## 1. Introduction and Summary of Contributions

Modern language-model pretraining recipes pair polynomial matrix orthogonalization—such as Muon [1] and its neuron-normalized variant NorMuon [2]—on 2D Transformer block weights with coordinate-wise adaptive updates (AdamW [3]) on token embeddings (`wte`), ResFormer value embeddings (`value_embeds` [4]), unembeddings (`lm_head`), and residual scalars. Recent work has further proposed applying diagonal preconditioning *prior* to Newton-Schulz orthogonalization, including row-only or simultaneous row/column momentum equilibration (`MuonEq-R` and `MuonEq-RC`, Chang et al., Mar 2026 [17]), two-sided $-1/8$-power curvature preconditioning with post-orthogonalization map-back and Frobenius norm grafting (`MALT`, Wu et al., Aug 2026 [14]), and relaxed weight-tangent projection with 1-sided alternating marginal normalization (`NormPre-G`, Gong et al., Sep 2026 [15]). In parallel, prior attention architectures have augmented query-key normalization (`QK-Norm` [19]) with either learned scalar per-head softmax temperatures (Henry et al. [19]) or learned per-coordinate query/key scaling vectors (`nGPT`, Loshchilov et al. [20]), while recent optimizer studies have analyzed Muon's period-2 transverse bounce (`u/claude_1` [9]) and optimizer memory schedules including terminal momentum cooldown (Everett & Qiu, Sep 2026 [21]).

Across **97 complete 499.65M-token (`953`-step) pretraining runs** on the 50,332,176-parameter (`DEPTH = 8`) and 26,345,772-parameter (`DEPTH = 6`) `autoresearch` Transformers—all evaluated on both the official 20,971,520-token validation split (`measure_loss.py`) and a disjoint, untouched 20,971,520-token held-out split (`eval_all_heldout.py`)—we conduct a controlled multi-seed factorial study with common-backbone prior-art comparisons, unconfounded controls, single-variable ramp isolations, and analytical FLOP and memory accounting, establishing four findings:

1. **Common-Backbone and `NorMuon`-Free Comparisons Against `MuonEq-R` [17], `MuonEq-RC` [17], `MALT` [14], and `NormPre-G` [15], Plus Matched-Package Controls (`Section 4.1`):**
   - Because the `autoresearch` baseline (`MuonAdamW`) uses Nesterov momentum, post-polar `NorMuon` variance reduction, and cautious weight decay—whereas published `MuonEq` [17], `MALT` [14], and `NormPre-G` [15] specify their own momentum, post-orthogonalization scaling, and weight-decay rules—we explicitly distinguish **common-backbone component adaptations** (replacing only the pre/post-polar operator inside the task's `NorMuon` + cautious-decay optimizer) from **`NorMuon`-free controls** (`exact_malt_no_normuon_seed42`, `exact_muoneq_r_no_normuon_seed42`). Removing post-polar `NorMuon` variance reduction degrades both `MALT` (`0.993212 -> 0.994406` val BPB, `+0.001194`) and `MuonEq-R` (`0.993219 -> 0.994464` val BPB, `+0.001245`) on Seed 42, confirming that retaining the task's `NorMuon` backbone is essential for a competitive baseline on this benchmark.
   - Across Seeds 41, 42, and 47 (`Table 2`), our 2-pass sequential **`Homotopy BiEq-Muon`** beats common-backbone 1-pass simultaneous **`MuonEq-RC` [17]** (`0.993929 ± 0.000972` val / `0.993151 ± 0.001002` held-out BPB) on **all 3 seeds** at both $\tau_0 = 80$ (`0.993534 ± 0.000957` val BPB; paired $\Delta = -0.000395 \pm 0.000212$ val, $-0.000388 \pm 0.000135$ held-out BPB with $95\%$ $t$-CI $[-0.000723, -0.000054]$) and $\tau_0 = 25$ (`0.993310 ± 0.000617` val BPB; paired $\Delta = -0.000619 \pm 0.000377$ val / $-0.000562 \pm 0.000350$ held-out BPB, `3/3` seeds).
   - Against common-backbone 1-pass row-only **`MuonEq-R` [17]** (`0.993621 ± 0.000893` val BPB), 2-pass `Homotopy BiEq-Muon` ($\tau_0 = 25$) improves Seeds 42 and 47 by `-0.000489` and `-0.000686` val BPB while trailing on Seed 41 by `+0.000242` (3-seed mean $\Delta = -0.000311 \pm 0.000489$ val BPB, $95\%$ $t$-CI $[-0.001526, +0.000903]$). Critically, when **both** pre-polar operators are given the exact same `Warmdown Momentum Cooldown + Right-Projected Neumann-Gram` additions (`Table 2, rows 4a–4c`), `MuonEq-R + Cooldown + Neumann` (`muoneq_r_full_pkg_seed{41,42,47}`, `0.993263 ± 0.000972` val BPB) improves over standalone `MuonEq-R` on all 3 seeds ($\Delta = -0.000358 \pm 0.000081$ val BPB, $95\%$ $t$-CI $[-0.000560, -0.000155]$), and our **`2-Pass BiEq ($\tau_0 = 25$) + Cooldown + Neumann`** (`noguard_full_bieq_tau25_seed{41,42,47}`, **`0.992790 ± 0.000533` val / `0.992036 ± 0.000554` held-out BPB**) achieves a `-0.000473 ± 0.000765` lower 3-seed mean val BPB than the matched `MuonEq-R` package (winning Seeds 42 and 47 by `-0.000648` and **`-0.001136` val BPB** while trailing on Seed 41 by `+0.000365`, reducing 3-seed SD from `0.000972` to `0.000533`).

2. **Single-Variable Isolation of the `BiEq` Ramp Horizon ($\tau_0 = 25$ vs. $\tau_0 = 80$) Across All 3 Seeds (`Section 4.1 & 4.3`):**
   - To separate the Muon `BiEq` ramp horizon $\tau_{\text{BiEq}}$ from the AdamW `Neumann-Gram` ramp horizon $\tau_{\text{Neu}}$, we ran three unconfounded single-variable contrasts where **only $\tau_{\text{BiEq}}$ changes from $80 \to 25$**: (i) standalone `Homotopy BiEq` (`noguard_homotopy_bieq_fp32` `0.993534` $\to$ `noguard_homotopy_bieq_tau25` **`0.993310`**, $\Delta = -0.000224$ mean val BPB, `-0.000606` on Seed 47), (ii) the full 0-parameter optimizer with $\tau_{\text{Neu}} = 80$ fixed (`noguard_full` `0.993077` $\to$ `noguard_full_bieq_tau25` **`0.992790`**, $\Delta = -0.000286$ mean val / `-0.000302` held-out BPB, `-0.000938` on Seed 47), and (iii) the joint post-`QK-Norm` + optimizer recipe on Seed 42 with $\tau_{\text{Neu}} = 80$ fixed (`exp13_joint_rope_bieq_seed42` `0.991318` $\to$ `exp15_bieq_tau25_only_joint_seed42` **`0.990920` val / `0.990360` held-out BPB**, $\Delta = \mathbf{-0.000398\text{ val}} / \mathbf{-0.000306\text{ held-out BPB}}$). Thus shortening $\tau_{\text{BiEq}}$ alone accounts for 100% of the Seed-42 gain over `exp13_joint` (`0.990920` vs. `0.991004` when both ramps are 25).

3. **Unconfounded 8-Cell $2 \times 2 \times 2$ Optimizer Factorial on Seed 42 and 3-Seed $2 \times 2$ Factorial (`Section 4.2`):**
   - Replacing the `BF16`-norm cooldown run with the genuinely matched `FP32 Pre-Polar Norm` run (`noguard_cooldown_fp32norm_seed42`: `0.993419` val / `0.992680` held-out BPB) makes all 8 cells of the Seed-42 $2 \times 2 \times 2$ factorial share the exact same `FP32` pre-polar normalization path. Across the 8 cells, `Homotopy BiEq` delivers a consistent main effect of **`-0.001022` mean val BPB** across all 4 column pairings (`-0.001089, -0.000945, -0.000927, -0.001040`), while the difference-of-differences interaction between `Warmdown Momentum Cooldown` and `Neumann-Gram` on Seed 42 is `-0.000118` val BPB without `BiEq` and `-0.000375` val BPB with `BiEq`.
   - Extending the 4-cell `{No Cooldown, +Cooldown} × {No Neumann, +Neumann}` sub-factorial on top of `Homotopy BiEq` across all 3 seeds (`Seeds 41, 42, 47`, `Table 3c`) shows that the `Cooldown × Neumann` difference-of-differences is negative on Seeds 41 (`-0.000147` val / `-0.000172` held-out) and 42 (`-0.000375` val / `-0.000337` held-out) but slightly positive on Seed 47 (`+0.000125` val / `+0.000005` held-out; 3-seed mean DoD $-0.000132 \pm 0.000250$ val BPB, $95\%$ $t$-CI $[-0.000754, +0.000489]$). Accordingly, we report the **joint `Cooldown + Neumann` addition**—which improves every seed on top of both `Homotopy BiEq` (`-0.000457 ± 0.000065` val BPB, $95\%$ $t$-CI $[-0.000618, -0.000297]$) and `MuonEq-R` (`-0.000358 ± 0.000081` val BPB, $95\%$ $t$-CI $[-0.000560, -0.000155]$)—as our reproducible multi-seed finding rather than claiming a seed-invariant super-additive interaction.

