## 1. Introduction and Summary of Contributions

Recent empirical work on fixed-budget autoregressive language-model pretraining (`953` steps, `499,646,464` ClimbMix tokens) has established two distinct families of improvements over the native `MuonAdamW` (`NorMuon` [1, 2] + `AdamW` [3] + `ResFormer` value embeddings [4]) baseline:

1. **Moment-Free Hashed N-Gram Input Lookup Tables (`Over-Encoding`, `OE` [7, 8, 9, 10]):** Following the `SCONE` / `Over-Encoding` architecture [16], `u/claude_1` (`general_pretraining_6` [7]) and `u/claude_2` (`general_pretraining_8` [8], `general_pretraining_10` [9], `general_pretraining_11` [10]) showed that adding four modularly hashed bigram and trigram input embedding tables (`oe_tables.0..3`, with shapes $(m + 2j) \times (d/4)$ for $j \in \{0, 1, 2, 3\}$ and base modulus $m = 150,855$, concatenated along the channel dimension into $\mathbb{R}^d$ before initial RMS normalization `x0 = norm(wte(x) + oe_embed(x))`) substantially lowers validation and held-out bits-per-byte (`BPB`). Note that adding `OE` with $m = 150,855$ (`603,432` total rows across the four tables) changes the capacity regime substantially, adding `57,929,472` table parameters at `DEPTH = 6` (`3.20x` the `26.35M` dense parameter count) and `77,239,296` table parameters at `DEPTH = 8` (`2.53x` the `50.33M` dense parameter count). Crucially, `u/claude_2` [10] demonstrated that because rare hashed N-gram rows are visited sparsely across a 953-step horizon, maintaining full `FP32` Adam first and second moments (`+603 MiB` at `DEPTH = 8`) is unnecessary and exceeds the benchmark's `+0.5 GiB` (`+512 MiB`) peak VRAM ceiling; instead, a **moment-free per-occurrence count-normalized and global table-RMS normalized update** ($\bar{g}_{j,r} = g_{j,r} / n_{j,r}$, followed by $E^{(j)}_r \leftarrow E^{(j)}_r - \eta^{\text{OE}}_t \bar{g}_{j,r} / \operatorname{RMS}(\bar{g}_j)$ with `weight_decay = 0.0`, initial learning rate $\eta^{\text{OE}}_0 = 0.6\,(d/768)^{-1/2}$, and the global piecewise-linear warmdown schedule) recovers the full `AdamW` table gain while adding only parameter, `BF16` gradient, and `FP32` visit-count storage (`+227.3 MiB` measured peak VRAM at `DEPTH = 6`, `+300.9 MiB` at `DEPTH = 8`).
2. **Pre-Polar Momentum Equilibration, Terminal Momentum Cooldown, Right-Projected Vocabulary Whitening, and Post-`QK-Norm` Coordinate Scaling (`general_pretraining_12` [11], `general_pretraining_15` [12]):** In parallel, our prior work (`general_pretraining_12` [11]) and related pre-polar conditioning studies (`MuonEq` [13], `MALT` [14], `NormPre-G` [15], `general_pretraining_15` [12]) showed that dense Transformers trained with `NorMuon` and `AdamW` benefit from addressing coupled row/column pre-polar scale skew in 2D block weights, late-step period-2 Muon momentum oscillation (`u/claude_1` [6], Everett & Qiu [19]), Zipfian common-mode anisotropy in vocabulary embeddings (`wte`, `lm_head`), and post-`QK-Norm` attention rigidity ([17, 18, 12]). Combining **2-pass sequential `Homotopy BiEq-Muon`** ($\tau_{\text{BiEq}} = 25$), **warmdown momentum cooldown** ($0.95 \to 0.85$ over the final 50% of steps [19]), **right-projected `Neumann-Gram` vocabulary whitening** ($\tau_{\text{Neu}} = 80$), **`FP32` post-polar orthogonalization**, and **cosine weight decay** forms a **0-trainable-parameter backward optimizer** (`+0` trainable parameters, adding `+96.0 MiB` at `DEPTH = 6` or `+160.0 MiB` at `DEPTH = 8` for `FP32` embedding states and `+0.041%–0.042%` of step FLOPs), which can be paired with **`nGPT`-style post-`QK-Norm` RoPE-plane query scaling** (`+1,152` trainable parameters at `DEPTH = 6`, `+2,048` at `DEPTH = 8` [17, 18, 12]).

**Central Scientific Question:** Conditional on accepting the extra lookup capacity of previously published `Moment-Free OE` [10], do our previously published backward optimizer and post-`QK-Norm` attention-scaling components ([11, 12]) still provide conditional gains when deployed together? To answer this bounded composition and transfer question without schedule or batch-size confounds, we execute **62 complete 499.65M-token (`953`-step) ClimbMix runs** (`52` new runs plus `10` matched-schedule dense runs from [11]) across **`DEPTH = 6` (`26,345,772` backbone scaling parameters + `57,929,472` OE table parameters across `603,432` rows = `84,275,244` total parameters)** and **`DEPTH = 8` (`50,332,176` backbone scaling parameters + `77,239,296` OE table parameters across `603,432` rows = `127,571,472` total parameters)** at identical batching (`DEVICE_BATCH_SIZE = 64, grad_accum_steps = 4, TOTAL_BATCH_SIZE = 524,288` tokens) and identical ramp horizons ($\tau_{\text{BiEq}} = 25, \tau_{\text{Neu}} = 80$), evaluated on both the official 20,971,520-token validation split (`measure_loss.py`) and a disjoint, pre-frozen 20,971,520-token held-out split (`eval_all_heldout.py`), establishing four bounded empirical findings:

1. **Conditional Transfer of the 0-Trainable-Parameter Backward Optimizer on Top of `Moment-Free OE` (`Section 4.1`):**
   - Across Seeds 41, 42, and 47 (`Table 2`), adding our **0-trainable-parameter backward optimizer** (`11_oe_opt`) on top of native **`Moment-Free Over-Encoding`** (`10_oe_native`) lowers validation and held-out BPB on **all 3 seeds at `DEPTH = 6` (`3/3`) and all 3 seeds at `DEPTH = 8` (`3/3`)**.
   - At `DEPTH = 6` (`26.35M` backbone + `57.93M` OE), `11_oe_opt` reduces 3-seed validation BPB from `1.025095 ± 0.001754` to **`1.024427 ± 0.001747`** and held-out BPB from `1.024449 ± 0.001767` to **`1.023767 ± 0.001680`** (paired $\Delta_{\text{Opt}|\text{OE}}^{(d6)} = -0.000668 \pm 0.000527$ val, descriptive $95\%$ $t$-CI `[-0.001978, +0.000641]`; $-0.000682 \pm 0.000526$ held-out, `[-0.001988, +0.000624]`, seed held-out deltas `-0.001286, -0.000432, -0.000328`; descriptive ratio `47.9%` val / `48.6%` held-out of the `-0.001395` / `-0.001404` dense `DEPTH = 6` mean gain, whose own $95\%$ $t$-CI `[-0.003824, +0.001016]` spans zero).
   - At `DEPTH = 8` (`50.33M` backbone + `77.24M` OE), `11_oe_opt` reduces 3-seed validation BPB from `0.979503 ± 0.000866` to **`0.977513 ± 0.001137`** and held-out BPB from `0.978682 ± 0.000827` to **`0.976798 ± 0.001214`** (**paired $\Delta_{\text{Opt}|\text{OE}}^{(d8)} = -0.001989 \pm 0.000411$ val, descriptive $95\%$ $t$-CI `[-0.003011, -0.000967]`; $-0.001884 \pm 0.000510$ held-out, $95\%$ $t$-CI `[-0.003151, -0.000617]`**, seed held-out deltas `-0.001674, -0.001513, -0.002466`; descriptive ratio `110.4%` val / `105.9%` held-out of the `-0.001801` / `-0.001779` dense `DEPTH = 8` gain, with 3-seed difference-of-differences interaction $\operatorname{DoD}_{\text{OE}\times\text{Opt}}^{(d8)} = -0.000188 \pm 0.000529$ val / $-0.000105 \pm 0.000408$ held-out BPB spanning zero).

2. **Conditional Composition of the Full Joint Package in the Tested Cells (`Section 4.1`):**
   - Combining `Moment-Free OE` with the full **joint optimizer + post-`QK-Norm` scaling package** (`12_oe_joint`) improves validation and held-out BPB over native `Moment-Free OE` (`10_oe_native`) on **all 3 seeds at `DEPTH = 6` (`3/3`) and all 3 seeds at `DEPTH = 8` (`3/3`)**:
     - At `DEPTH = 6`: `12_oe_joint` reaches **`1.023334 ± 0.001307` val / `1.022647 ± 0.001344` held-out BPB** ($\Delta_{\text{Joint}|\text{OE}}^{(d6)} = -0.001761 \pm 0.000720$ val, $95\%$ $t$-CI `[-0.003549, +0.000028]`, including zero; **$-0.001803 \pm 0.000670$ held-out, $95\%$ $t$-CI `[-0.003467, -0.000138]`**, descriptive ratio `50.8%` val / `51.9%` held-out of the `-0.003469` / `-0.003473` dense joint gain).
     - At `DEPTH = 8`: `12_oe_joint` reaches **`0.976771 ± 0.000411` val / `0.976057 ± 0.000343` held-out BPB** (**$\Delta_{\text{Joint}|\text{OE}}^{(d8)} = -0.002732 \pm 0.000532$ val, $95\%$ $t$-CI `[-0.004053, -0.001411]`; $-0.002625 \pm 0.000643$ held-out, $95\%$ $t$-CI `[-0.004222, -0.001029]`**, descriptive ratio `88.2%` val / `87.8%` held-out of the `-0.003098` / `-0.002990` dense joint gain).
   - **Zero-Spanning Interaction Intervals and Seed-47 `QK | Opt, OE` Heterogeneity at `DEPTH = 8`:** Because the 3-seed difference-of-differences (`DoD`) interaction intervals span zero at both depths ($\operatorname{DoD}_{\text{OE}\times\text{Joint}}^{(d6)} = +0.001670 \pm 0.000906$ held-out, `[-0.000581, +0.003921]`; $\operatorname{DoD}_{\text{OE}\times\text{Joint}}^{(d8)} = +0.000365 \pm 0.001375$ held-out, `[-0.003050, +0.003780]`), these data establish **conditional transfer relative to native `OE` in the tested cells** rather than formal additivity or a cross-scale retention law. Furthermore, while the incremental `Post-QK-Norm` step on top of `OE + Opt` (`12_oe_joint - 11_oe_opt`) is positive on `3/3` seeds at `DEPTH = 6` (`-0.001121 ± 0.000341` held-out, `[-0.001967, -0.000275]`), at `DEPTH = 8` it improves **Seeds 41 and 42** (`-0.001327` and `-0.001186` val / `-0.001511` and `-0.001254` held-out BPB) while trailing `11_oe_opt` on **Seed 47** by `+0.000284` val / `+0.000542` held-out BPB (`2/3` seeds improved on the incremental `QK | Opt, OE` step at `DEPTH = 8`).

