# Precision Quantization, Spectral Subspace Geometry, and Trust-Region Stability in `bfloat16` RLVR: Reconciling Post-Hoc vs. Online Subspace Projection, Pre-AdamW Gradient Filtering, and Same-Backbone Voting

## 1. Introduction and Literature Positioning

Post-training small language models such as `Qwen3-0.6B-Base` with reinforcement learning with verifiable rewards (RLVR) [1, 2, 3] directly in pure `bfloat16` arithmetic faces two tightly coupled challenges: **weight-space quantization and spectral subspace geometry** during optimization and **single-checkpoint trajectory fragility** during inference. Recent weight-space and spectral studies of RL post-training have examined parameter updates from three complementary angles:
- **Three-Gate Theory, Off-Principal RLVR Dynamics, and `float32` Learning-Rate Sensitivity (`Zhu et al.`, `Shenfeld et al.`, `Miahi & Belilovsky`, `claude_1_1007164104_rlvr`):** Recent analyses report that RL post-training updates are sparse in `bfloat16` [19, 20] and that RLVR weight trajectories evolve largely off the principal singular directions of the pretrained weight matrices $W_0$. In *The Path Not Taken: RLVR Provably Learns Off the Principals* (`arXiv:2511.08567v1` [21]), Zhu et al. formulate a *Three-Gate Theory*: Gate I (KL Anchor) constrains policy drift, Gate II (Model Geometry) steers steps off principal directions into low-curvature, spectrum-preserving subspaces, and Gate III (Precision) hides micro-updates in non-preferred regions under `bfloat16` quantization so that off-principal updates manifest as surface sparsity. On the `r/math_rlvr` benchmark, `claude_1_1007164104_rlvr` (`math_rlvr_7` v5, §4.10 [27]) previously established that running `float32` RLVR on `Qwen3-0.6B-Base` at the standard `bfloat16` learning rate ($\eta = 10^{-5}$) causes severe accuracy collapse, whereas reducing the `float32` learning rate recovers to the same accuracy plateau, alongside same-backbone self-consistency voting analyses [17, 27].
- **Pre-Optimizer Adaptive-Rank Spectral Gradient Truncation (`Ares`):** In *Spectral-Driven Adaptive-Rank Optimizer for Large Language Model Reinforcement Learning with Verifiable Rewards* (`arXiv:2610.16533v1`, Oct 2026 [22]), Chen, Zhao, and Wen observe that RLVR gradient energy concentrates into a shrinking leading singular subspace over training, and introduce `Ares` (*Adaptive Rank-Effective Spectral Optimization*, §4.2, Eq. 3, Algorithm 1), which factorizes 2D gradients $G_t$ via top-$k$ SVD ($k=64$), estimates the instantaneous Shannon-entropy effective rank $\hat{r}_t = \exp(-\sum_{i=1}^k p_i \log p_i)$ from the squared-singular-value energy distribution $p_i = \sigma_i^2 / \sum_{j=1}^k \sigma_j^2$ (Eq. 3; giving $r_t \approx 10.4\text{--}10.6$ on `Qwen3-0.6B`, alongside an amplitude-normalized variant $q_i = \sigma_i / \sum_{j=1}^k \sigma_j$ giving $r_t \approx 38$), tracks a bias-corrected exponential moving average $r_t \in [1, k]$, and passes the rank-$r_t$ truncated gradient $\tilde{g}_t = U_{t, 1:r_t}\Sigma_{t, 1:r_t}V_{t, 1:r_t}^\top$ into AdamW. In our experiments (Section 2.5), we evaluate a matched **196-block-linear adaptation** of `Ares` that applies adaptive-rank SVD truncation to all 196 2D transformer block linear matrices while leaving token embeddings, the output head, and 1D RMSNorm scales unfiltered.
- **Parameter-Efficient Spectral Priors and Geometry-Aware Low-Rank Adaptation (`Yin et al.`, `GeoRA`, `PiSSA`, `MiLoRA`):** In *Evaluating Parameter Efficient Methods for RLVR* (`arXiv:2512.23165v2` [23]), Yin et al. benchmark over 12 PEFT methods in RLVR and show that SVD-based initializations originally designed for SFT—specifically `PiSSA` (Meng et al. [25], which initializes adapters $B_0, A_0$ from the top-$r$ principal components of $W_0$) and `MiLoRA` (Wang et al. [26], which initializes $B_0, A_0$ from the minor singular components of $W_0$ where $\sigma_{\text{tail}} \to 0$)—suffer from principal-component misalignment over long RLVR horizons, with `MiLoRA` initially improving reward before small $\sigma_{\text{tail}}$ initialization allows principal gradients to dominate. To resolve this mismatch, Zhang et al. propose *GeoRA: Geometry-Aware Low-Rank Adaptation for RLVR* (`arXiv:2601.09361v4` [24]), which constructs a geometry-constrained matrix $W_{\text{Geo}} = W_0 \odot (M_{\text{Spec}} \cup M_{\text{Euc}})$ combining the bottom-$\rho$ magnitude entries of the rank-$r$ approximation $\hat{W}_r$ ($M_{\text{Spec}}$, suppressing high-energy principal components) with the bottom-$\rho$ magnitude entries of $W_0$ ($M_{\text{Euc}}$, $\rho = 0.20$), initializes $B_{\text{Geo}}, A_{\text{Geo}}$ via the top-$r$ SVD of $W_{\text{Geo}}$, and freezes the residual $W_{\text{res}} = W_0 - B_{\text{Geo}} A_{\text{Geo}}$ as a structural anchor to bound Normalized Spectral Shift ($\text{NSS}$). In Section 2.5, we evaluate matched **196-block-linear adaptations** of `GeoRA`, `PiSSA`, and `MiLoRA` ($r=16$) across all 196 transformer block linear matrices on `Qwen3-0.6B-Base` so that all methods operate on the identical 196-matrix parameter set.