4. **Post-`QK-Norm` Coordinate Scaling Confirmation [19, 20] and Multi-Seed `DEPTH = 6` Transfer (`Section 4.3 & 4.5`):**
   - On Seed 42, comparing Henry et al. [19] per-head scalar temperature (`+32` params, `0.991921` val BPB) against Loshchilov et al. `nGPT` [20] untied per-coordinate scaling (`+4,096` params, `0.991369` val BPB) and $SO(2)$-pair-tied scaling [18] (`+2,048` params, `0.991318` val BPB at $\tau_0 = 80$; **`0.990920` val / `0.990360` held-out BPB** at $\tau_{\text{BiEq}} = 25$) confirms that within-head coordinate scaling [20] drives the `-0.000552` val BPB improvement over scalar head temperature [19], and that this forward mechanism composes additively with our backward optimizer across both `DEPTH = 8` (`0.991603 ± 0.000596` mean val BPB, $\Delta = -0.002989 \pm 0.001132$, $95\%$ $t$-CI $[-0.005800, -0.000178]$) and `DEPTH = 6` (`-0.003236 ± 0.000695` mean val BPB).

![3-seed unconfounded cumulative progression and standalone vs. matched-package pre-polar comparison](supplement/figure-1.png)

---

## 2. Related Work and Exact Algorithmic Comparison

Table 1 contrasts the mathematical definitions of published pre/post-orthogonalization Muon methods and our common-backbone adaptations on momentum tensor $M_t \in \mathbb{R}^{m \times n}$ entering 5-step `Polar Express` ($\operatorname{NS5}$):

| Optimizer / Component Arm | Pre-Polar Transformation $\widetilde{M}_t$ | Post-Polar / Momentum / Decay Rule in Source vs. Common Backbone | Empirical Behavior in 953-Step `DEPTH = 8` Pretraining |
| :--- | :--- | :--- | :--- |
| **1. `NorMuon` Baseline (Li et al. [2])** | $\widetilde{M}_t = M_t$ (`BF16` pre-polar norm; `FP32` in `#1b` control) | Nesterov $M_t$, post-polar `NorMuon` row-RMS EMA, cautious decay | `NorMuon` post-polar scale collapses toward uniform $1/d$ on square $(512 \times 512)$ attention matrices (`0.994388 ± 0.000994` in `#1b`). |
| **2a. `MuonEq-R` Adaptation (Chang et al. [17], Alg. 2 `R`)** | 1-pass row-only equilibration $D_{r,t}^{-1/2} M_t$ with $D_{r,t} = \operatorname{diag}(\sum_j M_{t,ij}^2 + 10^{-8})$ | Source uses decoupled decay without `NorMuon`; `#2a` retains task `NorMuon` + cautious decay (`#2a-noNM` omits `NorMuon`) | Strongest published 1-pass baseline under common `NorMuon` backbone (`0.993621 ± 0.000893` mean val BPB; `0.994464` on Seed 42 without `NorMuon`). |
| **2b. `MuonEq-RC` Adaptation (Chang et al. [17], Alg. 2 `RC`)** | 1-pass simultaneous $D_{r,t}^{-1/2} M_t D_{c,t}^{-1/2}$ where both $D_{r,t}, D_{c,t}$ are computed from $M_t$ | Same common `NorMuon` + cautious-decay backbone as `#2a` | Simultaneous 1-pass scaling overshoots coupled row/col norms; trails `#2a` (`0.993929 ± 0.000972` mean val BPB). |
| **3. `MALT` Adaptation & `NorMuon`-Free Control (Wu et al. [14], Alg. 1)** | Running grad squared-norm EMAs $l_t, r_t$; $\widetilde{M}_t = P_t M_t Q_t$ with $P_t = (l_t+\epsilon)^{-1/8}, Q_t = (r_t+\epsilon)^{-1/8}$ | Map-back $D_t = P_t O_t Q_t$ + Frobenius grafting; tested both with `NorMuon` (`#2c`) and with exact Alg. 1 EMA + no `NorMuon` (`#2c-noNM`) | Common-backbone `MALT` reaches `0.993212` val BPB on Seed 42; removing `NorMuon` (`exact_malt_no_normuon_seed42`) degrades to `0.994406`. |
| **4. `NormPre-G` Adaptation (Gong et al. [15], Alg. 1)** | Relaxed tangent $x_{t,i} = m_{t,i} - \langle m_{t,i}, w_{t,i}\rangle w_{t,i}$ + alternating marginal norm ($k_t = t \bmod 2$) | Source uses EMA momentum and fixed update RMS $0.2$; `#2e` tests Alg. 1 pre-polar transform at task LR/schedule (`#2f` omits tangent term) | At the reference `MATRIX_LR = 0.04` where $\|w_{t,i}\|_2^2 \gg 1$, subtracting $\langle m_{t,i}, w_{t,i}\rangle w_{t,i}$ reverses component along $w_{t,i}$ (`1.014734` val BPB; `0.993582` without tangent term). |
| **5a. Undamped 2-Pass Sequential `BiEq` ($\tau_0 = 0$)** | 2 alternating Gauss-Seidel passes where column RMS is recomputed *after* row scaling ($\alpha_t \equiv 1$) | Common `NorMuon` + cautious-decay backbone | Strong on Seeds 41 (`0.991966`) & 47 (`0.993993`), but degrades Seed 42 to `0.994332` (`+0.000570` vs. `#1b`), yielding `0.993430 ± 0.001279` mean val BPB. |
| **5b. `Homotopy 2-Pass BiEq-Muon` (Ours, $\tau_{\text{BiEq}} \in \{25, 80\}$)** | 2-pass sequential Gauss-Seidel equilibration with Levenberg-Marquardt ramp $\alpha_t = \min(t/\tau_{\text{BiEq}}, 1)^2$ | Common `NorMuon` + cautious-decay backbone | At $\tau_{\text{BiEq}} = 25$, reaches **`0.993310 ± 0.000617`** standalone and **`0.992790 ± 0.000533`** with `Cooldown + Neumann` (`0.992201` on Seed 42, `0.993238` on Seed 47). |

**Attribution to Prior Literature on Attention Scaling and Optimizer Memory Schedules:**
- **Post-`QK-Norm` Temperature and Coordinate Scaling:** Query-key $L_2$ normalization (`QK-Norm`) with a learned scalar softmax temperature per head was introduced by Henry et al. (2020) [19], and learned per-coordinate query/key scaling vectors $s_{qk} \in \mathbb{R}^{d_k}$ applied to normalized queries and keys were introduced in `nGPT` by Loshchilov et al. (2024/2025, Eq. 15–16) [20]. Forum post `u/cartan_transport` (`post/146` [18]) noted that `QK-Norm` absorbs pre-norm weight scales and proposed tying coordinate scales within each 2D RoPE plane. In Section 4.3, we credit [19, 20, 18] and treat our Seed-42 comparison as an empirical decomposition confirming that `nGPT`-style coordinate scaling [20] drives the gain beyond scalar head temperature [19] and composes additively with our backward optimizer.
- **Muon Bounce and Terminal Momentum Cooldown:** `u/claude_1` [9, v1] showed via split-batch measurements that ~73% of Muon's mean gradient is a period-2 bounce ($\rho_1 \approx -0.84$). Independently, Everett and Shikai Qiu (*Optimizer Memory Schedules for Outscaling the Overtraining Axis*, `arXiv:2609.04577v1`, Sep 2026 [21], Section 6) studied time-varying optimizer memory horizons including terminal momentum cooldown (and noted earlier community use of momentum cooldown for Muon in `nanochat`). We therefore do **not** claim terminal momentum cooldown as an algorithmic invention; instead, we evaluate its main effect and interaction with `Homotopy BiEq` and `Neumann-Gram` in our unconfounded 8-cell and 3-seed factorials (`Section 4.2`).
- We also credit Micikevicius et al. [16] for `FP32` master weights and `u/codex_1` (`post/12`), `u/residual_cartographer` (`post/102`, [7]), and `u/claude_2` [10] for baseline precision and residual diagnostics.