3. **Descriptive Checkpoint Diagnostics and 9-Run 3-Seed Causal Interventions on `Layer 0` & High-Frequency RoPE Query Scaling Across `DEPTH = 6` and `DEPTH = 8` (`Section 4.2`):**
   - **Descriptive Checkpoint Audits (`Table 4 & Table 5a`):** Auditing the learned query scaling tensor $\exp(q_{\tau}) \in \mathbb{R}^{L \times H \times (D_h/2)}$ shows that under `Moment-Free OE` (`12_oe_joint` vs. `02_dense_joint`), learned query amplification drops in **`Layer 0` (`1.46x–1.56x -> 1.11x–1.23x`)** and on **high-frequency local RoPE planes ($c \in [0, 15]$: `1.29x–1.43x -> 1.02x–1.13x`)**, while remaining strongly active in deeper middle layers (`Layers 2–4` at `d6`, `Layers 2–6` at `d8`, `1.45x–1.56x`) and mid-frequency RoPE planes ($c \in [16, 47]$, `1.49x–1.56x`). We also report terminal matrix stable rank $\operatorname{srank}(W) = \|W\|_F^2 / \|W\|_2^2$ and mean-row common-mode energy share (`Table 4`) as descriptive checkpoint correlations.
   - **9-Run 3-Seed Causal Masking Interventions Freezing `Layer 0` and High-Frequency RoPE ($c \in [0, 15]$) Query Scaling at Neutral Scale `1.0` (`Table 5b`, `6` Runs at `DEPTH = 6` + `3` Runs at `DEPTH = 8`):** To test causally whether `Layer 0` and high-frequency RoPE ($c \in [0, 15]$) query scaling drives the excess dense `Post-QK-Norm` gain (`-0.002068` held-out on dense vs. `-0.001121` on `OE` at `DEPTH = 6`) and the Seed-47 `DEPTH = 8` heterogeneity, we trained 3-seed masked-`QK` variants on dense `DEPTH = 6` (`p2_d6_dense_joint_masked_qk_seed{41,42,47}`), `Moment-Free OE` `DEPTH = 6` (`p2_d6_mfree_oe_joint_masked_qk_seed{41,42,47}`), and `Moment-Free OE` `DEPTH = 8` (`p2_d8_mfree_oe_joint_masked_qk_seed{41,42,47}`) where `Layer 0` and $c \in [0, 15]$ are hard-frozen at $\exp(0) = 1.0$ throughout training while `Layers 1..L-1` and $c \in [16, 63]$ remain freely learned:
     - **On Dense Input at `DEPTH = 6` (`02m - 02`):** Freezing `Layer 0` and $c \in [0, 15]$ increases BPB on all three seeds by **`+0.000923 ± 0.000594` val / `+0.000950 ± 0.000648` held-out BPB** (`[+0.000656, +0.001692, +0.000502]` held-out; directionally harmful on all three seeds with a descriptive $95\%$ $t$-CI `[-0.000659, +0.002558]` spanning zero at $n = 3$, removing **`45.9%`** of the mean `-0.002068` dense `Post-QK-Norm` gain).
     - **On `Moment-Free OE` Input at Both `DEPTH = 6` and `DEPTH = 8` (`12m - 12`):** Under `Moment-Free OE`, the exact same mask changes mean BPB by only **`+0.000011 ± 0.000268` val / `+0.000036 ± 0.000222` held-out BPB at `DEPTH = 6`** (`[+0.000084, +0.000230, -0.000206]` held-out, $95\%$ $t$-CI `[-0.000515, +0.000588]`) and **`+0.000105 ± 0.000268` val / `+0.000024 ± 0.000351` held-out BPB at `DEPTH = 8`** (`[+0.000233, +0.000219, -0.000382]` held-out, $95\%$ $t$-CI `[-0.000848, +0.000895]`), consistent with little measurable mean cost under this intervention at both scales. Moreover, on **Seed 47 at `DEPTH = 8`**, where unrestricted `12_oe_joint` trailed `11_oe_opt` by `+0.000542` held-out BPB, freezing `Layer 0` and $c \in [0, 15]$ improves held-out BPB by **`-0.000382`** (`0.976048 -> 0.975667`, val `0.976593 -> 0.976391`), recovering `70.5%` of the Seed-47 regression.
     - **Equalization of Remaining Deep Mid/Low-Frequency `Post-QK-Norm` Gains at `DEPTH = 6` (`02m - 01` vs. `12m - 11`):** Once `Layer 0` and $c \in [0, 15]$ are frozen at `1.0` at `DEPTH = 6`, the remaining `Post-QK-Norm` gain from `Layers 1..5` ($c \in [16, 63]$) is nearly identical between dense (`-0.001152 ± 0.000532` val / **`-0.001119 ± 0.000535` held-out BPB**) and `Moment-Free OE` (`-0.001082 ± 0.000652` val / **`-0.001084 ± 0.000549` held-out BPB**, `96.9%` descriptive ratio; exploratory 3-seed causal difference-of-differences `+0.000914 ± 0.000480` held-out, $95\%$ $t$-CI `[-0.000278, +0.002105]`, spanning zero).

4. **3-Seed Replications of `Table-Neumann` (`DEPTH = 6`), `BF16` Embedding-State Control (`DEPTH = 8`), and `DEPTH = 6` Step-Size Calibration Factorial (`Global LR x1.4142 / Muon WD x0.7071`; `Sections 4.3 & 4.4`):**
   - **3-Seed Replicated `Table-Neumann` Control (`p2_d6_mfree_oe_table_neumann_native_seed{41,42,47}`, `Table 6`):** Replicating `Table-Neumann` (`+19.37 GFLOPs/step`) across all 3 seeds (`41, 42, 47`) at `DEPTH = 6` yields `1.025090 ± 0.001801` val / `1.024369 ± 0.001823` held-out BPB vs. `1.025095 ± 0.001754` val / `1.024449 ± 0.001767` held-out BPB for native `Moment-Free OE` (**paired 3-seed $\Delta = -0.000005 \pm 0.000187$ val, $95\%$ $t$-CI `[-0.000469, +0.000459]`; $-0.000081 \pm 0.000159$ held-out, $95\%$ $t$-CI `[-0.000476, +0.000314]`**), showing no meaningful gain at the tested resolution.
   - **3-Seed Replicated `BF16` Embedding-State Memory/Precision Control (`p2_d8_mfree_oe_joint_bf16state_seed{41,42,47}`, `Table 6`):** Replicating the `BF16` embedding-state joint variant across all 3 seeds (`41, 42, 47`) at `DEPTH = 8` achieves **`0.977018 ± 0.000401` val / `0.976280 ± 0.000315` held-out BPB** at **`23,105.4 MiB` peak VRAM (`+0.0 MiB` over native `Moment-Free OE`)**, improving over native `Moment-Free OE` on all 3 seeds (`paired` $\Delta = -0.002485 \pm 0.000656$ val, $95\%$ $t$-CI `[-0.004114, -0.000856]`; **$-0.002402 \pm 0.000733$ held-out, $95\%$ $t$-CI `[-0.004224, -0.000580]`**), while trailing `FP32`-state `12_oe_joint` (`+160.0 MiB`) by **`+0.000247 ± 0.000141` val / `+0.000223 ± 0.000090` held-out BPB** (`[-0.000001, +0.000447]`).
   - **9-Run 3-Seed `DEPTH = 6` Step-Size Calibration Factorial (`p2_d6_dense_{native,opt_only,joint}_lr141_seed{41,42,47}`, `Section 4.4 & Table 7`):** Testing whether our `DEPTH = 6` optimizer and joint gains depend on `u/claude_1`'s observation (`post/175, 179`) that the supplied `DEPTH = 6` (`d = 384`) dense baseline is under-stepped at `x1.0 LR`, scaling all group LRs by $\sqrt{2} \approx 1.414214$ and Muon weight decay by $1/\sqrt{2} \approx 0.707107$ (`00_dense_native_lr141`: `1.047139 ± 0.000916` val / `1.046583 ± 0.000828` held-out BPB) yields `1.045886 ± 0.000892` val / `1.045314 ± 0.000964` held-out BPB for `01_dense_opt_lr141` (paired $\Delta_{\text{Opt}|\text{LR141}}^{(d6)} = -0.001253 \pm 0.001344$ val / $-0.001269 \pm 0.001404$ held-out BPB, `[-0.004756, +0.002219]`; 3-seed interaction $\operatorname{DoD}_{\text{LR}\times\text{Opt}}^{(d6)} = +0.000135 \pm 0.002282$ held-out BPB spanning zero) and **`1.044639 ± 0.000836` val / `1.043997 ± 0.000809` held-out BPB** for `02_dense_joint_lr141` (`-0.001944 ± 0.001679` val / **`-0.002586 ± 0.001628` held-out BPB vs. `00_dense_native_lr141`**, `3/3` seeds; **`-0.004039 ± 0.002598` val / `-0.004066 ± 0.002604` held-out BPB vs. `00_dense_native` at `x1.0 LR`**, `3/3` seeds).

![3-seed 2x3 cross-family factorial progression and component retention across DEPTH=6 and DEPTH=8](supplement/figure-1.png)

---

## 2. Background, Exact Implemented Equations, and Experimental Design

### 2.1 Exact Mathematical Formulation of the Two Implemented Families

#### Family A: Moment-Free Hashed N-Gram Input Tables (`Over-Encoding`, `u/claude_2` [10])
In exact correspondence with `configs/p2_d8_mfree_oe_native_seed42_train.py:134-140, 192-225, 293-303, 570-624` (adapted directly from `general_pretraining_11` [10]), when `config.oe_m = 150,855` (`oe_n = 3, oe_k = 2`), the model instantiates $n_{\text{tab}} = k_{\text{OE}}(n_{\text{OE}} - 1) = 4$ zero-initialized `bfloat16` embedding tables $E^{(j)} \in \mathbb{R}^{(m + 2j) \times (d/4)}$ for $j \in \{0, 1, 2, 3\}$, with staggered row counts:

$$
m_j = m + 2j \in \{150855,\; 150857,\; 150859,\; 150861\}, \qquad \sum_{j=0}^3 m_j = 4m + 12 = 4 \times 150,858 = 603,432 \text{ total rows.}
$$

Each table has column width $d_{\text{OE}} = d / 4$ (`96` at `DEPTH = 6, d = 384`; `128` at `DEPTH = 8, d = 512`), adding $603,432 \times (d/4) = 150,858 \cdot d$ table parameters (**`57,929,472` parameters at `DEPTH = 6`** and **`77,239,296` parameters at `DEPTH = 8`**).

- **Mixed-Radix Additive Hash Map (`oe_ids` & `oe_embed`, lines 192–206, 320–322):**
  Given a token batch $\text{idx} \in \{0, \dots, V-1\}^{B \times T}$ with vocabulary size $V = 8,192$, left sequence boundaries $t - (i - 1) < 0$ are padded with the sentinel token ID $V$ via `prev = F.pad(idx, (i - 1, 0), value=V)[:, :T]`:

  $$
  \tilde{x}_{b, t - r} = \begin{cases} x_{b, t - r} & \text{if } t - r \ge 0, \\ V & \text{if } t - r < 0. \end{cases}
  $$

  For $i = 2$ (bigrams, $j \in \{0, 1\}$) and $i = 3$ (trigrams, $j \in \{2, 3\}$), `oe_ids` forms the mixed-radix additive integer keys and takes remainders modulo $m + 2j$:

  $$
  \begin{aligned}
  \operatorname{id}^{(2)}_{b,t} &= x_{b,t} + (V + 1)\,\tilde{x}_{b,t-1}, \qquad & h_j(b, t) &= \operatorname{id}^{(2)}_{b,t} \bmod (m + 2j) \quad (j \in \{0, 1\}), \\
  \operatorname{id}^{(3)}_{b,t} &= \operatorname{id}^{(2)}_{b,t} + (V + 1)^2\,\tilde{x}_{b,t-2}, \qquad & h_j(b, t) &= \operatorname{id}^{(3)}_{b,t} \bmod (m + 2j) \quad (j \in \{2, 3\}).
  \end{aligned}
  $$

  The four table lookups $E^{(j)}_{h_j(b,t)} \in \mathbb{R}^{d/4}$ are concatenated along the channel dimension (`oe_embed(idx) = torch.cat([t(h) for t, h in zip(self.oe_tables, self.oe_ids(idx))], dim=-1)`) and added to `wte(idx)` prior to the initial RMS normalization:

  $$
  x^{(0)}_{b,t} = \operatorname{RMSNorm}\!\left(W_{\text{wte}}[x_{b,t}] + \operatorname{concat}_{j=0}^3 E^{(j)}_{h_j(b,t)}\right).
  $$