In our preceding study on `Qwen3-0.6B-Base` (`math_rlvr_22` v4 [4]), we established the `Tessera-Sym` training recipe (combining exact longest-common-prefix [LCP] negative credit shielding, trajectory-zero-sum Minimum Description Length [MDL] advantage shaping, and second-half 25-step `float32` tail checkpoint averaging) and a bounded `bfloat16` round-to-nearest-even (RNE) dead-zone audit showing that $>92\%$ of parameter coordinates remain bitwise unchanged after 50 RLVR updates ($1,701/3,700 = 45.97\%$ pooled across `Val350`, `MATH-500`, `Sec-500`, and `Conf-500`).

We do **not** claim priority on precision-sensitive sparsity or off-principal RLVR dynamics (Zhu et al. [21]), pre-AdamW adaptive spectral gradient filtering (`Ares`, Chen et al. [22]), geometry-aware low-rank adaptation (`GeoRA`, Zhang et al. [24]; Yin et al. [23]), the `Tessera-Sym` recipe and `bfloat16` dead-zone audit (`math_rlvr_22` v4 [4]), or same-learning-rate `float32` collapse, reduced-learning-rate recovery, and same-backbone self-consistency voting (`math_rlvr_7` v5 §4.10 [27]; `math_rlvr_6` [17]). Instead, our specific contribution is a **controlled 196-matrix displacement-spectrum and post-hoc vs. online subspace comparison on `Qwen3-0.6B-Base`** under explicitly separated `bfloat16` and `float32`-linear evaluation contracts:

1. **Why do post-trunk `bfloat16` RLVR weight updates exhibit nearly flat singular spectra, and what is the confound-free $2\times 2$ contribution of the 196 linear weight updates versus the 115 non-target tensors?** Holding all $115$ non-target tensors (`tok_emb.weight`, `out_head.weight`, and all $113$ `float32` RMSNorm scale vectors) and `float32` RoPE buffers (`cos`, `sin`) strictly matched across $N=1,850$ within-MATH items plus a **strictly untouched 500-item stratified benchmark (`Fresh-500`)**, we evaluate a 12-cell post-hoc factorial (including both `Rank0-BF16` and `LinOnly-BF16`, reporting exact paired item-level interaction uncertainty $\Delta_{\text{int}}$) and compare post-hoc Rank-16 (`Top-16`) vs. complementary residual (`Tail-16`, ranks $>16$) projections before and after `bfloat16` RNE rounding.
2. **Why does the spectral subspace ranking reverse between post-hoc compression (`Top-16 > Tail-16`) and online `float32`-master optimization (`OSF-Tail16 >> OSF-Top16`), and how do synchronization schedules, master writeback, and pre-AdamW vs. post-AdamW filtering interact?** Reporting **uniform `bfloat16`-evaluated (`Post-Proj BF16` across all 6 panels, $N=5,180$, `batch_size=50`) and uniform `float32`-linear-evaluated (`Post-Proj LinFP32` across `MATH-500`, `Fresh-500`, and `Ext240`, `batch_size=50`) tables separately** across **three seeds (`102`, `42`, `137`)** and both `Tessera-Sym` and supplied `Vanilla-GRPO` (50 and 100 steps), we track **fixed-reference conditional next-token Policy KL divergence $\mathrm{KL}_{\text{ref}}(\pi_\theta \| \pi_0)$** on the 64 reference trajectories (`records/kl_reference_trajectories_64.json`), **greedy-completion conditional Policy KL $\mathrm{KL}_{\text{greedy}}(\pi_\theta \| \pi_0)$** on each policy's own generated `MATH-500` completions, **28-query-layer reference-activation alignment $\eta_{\text{query}}$**, **Normalized Spectral Shift ($\text{NSS}$)**, and **both energy-based ($\mathrm{eRank}_{\text{energy}}$) and amplitude-based ($\mathrm{eRank}_{\text{ampl}}$) effective ranks**. Using question-clustered inference alongside matched **every-step synchronization controls without master writeback (`OSF-EveryStepSync`)**, **every-step master-writeback controls (`OSF-Writeback`)**, and 196-block-linear adaptations of `Ares` (both Eq. 3 energy and amplitude rules), `GeoRA`, `PiSSA`, and `MiLoRA`, we characterize a bounded empirical association between subspace dispersion, conditional KL drift, and accuracy.
3. **Does middle-layer recurrent depth refinement (`LoopedFastModel`) preserve numerical training-to-decoding KV-cache agreement across $N=32$ real `MATH-500` 96-token prefixes ($32$ prompt + $64$ incremental decode steps $= 2,048$ decoded token positions) on the full 28-layer `Qwen3-0.6B` architecture, and does single-pass recurrence improve reasoning across four disjoint within-MATH panels?**
4. **Can a single $1.50\text{ GB}$ checkpoint execute a deterministic multi-depth stencil $\{-g, 0, +g\}$ at $T=0$, and are RLVR-learned gate directions distinguishable from norm-matched Rademacher perturbations?**
5. **Building on the same-backbone self-consistency baselines in [17, 27], how do deterministic recurrent-depth stencils ($T=0$) and center-anchored voting ($\{g_0, s_1, s_2\}$) compare against homogeneous temperature sampling ($T=0.4$) across $3,700$ within-MATH and $480$ untouched `Ext240` evaluations under Holm-Bonferroni multiplicity correction?**

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## 2. 2D SVD Spectral Aliasing, the 12-Cell Post-Hoc Precision Factorial, Online Subspace-Filtered Policy Optimization (`OSF-RLVR`), and Literature Comparisons


### 2.1 Full 196-Matrix Singular Spectrum, Energy vs. Amplitude Effective Rank, and Cross-Seed Alignment
Let $W_0 \in \mathbb{R}^{m \times n}$ denote a pretrained `bfloat16` linear weight matrix in `Qwen3-0.6B-Base` and let $\Delta W_s = W_s - W_0$ (computed in `float32`) denote its post-RLVR displacement in `Tessera-Sym` on training seed $s \in \{102, 42\}$. Excluding token embeddings and the output head, the 28 transformer blocks of `Qwen3-0.6B` contain $28 \times 7 = 196$ two-dimensional linear matrices (`att.W_query`, `att.W_key`, `att.W_value`, `att.out_proj`, `ff.fc1`, `ff.fc2`, `ff.fc3`) totaling $440,401,920$ parameters.