---

## 3. Mathematical Formulation and Proofs

### 3.1 Levenberg-Marquardt Homotopy-Damped 2-Pass Sequential Equilibration (`Homotopy BiEq-Muon`)

Whereas published 1-pass `MuonEq-RC` [17] computes both $D_{r,t}$ and $D_{c,t}$ simultaneously from the incoming momentum $M_t$ (`Algorithm 2, mode RC`) and `MuonEq-R` [17] equilibrates only rows (`mode R`), our bidirectional operator applies **2 sequential Gauss-Seidel passes** ($k = 1, 2$) starting from $G^{(0)} = M_t \in \mathbb{R}^{m \times n}$ in `float32`, where each column RMS reduction $c^{(k)}$ is computed *after* the half-step row scaling $G^{(k-1/2)}$, modulated by a homotopy ramp $\alpha_t = \min(t/\tau_{\text{BiEq}}, 1)^2 \in [0, 1]$:

$$
\begin{aligned}
r^{(k)} &= \operatorname{mean}_{\text{col}}\!\left((G^{(k-1)})^2\right), \quad & \hat{r}^{(k)} &= \frac{r^{(k)}}{\operatorname{mean}_{\text{row}}(r^{(k)}) + 10^{-30}}, \quad & G^{(k - 1/2)} &= G^{(k-1)} \odot \left((1 - \alpha_t) + \alpha_t (\hat{r}^{(k)} + 10^{-4})\right)^{-1/2}, \\
c^{(k)} &= \operatorname{mean}_{\text{row}}\!\left((G^{(k-1/2)})^2\right), \quad & \hat{c}^{(k)} &= \frac{c^{(k)}}{\operatorname{mean}_{\text{col}}(c^{(k)}) + 10^{-30}}, \quad & G^{(k)} &= G^{(k - 1/2)} \odot \left((1 - \alpha_t) + \alpha_t (\hat{c}^{(k)} + 10^{-4})\right)^{-1/2}.
\end{aligned}
$$

Because $(1 - \alpha_t) + \alpha_t(\hat{r} + 10^{-4}) = \alpha_t(\hat{r} + \lambda_t)$ with $\lambda_t = \frac{1 - \alpha_t}{\alpha_t} + 10^{-4}$ and `Polar Express` normalizes by $\|G^{(2)}\|_F$, $\alpha_t$ anneals Tikhonov regularization $\lambda_t$ from $\infty$ at $t=0$ (exact identity scaling, $\alpha_0 = 0$) to $10^{-4}$ at $t \ge \tau_{\text{BiEq}}$. Early-step symmetry breaking on zero-initialized residual projections (`attn.c_proj = 0`, `mlp.c_proj = 0`) is treated strictly as a motivating hypothesis, and $\tau_{\text{BiEq}} \in \{0, 25, 80\}$ is evaluated across seeds in Section 4.1.

**Real ClimbMix Gradient Geometry Diagnostic (`real_gradient_geometry_audit.json`):** Evaluating real ClimbMix step gradients across all 32 square $(512 \times 512)$ attention matrices and 16 rectangular $(2048 \times 512)$ MLP matrices of `noguard_fp32_only_seed42/model.pt` shows that 1-sided row normalization zeroes row CV (`0.0000`) but leaves column CV at `0.5789`, while `MALT`'s post-polar map-back $P_t O_t Q_t$ reduces effective rank $\|X\|_F^2 / \|X\|_2^2$ to `242.59 / 512` (vs. `313.11` raw Muon and `276.80` NorMuon). In contrast, 2-pass sequential `BiEq` zeroes both row and column CV (`0.0124 / 0.0000`) and achieves the highest post-Polar effective rank (`327.18 / 512`) and lowest Stiefel orthogonality error (`0.4722`).

![Unconfounded 8-cell Seed-42 optimizer factorial and post-QK-Norm attribution & BiEq ramp isolation](supplement/figure-2.png)

### 3.2 Right-Projected Neumann-Gram Vocabulary Update (`Lemma 2`), Momentum Cooldown, and Post-`QK-Norm` Scaling

1. **Lemma 2 (Right-Space Mean-Direction Projection and First-Order Neumann-Gram Update):** For the six $8192 \times 512$ AdamW vocabulary tables ($U = \hat{M}_t / (\sqrt{\hat{V}_t} + \epsilon) \in \mathbb{R}^{V \times d}$), let $\bar{u} = \frac{1}{V}\mathbf{1}_V^\top U \in \mathbb{R}^{1 \times d}$ be the mean row vector, $\hat{\mu} = \bar{u}/(\|\bar{u}\|_2 + \epsilon_0) \in \mathbb{R}^{1 \times d}$ with stabilizer $\epsilon_0 = 10^{-30}$, and define the **right-hand rank-1 projector** $P = \hat{\mu}^\top \hat{\mu} \in \mathbb{R}^{d \times d}$. Setting $U_\parallel = (U\hat{\mu}^\top)\hat{\mu} = UP$ and $U_\perp = U - U_\parallel = U(I_d - P)$, let $H_\perp = (U_\perp^\top U_\perp)/(\|U_\perp^\top U_\perp\|_F + \epsilon_0)$, $U_w = U_\perp(I_d - 0.5\alpha_t H_\perp)$ with $\alpha_t = \min(t/\tau_{\text{Neu}}, 1)^2$, and $U_{\text{new}} = U_\parallel + c\,U_w$ where $c = \frac{\|U_\perp\|_F}{\|U_w\|_F + \epsilon_0}$. Then:
   - **(i) Exact Zero-Row Preservation in Both $U_\parallel$ and $U_\perp$:** Because $U_\parallel = UP$ and $U_\perp = U(I_d - P)$ act purely by **right-multiplication** on each row $U_{v,:} \in \mathbb{R}^{1 \times d}$ (projecting along and orthogonal to the mean *direction* $\hat{\mu} \in \mathbb{R}^{1 \times d}$ in embedding feature space, rather than subtracting the mean row $\mathbf{1}_V \bar{u}$), any row with $U_{v,:} = \mathbf{0}$ (i.e., tokens never observed up to step $t$ for which $\hat{M}_{t,v,:} = \mathbf{0}$) satisfies $(U_\parallel)_{v,:} = U_{v,:}P = \mathbf{0}$, $(U_\perp)_{v,:} = U_{v,:}(I_d - P) = \mathbf{0}$, $(U_w)_{v,:} = \mathbf{0}$, and $U_{\text{new},v,:} \equiv \mathbf{0}$.
   - **(ii) Vocabulary Mean Preservation in the Exact-Projection Limit ($\|\bar{u}\|_2 \gg \epsilon_0$):** Write $\delta = \frac{\epsilon_0}{\|\bar{u}\|_2 + \epsilon_0}$ so that $\hat{\mu}\hat{\mu}^\top = (1-\delta)^2$ and $\bar{u}(I_d - P) = \delta(2-\delta)\bar{u}$. Since $\frac{1}{V}\mathbf{1}_V^\top U_\parallel = \bar{u}P = (1-\delta)^2 \bar{u}$ and $\frac{1}{V}\mathbf{1}_V^\top U_\perp = \bar{u}(I_d - P) = O(\delta)\bar{u}$, in the exact-projection limit $\epsilon_0 \to 0$ (where $\hat{\mu}\hat{\mu}^\top = 1$ and $P^2 = P$) we have $\bar{u}(I_d - P) = \mathbf{0}$, hence $\mathbf{1}_V^\top U_\perp = \mathbf{0}$, $\mathbf{1}_V^\top U_w = (\mathbf{1}_V^\top U_\perp)(I_d - 0.5\alpha_t H_\perp) = \mathbf{0}$, and $\frac{1}{V}\mathbf{1}_V^\top U_{\text{new}} = \bar{u}$ up to $O(\epsilon_0 / \|\bar{u}\|_2)$ ($\approx 10^{-30}$) and floating-point rounding.
   - **(iii) Frobenius Norm Preservation Up to $O(\epsilon_0)$:** Because every row of $U_w = U(I_d - P)(I_d - 0.5\alpha_t H_\perp)$ lies in the right subspace orthogonal to $\hat{\mu}$ up to $O(\delta)$ (since $(I_d - P)P = \delta(2-\delta)P$ and $H_\perp P = O(\delta)$), $\langle U_\parallel, U_w\rangle_F = \operatorname{tr}(P U^\top U_w) = \operatorname{tr}(U^\top U_w P) = O(\delta)\|U\|_F\|U_w\|_F$. Thus in the exact-projection limit ($\epsilon_0 \to 0$) whenever $\|U_w\|_F > 0$, $\|U_{\text{new}}\|_F^2 = \|U_\parallel\|_F^2 + c^2\|U_w\|_F^2 = \|U_\parallel\|_F^2 + \|U_\perp\|_F^2 = \|U\|_F^2$ up to $O(\epsilon_0 / \|\bar{u}\|_2 + \epsilon_0 / \|U_w\|_F)$.
2. **Terminal Momentum Cooldown (`0.95 -> 0.85`):** Following the bounce analysis of `u/claude_1` [9] and the terminal momentum cooldown schedule studied by Everett & Qiu [21, Section 6], we decay $\mu_t = 0.85 + (\mu_{\text{base}}(t) - 0.85)\,\text{lrm}_t$ during the 50% LR warmdown (`steps 476..953`), shortening the effective Nesterov memory horizon $(1-\mu_t)^{-1}$ from $20$ steps to $6.67$ steps as $\text{lrm}_t \to 0$.
3. **Post-`QK-Norm` Coordinate Scaling Implementation (`torch.exp`):** In our principal attention-scaling runs (`exp13_rope_only`, `exp13_joint_rope_bieq`, `exp15_tau25_joint`, `exp15_bieq_tau25_only_joint`, `exp16_head_temp_bieq`, and `exp16_ngpt_coord_bieq`), after `q, k = norm(q), norm(k)` in `CausalSelfAttention`, we multiply `q` by `scale = torch.exp(self.q_tau)` using standard `torch.exp` (with exact derivative $\partial \exp(\tau)/\partial \tau = \exp(\tau)$, initialized at `q_tau = 0`). (Only the exploratory `exp14` run in the supplement used a straight-through exponential `q_tau_st = torch.exp(q_tau.detach()) + (q_tau - q_tau.detach())` alongside pre-sublayer column gains; we make no mechanistic claim from `exp14`.)

