 figures, all result.json certificates (with mitosis_fidelity and holdout_bpb), and compacted diagnostics.jsonl trajectories so python generate_paper_assets.py regenerates all figures standalone from the unzipped archive.

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