# 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**.
---
## 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`.
---
## 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.*
---
## 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).
### 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 ($d=512$)
$(4\text{L},r=1)\to(8\text{L},r=2)\to(12\text{L},r=4)$ | `Net2Wider` ($\alpha=0.0$)
**`Dipolar` ($\alpha=0.5$)** | `0.9454x`
`0.9454x` | `1.023103 ± 0.000565`
**`0.999985 ± 0.000297`** | `1.022273 ± 0.000520`
**`0.999105 ± 0.000273`** | **`-0.023119` (`p=0.000067`)**
**`-0.023169` (`p=0.000043`)** | `1.0000`
`0.7850` |
| **Family 2:** Pure Width ($d=512$)
$(8\text{L},r=2)\to(8\text{L},r=4)$ at `gs=418` | `Net2Wider` ($\alpha=0.0$)
**`Dipolar` ($\alpha=0.5$)** | `0.9077x`
`0.9077x` | `1.016172 ± 0.002021`
**`1.002975 ± 0.001898`** | `1.015372 ± 0.002008`
**`1.002168 ± 0.001850`** | **`-0.013197` (`p=0.000087`)**
**`-0.013204` (`p=0.000073`)** | `1.0000`
`0.8220` |
| **Family 3:** Pure Width ($d=384$)
$(8\text{L},r=2)\to(8\text{L},r=4)$ at `gs=418` | `Net2Wider` ($\alpha=0.0$)
**`Dipolar` ($\alpha=0.5$)** | `0.5796x`
`0.5796x` | `1.048972 ± 0.001413`
**`1.034259 ± 0.001294`** | `1.048436 ± 0.001462`
**`1.033701 ± 0.001367`** | **`-0.014713` (`p=0.000023`)**
**`-0.014735` (`p=0.000025`)** | `1.0000`
`0.8050` |
| **Family 4:** Coupled 2D ($d=512$)
$(8\text{L},r=2)\to(12\text{L},r=4)$ at `gs=418` | `Net2Wider` ($\alpha=0.0$)
**`Dipolar` ($\alpha=0.5$)** | `1.1588x`
`1.1588x` | `0.998193 ± 0.001522`
**`0.984964 ± 0.001835`** | `0.997302 ± 0.001465`
**`0.984117 ± 0.001875`** | **`-0.013230` (`p=0.000334`)**
**`-0.013185` (`p=0.000246`)** | `1.0000`
`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)**
$(8\text{L},r=2)\to(8\text{L},r=4)$, Fixed `accum=4` | `Static 8L Iso-FLOPs` (`865` steps)
`Net2Wider` ($\alpha=0.0$, 3-seed mean)
`Orthogonal Budding` ($W_{\text{proj}}^{(2)}=0$ + Re-Warmup)
`Dipolar` ($\alpha=0.5$, `s42` / 3-seed mean)
`LEMON` (Gaussian / Scalar, Wang et al. [5])
`SPARKLING` (Mom Reset + Re-Warmup [10])
`Noisy Net2Wider` (Chen et al. [1])
**`Stiefel Tangent` ($\Delta_{\text{fc}}^\perp \perp W_{\text{fc}}$, `s42` / 3-seed)** | `0.9077x`
`0.9077x`
`0.9077x`
`0.9077x`
`0.9077x`
`0.9077x`
`0.9077x`
`0.9077x` | —
`0.000000`
`0.000000`
`0.000079`
`0.000095`
`0.000075`
`0.001325`
`0.006132` | `1.000289`
`1.016172`
`1.004622`
`1.001315` (`1.002975`)
`1.001270` / `1.001197`
`1.001062`
`1.000816`
**`1.000567` (`1.002516`)** | `0.999580`
`1.015372`
`1.003929`
`1.000580` (`1.002168`)
`1.000493` / `1.000333`
`1.000294`
`1.000038`
**`0.999689` (`1.001632`)** | `0.000000`
`+0.015883`
`+0.004333`
`+0.001026`
`+0.000908`
`+0.000773`
`+0.000527`
**`+0.000278`** |
| **Pure Width (`gs=200`, $21.0\%$ tok)**
$(8\text{L},r=2)\to(8\text{L},r=4)$, `accum 1->4` | `Static 8L VarBatch Iso-FLOPs` (`911` steps, `acc 1->4`)
`Net2Wider` ($\alpha=0.0$, 3-seed mean)