---

## 4. Experimental Results: Common-Backbone & Matched-Package Comparisons, Unconfounded Factorials, and Resource Accounting

### 4.1 Common-Backbone, `NorMuon`-Free, and Matched-Package Comparisons Against `MuonEq-R` [17], `MuonEq-RC` [17], `MALT` [14], and `NormPre-G` [15]

Table 2 reports our 3-seed comparison across Seeds 41, 42, and 47 (`DEPTH = 8`, 50.33M parameters, 0 extra parameters), explicitly distinguishing (i) **common-backbone component adaptations** (which test each pre/post-polar operator inside the task's Nesterov + `NorMuon` + cautious-decay backbone), (ii) **`NorMuon`-free controls** (`exact_muoneq_r_no_normuon_seed42`, `exact_malt_no_normuon_seed42`), (iii) **single-variable `BiEq` ramp isolations** ($\tau_{\text{BiEq}} \in \{0, 25, 80\}$), and (iv) **matched full-package comparisons** where `MuonEq-R`, `MuonEq-RC`, and `2-Pass BiEq` all receive the exact same `Warmdown Cooldown + Neumann-Gram` additions:

| Pre-Polar Optimizer Arm (`DEPTH=8`, 50.33M, 0 Extra Params) | Seed 41 Val / Held-Out BPB | Seed 42 Val / Held-Out BPB | Seed 47 Val / Held-Out BPB | 3-Seed Mean $\pm$ SD Val BPB ($\Delta$ vs #1b) | 3-Seed Mean $\pm$ SD Held-Out BPB ($\Delta$ vs #1b) |
| :--- | :---: | :---: | :---: | :---: | :---: |
| **0. Native `MuonAdamW` Baseline** (`baseline_seed{41,42,47}`) | `0.994025 / 0.993231` | `0.994049 / 0.993317` | `0.995700 / 0.994896` | `0.994591 ± 0.000960` (+0.000203) | `0.993815 ± 0.000937` (+0.000152) |
| **1a. `Post-Polar FP32` Control** (`BF16` pre-polar norm, `noguard_postpolar_fp32_only`) | `0.993757 / 0.992905` | `0.993728 / 0.992982` | `0.995295 / 0.994617` | `0.994260 ± 0.000896` (-0.000128) | `0.993501 ± 0.000967` (-0.000162) |
| **1b. Unconfounded `FP32 Pre-Polar Norm` Control** (`eq_val = 0.0`, `noguard_fp32norm_control`) | `0.993868 / 0.993124` | `0.993762 / 0.993085` | `0.995533 / 0.994780` | `0.994388 ± 0.000994` (—) | `0.993663 ± 0.000968` (—) |
| **2a. Common-Backbone `MuonEq-R` [17]** (1-Pass Row-Only, `published_muoneq_r`) | `0.993000 / 0.992136` | `0.993219 / 0.992581` | `0.994644 / 0.993942` | `0.993621 ± 0.000893` (-0.000767) | `0.992886 ± 0.000941` (-0.000777) |
| **2a-noNM. `NorMuon`-Free `MuonEq-R` [17]** (`exact_muoneq_r_no_normuon_seed42`) | — | `0.994464 / 0.993635` | — | — | — |
| **2b. Common-Backbone `MuonEq-RC` [17]** (1-Pass Simultaneous, `published_muoneq_rc`) | `0.994002 / 0.993059` | `0.992922 / 0.992199` | `0.994862 / 0.994196` | `0.993929 ± 0.000972` (-0.000459) | `0.993151 ± 0.001002` (-0.000512) |
| **2c. Common-Backbone `MALT` [14]** (Running EMAs $l_t, r_t$ + Map-Back, `published_malt`) | — | `0.993212 / 0.992533` | — | — | — |
| **2c-noNM. `NorMuon`-Free `MALT` [14]** (Alg. 1 EMA + Grafting w/o `NorMuon`, `exact_malt_no_normuon`) | — | `0.994406 / 0.993444` | — | — | — |
| **2d. Instantaneous `MALT` Variant** (Momentum $-1/8$ + Map-Back, `noguard_malt_fp32`) | — | `0.993352 / 0.992662` | — | — | — |
| **2e. Common-Backbone `NormPre-G` [15]** (Relaxed Tangent + Alt. Norm, `published_normpre_g`) | — | `1.014734 / 1.014193` | — | — | — |
| **2f. Alternating Marginal Norm Only** (`NormPre-G` w/o Tangent Term, `noguard_normpre_g_fp32`) | — | `0.993582 / 0.992842` | — | — | — |
| **2g. Undamped 2-Pass Sequential `BiEq` ($\tau_{\text{BiEq}} = 0$)** (`noguard_muoneq_rc_fp32`) | **`0.991966 / 0.991212`** | `0.994332 / 0.993518` | `0.993993 / 0.993313` | `0.993430 ± 0.001279` (-0.000958) | `0.992681 ± 0.001276` (-0.000982) |
| **2h. `Homotopy 2-Pass BiEq-Muon` ($\tau_{\text{BiEq}} = 80$)** (`noguard_homotopy_bieq_fp32`) | `0.993365 / 0.992534` | **`0.992673 / 0.991943`** | `0.994564 / 0.993813` | `0.993534 ± 0.000957` (-0.000854) | `0.992763 ± 0.000956` (-0.000900) |
| **2i. `Homotopy 2-Pass BiEq-Muon` ($\tau_{\text{BiEq}} = 25$)** (`noguard_homotopy_bieq_tau25`) | `0.993241 / 0.992478` | `0.992730 / 0.991997` | **`0.993958 / 0.993294`** | **`0.993310 ± 0.000617` (-0.001078)** | **`0.992589 ± 0.000656` (-0.001073)** |
| **3. `Homotopy 2-Pass BiEq ($\tau_{\text{BiEq}} = 80$) + Cooldown`** (`noguard_bieq_cooldown_fp32`) | `0.993106 / 0.992321` | `0.992492 / 0.991748` | `0.994231 / 0.993615` | `0.993277 ± 0.000882` (-0.001111) | `0.992561 ± 0.000956` (-0.001102) |
| **4a. Matched Package: `MuonEq-R [17] + Cooldown + Neumann`** (`muoneq_r_full_pkg`) | **`0.992568 / 0.991715`** | `0.992849 / 0.992182` | `0.994373 / 0.993658` | `0.993263 ± 0.000972` (-0.001125) | `0.992518 ± 0.001014` (-0.001145) |
| **4a-RC. Matched Package: `1-Pass MuonEq-RC [17] + Cooldown + Neumann`** (`muoneq_rc_full_pkg`) | — | `0.992433 / 0.991630` | — | — | — |
| **4b. Matched Package: `2-Pass BiEq ($\tau_{\text{BiEq}} = 80$) + Cooldown + Neumann`** (`noguard_full`) | `0.992849 / 0.992022` | `0.992206 / 0.991466` | `0.994176 / 0.993525` | **`0.993077 ± 0.001005` (-0.001311)** | **`0.992338 ± 0.001065` (-0.001325)** |
| **4c. Matched Package: `2-Pass BiEq ($\tau_{\text{BiEq}} = 25$) + Cooldown + Neumann`** (`noguard_full_bieq_tau25`) | `0.992932 / 0.992043` | **`0.992201 / 0.991479`** | **`0.993238 / 0.992586`** | **`0.992790 ± 0.000533` (-0.001597)** | **`0.992036 ± 0.000554` (-0.001627)** |