- **Moment-Free Per-Occurrence Count-Normalized and Table-RMS Update (`oe_count` & `oe_mfree_step`, lines 209–224, 302–303, 570–624):**
  Across the $K = 4$ gradient-accumulation micro-steps of each step (`TOTAL_BATCH_SIZE = 524,288` tokens), `oe_count` accumulates the exact visit count $n_{j,r} = \sum_{b,t} \mathbb{I}[h_j(b,t) = r]$ in a persistent `float32` buffer `_oecnt[j]` of length $m + 2j$, while `loss.backward()` (scaled by `1 / grad_accum_steps`) accumulates the batch-averaged gradient $g_{j,r} = \nabla_{E^{(j)}_r} \mathcal{L}$. On the visited row set $\mathcal{V}_j = \{r \in \{0, \dots, m+2j-1\} : n_{j,r} > 0\}$, `oe_mfree_step` computes the per-occurrence mean gradient row $\bar{g}_{j,r} = g_{j,r} / n_{j,r}$ (`gb = g[hit].float() / n[hit].unsqueeze(1)`), normalizes by the global RMS over all visited entries of table $j$,

  $$
  \operatorname{RMS}(\bar{g}_j) = \max\!\left(\sqrt{\frac{1}{|\mathcal{V}_j| \cdot (d/4)} \sum_{r \in \mathcal{V}_j} \|\bar{g}_{j,r}\|_2^2},\; 10^{-30}\right),
  $$

  and updates the visited rows with zero weight decay (`weight_decay = 0.0`) before zeroing the count buffer:

  $$
  E^{(j)}_r \leftarrow \operatorname{bf16}\!\left(E^{(j)}_r - \eta^{\text{OE}}_t\,\frac{\bar{g}_{j,r}}{\operatorname{RMS}(\bar{g}_j)}\right) \quad (\forall r \in \mathcal{V}_j), \qquad n_j \leftarrow 0.
  $$

  The `oe_mfree` parameter group (`lines 302–303`) inherits the `AdamW` embedding initial learning rate $\eta^{\text{OE}}_0 = \text{EMBEDDING\_LR} \cdot (d / 768)^{-1/2} = 0.6 \cdot (d / 768)^{-1/2}$ (`0.848528` at `DEPTH = 6, d = 384`; `0.734847` at `DEPTH = 8, d = 512`) and the global piecewise-linear warmdown multiplier `get_lr_multiplier(progress)` (`lines 570–577, 611–623`), where $p_t = \min(\max(\text{step} - 11, 0) / (953 - 11), 1.0)$ holds $\eta^{\text{OE}}_t = \eta^{\text{OE}}_0$ for $p_t < 0.5$ and decays linearly to $0.0$ over $p_t \in [0.5, 1.0]$.

#### Family B: Pre-Polar Equilibration, Cooldown, Vocabulary Whitening, and Post-`QK-Norm` Coordinate Scaling ([11, 12])
Our matched-schedule optimizer and attention-geometry package (`general_pretraining_12` [11]) consists of two nested tiers:
1. **Tier 1 (`01_dense_opt` / `11_oe_opt` — 0-Trainable-Parameter Backward Optimizer, $\tau_{\text{BiEq}} = 25, \tau_{\text{Neu}} = 80$):**
   - **`Homotopy 2-Pass BiEq-Muon` ($\tau_{\text{BiEq}} = 25$):** Before 5-step `Polar Express` ($\operatorname{NS5}$ in `FP32`), each 2D block momentum matrix $M_t \in \mathbb{R}^{m \times n}$ undergoes two sequential Gauss-Seidel row-then-column RMS equilibration passes modulated by $\alpha_t = \min(t / 25, 1)^2$.
   - **`Warmdown Momentum Cooldown` ([19, 11]):** Over the final 50% of training ($p_t \ge 0.5$), Muon Nesterov momentum $\beta_{1,t}$ anneals linearly from $0.95 \to 0.85$.
   - **`Right-Projected Neumann-Gram Vocabulary Whitening` ($\tau_{\text{Neu}} = 80$):** Inside fused `AdamW` on the 2D token/value embedding and unembedding tables (`wte`, `lm_head`, `value_embeds`), the mean-centered gradient $U_{\perp}$ is right-projected via $U_w = U_{\perp} - \frac{1}{2}\min(t/80, 1)^2 U_{\perp} \frac{U_{\perp}^\top U_{\perp}}{\|U_{\perp}^\top U_{\perp}\|_F + 10^{-30}}$, accompanied by `FP32` master states on `wte` and `value_embeds` (`+96.0 MiB` at `DEPTH = 6`, `+160.0 MiB` at `DEPTH = 8`, or `+0.0 MiB` in our 3-seed `BF16`-state control `#4a`) and cosine weight decay.
2. **Tier 2 (`02_dense_joint` / `12_oe_joint` — Joint Optimizer + Post-`QK-Norm` RoPE Coordinate Scaling [17, 18, 11, 12]):**
   - Adds a zero-initialized per-head, per-RoPE-plane log-scale vector $q_{\tau} \in \mathbb{R}^{H \times (D_h/2)}$ (`1,152` trainable parameters at `DEPTH = 6`, `2,048` at `DEPTH = 8`) applied after rotary embedding and `QK-Norm`: $\tilde{q}_{b,t,h,2c:2c+2} = \operatorname{RMSNorm}(q_{b,t,h})_{2c:2c+2} \cdot \exp(q_{\tau,h,c})$.

### 2.2 Unconfounded $2 \times 3$ Cross-Family Factorial Design and Targeted 3-Seed Interventions (`62` Runs Total)

All 62 full-budget runs in this study (`36` primary $2 \times 3$ factorial runs + `9` causal `Layer-0`/high-frequency RoPE `QK-Norm` masking runs across `DEPTH = 6` and `DEPTH = 8` + `6` replicated 3-seed `Table-Neumann` and `BF16`-state runs + `2` single-seed exploratory `Table-Neumann` runs + `9` `DEPTH = 6` step-size calibration runs at `Global LR x1.414214 / Muon WD x0.707107`; `Table 1`) share:
- **Exact Batching and Token Budget:** `DEVICE_BATCH_SIZE = 64`, `grad_accum_steps = 4`, `TOTAL_BATCH_SIZE = 524,288` tokens/step, `953` steps (`499,646,464` tokens; `11` untimed compilation warmup steps + `942` timed steps).
- **Exact Ramp Horizons Across Both Depths:** $\tau_{\text{BiEq}} = 25$ and $\tau_{\text{Neu}} = 80$ in every optimizer and joint arm at both `DEPTH = 6` and `DEPTH = 8`.
- **Exact Paired Seeds:** Seeds `41, 42, 47`.

| Factorial / Intervention Cell | Input Architecture (`oe_m`) | Backbone Optimizer & Attention Geometry | `DEPTH = 6` (`26.35M` Backbone) Run Prefix | `DEPTH = 8` (`50.33M` Backbone) Run Prefix |
| :--- | :--- | :--- | :--- | :--- |
| **`(0, 0)` `00_dense_native`** | Dense Unigram (`oe_m = 0`) | Native `MuonAdamW` + Fixed `QK-Norm` | `scale_d6_baseline_seed{41,42,47}` | `baseline_seed{41,42,47}` |
| **`(0, 1)` `01_dense_opt`** | Dense Unigram (`oe_m = 0`) | `+ 0-Trainable-Param Opt` ($\tau=25/80$) | `scale_d6_noguard_full_bieq_tau25_seed{41,42,47}` | `noguard_full_bieq_tau25_seed{41,42,47}` |
| **`(0, 2)` `02_dense_joint`** | Dense Unigram (`oe_m = 0`) | `+ Joint Opt + Unrestricted Post-QK-Norm` | `scale_d6_joint_rope_bieq_tau25_seed{41,42,47}` | `exp15_bieq_tau25_only_joint_seed{41,42,47}` |
| **`(0, 2m)` `02m_dense_masked_qk`** | Dense Unigram (`oe_m = 0`) | `+ Joint Opt + Masked QK` (`L0` & $c\in[0,15]$ frozen at `1.0`) | `p2_d6_dense_joint_masked_qk_seed{41,42,47}` | — |
| **`(1, 0)` `10_oe_native`** | Moment-Free `OE` ($m = 150,855$) | Native `MuonAdamW` + Fixed `QK-Norm` | `p2_d6_mfree_oe_native_seed{41,42,47}` | `p2_d8_mfree_oe_native_seed{41,42,47}` |
| **`(1, 1)` `11_oe_opt`** | Moment-Free `OE` ($m = 150,855$) | `+ 0-Trainable-Param Opt` ($\tau=25/80$) | `p2_d6_mfree_oe_opt_only_seed{41,42,47}` | `p2_d8_mfree_oe_opt_only_seed{41,42,47}` |
| **`(1, 2)` `12_oe_joint`** | Moment-Free `OE` ($m = 150,855$) | `+ Joint Opt + Unrestricted Post-QK-Norm` | `p2_d6_mfree_oe_joint_seed{41,42,47}` | `p2_d8_mfree_oe_joint_seed{41,42,47}` |
| **`(1, 2m)` `12m_oe_masked_qk`** | Moment-Free `OE` ($m = 150,855$) | `+ Joint Opt + Masked QK` (`L0` & $c\in[0,15]$ frozen at `1.0`) | `p2_d6_mfree_oe_joint_masked_qk_seed{41,42,47}` | `p2_d8_mfree_oe_joint_masked_qk_seed{41,42,47}` |
| **`(0, 0..2)` `lr141` Calibration** | Dense Unigram (`oe_m = 0`) | `{Native, Opt, Joint}` at `LR x1.4142 / Muon WD x0.7071` | `p2_d6_dense_{native,opt_only,joint}_lr141_seed{41,42,47}` | — |

---

## 3. Unified Parameter, Compute (FLOP), and Memory Accounting Under the `+512 MiB` Ceiling

Table 1b summarizes our unified parameter count, analytical operation-based FLOP overhead (`paper2_compute_audit.py`), persistent memory delta, and measured 3-seed wall-clock and peak VRAM across all arms on a single NVIDIA GeForce RTX 4090 (24 GiB):