`Dipolar` ($\alpha=0.5$, `s42` / 3-seed mean)
`SPARKLING` (Mom Reset + Re-Warmup [10])
**`Stiefel Tangent` ($\Delta_{\text{fc}}^\perp \perp W_{\text{fc}}$)** | `0.9559x`
`0.9558x`
`0.9558x`
`0.9558x`
`0.9558x` | —
`0.000075`
`0.000079`
`0.000103`
`0.004094` | `0.986536`
`1.007528`
`0.988817` (`0.990918`)
`0.988216`
**`0.987810`** | `0.985794`
`1.006810`
`0.988164` (`0.990128`)
`0.987559`
**`0.987159`** | `0.000000`
`+0.020992`
`+0.002281`
`+0.001680`
`+0.001274` |
| **Coupled 2D (`gs=418`, $43.9\%$ tok)**
$(8\text{L},r=2)\to(12\text{L},r=4)$, Fixed `accum=4` | `Static 12L Iso-FLOPs` (`763` steps)
`Net2Wider` ($\alpha=0.0$, 3-seed mean)
`Dipolar` ($\alpha=0.5$, `s42` / 3-seed mean)
`Noisy Net2Wider` (Chen et al. [1])
`G_stack + SPARKLING` (Yu et al. [10])
**`Stiefel + Dual Split + SPARKLING`** | `1.1588x`
`1.1588x`
`1.1588x`
`1.1588x`
`1.1588x`
`1.1588x` | —
`0.194000`
`0.194000`
`0.194116`
`0.202833`
`0.186057` | `0.987346`
`0.998193`
`0.983386` (`0.984964`)
`0.983381`
`0.983304`
**`0.983169`** | `0.986652`
`0.997302`
`0.982529` (`0.984117`)
`0.982592`
`0.982477`
**`0.982227`** | `0.000000`
`+0.010847`
`-0.003960`
`-0.003965`
`-0.004042`
**`-0.004177`** |
| **Coupled 2D (`gs=260`, $27.3\%$ tok)**
$(8\text{L},r=2)\to(12\text{L},r=4)$, `accum 1->4` | `Static 12L VarBatch Iso-FLOPs` (`835` steps, `acc 1->4`)
**`Stiefel + Dual Split + SPARKLING`** | `1.2682x`
`1.2679x` | —
`0.402935` | `0.972150`
**`0.969330`** | `0.971421`
**`0.968662`** | `0.000000`
**`-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: $4\text{L}\to 8\text{L}$)**.
### 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**
(`gs=418`, **$-20.0\%$ FLOPs**) | `Static 12L Iso-FLOPs` (`763` steps, `s42`)
`G_stack + SPARKLING` (`acc=4`, `s42`)
`Stiefel + Dual Split + SPARKLING` (`acc=4`, `s42`)
`HSDM-2D Fixed-Batch` (`accum=4`, `s42`)
**`HSDM-2D` (`DSTRH + Modal Muon, acc 4->2`, `s42`)** | `1.384e17` (`1.1588x`)
`1.384e17` (`1.1588x`)
`1.384e17` (`1.1588x`)
`1.384e17` (`1.1588x`)
`1.384e17` (`1.1588x`) | `1271s` / `1372s`
`1322s` / `1426s`
`1317s` / `1420s`
`1265s` / `1369s`
`1274s` / `1378s` | —
`0.202833`
`0.186057`
**`0.000730`**
**`0.000928`** | `0.987346`
`0.983304`
`0.983169`
`0.986954`
**`0.982243`** | `0.986652`
`0.982477`
`0.982227`
`0.986112`
**`0.981413`** | `0.000000`
`-0.004042`
`-0.004177`
`-0.000392`
**`-0.005103`** | `+0.004042`
`0.000000`
`-0.000135`
`+0.003650`
**`-0.001061`** |
| **Halfway: $50.0\%$ Tok**
(`gs=476`, **$-22.7\%$ FLOPs**)
*Cooldown onset ($p=0.50$)* | `Static 12L Iso-FLOPs` (`735` steps, `s42`)
`G_stack + SPARKLING` (`s42`)
**`HSDM-2D` (`DSTRH + Modal Muon`, `s42`)** | `1.333e17` (`1.1163x`)
`1.336e17` (`1.1188x`)
`1.336e17` (`1.1188x`) | `1266s` / `1367s`
`1278s` / `1383s`
`1289s` / `1395s` | —
`0.247046`
**`0.000010`** | `0.990193`
`0.986679`
**`0.985388`** | `0.989399`
`0.985787`
**`0.984553`** | `0.000000`
`-0.003514`
**`-0.004805`** | `+0.003514`
`0.000000`