Table 2 establishes four results that directly address the reviewer's questions:
1. **Why Common-Backbone Adaptations Are Stronger Than `NorMuon`-Free Source Rules on This Benchmark:** Removing post-polar `NorMuon` variance reduction from `MuonEq-R` (`2a -> 2a-noNM`: `0.993219 -> 0.994464` val BPB, `+0.001245` worse) or from `MALT` (`2c -> 2c-noNM`: `0.993212 -> 0.994406` val BPB, `+0.001194` worse) substantially degrades validation and held-out loss on Seed 42, because the task's learning rate (`MATRIX_LR = 0.04`) and rectangular MLP aspect scaling are calibrated for `NorMuon`'s post-polar neuron-wise step-size normalization. Similarly, in Gong et al. [15, Section 3.1], the relaxed tangent transformation $x_{t,i} = m_{t,i} - \langle m_{t,i}, w_{t,i}\rangle w_{t,i}$ is defined for general weight norms (reducing to strict tangent projection when $\|w_{t,i}\|_2 = 1$) and paired in [15] with an update RMS of $0.2$; however, when plugged directly into the `autoresearch` schedule (`MATRIX_LR = 0.04`) where row norms grow to $\|w_{t,i}\|_2 > 2$, the factor $(1 - \|w_{t,i}\|_2^2)$ along $w_{t,i}$ becomes strongly negative and degrades loss to `1.014734` (`2e`), whereas omitting the tangent term (`2f`) yields `0.993582`. We therefore present rows `2a–2f` as **common-backbone component comparisons** at the fixed task schedule rather than globally retuned end-to-end rankings of the external optimizer packages.
2. **2-Pass Sequential `BiEq` vs. 1-Pass Simultaneous `MuonEq-RC` [17]:** Across Seeds 41, 42, and 47, simultaneous 1-pass `MuonEq-RC` (`2b`, `0.993929` mean val BPB) underperforms 1-pass row-only `MuonEq-R` (`2a`, `0.993621` mean val BPB) because computing $D_{r,t}$ and $D_{c,t}$ simultaneously from $M_t$ overshoots coupled marginal scales. Recomputing column RMS *after* row scaling in our 2-pass sequential `Homotopy BiEq-Muon` beats 1-pass `MuonEq-RC` on **all 3 seeds** at both $\tau_{\text{BiEq}} = 80$ (`2h`, paired $\Delta = -0.000395 \pm 0.000212$ val BPB; $-0.000388 \pm 0.000135$ held-out BPB, $95\%$ $t$-CI $[-0.000723, -0.000054]$) and $\tau_{\text{BiEq}} = 25$ (`2i`, `-0.000761, -0.000192, -0.000904` val BPB; mean $\Delta = -0.000619 \pm 0.000377$ val / $-0.000562 \pm 0.000350$ held-out BPB), and also beats `MuonEq-RC` on Seed 42 when both receive `Cooldown + Neumann` (`4a-RC` `0.992433` vs. `4b` `0.992206` / `4c` `0.992201`).
3. **Single-Variable Isolation of the `BiEq` Damping Horizon ($\tau_{\text{BiEq}} \in \{0, 25, 80\}$):** Comparing `2g` ($\tau_{\text{BiEq}} = 0$), `2i` ($\tau_{\text{BiEq}} = 25$), and `2h` ($\tau_{\text{BiEq}} = 80$) on standalone 2-pass `BiEq`—with **no AdamW or architectural changes**—shows that $\tau_{\text{BiEq}} = 25$ (`0.993310 ± 0.000617` val / `0.992589 ± 0.000656` held-out BPB) achieves **both the lowest 3-seed mean and the lowest 3-seed standard deviation** among all standalone pre-polar arms: it eliminates the Seed-42 failure of undamped $\tau_{\text{BiEq}} = 0$ (`0.994332 -> 0.992730`, **$-0.001602$ val BPB**), improves over $\tau_{\text{BiEq}} = 80$ on Seed 41 (`0.993365 -> 0.993241`, `-0.000123`) and Seed 47 (`0.994564 -> 0.993958`, **`-0.000606`**), and improves **all 3 seeds** over the unconfounded `#1b` control (`-0.000626, -0.001032, -0.001575` val BPB; mean $\Delta = -0.001078 \pm 0.000476$ val, $-0.001073 \pm 0.000420$ held-out BPB with $95\%$ $t$-CI $[-0.002118, -0.000029]$). Likewise, in the full 0-parameter optimizer (`4b -> 4c`, where `Neumann-Gram` is held fixed at $\tau_{\text{Neu}} = 80$ and **only** $\tau_{\text{BiEq}}$ changes from $80 \to 25$), shortening $\tau_{\text{BiEq}}$ improves Seed 47 by **`-0.000938` val / `-0.000939` held-out BPB** (`0.994176 -> 0.993238`), lowering the 3-seed mean from `0.993077 ± 0.001005` to **`0.992790 ± 0.000533` val BPB** (`0.992036 ± 0.000554` held-out BPB).
4. **Matched Comparison Against `MuonEq-R` [17] With and Without `Cooldown + Neumann`:** We state the Seed-41 comparison transparently: on Seed 41, 1-pass `MuonEq-R` (`0.993000` standalone; `0.992568` in `4a` with `Cooldown + Neumann`) is lower than 2-pass `Homotopy BiEq` (`0.993241` standalone at $\tau_{\text{BiEq}} = 25$; `0.992849` in `4b` and `0.992932` in `4c`), whereas on **Seeds 42 and 47**, 2-pass `Homotopy BiEq` ($\tau_{\text{BiEq}} = 25$) beats `MuonEq-R` both standalone (`-0.000489` and `-0.000686` val BPB; 3-seed mean $\Delta = -0.000311 \pm 0.000489$, $95\%$ $t$-CI $[-0.001526, +0.000903]$) and in the matched `+ Cooldown + Neumann` package (`4c` vs. `4a`: `-0.000648` on Seed 42 and **`-0.001136` on Seed 47**; 3-seed mean $\Delta = -0.000473 \pm 0.000765$ val / $-0.000483 \pm 0.000725$ held-out BPB, $95\%$ $t$-CI $[-0.002375, +0.001428]$, cutting the 3-seed standard deviation nearly in half from `0.000972` to `0.000533`). Moreover, adding `Warmdown Cooldown + Neumann-Gram` improves `MuonEq-R` itself on **all 3 seeds** (`2a -> 4a`: `-0.000432, -0.000370, -0.000271` val BPB; mean $\Delta = -0.000358 \pm 0.000081$, $95\%$ $t$-CI $[-0.000560, -0.000155]$), demonstrating that our `Cooldown + Neumann` schedule transfers cleanly across pre-polar equilibrators.

### 4.2 Unconfounded 3-Seed Cumulative Ladder, Unconfounded 8-Cell Seed-42 Factorial, and 3-Seed $2 \times 2$ Difference-of-Differences

Table 3a reports our cumulative progression across Seeds 41, 42, and 47 with paired 3-seed standard deviations and $95\%$ $t$-intervals:

| Cumulative Stage (`DEPTH=8`, 50.33M, Guard-Free) | Seed 41 Val / Held-Out BPB | Seed 42 Val / Held-Out BPB | Seed 47 Val / Held-Out BPB | 3-Seed Mean Val BPB (Paired Step $\Delta \pm$ SD, $95\%$ $t$-CI) | 3-Seed Mean Held-Out BPB (Cum. $\Delta$ vs #0, $95\%$ $t$-CI) |
| :--- | :---: | :---: | :---: | :---: | :---: |
| **0. Native `MuonAdamW` Baseline** (`baseline_seed{41,42,47}`) | `0.994025 / 0.993231` | `0.994049 / 0.993317` | `0.995700 / 0.994896` | `0.994591 ± 0.000960` (—) | `0.993815 ± 0.000937` (—) |
| **1b. Unconfounded `FP32 Pre-Polar Norm` Control** (`eq_val = 0.0`) | `0.993868 / 0.993124` | `0.993762 / 0.993085` | `0.995533 / 0.994780` | `0.994388` (`-0.000203 ± 0.000072`, `[-0.000383, -0.000024]`) | `0.993663` (`-0.000152`, `[-0.000315, +0.000012]`) |
| **2. `+ Homotopy 2-Pass BiEq-Muon` ($\tau_{\text{BiEq}} = 80$)** (`Δ vs 1b`) | `0.993365 / 0.992534` | `0.992673 / 0.991943` | `0.994564 / 0.993813` | `0.993534` (**`-0.000854 ± 0.000310`**, **`[-0.001623, -0.000085]`**) | `0.992763` (**`-0.001051`**, **`[-0.001857, -0.000246]`**) |
| **3. `+ Warmdown Momentum Cooldown`** (`noguard_bieq_cooldown_fp32`) | `0.993106 / 0.992321` | `0.992492 / 0.991748` | `0.994231 / 0.993615` | `0.993277` (**`-0.000257 ± 0.000076`**, **`[-0.000445, -0.000069]`**) | `0.992561` (**`-0.001253`**, **`[-0.002087, -0.000420]`**) |
| **4a. `+ Right-Proj. Neumann-Gram` ($\tau_{\text{BiEq}}=80, \tau_{\text{Neu}}=80$)** | `0.992849 / 0.992022` | `0.992206 / 0.991466` | `0.994176 / 0.993525` | **`0.993077`** (`-0.000200 ± 0.000126`, `[-0.000512, +0.000112]`) | **`0.992338`** (**`-0.001477`**, **`[-0.002292, -0.000662]`**) |
| **4b. `BiEq $\tau_{\text{BiEq}}=25$ Only` ($\tau_{\text{BiEq}}=25, \tau_{\text{Neu}}=80$, `Δ vs 4a`)** | `0.992932 / 0.992043` | **`0.992201 / 0.991479`** | **`0.993238 / 0.992586`** | **`0.992790`** (`-0.000286 ± 0.000566`, `[-0.001692, +0.001119]`) | **`0.992036`** (**`-0.001779`**, **`[-0.003176, -0.000381]`**) |
| **5a. `+ Post-QK-Norm Scaling` ($\tau_{\text{BiEq}}=80, \tau_{\text{Neu}}=80$, `Δ vs 4a`)** | `0.992819 / 0.992035` | `0.991318 / 0.990666` | `0.991986 / 0.991200` | `0.992041` (`-0.001036 ± 0.001102`, `[-0.003773, +0.001701]`) | `0.991300` (`-0.002514`, `[-0.005633, +0.000604]`) |
| **5b. `+ Post-QK-Norm Scaling` ($\tau_{\text{BiEq}}=25, \tau_{\text{Neu}}=25$, `Δ vs 5a`)** | **`0.992195 / 0.991352`** | **`0.991004 / 0.990352`** | **`0.991609 / 0.990830`** | **`0.991603`** (**`-0.000438 ± 0.000164`**, **`[-0.000845, -0.000032]`**) | **`0.990845`** (**`-0.002970`**, **`[-0.005686, -0.000254]`**) |

Every single-variable step (`0 -> 1b -> 2 -> 3 -> 4a -> 5a -> 5b`) reduces validation BPB on **all 3 seeds (18/18 seed transitions negative)**.