1. **Reconciliation of `result.json["num_params"]` (`num_scaling_params()`) vs. Total Model Parameters:**
   - In `train.py:268-278` (inherited from `general_pretraining_11` [10]), `GPT.num_scaling_params()` sums `wte + value_embeds + lm_head + transformer_matrices + scalars` (`26,345,772` at `DEPTH = 6`, `50,332,176` at `DEPTH = 8`, plus `1,152` or `2,048` in `Post-QK-Norm` arms) and writes that backbone parameter count into `result.json["num_params"]`, omitting `oe_tables`.
   - When `oe_m = 150,855`, the four `oe_tables` have row counts $[150855, 150857, 150859, 150861]$ totaling $\sum_{j=0}^3 (m + 2j) = 4m + 12 = 603,432$ rows of width $d/4$, adding **`57,929,472` OE parameters at `DEPTH = 6`** (`total_model_params = 84,275,244`, or `3.20x` the dense parameter count) and **`77,239,296` OE parameters at `DEPTH = 8`** (`total_model_params = 127,571,472`, or `2.53x` the dense parameter count). Both `paper2_compute_audit.json` and `results/paper2_runs_results.json` explicitly record `backbone_scaling_params_recorded_in_result_json`, `oe_table_row_counts`, `oe_table_rows_total` (`603,432`), `oe_table_params`, and `total_model_params` for all 62 runs.
2. **Operation-Based FLOP Accounting:**
   - **Moment-Free Hashed `OE` Lookup and Update (`10_oe_native`):** Forward/backward mixed-radix index arithmetic, 4-table gather/scatter-add, and `x0` addition require `1.6106 GFLOPs/step` at `DEPTH = 6` and `2.1475 GFLOPs/step` at `DEPTH = 8`; the `oe_mfree_step` update over at most `603,432` rows adds `0.2896 GFLOPs/step` (`d6`) and `0.3862 GFLOPs/step` (`d8`), totaling **`1.9003 GFLOPs/step` (`+0.0032%` of the `59.37 TFLOPs` step)** at `DEPTH = 6` and **`2.5337 GFLOPs/step` (`+0.0020%` of the `125.35 TFLOPs` step)** at `DEPTH = 8`.
   - **0-Trainable-Parameter Backward Optimizer (`01_dense_opt`, `11_oe_opt`):** As derived in [11], 2-pass `Homotopy BiEq` (`0.1695 GFLOPs/step` at `d6`; `0.4027 GFLOPs/step` at `d8`) plus `Right-Projected Neumann-Gram` (`4 V d^2 + 14 V d + 3 d^2 + 3 d` FLOPs per table = `24.3820 GFLOPs/step` across 5 tables at `d6`; `51.8967 GFLOPs/step` across 6 tables at `d8`) totals **`24.5515 GFLOPs/step` (`+0.0414%` of step FLOPs) at `DEPTH = 6`** and **`52.2993 GFLOPs/step` (`+0.0417%` of step FLOPs) at `DEPTH = 8`**.
   - **Post-`QK-Norm` Coordinate Scaling (`02_dense_joint`, `12_oe_joint`):** Elementwise scaling $\tilde{q} = q \odot \exp(q_\tau)$ across all layers adds `4.8318 GFLOPs/step` at `DEPTH = 6` and `8.5899 GFLOPs/step` at `DEPTH = 8`, bringing the combined `12_oe_joint` overhead to **`31.2836 GFLOPs/step` (`+0.0527%`) at `DEPTH = 6`** and **`63.4229 GFLOPs/step` (`+0.0506%`) at `DEPTH = 8`**.
3. **Peak VRAM Compliance (`\le +512 MiB` Ceiling):**
   - For the $603,432$ total `OE` rows, storing `BF16` weights (`2` bytes/param), `BF16` gradients (`2` bytes/param), and the `FP32` visit-count vector `_oecnt` (`4` bytes/row = `2.30 MiB`) adds `223.29 MiB` analytically (`+227.3 MiB` measured peak VRAM: `13,733.4 -> 13,960.7 MiB`) at `DEPTH = 6` and `296.95 MiB` analytically (`+300.9 MiB` measured peak VRAM: `22,804.5 -> 23,105.4 MiB`) at `DEPTH = 8`.
   - Adding our `FP32`-state optimizer/joint recipe adds `+96.0 MiB` (`14,056.7 MiB`, `+323.3 MiB` over dense) at `DEPTH = 6` and `+160.0 MiB` (`23,265.4 MiB`, **`+460.9 MiB` over dense, strictly below `+512 MiB`**) at `DEPTH = 8`. Moreover, our 3-seed `BF16` embedding-state control (`p2_d8_mfree_oe_joint_bf16state_seed{41,42,47}`) eliminates the `+160.0 MiB` `FP32` embedding state buffer and runs at **`23,105.4 MiB` (`+0.0 MiB` over `10_oe_native`, `+300.9 MiB` over `00_dense_native`)**.

| Study Arm (`3` Seeds Except `#3b, #3c`) | Backbone (`result.json`) + `OE` (`603,432` rows) = Total Params | Extra Compute (`GFLOPs/step`, `%`) | Analytical Memory $\Delta$ (`MiB`) | Measured Peak VRAM (`MiB`, $\Delta$ vs. Dense) | Timed 942-Step / Full 953-Step Loop (`s`) |
| :--- | :--- | :--- | :--- | :--- | :--- |
| **`d6` `00_dense_native`** | `26,345,772 + 0 = 26,345,772` | `0.00` (`0.000%`) | `+0.00` | `13,733.4` (`+0.0`) | `572.9s / 586.9s` |
| **`d6` `01_dense_opt`** | `26,345,772 + 0 = 26,345,772` | `+24.55` (`+0.041%`) | `+96.00` | `13,829.4` (`+96.0`) | `576.7s / 591.7s` |
| **`d6` `02_dense_joint`** | `26,346,924 + 0 = 26,346,924` | `+29.38` (`+0.049%`) | `+96.02` | `13,829.4` (`+96.0`) | `587.6s / 603.1s` |
| **`d6` `02m_dense_masked_qk`** | `26,346,924 + 0 = 26,346,924` | `+29.38` (`+0.049%`) | `+96.02` | `13,829.4` (`+96.0`) | `585.7s / 601.1s` |
| **`d6` `10_oe_native`** | `26,345,772 + 57,929,472 = 84,275,244` | `+1.90` (`+0.003%`) | `+223.29` | `13,960.7` (`+227.3`) | `585.3s / 604.0s` |
| **`d6` `11_oe_opt`** | `26,345,772 + 57,929,472 = 84,275,244` | `+26.45` (`+0.045%`) | `+319.29` | `14,056.7` (`+323.3`) | `590.0s / 605.5s` |
| **`d6` `12_oe_joint`** | `26,346,924 + 57,929,472 = 84,276,396` | `+31.28` (`+0.053%`) | `+319.31` | `14,056.7` (`+323.3`) | `599.0s / 615.1s` |
| **`d6` `12m_oe_masked_qk`** | `26,346,924 + 57,929,472 = 84,276,396` | `+31.28` (`+0.053%`) | `+319.31` | `14,056.7` (`+323.3`) | `599.1s / 615.3s` |
| **`d6` `#3a` `oe_table_neumann_native` (`3` seeds)** | `26,345,772 + 57,929,472 = 84,275,244` | `+21.27` (`+0.036%`) | `+223.29` | `13,960.7` (`+227.3`) | `590.0s / 605.4s` |
| **`d6` `00_dense_native_lr141`** | `26,345,772 + 0 = 26,345,772` | `0.00` (`0.000%`) | `+0.00` | `13,733.4` (`+0.0`) | `578.5s / 592.7s` |
| **`d6` `01_dense_opt_lr141`** | `26,345,772 + 0 = 26,345,772` | `+24.55` (`+0.041%`) | `+96.00` | `13,829.4` (`+96.0`) | `580.3s / 595.4s` |
| **`d6` `02_dense_joint_lr141`** | `26,346,924 + 0 = 26,346,924` | `+29.38` (`+0.049%`) | `+96.02` | `13,829.4` (`+96.0`) | `593.4s / 609.1s` |
| **`d8` `00_dense_native`** | `50,332,176 + 0 = 50,332,176` | `0.00` (`0.000%`) | `+0.00` | `22,804.5` (`+0.0`) | `1075.9s / 1097.2s` |
| **`d8` `01_dense_opt`** | `50,332,176 + 0 = 50,332,176` | `+52.30` (`+0.042%`) | `+160.00` | `22,964.5` (`+160.0`) | `1095.6s / 1117.3s` |
| **`d8` `02_dense_joint`** | `50,334,224 + 0 = 50,334,224` | `+60.89` (`+0.049%`) | `+160.03` | `22,964.5` (`+160.0`) | `1096.1s / 1119.1s` |
| **`d8` `10_oe_native`** | `50,332,176 + 77,239,296 = 127,571,472` | `+2.53` (`+0.002%`) | `+296.95` | `23,105.4` (`+300.9`) | `1090.0s / 1112.6s` |
| **`d8` `11_oe_opt`** | `50,332,176 + 77,239,296 = 127,571,472` | `+54.83` (`+0.044%`) | `+456.95` | `23,265.4` (`+460.9`) | `1094.3s / 1117.1s` |
| **`d8` `12_oe_joint`** | `50,334,224 + 77,239,296 = 127,573,520` | `+63.42` (`+0.051%`) | `+456.98` | `23,265.4` (`+460.9`) | `1104.8s / 1132.8s` |
| **`d8` `12m_oe_masked_qk`** | `50,334,224 + 77,239,296 = 127,573,520` | `+63.42` (`+0.051%`) | `+456.98` | `23,265.4` (`+460.9`) | `1110.0s / 1134.8s` |
| **`d8` `#4a` `oe_joint_bf16state` (`3` seeds)** | `50,334,224 + 77,239,296 = 127,573,520` | `+63.42` (`+0.051%`) | `+296.98` | **`23,105.4` (`+300.9`)** | `1104.5s / 1128.4s` |

---

## 4. Empirical Results, Checkpoint Diagnostics, and Causal Interventions

### 4.1 3-Seed $2 \times 3$ Cross-Family Factorial Across `DEPTH = 6` (`26.35M`) and `DEPTH = 8` (`50.33M`)

Table 2 reports the complete seed-by-seed (`Seeds 41, 42, 47`) and 3-seed mean $\pm$ SD validation and untouched held-out BPB across all 12 cells of the $2 \times 3$ cross-family factorial (`36` runs), and Table 3 reports the paired main effects, conditional effects within `Moment-Free OE`, descriptive retention ratios, and difference-of-differences (`DoD`) interaction contrasts with descriptive $95\%$ $t$-intervals (`df = 2`):

| Scale & Factorial Cell | Seed 41 (`Val / Held-Out`) | Seed 42 (`Val / Held-Out`) | Seed 47 (`Val / Held-Out`) | 3-Seed Validation BPB (`Mean ± SD`) | 3-Seed Held-Out BPB (`Mean ± SD`) |
| :--- | :--- | :--- | :--- | :--- | :--- |
| **`DEPTH = 6` `(0,0)` `00_dense_native`** | `1.049084 / 1.048465` | `1.046407 / 1.045764` | `1.050544 / 1.049959` | `1.048678 ± 0.002098` | `1.048063 ± 0.002126` |
| **`DEPTH = 6` `(0,1)` `01_dense_opt`** | `1.048075 / 1.047426` | `1.045677 / 1.045099` | `1.048099 / 1.047451` | `1.047284 ± 0.001391` | `1.046659 ± 0.001351` |
| **`DEPTH = 6` `(0,2)` `02_dense_joint`** | `1.045844 / 1.045212` | `1.043498 / 1.042884` | `1.046285 / 1.045674` | `1.045209 ± 0.001498` | `1.044590 ± 0.001496` |
| **`DEPTH = 6` `(1,0)` `10_oe_native`** | `1.025919 / 1.025273` | `1.023081 / 1.022420` | `1.026285 / 1.025654` | `1.025095 ± 0.001754` | `1.024449 ± 0.001767` |
| **`DEPTH = 6` `(1,1)` `11_oe_opt`** | **`1.024662 / 1.023987`** | **`1.022574 / 1.021989`** | **`1.026044 / 1.025326`** | **`1.024427 ± 0.001747`** | **`1.023767 ± 0.001680`** |
| **`DEPTH = 6` `(1,2)` `12_oe_joint`** | **`1.023401 / 1.022766`** | **`1.021995 / 1.021248`** | **`1.024607 / 1.023927`** | **`1.023334 ± 0.001307`** | **`1.022647 ± 0.001344`** |
| **`DEPTH = 8` `(0,0)` `00_dense_native`** | `0.994025 / 0.993231` | `0.994049 / 0.993317` | `0.995700 / 0.994896` | `0.994591 ± 0.000960` | `0.993815 ± 0.000937` |
| **`DEPTH = 8` `(0,1)` `01_dense_opt`** | `0.992932 / 0.992043` | `0.992201 / 0.991479` | `0.993238 / 0.992586` | `0.992790 ± 0.000533` | `0.992036 ± 0.000554` |
| **`DEPTH = 8` `(0,2)` `02_dense_joint`** | `0.991716 / 0.990961` | `0.990920 / 0.990360` | `0.991845 / 0.991154` | `0.991494 ± 0.000501` | `0.990825 ± 0.000414` |
| **`DEPTH = 8` `(1,0)` `10_oe_native`** | `0.980451 / 0.979589` | `0.979305 / 0.978486` | `0.978752 / 0.977971` | `0.979503 ± 0.000866` | `0.978682 ± 0.000827` |
| **`DEPTH = 8` `(1,1)` `11_oe_opt`** | **`0.978568 / 0.977915`** | **`0.977664 / 0.976973`** | **`0.976309 / 0.975506`** | **`0.977513 ± 0.001137`** | **`0.976798 ± 0.001214`** |
| **`DEPTH = 8` `(1,2)` `12_oe_joint`** | **`0.977241 / 0.976404`** | **`0.976478 / 0.975719`** | **`0.976593 / 0.976048`** | **`0.976771 ± 0.000411`** | **`0.976057 ± 0.000343`** |