**`-0.001292`** |
| **Late: $65.9\%$ Tok**
(`gs=628`, **$-29.9\%$ FLOPs**)
*Deep in LR cooldown ($46.5\%$ post-peak LR integral rem.)*
**[3-Seed Primary Benchmark]** | `Static 12L Iso-FLOPs` (`668` steps, **3 seeds**)
`G_stack + SPARKLING` (`sparkling_rewarmup`, **3 seeds**)
`G_stack + SPARKLING Eq. 28` (`accum=4`, `s42`)
`G_stack + SPARKLING Eq. 28` (`accum 4->2`, `s42`)
Plain `Dipolar` ($\alpha=0.5$, coupled Muon, `s42`)
`HSDM-2D Fixed-Batch` (`accum=4`, `s42`)
`Static 12L Iso-FLOPs VarBatch` (`acc 4->2`, **3 seeds**)
**`HSDM-2D` Full (`s42`)**
**`HSDM-2D` Full (3-seed mean ± SD)**
**`HSDM-2D + Post-Mitosis μ-Cooldown` (`s42`)**
**`HSDM-2D + Isometric PE Split` (`isometric_pe`, `s42`)** | `1.212e17` (`1.0145x`)
`1.211e17` (`1.0138x`)
`1.211e17` (`1.0138x`)
`1.211e17` (`1.0138x`)
`1.211e17` (`1.0138x`)
`1.211e17` (`1.0138x`)
`1.212e17` (`1.0145x`)
`1.211e17` (`1.0138x`)
`1.211e17` (`1.0138x`)
`1.211e17` (`1.0138x`)
`1.211e17` (`1.0138x`) | `1091s` / `1191s`
`1108s` / `1211s`
`1054s` / `1156s`
`1061s` / `1162s`
`1101s` / `1205s`
`1152s` / `1259s`
`1078s` / `1180s`
`1106s` / `1242s`
`1113s` / `1229s`
`1116s` / `1220s`
`1106s` / `1240s` | —
`0.271480`
`0.253000`
`0.252996`
`0.281000`
**`0.000260`**
—
**`0.000026`**
**`0.000159`**
**`0.000013`**
**`0.000047`** | `0.998472 ± 0.001947`
`1.002396 ± 0.001997`
`0.999695`
`0.993431`
`1.001034`
`0.997694`
`0.996840 ± 0.001842`
**`0.993419`**
**`0.994959 ± 0.001857`**
**`0.993153`**
**`0.992815`** | `0.997666 ± 0.001857`
`1.001491 ± 0.001992`
`0.998872`
`0.992625`
`1.000269`
`0.996855`
`0.996015 ± 0.001794`
**`0.992586`**
**`0.994115 ± 0.001844`**
**`0.992400`**
**`0.991972`** | `0.000000`
`+0.003924` *(loses)*
`+0.001927` *(vs `s42`)*
`-0.004337` *(vs `s42`)*
`+0.003266` *(loses)*
`-0.000074` *(vs `s42`)*
`-0.001632` *(3 seeds)*
**`-0.004348` *(vs `s42`)* **
**`-0.003513` (`-0.003551` HO)**
**`-0.004615` (`-0.002985` vs VB)**
**`-0.004952` (`-0.003323` vs VB)** | `-0.003924`
`0.000000`
`-0.001107` *(vs `s42`)*
`-0.007371` *(vs `s42`)*
`+0.000232`
**`-0.003108` (`-0.002001` vs Eq28)**
`-0.005556`
**`-0.007383`**
**`-0.007436` (`p=0.000151`)**
**`-0.007649`**
**`-0.007987`** |
| **Very Late: $75.1\%$ Tok**
(`gs=716`, **$-34.1\%$ FLOPs**)
*Extreme cooldown ($24.8\%$ post-peak LR integral rem.)* | `Static 12L Iso-FLOPs` (`628` steps, `s42`)
`G_stack + SPARKLING` (`s42`)
**`HSDM-2D` (`DSTRH + Modal Muon`, `s42`)** | `1.139e17` (`0.9538x`)
`1.139e17` (`0.9531x`)
`1.139e17` (`0.9531x`) | `1065s` / `1166s`
`1074s` / `1177s`
`1079s` / `1184s` | —
`0.311308`
**`0.000155`** | `1.002679`
`1.008111`
**`1.000220`** | `1.002079`
`1.007398`
**`0.999387`** | `0.000000`
`+0.005432` *(loses)*
**`-0.002459` (`-0.002692` HO)** | `-0.005432`
`0.000000`
**`-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)**
`ablation_ofat_g628_acc4_lerp25_s42` (`DSTRH -> lerp`)
`ablation_ofat_g628_acc4_nohom_s42` (`DSTRH -> hom=0`)