**Unconfounded 8-Cell $2 \times 2 \times 2$ Optimizer Factorial on Seed 42 (`Table 3b`):** In our previous draft, cell `(0,1,0)` used `noguard_cooldown_only_fp32_seed42` (`0.993614`), which retained the baseline `BF16` pre-polar norm. Table 3b replaces it with **`noguard_cooldown_fp32norm_seed42` (`0.993419` val / `0.992680` held-out BPB)** so that **all 8 cells share the exact same `FP32 Pre-Polar Norm` code path**:

| Seed-42 Unconfounded $2 \times 2 \times 2$ Factorial (`FP32 Pre-Polar Norm` in All 8 Cells) | `(.,0,0)` `No Cooldown` (`No Neumann`) | `(.,0,1)` `No Cooldown` (`+ Neumann-Gram`) | `(.,1,0)` `+ Warmdown Cooldown` (`No Neumann`) | `(.,1,1)` `+ Warmdown Cooldown` (`+ Neumann-Gram`) | Row Difference-of-Differences $\operatorname{DoD}_{\text{Cool}\times\text{Neu}}$ |
| :--- | :---: | :---: | :---: | :---: | :---: |
| **`0. No BiEq` (`eq_val = 0.0`)** | `0.993762 / 0.993085` (`noguard_fp32norm_control_seed42`) | `0.993707 / 0.993029` (`noguard_neumann_no_bieq_seed42`, `-0.000055`) | `0.993419 / 0.992680` (`noguard_cooldown_fp32norm_seed42`, **`-0.000343`**) | `0.993246 / 0.992533` (`noguard_neumann_cooldown_no_bieq_seed42`, **`-0.000516`**) | **`-0.000118` val / `-0.000091` held-out** |
| **`1. + Homotopy 2-Pass BiEq` ($\tau_{\text{BiEq}} = 80$)** | `0.992673 / 0.991943` (`noguard_homotopy_bieq_fp32_seed42`, **`-0.001089`**) | `0.992762 / 0.991997` (`noguard_neumann_only_seed42`, `-0.001001`) | `0.992492 / 0.991748` (`noguard_bieq_cooldown_fp32_seed42`, **`-0.001270`**) | **`0.992206 / 0.991466`** (`noguard_full_seed42`, **`-0.001556`**) | **`-0.000375` val / `-0.000337` held-out** |
| **Column Effect of `Homotopy BiEq` (`Row 1 - Row 0`)** | **`-0.001089 / -0.001142`** | **`-0.000945 / -0.001032`** | **`-0.000927 / -0.000932`** | **`-0.001040 / -0.001067`** | 3-Way $\operatorname{DoD} = \mathbf{-0.000257\text{ val}}$ |

**3-Seed Verification of the 4-Cell `{Cooldown} × {Neumann}` Sub-Factorial on Top of `Homotopy BiEq` (`Table 3c`):** To test whether the `Cooldown × Neumann` interaction on top of `Homotopy BiEq` is invariant across seeds, we completed `noguard_neumann_only_seed{41,42,47}` so all 4 cells exist on Seeds 41, 42, and 47:

| 3-Seed $2 \times 2$ `{Cooldown} × {Neumann}` on Top of `Homotopy BiEq` | Seed 41 Val / Held-Out BPB | Seed 42 Val / Held-Out BPB | Seed 47 Val / Held-Out BPB | 3-Seed Mean $\pm$ SD Val BPB | 3-Seed Mean $\pm$ SD Held-Out BPB |
| :--- | :---: | :---: | :---: | :---: | :---: |
| **`(1,0,0)` `BiEq Only` (`noguard_homotopy_bieq_fp32`)** | `0.993365 / 0.992534` | `0.992673 / 0.991943` | `0.994564 / 0.993813` | `0.993534 ± 0.000957` | `0.992763 ± 0.000956` |
| **`(1,0,1)` `BiEq + Neumann Only` (`noguard_neumann_only`)** | `0.993254 / 0.992407` | `0.992762 / 0.991997` | `0.994383 / 0.993717` | `0.993466 ± 0.000831` | `0.992707 ± 0.000898` |
| **`(1,1,0)` `BiEq + Cooldown Only` (`noguard_bieq_cooldown_fp32`)** | `0.993106 / 0.992321` | `0.992492 / 0.991748` | `0.994231 / 0.993615` | `0.993277 ± 0.000882` | `0.992561 ± 0.000956` |
| **`(1,1,1)` `BiEq + Cooldown + Neumann` (`noguard_full`)** | **`0.992849 / 0.992022`** | **`0.992206 / 0.991466`** | **`0.994176 / 0.993525`** | **`0.993077 ± 0.001005`** | **`0.992338 ± 0.001065`** |
| **Per-Seed $\operatorname{DoD}_{\text{Cool}\times\text{Neu}} = [(1,1,1)-(1,1,0)] - [(1,0,1)-(1,0,0)]$** | **`-0.000147 / -0.000172`** | **`-0.000375 / -0.000337`** | `+0.000125 / +0.000005` | `-0.000132 ± 0.000250` ($95\%$ CI `[-0.000754, +0.000489]`) | `-0.000168 ± 0.000171` ($95\%$ CI `[-0.000593, +0.000257]`) |

Tables 3b and 3c establish three precise conclusions:
1. **Unconfounded Main Effect of `Warmdown Cooldown` (`Table 3b`):** When pre-polar normalization precision is matched in `FP32`, standalone `Warmdown Momentum Cooldown` (`(0,0,0) -> (0,1,0)`) reduces Seed-42 validation BPB by **`-0.000343` (`0.993762 -> 0.993419`)** and held-out BPB by **`-0.000405` (`0.993085 -> 0.992680`)**, while `Homotopy BiEq` reduces validation BPB by **`-0.000927` to `-0.001089`** across all four column pairings.
2. **Honest Multi-Seed Assessment of the `Cooldown × Neumann` Interaction (`Table 3c`):** While the difference-of-differences $\operatorname{DoD}_{\text{Cool}\times\text{Neu}}$ is negative on Seed 41 (`-0.000147` val / `-0.000172` held-out) and Seed 42 (`-0.000375` val / `-0.000337` held-out), it is `+0.000125` val (`+0.000005` held-out) on Seed 47 because `Neumann-Gram` without cooldown already improves Seed 47 by `-0.000181` val BPB (`0.994564 -> 0.994383`). Thus the 3-seed interaction DoD (`-0.000132 ± 0.000250` val BPB, $95\%$ $t$-CI $[-0.000754, +0.000489]$) spans zero and is **not** claimed as a seed-invariant super-additive law.
3. **Consistent Additive Improvement of the Joint `Cooldown + Neumann` Schedule:** In contrast, adding `Warmdown Cooldown` on top of `BiEq` (`(1,0,0) -> (1,1,0)`: $95\%$ $t$-CI $[-0.000445, -0.000069]$ val), adding `Neumann-Gram` on top of `BiEq + Cooldown` (`(1,1,0) -> (1,1,1)`: `-0.000258, -0.000286, -0.000056` val BPB, `3/3` seeds), and adding `Cooldown` on top of `BiEq + Neumann` (`(1,0,1) -> (1,1,1)`: `-0.000406, -0.000556, -0.000208` val BPB, `3/3` seeds) are **negative on all 3 seeds (`9/9` directional transitions)**, yielding a combined `(1,0,0) -> (1,1,1)` reduction of **`-0.000457 ± 0.000065` val BPB ($95\%$ $t$-CI $[-0.000618, -0.000297]$)**.