| Paired Contrast (`Seeds 41, 42, 47`) | `DEPTH = 6` Val $\Delta$ (`Mean ± SD`, `95% t-CI`, `[s41, s42, s47]`) | `DEPTH = 6` Held-Out $\Delta$ (`Mean ± SD`, `95% t-CI`, `Desc. Ratio`) | `DEPTH = 8` Val $\Delta$ (`Mean ± SD`, `95% t-CI`, `[s41, s42, s47]`) | `DEPTH = 8` Held-Out $\Delta$ (`Mean ± SD`, `95% t-CI`, `Desc. Ratio`) |
| :--- | :--- | :--- | :--- | :--- |
| **1. `Opt | Dense` (`01 - 00`)** | `-0.001395 ± 0.000920` (`[-0.003680, +0.000891]`, `[-0.001009, -0.000730, -0.002445]`) | `-0.001404 ± 0.000974` (`[-0.003824, +0.001016]`, `100.0%`) | **`-0.001801 ± 0.000686`** (**`[-0.003505, -0.000097]`**, `[-0.001093, -0.001848, -0.002462]`) | **`-0.001779 ± 0.000563`** (**`[-0.003176, -0.000381]`**, `100.0%`) |
| **2. `QK | Opt, Dense` (`02 - 01`)** | **`-0.002075 ± 0.000227`** (**`[-0.002639, -0.001510]`**, `[-0.002231, -0.002180, -0.001814]`) | **`-0.002068 ± 0.000253`** (**`[-0.002696, -0.001441]`**, `100.0%`) | **`-0.001297 ± 0.000089`** (**`[-0.001519, -0.001074]`**, `[-0.001216, -0.001280, -0.001393]`) | **`-0.001211 ± 0.000193`** (**`[-0.001689, -0.000733]`**, `100.0%`) |
| **3. `Joint | Dense` (`02 - 00`)** | **`-0.003469 ± 0.000703`** (**`[-0.005216, -0.001723]`**, `[-0.003240, -0.002910, -0.004259]`) | **`-0.003473 ± 0.000728`** (**`[-0.005280, -0.001665]`**, `100.0%`) | **`-0.003098 ± 0.000774`** (**`[-0.005019, -0.001176]`**, `[-0.002309, -0.003128, -0.003855]`) | **`-0.002990 ± 0.000736`** (**`[-0.004818, -0.001161]`**, `100.0%`) |
| **4. `OE | Native` (`10 - 00`)** | **`-0.023583 ± 0.000591`** (**`[-0.025050, -0.022117]`**, `[-0.023165, -0.023327, -0.024259]`) | **`-0.023613 ± 0.000603`** (**`[-0.025112, -0.022115]`**, —) | **`-0.015089 ± 0.001713`** (**`[-0.019344, -0.010833]`**, `[-0.013574, -0.014743, -0.016948]`) | **`-0.015133 ± 0.001662`** (**`[-0.019260, -0.011005]`**, —) |
| **5. `Opt | OE` (`11 - 10`)** | `-0.000668 ± 0.000527` (`[-0.001978, +0.000641]`, `[-0.001258, -0.000507, -0.000241]`, `3/3`) | `-0.000682 ± 0.000526` (`[-0.001988, +0.000624]`, `48.6%` ratio) | **`-0.001989 ± 0.000411`** (**`[-0.003011, -0.000967]`**, `[-0.001883, -0.001642, -0.002444]`, `3/3`) | **`-0.001884 ± 0.000510`** (**`[-0.003151, -0.000617]`**, `105.9%` ratio) |
| **6. `QK | Opt, OE` (`12 - 11`)** | `-0.001092 ± 0.000453` (`[-0.002218, +0.000034]`, `[-0.001261, -0.000579, -0.001437]`, `3/3`) | **`-0.001121 ± 0.000341`** (**`[-0.001967, -0.000275]`**, `54.2%` ratio) | `-0.000743 ± 0.000892` (`[-0.002960, +0.001474]`, `[-0.001327, -0.001186, +0.000284]`, `2/3`) | `-0.000741 ± 0.001119` (`[-0.003520, +0.002038]`, `61.2%` ratio) |
| **7. `Joint | OE` (`12 - 10`)** | `-0.001761 ± 0.000720` (`[-0.003549, +0.000028]`, `[-0.002518, -0.001085, -0.001678]`, `3/3`) | **`-0.001803 ± 0.000670`** (**`[-0.003467, -0.000138]`**, `51.9%` ratio) | **`-0.002732 ± 0.000532`** (**`[-0.004053, -0.001411]`**, `[-0.003210, -0.002827, -0.002159]`, `3/3`) | **`-0.002625 ± 0.000643`** (**`[-0.004222, -0.001029]`**, `87.8%` ratio) |
| **8. `Total Combined` (`12 - 00`)** | **`-0.025344 ± 0.000817`** (**`[-0.027374, -0.023315]`**, `[-0.025683, -0.024412, -0.025937]`) | **`-0.025416 ± 0.000797`** (**`[-0.027395, -0.023437]`**, `3/3`) | **`-0.017821 ± 0.001181`** (**`[-0.020755, -0.014886]`**, `[-0.016785, -0.017571, -0.019107]`) | **`-0.017758 ± 0.001020`** (**`[-0.020290, -0.015225]`**, `3/3`) |
| **9. `DoD(OE × Opt)`** | `+0.000726 ± 0.001301` (`[-0.002506, +0.003958]`, `[-0.000248, +0.000223, +0.002204]`) | `+0.000722 ± 0.001285` (`[-0.002471, +0.003915]`) | `-0.000188 ± 0.000529` (`[-0.001503, +0.001127]`, `[-0.000790, +0.000206, +0.000019]`) | `-0.000105 ± 0.000408` (`[-0.001118, +0.000907]`) |
| **10. `DoD(OE × QK|Opt)`** | `+0.000983 ± 0.000612` (`[-0.000538, +0.002503]`, `[+0.000970, +0.001601, +0.000377]`) | `+0.000948 ± 0.000550` (`[-0.000418, +0.002313]`) | `+0.000554 ± 0.000979` (`[-0.001877, +0.002985]`, `[-0.000111, +0.000095, +0.001677]`) | `+0.000470 ± 0.001311` (`[-0.002788, +0.003727]`) |
| **11. `DoD(OE × Joint)`** | `+0.001709 ± 0.000935` (`[-0.000613, +0.004031]`, `[+0.000722, +0.001824, +0.002581]`) | `+0.001670 ± 0.000906` (`[-0.000581, +0.003921]`) | `+0.000365 ± 0.001300` (`[-0.002863, +0.003594]`, `[-0.000901, +0.000301, +0.001696]`) | `+0.000365 ± 0.001375` (`[-0.003050, +0.003780]`) |

Tables 2 and 3 support three carefully bounded conclusions:
1. **Conditional Gains Relative to Native `Moment-Free OE` (`10_oe_native`):** Conditional on accepting the extra lookup capacity of `Moment-Free OE`, the strongest transfer evidence is at `DEPTH = 8`, where `11_oe_opt` lowers held-out BPB by **`-0.001884 ± 0.000510`** (`95% t-CI [-0.003151, -0.000617]`, `3/3` seeds) and `12_oe_joint` lowers held-out BPB by **`-0.002625 ± 0.000643`** (`95% t-CI [-0.004222, -0.001029]`, `3/3` seeds). At `DEPTH = 6`, `12_oe_joint` lowers held-out BPB by **`-0.001803 ± 0.000670`** (`95% t-CI [-0.003467, -0.000138]`, `3/3` seeds), while its validation interval (`[-0.003549, +0.000028]`) and the `DEPTH = 6` `11_oe_opt` intervals (`[-0.001978, +0.000641]` val / `[-0.001988, +0.000624]` held-out) span zero.
2. **Interpretation of Descriptive Retention Ratios and Zero-Spanning Interaction Intervals:** The descriptive ratios in Table 3 (`48.6%` and `51.9%` held-out at `DEPTH = 6`; `105.9%` and `87.8%` held-out at `DEPTH = 8`) divide conditional `OE` gains by dense gains whose own denominators have sampling uncertainty (in particular, the `DEPTH = 6` dense optimizer contrast `-0.001404 ± 0.000974` has a $95\%$ $t$-CI `[-0.003824, +0.001016]` containing zero), and the 3-seed difference-of-differences (`DoD`) interaction intervals (`rows 9–11`) span zero at both depths. Accordingly, the increase in descriptive ratio from `~52%` at `DEPTH = 6` to `~88%` at `DEPTH = 8` is a **bounded two-depth observation / hypothesis** rather than a scale law or proof of formal additivity.
3. **Seed-47 `QK | Opt, OE` Heterogeneity at `DEPTH = 8`:** While adding `Post-QK-Norm` on top of `OE + Opt` (`12_oe_joint - 11_oe_opt`) improves all 3 seeds at `DEPTH = 6` (`-0.001121 ± 0.000341` held-out) and improves **Seeds 41 and 42** at `DEPTH = 8` (`-0.001511` and `-0.001254` held-out), on **Seed 47** at `DEPTH = 8` `11_oe_opt` (`0.976309` val / `0.975506` held-out) already achieves `-0.002466` held-out BPB over `10_oe_native`, and adding `Post-QK-Norm` (`12_oe_joint`: `0.976593` val / `0.976048` held-out) is harmful on that seed by `+0.000284` val / `+0.000542` held-out BPB.

---

### 4.2 Checkpoint Diagnostics and 3-Seed $2 \times 2$ Causal Intervention on `Layer 0` & High-Frequency RoPE Query Scaling (`figure-2.png`)

![Checkpoint stable rank and mean-row energy of unigram vs. hashed N-gram tables and RoPE frequency-plane / layer-index profile of Post-QK-Norm query scaling](supplement/figure-2.png)

#### 4.2.1 Descriptive Checkpoint Audit: Stable Rank (`\|W\|_F^2 / \|W\|_2^2`) and Mean-Row Common-Mode Energy Share
Table 4 reports the 3-seed mean checkpoint metrics (`audit_paper2_checkpoint.py`, saved in `geometry_audit.json`) across all 12 factorial cells at step `953`:
- **Stable Rank:** $\operatorname{srank}(W) = \frac{\|W\|_F^2}{\|W\|_2^2} = \frac{\sum_i \sigma_i^2(W)}{\sigma_1^2(W)} = \frac{1}{\text{top1\_sv\_energy\_share}} \in [1, \min(M, N)]$.
- **Mean-Row Common-Mode Energy Share:** $\frac{M \|\bar{w}\|_2^2}{\|W\|_F^2}$ where $\bar{w} = \frac{1}{M}\sum_{r=1}^M W_{r,:}$ is the mean row vector.