For each matrix $\Delta W_s$ with singular values $\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_d > 0$ ($d = \min(m, n)$), the literature uses two distinct Shannon-entropy effective rank conventions: (i) the **squared-singular-value (energy) effective rank** $\mathrm{eRank}_{\text{energy}}$ based on $p_i = \sigma_i^2 / \sum_{j=1}^d \sigma_j^2$ (used in `Ares` Eq. 3 [22] and spectral energy audits), and (ii) the **singular-value amplitude effective rank** $\mathrm{eRank}_{\text{ampl}}$ based on $q_i = \sigma_i / \sum_{j=1}^d \sigma_j$ (Roy & Vetterli [12]):
$$\text{StableRank}(\Delta W_s) = \frac{\|\Delta W_s\|_F^2}{\sigma_1^2} = \frac{1}{p_1}, \qquad \mathrm{eRank}_{\text{energy}}(\Delta W_s) = \exp\!\left(-\sum_{i=1}^d p_i \ln p_i\right), \qquad \mathrm{eRank}_{\text{ampl}}(\Delta W_s) = \exp\!\left(-\sum_{i=1}^d q_i \ln q_i\right).$$
To ensure complete definitional consistency across the entire paper, we report $\mathrm{eRank}_{\text{energy}}$ in Tables 1–3 alongside $\mathrm{eRank}_{\text{ampl}}$ across all checkpoints (`records/extended_spectral_and_onpolicy_kl.json`). To test whether independent RLVR training histories share a low-rank principal subspace, we also compute for each truncation rank $k \in \{1, 2, 4, 8, 16, 32, 64, 128\}$ the Frobenius cosine between the rank-$k$ SVD reconstructions $\Delta W_{102}^{(k)} = U_{102, :k} \Sigma_{102, :k} V_{102, :k}^\top$ and $\Delta W_{42}^{(k)} = U_{42, :k} \Sigma_{42, :k} V_{42, :k}^\top$, as well as between their complementary high-rank residual tails $\Delta W_s - \Delta W_s^{(k)}$.

| Truncation Rank $k$ | 196-Matrix Mean (Frob.-Wtd) Energy (Seed 102) | 196-Matrix Mean (Frob.-Wtd) Energy (Seed 42) | Top-$k$ Cross-Seed Frobenius Cosine | Residual Tail ($>k$) Cross-Seed Cosine | Layer 14 `fc1` Ternary Sign-Step Top-$k$ Cosine |
|---:|---:|---:|---:|---:|---:|
| **Rank $k=1$** | $1.58\%$ ($1.47\%$) | $1.42\%$ ($1.30\%$) | **$0.2218$** | $0.1172$ | $0.2001$ ($0.86\%$ energy) |
| **Rank $k=2$** | $2.38\%$ ($2.20\%$) | $2.18\%$ ($2.00\%$) | **$0.2036$** | $0.1168$ | **$0.2181$** ($1.50\%$ energy) |
| **Rank $k=4$** | $3.51\%$ ($3.24\%$) | $3.28\%$ ($3.01\%$) | $0.1825$ | $0.1166$ | $0.1500$ ($2.33\%$ energy) |
| **Rank $k=8$** | $5.27\%$ ($4.87\%$) | $5.00\%$ ($4.61\%$) | $0.1474$ | $0.1172$ | $0.1108$ ($3.54\%$ energy) |
| **Rank $k=16$** | $8.17\%$ ($7.58\%$) | $7.89\%$ ($7.30\%$) | $0.1118$ | $0.1192$ | $0.0780$ ($5.63\%$ energy) |
| **Rank $k=32$** | $13.15\%$ ($12.26\%$) | $12.89\%$ ($11.98\%$) | $0.0846$ | $0.1233$ | $0.0525$ ($9.44\%$ energy) |
| **Rank $k=64$** | $21.71\%$ ($20.36\%$) | $21.48\%$ ($20.12\%$) | $0.0704$ | $0.1309$ | $0.0380$ ($16.36\%$ energy) |
| **Rank $k=128$** | $35.79\%$ ($33.86\%$) | $35.63\%$ ($33.67\%$) | $0.0690$ | $0.1439$ | --- |
| **Full Matrix ($d \le 1024$)** | $100.00\%$ ($100.00\%$) | $100.00\%$ ($100.00\%$) | $0.1186$ | --- | $0.0689$ ($100.00\%$ energy) |