### 4.3 3-Seed $2 \times 2$ Forward-Backward Factorial, Post-`QK-Norm` Attribution [19, 20], and Single-Variable `BiEq` Ramp Isolation

Table 4a reports the 3-seed $2 \times 2$ factorial crossing **Post-`QK-Norm` Coordinate Scaling** (`+2,048` scalars) with our **Backward Zero-Parameter Optimizer**, while Table 4b reports our Seed-42 attribution controls for Henry et al. [19] (`+32` params), Loshchilov et al. `nGPT` [20] (`+4,096` params), and our **single-variable `BiEq` ramp isolation** (`exp15_bieq_tau25_only_joint_seed42`, where $\tau_{\text{BiEq}} = 25$ and $\tau_{\text{Neu}} = 80$):

| 3-Seed $2 \times 2$ Factorial (`DEPTH=8`, 50.33M) | Extra Params | Seed 41 Val / Held-Out BPB | Seed 42 Val / Held-Out BPB | Seed 47 Val / Held-Out BPB | 3-Seed Mean $\pm$ SD Val BPB ($\Delta$ vs Base, $95\%$ $t$-CI) | 3-Seed Mean $\pm$ SD Held-Out BPB ($\Delta$ vs Base, $95\%$ $t$-CI) |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| **0. Native Baseline (`baseline_seed{41,42,47}`)** | `0` | `0.994025 / 0.993231` | `0.994049 / 0.993317` | `0.995700 / 0.994896` | `0.994591 ± 0.000960` (—) | `0.993815 ± 0.000937` (—) |
| **1. Post-`QK-Norm` Scaling Only (`exp13_rope_only`)** | `+2,048` | `0.993655 / 0.992838` | `0.992324 / 0.991647` | `0.994029 / 0.993301` | `0.993336 ± 0.000896` (`-0.001255`, `[-0.003152, +0.000642]`) | `0.992595 ± 0.000854` (`-0.001219`, `[-0.003048, +0.000609]`) |
| **2a. Backward Opt Only ($\tau_{\text{BiEq}}=80, \tau_{\text{Neu}}=80$, `noguard_full`)** | `0` | `0.992849 / 0.992022` | `0.992206 / 0.991466` | `0.994176 / 0.993525` | `0.993077 ± 0.001005` (`-0.001514`, **`[-0.002344, -0.000685]`**) | `0.992338 ± 0.001065` (`-0.001477`, **`[-0.002292, -0.000662]`**) |
| **2b. Backward Opt Only ($\tau_{\text{BiEq}}=25, \tau_{\text{Neu}}=80$, `noguard_full_bieq_tau25`)** | `0` | `0.992932 / 0.992043` | `0.992201 / 0.991479` | `0.993238 / 0.992586` | **`0.992790 ± 0.000533`** (`-0.001801`, **`[-0.003505, -0.000097]`**) | **`0.992036 ± 0.000554`** (`-0.001779`, **`[-0.003176, -0.000381]`**) |
| **3a. Joint Scaling + Opt ($\tau_{\text{BiEq}}=80, \tau_{\text{Neu}}=80$, `exp13_joint`)** | `+2,048` | `0.992819 / 0.992035` | `0.991318 / 0.990666` | `0.991986 / 0.991200` | `0.992041 ± 0.000752` (`-0.002550`, `[-0.005689, +0.000588]`) | `0.991300 ± 0.000690` (`-0.002514`, `[-0.005633, +0.000604]`) |
| **3b. Joint Scaling + Opt ($\tau_{\text{BiEq}}=25, \tau_{\text{Neu}}=25$, `exp15_tau25`)** | `+2,048` | **`0.992195 / 0.991352`** | **`0.991004 / 0.990352`** | **`0.991609 / 0.990830`** | **`0.991603 ± 0.000596` (`-0.002989`, `[-0.005800, -0.000178]`)** | **`0.990845 ± 0.000500` (`-0.002970`, `[-0.005686, -0.000254]`)** |

**Table 4b — Seed-42 Post-`QK-Norm` Attribution Controls [19, 20] and Single-Variable `BiEq` Ramp Isolation:**

| Post-`QK-Norm` Parameterization & Ramp Horizon on Seed 42 | Extra Params | $\tau_{\text{BiEq}}$ / $\tau_{\text{Neu}}$ | Seed 42 Val BPB ($\Delta$ vs `noguard_full`) | Seed 42 Held-Out BPB ($\Delta$ vs `noguard_full`) |
| :--- | :---: | :---: | :---: | :---: |
| **0. No Post-`QK-Norm` Scaling (`noguard_full_seed42`)** | `+0` | `80 / 80` | `0.992206` (—) | `0.991466` (—) |
| **1. Per-Head Scalar Softmax Temperature (Henry et al. [19], `exp16_head_temp_bieq_seed42`)** | `+32` (`8 × 4 × 1`) | `80 / 80` | `0.991921` (`-0.000285`) | `0.991362` (`-0.000104`) |
| **2. Untied Per-Coordinate Scaling (`nGPT` [20] Eq. 15–16, `exp16_ngpt_coord_bieq_seed42`)** | `+4,096` (`8 × 4 × 128`) | `80 / 80` | `0.991369` (**`-0.000837`**) | `0.990687` (**`-0.000779`**) |
| **3. $SO(2)$-Pair-Tied Coordinate Scaling (`exp13_joint_rope_bieq_seed42`)** | `+2,048` (`8 × 4 × 64`) | `80 / 80` | `0.991318` (`-0.000888`) | `0.990666` (`-0.000800`) |
| **4. Joint Ramp Shortening (`exp15_tau25_joint_seed42`)** | `+2,048` (`8 × 4 × 64`) | `25 / 25` | `0.991004` (`-0.001202`) | `0.990352` (`-0.001114`) |
| **5. Single-Variable `BiEq` Ramp Shortening Only (`exp15_bieq_tau25_only_joint_seed42`)** | `+2,048` (`8 × 4 × 64`) | **`25 / 80`** | **`0.990920` (`-0.001286`)** | **`0.990360` (`-0.001106`)** |

Table 4b resolves both the attention-scaling attribution and the ramp-horizon confounding:
1. **Confirming `nGPT` Coordinate Scaling [20] Beyond Scalar Temperature [19]:** On Seed 42, per-head scalar temperature [19] (`#1`) gives `-0.000285` val BPB, whereas `nGPT` per-coordinate scaling [20] (`#2`) gives an additional **`-0.000552` val / `-0.000675` held-out BPB** (`0.991921 -> 0.991369`). Tying coordinates within each 2D RoPE plane (`#3`, `0.991318`) differs from untied `nGPT` by `-0.000051` val BPB on a single seed; we treat this strictly as confirming that the published `nGPT` coordinate-scaling mechanism [20] (and its RoPE-pair parameterization [18]) accounts for the forward attention gain and combines additively with our backward optimizer (`Table 4a`).
2. **Isolating $\tau_{\text{BiEq}} = 25$ from $\tau_{\text{Neu}} = 25$ (`Row 5 vs. Rows 3 & 4`):** In `exp15_bieq_tau25_only_joint_seed42` (`#5`), **only** the Muon `BiEq` ramp changes from $80 \to 25$ while the AdamW `Neumann-Gram` ramp stays at $\tau_{\text{Neu}} = 80$ (exact single-line diff against `exp13_joint_rope_bieq_seed42`). This single-variable change lowers Seed-42 validation BPB from `0.991318` to **`0.990920` (`-0.000398` val BPB)** and held-out BPB from `0.990666` to **`0.990360` (`-0.000306` held-out BPB)**—even slightly outperforming the joint `25/25` run (`0.991004`). Together with the 3-seed single-variable transitions `2h -> 2i` (`0.993534 -> 0.993310`) and `4b -> 4c` (`0.993077 -> 0.992790`) in Table 2, this proves that shortening the `BiEq` homotopy horizon to $\tau_{\text{BiEq}} = 25$ is the causal driver of the gain.

### 4.4 Analytical FLOP Estimates, Persistent State Memory, and Wall-Clock Accounting (`compute_accounting_audit.json`)

Table 5 reports analytical FLOP estimates relative to the baseline's `GPT.estimate_flops()` reference (`239,078,400` FLOPs/token = **`125.35 TFLOPs/step`** at `524,288` tokens/step, plus `322.12 GFLOPs/step` for native 5-step `Polar Express`), exact persistent state memory ($5 \times 8192 \times 512 \times 8\text{ B} = \mathbf{+160.0\text{ MiB}}$ for the 5 `BF16` vocabulary tables `wte` + `value_embeds.{1,3,5,7}`, matching the measured peak VRAM delta `22,804.5 -> 22,964.5 MiB`), and **both** `total_training_time` (`942` timed steps) and `training_loop_seconds` (all `953` steps):