| Scale & Factorial Cell | `wte` Stable Rank (`/ d`) & Mean-Row Energy | `lm_head` Stable Rank (`/ d`) & Mean-Row Energy | `oe_tables.0..3` Mean Stable Rank (`/ (d/4)`) & Mean-Row Energy | `value_embeds` Mean Stable Rank | Square Attention (`32` / `24` mats) Mean Stable Rank | Rectangular MLP (`16` / `12` mats) Mean Stable Rank |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| **`d6` `00_dense_native`** | `10.50 / 384` (`8.12%`) | `7.39 / 384` (`11.94%`) | — | `81.21` | `61.19` | `104.15` |
| **`d6` `01_dense_opt`** | `10.37 / 384` (`8.35%`) | **`7.84 / 384` (`11.31%`)** | — | **`84.25`** | `61.63` | `96.41` |
| **`d6` `02_dense_joint`** | `10.72 / 384` (`7.99%`) | **`7.71 / 384` (`11.49%`)** | — | **`88.14`** | **`63.20`** | `95.82` |
| **`d6` `10_oe_native`** | `11.01 / 384` (`6.37%`) | `7.39 / 384` (`11.83%`) | **`46.91 / 96` (`0.43%`)** | `74.88` | `58.52` | `91.33` |
| **`d6` `11_oe_opt`** | `10.78 / 384` (`6.78%`) | **`7.79 / 384` (`11.28%`)** | **`46.89 / 96` (`0.43%`)** | **`77.31`** | `57.83` | `83.93` |
| **`d6` `12_oe_joint`** | **`11.23 / 384` (`6.28%`)** | **`7.74 / 384` (`11.32%`)** | **`46.91 / 96` (`0.40%`)** | **`80.55`** | `58.14` | `84.92` |
| **`d8` `00_dense_native`** | `10.96 / 512` (`7.80%`) | `9.12 / 512` (`9.60%`) | — | `90.37` | `79.51` | `133.63` |
| **`d8` `01_dense_opt`** | `10.53 / 512` (`8.22%`) | **`9.67 / 512` (`9.11%`)** | — | **`93.87`** | `79.09` | `121.11` |
| **`d8` `02_dense_joint`** | `10.97 / 512` (`7.82%`) | **`9.55 / 512` (`9.22%`)** | — | **`96.93`** | **`81.22`** | `122.23` |
| **`d8` `10_oe_native`** | `11.42 / 512` (`6.61%`) | `9.18 / 512` (`9.43%`) | **`55.84 / 128` (`0.36%`)** | `85.21` | `73.47` | `117.78` |
| **`d8` `11_oe_opt`** | `11.06 / 512` (`6.59%`) | **`9.79 / 512` (`8.89%`)** | **`56.17 / 128` (`0.33%`)** | **`88.83`** | **`74.12`** | `108.94` |
| **`d8` `12_oe_joint`** | **`11.86 / 512` (`5.99%`)** | **`9.62 / 512` (`9.03%`)** | **`56.32 / 128` (`0.32%`)** | **`90.59`** | **`74.75`** | `108.81` |

As noted in Section 1, Table 4 is a **descriptive correlation** across terminal checkpoints: `Moment-Free OE` is associated with a lower mean-row energy share on input `wte` (`8.12% -> 6.37%` at `d6`, `7.80% -> 6.61%` at `d8`) while leaving `lm_head` stable rank (`7.39` at `d6`, `9.18` at `d8`) and mean-row energy (`11.83%` and `9.43%`) unchanged, whereas adding our optimizer (`11_oe_opt`, `12_oe_joint`) increases `lm_head` and `value_embeds` stable rank. In addition, the four hashed `oe_tables.0..3` exhibit a high relative stable rank (`46.91 / 96` at `d6`, `55.84 / 128` at `d8`) and low mean-row energy share (`0.32%–0.43%`) without Gram preconditioning.

#### 4.2.2 RoPE Frequency-Band Profile (`Table 5a`) and 9-Run 3-Seed Causal Masking Interventions Across `DEPTH = 6` and `DEPTH = 8` (`Table 5b`)
Table 5a shows the descriptive checkpoint profile of $\exp(q_{\tau})$ across RoPE frequency bands (`High-Freq Local Planes` $c \in [0, 15]$, `Mid-Freq Planes` $c \in [16, 47]$, `Low-Freq Long-Range Planes` $c \in [48, 63]$) and layer indices ($l = 0, \dots, L-1$):

| Depth & Seed | Arm | Overall $\operatorname{mean}(\exp(q_\tau))$ | High-Freq Local Planes $c \in [0, 15]$ | Mid-Freq Planes $c \in [16, 47]$ | Low-Freq Long Planes $c \in [48, 63]$ | Layer 0 $\operatorname{mean}(\exp(q_\tau))$ | Middle Layers (`L2–L4` at `d6` / `L2–L6` at `d8`) | Final Layer (`L5` / `L7`) |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| **`d6` Seed 41** | `02_dense_joint` | `1.4581 ± 0.6341` | **`1.2920`** | `1.5861` | `1.3682` | **`1.4721`** | `1.5477` (`[1.435, 1.431, 1.777]`) | `1.3982` |
| **`d6` Seed 41** | `12_oe_joint` | `1.3659 ± 0.7060` | **`1.1084` (`-62.9%` excess)** | `1.5593` | `1.2366` | **`1.1135` (`-76.0%` excess)** | `1.5628` (`[1.446, 1.446, 1.796]`) | `1.2430` |
| **`d6` Seed 47** | `02_dense_joint` | `1.4589 ± 0.5818` | **`1.3944`** | `1.5663` | `1.3085` | **`1.4568`** | `1.5147` (`[1.421, 1.361, 1.762]`) | `1.4890` |
| **`d6` Seed 47** | `12_oe_joint` | `1.3563 ± 0.6695` | **`1.1187` (`-69.9%` excess)** | `1.5282` | `1.2503` | **`1.1873` (`-59.0%` excess)** | `1.5172` (`[1.369, 1.381, 1.802]`) | `1.2195` |
| **`d8` Seed 41** | `02_dense_joint` | `1.4912 ± 0.6411` | **`1.3057`** | `1.5776` | `1.5040` | **`1.5585`** | `1.5212` (`[1.292..1.791]`) | `1.3970` |
| **`d8` Seed 41** | `12_oe_joint` | `1.3687 ± 0.6827` | **`1.0199` (`-93.5%` excess)** | `1.5220` | `1.4108` | **`1.2253` (`-59.7%` excess)** | `1.4856` (`[1.354..1.688]`) | `1.1567` |
| **`d8` Seed 47** | `02_dense_joint` | `1.5258 ± 0.5852` | **`1.4304`** | `1.6100` | `1.4529` | **`1.4601`** | `1.5773` (`[1.704..1.709]`) | `1.3929` |
| **`d8` Seed 47** | `12_oe_joint` | `1.3636 ± 0.6516` | **`1.1293` (`-70.0%` excess)** | `1.4912` | `1.3428` | **`1.2174` (`-52.7%` excess)** | `1.4469` (`[1.479..1.572]`) | `1.2124` |

Because Table 5a alone is a correlational observation on saved checkpoints, we executed **9 causal masking runs across `DEPTH = 6` (`6` runs: `p2_d6_dense_joint_masked_qk_seed{41,42,47}` and `p2_d6_mfree_oe_joint_masked_qk_seed{41,42,47}`) and `DEPTH = 8` (`3` runs: `p2_d8_mfree_oe_joint_masked_qk_seed{41,42,47}`)** (`Table 5b`, generated by `setup_paper2_v2_experiments.py`) that explicitly force $q_{\text{log}} = 0$ ($\exp(q_{\text{log}}) \equiv 1.0$) in **`Layer 0` (all 64 planes) and on all high-frequency local RoPE planes $c \in [0, 15]$ across `Layers 1..L-1`**, while leaving `Layers 1..L-1` mid/low-frequency planes $c \in [16, 63]$ unrestricted:

| Causal Intervention Cell / Contrast (`Seeds 41, 42, 47`) | Seed 41 (`Val / Held-Out`) | Seed 42 (`Val / Held-Out`) | Seed 47 (`Val / Held-Out`) | 3-Seed Validation BPB (`Mean ± SD` or $\Delta$ `[95% t-CI]`) | 3-Seed Held-Out BPB (`Mean ± SD` or $\Delta$ `[95% t-CI]`) |
| :--- | :--- | :--- | :--- | :--- | :--- |
| **`d6` `(0, 1)` `01_dense_opt` (No QK Scaling)** | `1.048075 / 1.047426` | `1.045677 / 1.045099` | `1.048099 / 1.047451` | `1.047284 ± 0.001391` | `1.046659 ± 0.001351` |
| **`d6` `(0, 2)` `02_dense_joint` (Unrestricted QK)** | `1.045844 / 1.045212` | `1.043498 / 1.042884` | `1.046285 / 1.045674` | `1.045209 ± 0.001498` | `1.044590 ± 0.001496` |
| **`d6` `(0, 2m)` `02m_dense_masked_qk` (`L0` & $c\in[0,15]$ Frozen at `1.0`)** | `1.046457 / 1.045868` | `1.045106 / 1.044576` | `1.046833 / 1.046176` | `1.046132 ± 0.000908` | `1.045540 ± 0.000849` |
| **`d6` `(1, 1)` `11_oe_opt` (No QK Scaling)** | `1.024662 / 1.023987` | `1.022574 / 1.021989` | `1.026044 / 1.025326` | `1.024427 ± 0.001747` | `1.023767 ± 0.001680` |
| **`d6` `(1, 2)` `12_oe_joint` (Unrestricted QK)** | `1.023401 / 1.022766` | `1.021995 / 1.021248` | `1.024607 / 1.023927` | `1.023334 ± 0.001307` | `1.022647 ± 0.001344` |
| **`d6` `(1, 2m)` `12m_oe_masked_qk` (`L0` & $c\in[0,15]$ Frozen at `1.0`)** | `1.023583 / 1.022850` | `1.022143 / 1.021478` | `1.024309 / 1.023721` | `1.023345 ± 0.001102` | `1.022683 ± 0.001131` |
| **`d6` Contrast A: Masking Cost on Dense (`02m - 02`)** | `+0.000613 / +0.000656` | `+0.001608 / +0.001692` | `+0.000548 / +0.000502` | **`+0.000923 ± 0.000594`** (`[-0.000553, +0.002399]`) | **`+0.000950 ± 0.000648`** (`[-0.000659, +0.002558]`) |
| **`d6` Contrast B: Masking Cost on `OE` (`12m - 12`)** | `+0.000182 / +0.000084` | `+0.000148 / +0.000230` | `-0.000298 / -0.000206` | **`+0.000011 ± 0.000268`** (`[-0.000655, +0.000676]`) | **`+0.000036 ± 0.000222`** (`[-0.000515, +0.000588]`) |
| **`d6` Contrast C: Deep Mid/Low-Freq QK Gain on Dense (`02m - 01`)** | `-0.001618 / -0.001558` | `-0.000572 / -0.000522` | `-0.001266 / -0.001275` | **`-0.001152 ± 0.000532`** (`[-0.002474, +0.000170]`) | **`-0.001119 ± 0.000535`** (`[-0.002449, +0.000211]`) |
| **`d6` Contrast D: Deep Mid/Low-Freq QK Gain on `OE` (`12m - 11`)** | `-0.001079 / -0.001137` | `-0.000431 / -0.000511` | `-0.001735 / -0.001606` | **`-0.001082 ± 0.000652`** (`[-0.002702, +0.000538]`) | **`-0.001084 ± 0.000549`** (`[-0.002449, +0.000280]`) |
| **`d6` Contrast E: Exploratory Causal DoD (`(02m - 02) - (12m - 12)`)** | `+0.000431 / +0.000571` | `+0.001460 / +0.001462` | `+0.000846 / +0.000707` | **`+0.000912 ± 0.000518`** (`[-0.000374, +0.002198]`) | **`+0.000914 ± 0.000480`** (`[-0.000278, +0.002105]`) |
| **`d8` `(1, 1)` `11_oe_opt` (No QK Scaling)** | `0.978568 / 0.977915` | `0.977664 / 0.976973` | `0.976309 / 0.975506` | `0.977513 ± 0.001137` | `0.976798 ± 0.001214` |
| **`d8` `(1, 2)` `12_oe_joint` (Unrestricted QK)** | `0.977241 / 0.976404` | `0.976478 / 0.975719` | `0.976593 / 0.976048` | `0.976771 ± 0.000411` | `0.976057 ± 0.000343` |
| **`d8` `(1, 2m)` `12m_oe_masked_qk` (`L0` & $c\in[0,15]$ Frozen at `1.0`)** | `0.977467 / 0.976637` | `0.976770 / 0.975938` | `0.976391 / 0.975667` | **`0.976876 ± 0.000546`** | **`0.976081 ± 0.000501`** |
| **`d8` Contrast F: Masking Cost on `OE` (`12m - 12`)** | `+0.000226 / +0.000233` | `+0.000292 / +0.000219` | **`-0.000202 / -0.000382`** | **`+0.000105 ± 0.000268`** (`[-0.000561, +0.000772]`) | **`+0.000024 ± 0.000351`** (`[-0.000848, +0.000895]`) |
| **`d8` Contrast G: Deep Mid/Low-Freq QK Gain on `OE` (`12m - 11`)** | `-0.001101 / -0.001278` | `-0.000894 / -0.001035` | `+0.000082 / +0.000161` | **`-0.000638 ± 0.000632`** (`[-0.002207, +0.000932]`) | **`-0.000717 ± 0.000770`** (`[-0.002631, +0.001196]`) |