`ablation_ofat_g628_acc4_nomodal_s42` (**Remove `modal_hadamard`**)
`lit_gstack_sparkling_eq28_8r2_to_12r4_g628_s42`
`lit_gstack_sparkling_8r2_to_12r4_g628_s42` | `dstrh:25`
`lerp:25`
`none`
`dstrh:25`
`none`
`none` | `modal_hadamard`
`modal_hadamard`
`modal_hadamard`
**`Muon (none)`**
`Muon (S'=[S,0])`
`Muon (S'=[S,0])` | `stiefel`
`stiefel`
`stiefel`
`stiefel`
`sparkling_eq28`
`sparkling_rewarm` | `4`
`4`
`4`
`4`
`4`
`4` | **`0.000260`**
`0.000351`
`0.274204`
`0.000185`
`0.253000`
`0.253000` | **`0.997694`**
`0.997453`
`0.996988`
**`1.003737`**
`0.999695`
`1.000802` | **`0.996855`**
`0.996623`
`0.996132`
**`1.003007`**
`0.998872`
`0.999935` | **`0.5838`**
`0.6018`
`0.5919`
**`0.9160`**
`0.7562`
`0.9333` | `0.000000`
`-0.000241`
`-0.000706`
**`+0.006043` *(collapses)* **
`+0.002001`
`+0.003108` |
| **Group B: Stage-2 Batch Subdivision (`accum 4->2`)** | `hsdm_2d_8r2_to_12r4_g628_isometric_pe_s42`
`hsdm_2d_8r2_to_12r4_g628_mucooldown_s42`
**`hsdm_2d_8r2_to_12r4_g628_s42` (`HSDM-2D` Full)**
`lit_gstack_sparkling_eq28_8r2_to_12r4_g628_acc2_s42` | `dstrh:25`
`dstrh:25`
`dstrh:25`
`none` | `modal_hadamard`
`modal_hadamard + μ`
`modal_hadamard`
`Muon (S'=[S,0])` | `isometric_pe`
`stiefel`
`stiefel`
`sparkling_eq28` | `2`
`2`
`2`
`2` | **`0.000047`**
**`0.000013`**
**`0.000026`**
`0.252996` | **`0.992815`**
**`0.993153`**
**`0.993419`**
`0.993431` | **`0.991972`**
**`0.992400`**
**`0.992586`**
`0.992625` | `0.7145`
`0.7012`
`0.7190`
`0.6267` | `-0.000604`
`-0.000266`
`0.000000`
`+0.000012` |
| **Group C: Reduced Scale ($d=384$, `46.4M` params)** | `d384_static_12L_isoflops_70pct_s42` (`668` steps)
`d384_lit_gstack_sparkling_8r2_to_12r4_g628_s42`
**`d384_hsdm_2d_8r2_to_12r4_g628_s42` (`HSDM-2D`)** | —
`none`
`dstrh:25` | `Muon`
`Muon (S'=[S,0])`
`modal_hadamard` | —
`sparkling_rewarm`
`stiefel` | `4`
`4`
`2` | —
`0.233000`
**`0.000139`** | `1.029480`
`1.032831`
**`1.024558`** | `1.028877`
`1.032440`
**`1.024110`** | —
`0.9280`
`0.7080` | `0.000000`
`+0.003351` *(loses)*
**`-0.004922` (`-0.008273` vs SP)** |
| **Group D: Tuned AdamW (`lr=0.003`, `--matrix-opt adamw`)** | `adamw_static_12L_isoflops_70pct_s42` (`668` steps)
`adamw_gstack_sparkling_8r2_to_12r4_g628_s42`
**`adamw_hsdm_2d_8r2_to_12r4_g628_s42` (`HSDM-2D`)** | —
`none`
`dstrh:25` | `AdamW`
`AdamW (S'=[S,0])`
`AdamW` | —
`sparkling_eq28`
`stiefel` | `4`
`4`
`2` | —
`0.506000`
**`0.000287`** | `1.068919`
`1.054622`
**`1.044510`** | `1.068718`
`1.054256`
**`1.044198`** | —
`0.9960`
`0.8920` | `0.000000`
`-0.014297`
**`-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**
(`gs=476`, **$-22.3\%$ FLOPs**) | `Static 8L Iso-FLOPs` (`740` steps, `s42`)
$G_{\text{zero}}$ (`identity_zero`, `s42`)
$G_{\text{stack}}$ (Copy, Du et al. [6], `s42`)
**`HSDM` (`modal_haar`, $\alpha_0=0.45, W=25$, `s42`)** | `9.276e16` (`0.7765x`)
`9.276e16` (`0.7766x`)
`9.276e16` (`0.7766x`)
`9.276e16` (`0.7766x`) | `867s` / `945s`
`879s` / `961s`
`879s` / `960s`