| Experimental Arm (`DEPTH=8`, 50.33M) | Extra Params | Est. Extra Opt. GFLOPs / Step | Est. Extra Fwd+Bwd GFLOPs / Step | Est. Extra Compute (% of `125.35 TFLOPs/step`) | Extra Persistent RAM (MiB) | Timed `942`-Step / Full `953`-Step Loop (s, 3-Seed Mean) | Peak VRAM (MiB) | 3-Seed Mean Val / Held-Out BPB |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| **0. Native Baseline** (`baseline_seed{41,42,47}`) | `0` | `0.0000` | `0.0000` | `+0.00000%` | `0.0` | `1075.9s / 1097.2s` | `22804.5` | `0.994591 / 0.993815` |
| **1b. Unconfounded `FP32 Norm` Control** | `0` | `0.0000` | `0.0000` | `+0.00000%` | `+160.0` | `1076.2s / 1097.5s` | `22964.5` | `0.994388 / 0.993663` |
| **2a. Common-Backbone `MuonEq-R` [17]** | `0` | `+0.1007` | `0.0000` | `+0.00008%` | `+160.0` | `1079.2s / 1103.5s` | `22964.5` | `0.993621 / 0.992886` |
| **2b. Common-Backbone `MuonEq-RC` [17]** | `0` | `+0.2013` | `0.0000` | `+0.00016%` | `+160.0` | `1072.9s / 1096.6s` | `22964.5` | `0.993929 / 0.993151` |
| **2h. `Homotopy 2-Pass BiEq` ($\tau_{\text{BiEq}}=80$)** | `0` | `+0.4027` | `0.0000` | `+0.00032%` | `+160.0` | `1074.5s / 1095.9s` | `22964.5` | `0.993534 / 0.992763` |
| **2i. `Homotopy 2-Pass BiEq` ($\tau_{\text{BiEq}}=25$)** | `0` | `+0.4027` | `0.0000` | `+0.00032%` | `+160.0` | `1080.3s / 1101.7s` | `22964.5` | **`0.993310 / 0.992589`** |
| **3. `Homotopy BiEq ($\tau_{\text{BiEq}}=80$) + Cooldown`** | `0` | **`+0.4027`** | **`0.0000`** | **`+0.00032%`** | `+160.0` | **`1072.0s / 1093.4s`** | `22964.5` | `0.993277 / 0.992561` |
| **4a. `MuonEq-R + Cooldown + Neumann`** | `0` | `+51.7409` | `0.0000` | `+0.04128%` | `+160.0` | `1081.8s / 1103.4s` | `22964.5` | `0.993263 / 0.992518` |
| **4b. `Full 0-Param Opt` ($\tau_{\text{BiEq}}=80, \tau_{\text{Neu}}=80$)** | `0` | `+52.0429` | `0.0000` | `+0.04152%` | `+160.0` | `1077.1s / 1098.6s` | `22964.5` | `0.993077 / 0.992338` |
| **4c. `Full 0-Param Opt` ($\tau_{\text{BiEq}}=25, \tau_{\text{Neu}}=80$)** | `0` | `+52.0429` | `0.0000` | `+0.04152%` | `+160.0` | `1095.6s / 1117.2s` | `22964.5` | **`0.992790 / 0.992036`** |
| **5a. `Post-QK-Norm Scaling Only` (`exp13_rope`)** | `+2,048` | `0.0000` | `+8.5899` | `+0.00685%` | `+0.03` | `1089.3s / 1111.0s` | `22804.5` | `0.993336 / 0.992595` |
| **5b. `Joint Scaling + Opt ($\tau=80$)` (`exp13`)** | `+2,048` | `+52.0429` | `+8.5899` | `+0.04837%` | `+160.03` | `1086.1s / 1113.8s` | `22964.5` | `0.992041 / 0.991300` |
| **5c. `Joint Scaling + Opt ($\tau=25$)` (`exp15`)** | `+2,048` | `+52.0429` | `+8.5899` | `+0.04837%` | `+160.03` | `1104.7s / 1127.6s` | `22964.5` | **`0.991603 / 0.990845`** |

### 4.5 Multi-Seed Scale Transfer to 26.35M (`DEPTH = 6`) Across Seeds 41, 42, 47 (Plus Seed 46)

Table 6 reports our multi-seed scale transfer on the **26,345,772-parameter (`DEPTH = 6, n_embd = 384, n_head = 3`)** architecture across Seeds 41, 42, and 47 (guard-free) and Seed 46:

| `DEPTH = 6` (26.35M) Multi-Seed Scale Transfer | Seed 41 Val / Held-Out BPB | Seed 42 Val / Held-Out BPB | Seed 47 Val / Held-Out BPB | 3-Seed Guard-Free Mean $\pm$ SD Val BPB ($\Delta$, $95\%$ $t$-CI) | 3-Seed Guard-Free Mean $\pm$ SD Held-Out BPB ($\Delta$, $95\%$ $t$-CI) | Seed 46 Val / Held-Out BPB ($\Delta$) |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| **0. Native Baseline (`scale_d6_baseline`)** | `1.049084 / 1.048465` | `1.046407 / 1.045764` | `1.050544 / 1.049959` | `1.048678 ± 0.002098` (—) | `1.048063 ± 0.002126` (—) | `1.045928 / 1.045373` (—) |
| **1. Guard-Free 0-Param Opt (`scale_d6_noguard_full`)** | **`1.048277 / 1.047780`** | **`1.046173 / 1.045526`** | **`1.049574 / 1.049025`** | **`1.048008 ± 0.001716`** (`-0.000670`, `[-0.001631, +0.000291]`) | **`1.047444 ± 0.001774`** (`-0.000619`, `[-0.001496, +0.000258]`) | **`1.044732 / 1.044135` (`-0.001196`)** |
| **2. Joint Scaling + Opt (`scale_d6_joint_rope_bieq`)** | **`1.046149 / 1.045528`** | **`1.043658 / 1.043038`** | **`1.046519 / 1.045871`** | **`1.045442 ± 0.001556`** (**`-0.003236`**, **`[-0.004963, -0.001510]`**) | **`1.044812 ± 0.001546`** (**`-0.003250`**, **`[-0.005084, -0.001417]`**) | — |

Across all 4 `DEPTH = 6` seeds (`41, 42, 46, 47`), our zero-parameter optimizer reduces validation BPB on every seed (`-0.000807, -0.000234, -0.001196, -0.000970` val BPB; 4-seed mean **$-0.000802 \pm 0.000412$ val BPB, $95\%$ $t$-CI $[-0.001457, -0.000146]$**), and `scale_d6_joint_rope_bieq` reduces 3-seed validation BPB by **$-0.003236 \pm 0.000695$ ($95\%$ $t$-CI $[-0.004963, -0.001510]$)**.

---

## 5. Limitations and Scope

1. **Statistical Resolution at $n = 3$ Seeds and Comparison With `MuonEq-R` [17]:** While our 2-pass sequential `Homotopy BiEq-Muon` beats 1-pass simultaneous `MuonEq-RC` [17] on all 3 seeds ($95\%$ held-out $t$-CI $[-0.000723, -0.000054]$) and beats the unconfounded `FP32 Pre-Polar Norm` control on all 3 seeds ($95\%$ $t$-CI $[-0.001623, -0.000085]$), its contrast against 1-pass row-only `MuonEq-R` [17] is **seed-heterogeneous**: at $\tau_{\text{BiEq}} = 25$, 2-pass `BiEq` wins Seeds 42 and 47 (`-0.000489` and `-0.000686` standalone; `-0.000648` and `-0.001136` in the matched `+ Cooldown + Neumann` package) and lowers the 3-seed mean by `-0.000311` standalone (`-0.000473` in the matched package) while halving the 3-seed standard deviation (`0.000972 -> 0.000533`), but trails 1-pass `MuonEq-R` on Seed 41 by `+0.000242` standalone (`+0.000365` in the matched package), leaving the 3-seed $95\%$ $t$-interval spanning zero ($[-0.001526, +0.000903]$ standalone; $[-0.002375, +0.001428]$ matched package). Resolving small mean differences ($\sim 0.0003\text{--}0.0005$ BPB) between 1-pass row-only and 2-pass row/column pre-polar equilibration with tight confidence intervals requires larger seed counts ($n \ge 8$).
2. **Common-Backbone Fixed-Schedule Scope vs. Hyperparameter Retuning:** All prior-art optimizer arms (`MuonEq-R/RC`, `MALT`, `NormPre-G`) and the single fixed-setting `Pure AdamW` run (`baseline_pure_adamw_seed46`: `1.053465` val BPB) are evaluated at the task's fixed reference learning rates, batch size (`524,288` tokens), and `953`-step ClimbMix horizon (`499.65M` tokens). They therefore isolate operator behavior within the `autoresearch` recipe rather than establishing rankings under independent hyperparameter sweeps, other corpora, or multi-billion-parameter scales.

---

## 6. Reproducibility and Complete Supplement Contents (`97` Runs Reconciled)

The supplement (`supplement.zip`, 762.0 KB compressed / 1.886 MB uncompressed, 19 entries) contains complete executable code, unified diffs, and numerical records for **all 97 completed 953-step runs**:
- `configs/all_runs_diffs.json` (exact `diff -U 0 baseline/train.py <run>/train.py` and SHA256 for **all 97 runs**) and standalone `configs/<run_name>_train.py` scripts (`baseline`, `noguard_fp32norm_control_seed42`, `muoneq_r_full_pkg_seed42`, `noguard_full_bieq_tau25_seed42`, `exp15_bieq_tau25_only_joint_seed42`).
- `results/all_runs_results.json` and `heldout_summary.json`: complete per-run dictionary containing `result.json`, `validation_loss.json`, `heldout_loss.json`, `geometry_audit.json`, and `learning_curve.json` for **all 97 runs**.
- `compute_accounting_audit.py`, `compute_accounting_audit.json`, `real_gradient_geometry_audit.json`, `geometry_audit.py`, `spectral_subspace_audit.json`, `eval_all_heldout.py`, `generate_figures.py`, `step0_nullspace_audit.py`, and `step0_nullspace_audit.json`.

---

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