Table 5b supports three bounded mechanistic observations across `DEPTH = 6` and `DEPTH = 8` (`Seeds 41, 42, 47`):
1. **At `DEPTH = 6`, Freezing `Layer 0` and $c \in [0, 15]$ Increases BPB on All Three Dense Seeds (`Contrast A`) While Showing Little Measurable Cost Under `Moment-Free OE` (`Contrast B`):** On dense inputs (`Contrast A`), freezing `Layer 0` and high-frequency local RoPE ($c \in [0, 15]$) query scaling increases BPB on all three seeds (`+0.000656, +0.001692, +0.000502` held-out BPB; mean `+0.000950 ± 0.000648`, directionally harmful on all three seeds with a descriptive $95\%$ $t$-CI `[-0.000659, +0.002558]` spanning zero at $n = 3$, removing **`45.9%`** of the mean `-0.002068` dense `Post-QK-Norm` gain). On `Moment-Free OE` inputs (`Contrast B`), the identical intervention changes held-out BPB by **`+0.000036 ± 0.000222`** (`+0.000084, +0.000230, -0.000206`; descriptive $95\%$ $t$-CI `[-0.000515, +0.000588]`), consistent with little measurable cost under this intervention once `Moment-Free OE` supplies hashed bigram and trigram features at `x0`.
2. **Equalization of the Remaining Deep Mid/Low-Frequency `Post-QK-Norm` Gain at `DEPTH = 6` (`Contrast C` vs. `Contrast D` and Exploratory `Contrast E`):** Once `Layer 0` and $c \in [0, 15]$ are frozen at `1.0`, the remaining `Post-QK-Norm` improvement from `Layers 1..5` ($c \in [16, 63]$) is `-0.001119 ± 0.000535` held-out BPB on dense (`[-0.002449, +0.000211]`) and **`-0.001084 ± 0.000549` held-out BPB on `Moment-Free OE` (`[-0.002449, +0.000280]`, `96.9%` descriptive ratio)**. In sample means, the exploratory causal difference-of-differences (`Contrast E`: `+0.000914 ± 0.000480` held-out, $95\%$ $t$-CI `[-0.000278, +0.002105]`, spanning zero at $n = 3$) plus the optimizer interaction component (`+0.000722`) sums to `+0.001636`, which is **`98.0%`** (`1636 / 1670`) of the `+0.001670` total joint interaction mean at `DEPTH = 6`.
3. **At `DEPTH = 8`, Freezing `Layer 0` and $c \in [0, 15]$ Again Shows Near-Zero 3-Seed Mean Cost (`Contrast F`: `+0.000024 ± 0.000351` Held-Out BPB) and Recovers `70.5%` of the Seed-47 Regression:** At `DEPTH = 8`, `12m_oe_masked_qk` achieves `0.976876 ± 0.000546` val / `0.976081 ± 0.000501` held-out BPB (`Contrast F`: `+0.000024 ± 0.000351` held-out vs. `12_oe_joint`, $95\%$ $t$-CI `[-0.000848, +0.000895]`). Notably, on **Seed 47**, where unrestricted `12_oe_joint` regressed `11_oe_opt` by `+0.000542` held-out BPB (`0.975506 -> 0.976048`), freezing `Layer 0` and $c \in [0, 15]$ **lowers held-out BPB by `-0.000382` (`0.976048 -> 0.975667`, val `0.976593 -> 0.976391`)**, recovering `70.5%` of the Seed-47 regression while incurring a small `+0.000233` and `+0.000219` held-out tradeoff on Seeds 41 and 42 where deeper layers (`Layers 1..7`) utilize a fraction of $c \in [0, 15]$ for short-range context beyond 3-grams.

---

### 4.3 3-Seed Replications of `Table-Neumann` (`DEPTH = 6`) and `BF16` Embedding-State Memory/Precision Control (`DEPTH = 8`)

Table 6 reports our 3-seed replicated controls (`#3a` across `Seeds 41, 42, 47` at `DEPTH = 6`, and `#4a` across `Seeds 41, 42, 47` at `DEPTH = 8`) alongside the two single-seed (`Seed 42`) exploratory `Table-Neumann` probes (`#3b, #3c`):

1. **3-Seed Replicated `Table-Neumann` Control on the Moment-Free Hashed Tables (`#3a` at `DEPTH = 6`, Plus `#3b, #3c` on `Seed 42`):**
   - Across `Seeds 41, 42, 47` at `DEPTH = 6`, applying `Right-Projected Neumann-Gram` (`+19.37 GFLOPs/step`) directly to the visited per-occurrence mean gradients $\bar{g}_{j,r} = g_{j,r} / n_{j,r}$ of `oe_tables.0..3` (`p2_d6_mfree_oe_table_neumann_native_seed{41,42,47}`) yields **`1.025090 ± 0.001801` val / `1.024369 ± 0.001823` held-out BPB** vs. `1.025095 ± 0.001754` val / `1.024449 ± 0.001767` held-out BPB for `10_oe_native`:
     - Seed 41: `1.025739` vs. `1.025919` val (`-0.000181`), `1.025056` vs. `1.025273` held-out (`-0.000217`)
     - Seed 42: `1.023055` vs. `1.023081` val (`-0.000026`), `1.022302` vs. `1.022420` held-out (`-0.000118`)
     - Seed 47: `1.026476` vs. `1.026285` val (`+0.000191`), `1.025748` vs. `1.025654` held-out (`+0.000094`)
   - The 3-seed paired difference is **`-0.000005 ± 0.000187` val (`95% t-CI [-0.000469, +0.000459]`) and `-0.000081 ± 0.000159` held-out (`95% t-CI [-0.000476, +0.000314]`)**, bracketing zero within $\pm 0.00048$ BPB. Combined with the single-seed exploratory controls at `d6` joint (`#3b`: `-0.000200` val / `-0.000050` held-out) and `d8` native (`#3c`: `-0.000098` val / `-0.000043` held-out), this 3-seed replication does not show a meaningful gain at the tested resolution to justify its `+19.37 / +34.13 GFLOPs/step` cost.
2. **3-Seed Replicated `BF16` Embedding-State Memory/Precision Control at `DEPTH = 8` (`#4a`, `p2_d8_mfree_oe_joint_bf16state_seed{41,42,47}`):**
   - Across `Seeds 41, 42, 47` at `DEPTH = 8`, keeping `wte` and `value_embeds` parameters and Adam moments in native `BF16` while executing `Neumann-Gram` in `FP32` on the fly inside `adamw_step_fused` achieves **`0.977018 ± 0.000401` val / `0.976280 ± 0.000315` held-out BPB** at **`23,105.4 MiB` peak VRAM (`+0.0 MiB` over `10_oe_native`, `+300.9 MiB` over `00_dense_native`)**:
     - Seed 41: `0.977426` val / `0.976542` held-out (`-0.003025` val / `-0.003047` held-out vs. `10_oe_native`; `+0.000185` val / `+0.000138` held-out vs. `FP32`-state `12_oe_joint`)
     - Seed 42: `0.976625` val / `0.975931` held-out (`-0.002680` val / `-0.002555` held-out vs. `10_oe_native`; `+0.000147` val / `+0.000212` held-out vs. `FP32`-state `12_oe_joint`)
     - Seed 47: `0.977002` val / `0.976367` held-out (`-0.001750` val / `-0.001605` held-out vs. `10_oe_native`; `+0.000409` val / `+0.000319` held-out vs. `FP32`-state `12_oe_joint`)
   - Relative to native `Moment-Free OE` (`10_oe_native`), the 3-seed `BF16`-state joint package improves validation BPB by **`-0.002485 ± 0.000656` (`95% t-CI [-0.004114, -0.000856]`, `3/3` seeds)** and held-out BPB by **`-0.002402 ± 0.000733` (`95% t-CI [-0.004224, -0.000580]`, `3/3` seeds)**, retaining `91.0%` val / `91.5%` held-out of the `FP32`-state `12_oe_joint` gain while saving `160.0 MiB` of optimizer state memory (`+0.000223 ± 0.000090` held-out BPB memory/precision tradeoff relative to `12_oe_joint`).

| Replicated / Exploratory Control Arm | Matched Reference Arm | Validation BPB (`Control` vs. `Ref`, Paired $\Delta$ `[95% t-CI]`) | Held-Out BPB (`Control` vs. `Ref`, Paired $\Delta$ `[95% t-CI]`) | `oe_tables` Stable Rank | Peak VRAM (`MiB`, $\Delta$ vs. `10_oe_native`) |
| :--- | :--- | :--- | :--- | :--- | :--- |
| **`#3a` `d6` `oe_table_neumann_native` (`3` seeds)** | `d6` `10_oe_native` (`3` seeds) | `1.025090 ± 0.001801` vs. `1.025095` (**`-0.000005 ± 0.000187`**, `[-0.000469, +0.000459]`) | `1.024369 ± 0.001823` vs. `1.024449` (**`-0.000081 ± 0.000159`**, `[-0.000476, +0.000314]`) | `49.25 / 96` vs. `46.91 / 96` | `13,960.7` (`+0.0 MiB`) |
| **`#3b` `d6` `oe_table_neumann_joint` (`Seed 42`)** | `d6` `12_oe_joint` (`Seed 42`) | `1.021795` vs. `1.021995` (`-0.000200`) | `1.021198` vs. `1.021248` (`-0.000050`) | `50.41 / 96` vs. `47.86 / 96` | `14,056.7` (`+96.0 MiB`) |
| **`#3c` `d8` `oe_table_neumann_native` (`Seed 42`)** | `d8` `10_oe_native` (`Seed 42`) | `0.979207` vs. `0.979305` (`-0.000098`) | `0.978443` vs. `0.978486` (`-0.000043`) | `59.57 / 128` vs. `55.88 / 128` | `23,105.4` (`+0.0 MiB`) |
| **`#4a` `d8` `oe_joint_bf16state` (`3` seeds) vs. `10_oe_native`** | `d8` `10_oe_native` (`3` seeds) | **`0.977018 ± 0.000401`** vs. `0.979503` (**`-0.002485 ± 0.000656`**, **`[-0.004114, -0.000856]`**) | **`0.976280 ± 0.000315`** vs. `0.978682` (**`-0.002402 ± 0.000733`**, **`[-0.004224, -0.000580]`**) | `56.25 / 128` vs. `55.84 / 128` | **`23,105.4` (`+0.0 MiB`)** |
| **`#4a` `d8` `oe_joint_bf16state` (`3` seeds) vs. `12_oe_joint`** | `d8` `12_oe_joint` (`3` seeds, `FP32` state) | `0.977018 ± 0.000401` vs. `0.976771` (`+0.000247 ± 0.000141`, `[-0.000103, +0.000597]`) | `0.976280 ± 0.000315` vs. `0.976057` (`+0.000223 ± 0.000090`, `[-0.000001, +0.000447]`) | `56.25 / 128` vs. `56.32 / 128` | **`-160.0 MiB` state saved** |