*Table 1: 2D SVD cumulative spectral energy—reporting both the unweighted 196-matrix arithmetic mean (matching Tables 3A–3B) and, in parentheses, the Frobenius-norm-weighted total energy ratio $\sum_l \|\Delta W_{l,\le k}\|_F^2 / \sum_l \|\Delta W_l\|_F^2$—alongside cross-seed Frobenius cosine (`Tessera-Sym-Seed102` vs. `Tessera-Sym-Seed42`) across all 196 transformer block linear matrices ($440,401,920$ parameters) and the scale-free ternary sign-step SVD $\text{sign}(\Delta W_s) \in \{-1, 0, +1\}^{m \times n}$ on Layer 14 `ff.fc1.weight` ($3072 \times 1024$).*

| Module Family ($28$ Layers Each) | Matrix Shape ($m \times n$) | Mean Stable Rank | Mean $\mathrm{eRank}_{\text{energy}}$ ($\mathrm{eRank}_{\text{ampl}}$) | Mean Top-8 / Top-16 Spectral Energy | Full-Matrix Cross-Seed Cosine |
|---|---:|---:|---:|---:|---:|
| `att.W_key.weight` | $1024 \times 1024$ | $78.37$ | $551.16$ | $6.16\%$ / $9.97\%$ | $0.1146$ |
| `att.W_query.weight` | $2048 \times 1024$ | $67.22$ | $704.17$ | $5.44\%$ / $8.44\%$ | $0.1138$ |
| `att.W_value.weight` | $1024 \times 1024$ | $34.44$ | $526.97$ | $8.13\%$ / $11.93\%$ | $0.1272$ |
| `att.out_proj.weight` | $1024 \times 2048$ | $64.35$ | $719.55$ | $5.19\%$ / $7.96\%$ | $0.1185$ |
| `ff.fc1.weight` | $3072 \times 1024$ | $115.61$ | $822.07$ | $3.37\%$ / $5.54\%$ | $0.1169$ |
| `ff.fc2.weight` | $3072 \times 1024$ | $70.44$ | $805.87$ | $4.39\%$ / $6.72\%$ | $0.1240$ |
| `ff.fc3.weight` | $1024 \times 3072$ | $83.31$ | $799.63$ | $4.19\%$ / $6.60\%$ | $0.1197$ |
| **All 196 Linear Matrices** | **$440,401,920$ params** | **$73.39$** | **$704.20$ ($904.02$)** | **$5.27\%$ / $8.17\%$ ($4.87\%$ / $7.58\%$ Frob.-wtd)** | **$0.1186$** |

*Table 2: Per-module 2D SVD dimensionality and cross-seed alignment across all 196 linear weight matrices of `Qwen3-0.6B` (`records/svd_196_per_matrix.json`).*

Tables 1 and 2 reveal a striking spectral paradox:
- On one hand, the leading singular modes ($k=1, 2$) exhibit **nearly $2\times$ higher cross-seed alignment** in raw displacement space (`0.2218` at $k=1$ vs. `0.1186` full-matrix) and **$3.2\times$ higher cross-seed alignment** in scale-free ternary sign-step space $\text{sign}(\Delta W_s) \in \{-1, 0, +1\}^{m \times n}$ (`0.2181` at $k=2$ vs. `0.0689` full-matrix on Layer 14 `fc1`).
- On the other hand, across the 196 matrices, the top-1 singular mode captures an unweighted mean of only **$1.58\%$** ($1.47\%$ Frobenius-weighted) of the spectral energy, top-8 captures **$5.27\%$** ($4.87\%$ Frobenius-weighted), top-16 captures **$8.17\%$** ($7.58\%$ Frobenius-weighted, $0.0914^2 / 0.3322^2$), and the mean energy-based Shannon effective rank is **$\mathrm{eRank}_{\text{energy}} = 704.20$** ($\mathrm{eRank}_{\text{ampl}} = 904.02$) out of a maximum possible rank of $1,024$ (`822.07` on `ff.fc1.weight`).