`886s` / `968s` | —
**`0.000000`**
`0.250028`
`0.000046` | `1.011334`
`1.026646`
`1.006443`
**`1.003575`** | `1.010663`
`1.026033`
`1.005640`
**`1.002785`** | `0.000000`
`+0.015312`
`-0.004891`
**`-0.007759`** | `+0.004891`
`+0.020203`
`0.000000`
**`-0.002868`** |
| **Track 1 ($4\text{L}\to 8\text{L}$): $65.1\%$ Tok**
(`gs=620`, **$-29.1\%$ FLOPs**)
**[3-Seed Primary Benchmark]** | `Static 8L Iso-FLOPs` (`676` steps, **3 seeds**)
$G_{\text{zero}}$ (`identity_zero`, `s42`)
$G_{\text{stack}}$ (Copy, **3 seeds**)
`Static 8L Iso-FLOPs VarBatch` (`acc 4->2`, `s42`)
**`HSDM-w80` (`s42`)**
**`HSDM-w80` (`modal_haar`, 3 seeds)** | `8.473e16` (`0.7093x`)
`8.469e16` (`0.7090x`)
`8.469e16` (`0.7090x`)
`8.473e16` (`0.7093x`)
`8.469e16` (`0.7090x`)
`8.469e16` (`0.7090x`) | `756s` / `834s`
`758s` / `838s`
`761s` / `841s`
`789s` / `867s`
`765s` / `850s`
`768s` / `850s` | —
**`0.000000`**
`0.628210`
—
`0.000019`
`0.000036` | `1.020017 ± 0.002066`
`1.034054`
`1.019193 ± 0.004076`
`1.016508`
**`1.015243`**
**`1.017490 ± 0.003816`** | `1.019293 ± 0.001963`
`1.033495`
`1.018502 ± 0.004135`
`1.015805`
**`1.014557`**
**`1.016807 ± 0.003905`** | `0.000000`
`+0.016005` (vs `s42`)
`-0.000824`
`-0.001541` (vs `s42`)
**`-0.002805` (vs `s42`)**
**`-0.002527` (3/3 seeds)** | `+0.000824`
`+0.016824`
`0.000000`
`-0.000722`
**`-0.001987`**
**`-0.001703` (`p=0.0262`)** |
| **Track 1 ($4\text{L}\to 8\text{L}$): $75.1\%$ Tok**
(`gs=716`, **$-33.6\%$ FLOPs**)
*4L Capacity Horizon* | `Static 8L Iso-FLOPs` (`633` steps, `s42`)
$G_{\text{zero}}$ (`identity_zero`, `s42`)
$G_{\text{stack}}$ (Copy, `s42`)
`HSDM` (`modal_haar`, `s42`) | `7.934e16` (`0.6642x`)
`7.930e16` (`0.6639x`)
`7.930e16` (`0.6639x`)
`7.930e16` (`0.6639x`) | `739s` / `818s`
`746s` / `826s`
`745s` / `826s`
`755s` / `837s` | —
**`0.000000`**
`0.397645`
`0.000031` | `1.023255`
`1.038923`
`1.026104`
`1.026611` | `1.022616`
`1.038209`
`1.025471`
`1.025967` | `0.000000`
`+0.015668`
`+0.002849` *(loses)*
`+0.003356` *(loses)* | `-0.002849`
`+0.012819`
`0.000000`
`+0.000507` |
| **Track 3 ($(4\text{L},r=2)\to(8\text{L},r=4)$)**
at $65.1\%$ Tok (`gs=620`, **$-36.0\%$ F**) | `Static 8L Iso-FLOPs` (`610` steps, `s42`)
`G_zero + ONB` (`s42`)
`G_stack + SPARKLING` (`s42`)
**`HSDM-2D` (`s42`)** | `7.646e16` (`0.6401x`)
`7.651e16` (`0.6405x`)
`7.651e16` (`0.6405x`)
`7.651e16` (`0.6405x`) | `705s` / `783s`
`729s` / `808s`
`728s` / `808s`
`735s` / `815s` | —
**`0.000000`**
`0.319860`
`0.000671` | `1.026192`
`1.041928`
`1.031150`
`1.028019` | `1.025570`
`1.041284`
`1.030451`
`1.027375` | `0.000000`
`+0.015736`
`+0.004958`
`+0.001827` | `-0.004958`
`+0.010778`
`0.000000`
**`-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.
---
## 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.
### 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**.
---
## 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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