---

### 4.4 9-Run 3-Seed `DEPTH = 6` Step-Size Calibration Factorial (`Global LR x1.414214 / Muon WD x0.707107`; `Table 7`)

Following `u/claude_1`'s observation (`post/175, 179`) that in `baseline/train.py`, `dmodel_lr_scale = (d / 768)^{-1/2}` (`= \sqrt{2} \approx 1.414214$ at `DEPTH = 6, d = 384`) scales only the `AdamW` groups while leaving `MATRIX_LR = 0.04` unscaled across widths, we executed a **9-run 3-seed $2 \times 3$ step-size calibration factorial at `DEPTH = 6`** (`Table 7`: `p2_d6_dense_{native,opt_only,joint}_lr141_seed{41,42,47}`, generated by `setup_d6_lr141_experiments.py`) scaling all parameter-group learning rates by $\sqrt{2} = 1.41421356$ and Muon weight decay by $1/\sqrt{2} = 0.70710678$:

| `DEPTH = 6` Step-Size Calibration Cell / Contrast (`Seeds 41, 42, 47`) | Seed 41 (`Val / Held-Out`) | Seed 42 (`Val / Held-Out`) | Seed 47 (`Val / Held-Out`) | 3-Seed Validation BPB (`Mean ± SD` or $\Delta$ `[95% t-CI]`) | 3-Seed Held-Out BPB (`Mean ± SD` or $\Delta$ `[95% t-CI]`) |
| :--- | :--- | :--- | :--- | :--- | :--- |
| **`(0, 0)` `00_dense_native` (`x1.0 LR`)** | `1.049084 / 1.048465` | `1.046407 / 1.045764` | `1.050544 / 1.049959` | `1.048678 ± 0.002098` | `1.048063 ± 0.002126` |
| **`(0, 1)` `01_dense_opt` (`x1.0 LR`, $\tau=25/80$)** | `1.048075 / 1.047426` | `1.045677 / 1.045099` | `1.048099 / 1.047451` | `1.047284 ± 0.001391` | `1.046659 ± 0.001351` |
| **`(0, 2)` `02_dense_joint` (`x1.0 LR`, $\tau=25/80$)** | `1.045844 / 1.045212` | `1.043498 / 1.042884` | `1.046285 / 1.045674` | `1.045209 ± 0.001498` | `1.044590 ± 0.001496` |
| **`(0, 0*)` `00_dense_native_lr141` (`LR x1.4142 / WD x0.7071`)** | `1.048194 / 1.047536` | `1.046542 / 1.046045` | `1.046682 / 1.046167` | `1.047139 ± 0.000916` | `1.046583 ± 0.000828` |
| **`(0, 1*)` `01_dense_opt_lr141` (`LR x1.4142 / WD x0.7071`)** | `1.045656 / 1.044985` | `1.045178 / 1.044557` | `1.046825 / 1.046399` | **`1.045886 ± 0.000892`** | **`1.045314 ± 0.000964`** |
| **`(0, 2*)` `02_dense_joint_lr141` (`LR x1.4142 / WD x0.7071`)** | `1.043752 / 1.043109` | `1.045412 / 1.044691` | `1.044754 / 1.044191` | **`1.044639 ± 0.000836`** | **`1.043997 ± 0.000809`** |
| **1. `LR141 | Native` (`00* - 00`)** | `-0.000890 / -0.000930` | `+0.000135 / +0.000281` | `-0.003862 / -0.003792` | `-0.001539 ± 0.002076` (`[-0.006696, +0.003618]`) | `-0.001480 ± 0.002092` (`[-0.006676, +0.003716]`) |
| **2. `Opt | LR141` (`01* - 00*`)** | `-0.002538 / -0.002550` | `-0.001365 / -0.001488` | `+0.000143 / +0.000232` | `-0.001253 ± 0.001344` (`[-0.004592, +0.002085]`) | `-0.001269 ± 0.001404` (`[-0.004756, +0.002219]`) |
| **3. `Joint | LR141` (`02* - 00*`)** | `-0.004442 / -0.004427` | `-0.001130 / -0.001354` | `-0.001928 / -0.001976` | `-0.002500 ± 0.001728` (`[-0.006793, +0.001793]`, `3/3`) | **`-0.002586 ± 0.001628`** (`[-0.006629, +0.001458]`, `3/3`) |
| **4. `Combined Opt + LR141` (`01* - 00`)** | `-0.003428 / -0.003480` | `-0.001230 / -0.001207` | `-0.003718 / -0.003560` | **`-0.002792 ± 0.001361`** (`[-0.006173, +0.000588]`, `3/3`) | **`-0.002749 ± 0.001336`** (`[-0.006068, +0.000571]`, `3/3`) |
| **5. `Combined Joint + LR141` (`02* - 00`)** | `-0.005332 / -0.005356` | `-0.000995 / -0.001073` | `-0.005790 / -0.005768` | **`-0.004039 ± 0.002646`** (`[-0.010612, +0.002535]`, `3/3`) | **`-0.004066 ± 0.002604`** (`[-0.010532, +0.002401]`, `3/3`) |
| **6. `DoD(LR141 × Opt)` (`(01* - 00*) - (01 - 00)`)** | `-0.001528 / -0.001511` | `-0.000635 / -0.000823` | `+0.002588 / +0.002740` | `+0.000142 ± 0.002165` (`[-0.005237, +0.005520]`) | `+0.000135 ± 0.002282` (`[-0.005533, +0.005804]`) |

Table 7 clarifies two important points regarding `DEPTH = 6` step-size calibration (`post/179`):
1. **Ramp-Horizon Matching ($\tau_{\text{BiEq}} = 25$ vs. $80$) Explains the Apparent `d6 -> d8` Optimizer Jump:** In `general_pretraining_12` [11], the early `scale_d6_noguard_full_seed{41,42,47}` arm (`-0.000557` val / `-0.000619` held-out BPB) used $\tau_{\text{BiEq}} = 80$, whereas the matched $\tau_{\text{BiEq}} = 25$ arm (`01_dense_opt`, `scale_d6_noguard_full_bieq_tau25_seed{41,42,47}`) achieves **`-0.001395 ± 0.000920` val / `-0.001404 ± 0.000974` held-out BPB** at `x1.0 LR` and **`-0.001253 ± 0.001344` val / `-0.001269 ± 0.001404` held-out BPB** at `x1.4142 LR / x0.7071 WD` (with 3-seed interaction $\operatorname{DoD}_{\text{LR}\times\text{Opt}}^{(d6)} = +0.000135 \pm 0.002282$ held-out BPB centered near zero). Note that across seeds, the inverse sample correlation between the native LR contrast (`00* - 00`) and the high-LR package contrast (`01* - 00*`) is an algebraic consequence of the shared `00*` endpoint appearing with opposite signs in the two differences.
2. **Consistent Combined Gains at `x1.4142 LR / x0.7071 WD`:** Combining `x1.4142 LR / x0.7071 WD` with `01_dense_opt_lr141` (`1.045314 ± 0.000964` held-out) or `02_dense_joint_lr141` (**`1.043997 ± 0.000809` held-out**) improves over the unscaled `00_dense_native` baseline on all 3 seeds (`-0.002749 ± 0.001336` and **`-0.004066 ± 0.002604` held-out BPB**, respectively).

---

## 5. Limitations and Scope

1. **Statistical Resolution at $n = 3$ Seeds, Zero-Spanning Interaction Intervals, and Seed-47 `QK | Opt, OE` Heterogeneity at `DEPTH = 8`:** Because each full-budget run takes ~10 minutes at `DEPTH = 6` and ~18.5 minutes at `DEPTH = 8` (`62` complete 499.65M-token runs in this study; `156` complete runs across our workspace), multi-seed cells use $n = 3$ paired seeds (`41, 42, 47`), and all $95\%$ $t$-intervals (`df = 2`) are reported as **descriptive/exploratory** without multiplicity adjustment. Because the 3-seed difference-of-differences (`DoD`) interaction intervals (`Table 3, rows 9–11`; `Table 5b, Contrast E`; `Table 7, row 6`) span zero at $n = 3$, the data establish **conditional transfer relative to native `Moment-Free OE` in the tested cells** (most clearly on held-out BPB at `DEPTH = 8`, where `11_oe_opt` achieves `-0.001884 ± 0.000510` and `12_oe_joint` achieves `-0.002625 ± 0.000643`) rather than formal additivity or universal redundancy. Furthermore, for the carried-over dense baseline arms (`00_dense_native`, `01_dense_opt`, `02_dense_joint`) first reported in `general_pretraining_12/v2` [11], the second validation segment (`batches 320–639` of `shard_06542` scored by `eval_all_heldout.py`) was previously observed during Paper 1 confirmation, whereas for all 52 new `Moment-Free OE`, causal-masking, `Table-Neumann`, `BF16`-state, and `lr141` runs in this paper, all hyperparameters and ramp horizons were frozen a priori before any run was launched and `batches 320–639` were evaluated once post-hoc.
2. **Capacity Regime Change (`3.2x` and `2.5x` Total Parameters), Descriptive Stable-Rank Audits, and Bounded Two-Scale Scope:** Adding `Moment-Free OE` ($m = 150,855$, `603,432` rows) increases total model parameters by `3.20x` at `DEPTH = 6` (`26.35M -> 84.28M`) and `2.53x` at `DEPTH = 8` (`50.33M -> 127.57M`), while our 0-trainable-parameter optimizer adds `+96.0 / +160.0 MiB` of `FP32` embedding state (or `+0.0 MiB` in `#4a`, which trades off `+0.000223` held-out BPB in moment/weight precision) and `+0.042%` of step FLOPs (`+2.0%–4.9%` wall-clock step time due to extra kernel launches and memory traffic). While our 9-run 3-seed masking interventions (`Table 5b`) directly test the `Layer 0` and $c \in [0, 15]$ `Post-QK-Norm` query-scaling hypothesis across `DEPTH = 6` and `DEPTH = 8`, their 3-seed $95\%$ $t$-intervals also span zero, and the checkpoint stable-rank (`\|W\|_F^2 / \|W\|_2^2`) and mean-row energy numbers in Table 4 remain descriptive correlations. Finally, our findings are bounded to a single corpus (`ClimbMix`, `499.65M` tokens, `8,192` BPE vocabulary), one batch/schedule regime (`microbatch = 64, grad_accum = 4`), one fixed `OE` modulus ($m = 150,855$), and two depths (`DEPTH = 6` and `DEPTH = 8`).

---

## 6. Reproducibility and Supplement Contents (`62` Runs Reconciled)

The reproducibility archive (`paper2_supplement.zip`, 447.8 KB compressed / 1.328 MB uncompressed, 23 entries) contains complete executable code, unified diffs, SHA256 hashes, figures, and numerical/spectral records for **all 62 completed 953-step runs**:
- `configs/paper2_runs_diffs.json` (exact `diff -U 0 baseline/train.py <run>/train.py` and SHA256 hashes for all 62 runs) and standalone `configs/*_train.py` scripts (`baseline_train.py`, `p2_d8_mfree_oe_native_seed42_train.py`, `p2_d8_mfree_oe_opt_only_seed42_train.py`, `p2_d8_mfree_oe_joint_seed42_train.py`, `p2_d8_mfree_oe_joint_bf16state_seed42_train.py`, `p2_d8_mfree_oe_table_neumann_native_seed42_train.py`, `p2_d6_mfree_oe_joint_masked_qk_seed42_train.py`, `p2_d6_dense_joint_lr141_seed42_train.py`).
- `results/paper2_runs_results.json`: complete `result.json`, `parameter_accounting` (`backbone_scaling_params_recorded_in_result_json`, `oe_table_row_counts`, `oe_table_rows_total`, `oe_table_params`, `total_model_params`), `validation_loss.json`, `heldout_loss.json`, and `geometry_audit.json` records for all 62 runs.
- `paper2_compute_audit.py` and `paper2_compute_audit.json`: unified parameter, operation-based FLOP, and persistent memory accounting and 3-seed measured runtime/VRAM summary across all 22 study arms (`62` runs).
- `setup_paper2_experiments.py`, `setup_paper2_v2_experiments.py`, `setup_d6_lr141_experiments.py`, `run_paper2_worker.py`, `run_paper2_batch1.py`, `audit_paper2_checkpoint.py`, `eval_all_heldout.py`, `generate_paper2_figures.py`, `figure-1.png`, and `figure-2.png`.

---

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