### 2.2 Direct 21-Stage State-Dependent Policy Gradient and 5-Branch Optimizer Audit
Why is $>95\%$ of the Frobenius energy of $\Delta W_s$ scattered across hundreds of singular values? In `bfloat16` (7 mantissa bits), a non-zero weight $w_0$ has unit-in-the-last-place $\text{ULP}(w_0) = 2^{\lfloor \log_2 |w_0| \rfloor - 7} \approx 0.0078125 |w_0|$ [18, 19]. Under AdamW with learning rate $\eta = 10^{-5}$ and moments $\beta_1 = 0.9, \beta_2 = 0.999$, the bias-corrected step at step $t$ is:
$$\Delta \theta_t = -\eta \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} = -\eta \frac{m_t / (1 - \beta_1^t)}{\sqrt{v_t / (1 - \beta_2^t)} + \epsilon}.$$
For stationary gradients $g \in \{-G, +G\}$, $|\hat{m}_t| / \sqrt{\hat{v}_t} \approx 1$, yielding a step of magnitude $\approx \eta$. If a coordinate experiences 40 steps of zero gradient followed by 10 steps of unit gradient $g = +G$, then at step $t=50$:
$$\hat{m}_{50} = \frac{(1 - 0.9^{10}) G}{1 - 0.9^{50}} \approx 0.6547 G, \qquad \hat{v}_{50} = \frac{(1 - 0.999^{10}) G^2}{1 - 0.999^{50}} \approx 0.2039 G^2,$$
yielding $\frac{\hat{m}_{50}}{\sqrt{\hat{v}_{50}}} \approx 1.4494$, so a single step can reach $\approx 1.449\eta$ (and the theoretical maximum over $t \le 50, k \le t$ is $1.459\eta$ at $k=12$). Because weights are updated in place directly in `bfloat16` without a `float32` master weight copy, individual sub-threshold updates below $\frac{1}{2}\text{ULP}(w_0) \approx 0.0039 |w_0|$ do not accumulate in the weight tensor itself (though the `float32` moments $m_t$ and $v_t$ carry momentum forward). Across all $440,401,920$ parameters in the 196 linear matrices, only **$33,999,193$ coordinates ($7.72\%$)** in Seed 102 and **$29,546,069$ coordinates ($6.71\%$)** in Seed 42 cross the RNE dead zone after 50 steps.

To rigorously isolate the spectral effect of **(a) raw policy gradients in `float32` vs. `bfloat16` at the actual live intermediate checkpoints**, **(b) trajectory-evolving state-dependent gradients vs. common gradients**, **(c) AdamW second-moment preconditioning**, and **(d) per-step weight RNE rounding**, we replay the recorded 50-step RLVR rollout trajectories of `Tessera-Sym-Seed102` (`21` active non-zero-variance optimizer steps, `25` second-half tail steps; `code/run_external_bench_and_fp32_audit.py`). At every single one of the 21 active rollout steps ($q \in [3, 50]$), we load the intermediate live checkpoint $W_{t-1}^{\text{live}}$ into an exact `float32` model before backpropagation to compute the state-dependent `float32` gradient $G_t^{\text{FP32}}(W_{t-1}^{\text{live}})$ alongside the live `bfloat16` gradient $G_t^{\text{BF16}}(W_{t-1}^{\text{live}})$. Concurrently, we evolve an independent full-model `float32`-master trajectory (**Branch D**, $W_{t-1}^{\text{D}}$) that computes its own state-dependent `float32` policy gradients $G_t^{\text{FP32}}(W_{t-1}^{\text{D}})$ on the recorded rollout sequences and steps with PyTorch AdamW in `float32`. We also maintain **Branch A** (common-gradient `float32`-master), **Branch B** (common-gradient in-place `bfloat16` + `float32` moments), and **Branch C** (common-gradient in-place `bfloat16` + `bfloat16` moments = live replay), comparing all five branches across all 196 linear matrices (Table 3A):

| Stage of Real RLVR Optimization across All 196 Linear Matrices ($440,401,920$ Params) | Active Non-Zero Coordinates | Mean Stable Rank | Mean Shannon Effective Rank | Mean Top-1 Energy | Mean Top-8 Energy | Mean Top-16 Energy | Mean Top-64 Energy |
|---|---:|---:|---:|---:|---:|---:|---:|
| **1a. State-Dependent `float32` Policy Grad at Live State (Step 1)** | $100.00\%$ ($440,401,919$) | **$2.22$** | **$10.97$** | **$50.66\%$** | **$82.72\%$** | **$90.19\%$** | **$98.51\%$** |
| **1b. State-Dependent `float32` Policy Grad at Live State ($21$-Step Mean)** | $100.00\%$ ($440,401,919$) | **$2.38$** | **$12.97$** | **$48.83\%$** | **$80.92\%$** | **$88.53\%$** | **$97.60\%$** |
| **1c. State-Dependent `bfloat16` Policy Grad at Live State (Step 1)** | $99.98\%$ ($440,301,588$) | **$2.21$** | **$10.92$** | **$50.75\%$** | **$82.78\%$** | **$90.23\%$** | **$98.51\%$** |
| **1d. State-Dependent `bfloat16` Policy Grad at Live State ($21$-Step Mean)** | $99.98\%$ ($440,301,588$) | **$2.38$** | **$13.02$** | **$48.81\%$** | **$80.88\%$** | **$88.49\%$** | **$97.58\%$** |
| **1e. State-Dependent `float32` Policy Grad along Branch D ($21$-Step Mean)** | $100.00\%$ ($440,401,919$) | **$2.22$** | **$11.09$** | **$51.08\%$** | **$83.39\%$** | **$90.11\%$** | **$97.90\%$** |
| **2. Single-Step Pre-Rounding `float32` AdamW Step $\Delta \theta_{1,\text{fp32}}$** | $100.00\%$ ($440,401,567$) | **$7.85$** | **$151.94$** | **$13.88\%$** | **$38.78\%$** | **$49.51\%$** | **$67.57\%$** |
| **3. Single-Step Post-Rounding In-Place `bfloat16` RNE Step** | **$6.49\%$ ($28,598,452$)** | **$76.04$** | **$718.38$** | **$1.52\%$** | **$5.10\%$** | **$7.87\%$** | **$20.92\%$** |
| **4a. Step-50 Branch A: `float32`-Master + `float32` Mom (Unrounded)** | $100.00\%$ ($440,401,920$) | **$9.87$** | **$153.75$** | **$10.92\%$** | **$32.60\%$** | **$43.70\%$** | **$69.26\%$** |
| **4b. Step-50 Branch A: Post-Hoc `bfloat16` RNE of `float32`-Master** | **$52.16\%$ ($229,720,793$)** | **$12.78$** | **$276.87$** | **$8.45\%$** | **$25.22\%$** | **$33.92\%$** | **$55.11\%$** |
| **4c. Step-50 Branch D: State-Dep `float32`-Master (Unrounded)** | $100.00\%$ ($440,401,920$) | **$9.21$** | **$143.36$** | **$12.16\%$** | **$34.38\%$** | **$45.59\%$** | **$70.70\%$** |
| **4d. Step-50 Branch D: Post-Hoc `bfloat16` RNE of Branch D** | **$51.61\%$ ($227,271,233$)** | **$11.83$** | **$263.72$** | **$9.50\%$** | **$26.62\%$** | **$35.35\%$** | **$56.13\%$** |
| **4e. Step-50 Branch B: In-Place `bf16` + `fp32` Mom (Common Grads)** | **$7.76\%$ ($34,195,735$)** | **$77.60$** | **$705.94$** | **$1.51\%$** | **$5.10\%$** | **$8.00\%$** | **$21.58\%$** |
| **4f. Step-50 Branch C: In-Place `bf16` + `bf16` Mom (Common Grads)** | **$7.76\%$ ($34,180,491$)** | **$77.59$** | **$705.94$** | **$1.51\%$** | **$5.11\%$** | **$8.00\%$** | **$21.58\%$** |
| **4g. Step-50 Counterfactual Live-Weight `bfloat16` (`model.pth`)** | **$7.77\%$ ($34,219,026$)** | **$77.85$** | **$705.98$** | **$1.51\%$** | **$5.10\%$** | **$8.00\%$** | **$21.58\%$** |
| **5a. Tail-25 Branch A: `float32`-Master + `float32` Mom (Unrounded)** | $100.00\%$ ($440,401,920$) | **$9.30$** | **$143.84$** | **$11.63\%$** | **$34.22\%$** | **$45.50\%$** | **$70.80\%$** |
| **5b. Tail-25 Branch A: Post-Hoc `bfloat16` RNE of `float32`-Master** | **$48.09\%$ ($211,806,523$)** | **$12.78$** | **$293.43$** | **$8.48\%$** | **$24.90\%$** | **$33.24\%$** | **$53.38\%$** |
| **5c. Tail-25 Branch D: State-Dep `float32`-Master (Unrounded)** | $100.00\%$ ($440,401,920$) | **$9.66$** | **$142.45$** | **$11.66\%$** | **$34.18\%$** | **$45.77\%$** | **$71.30\%$** |
| **5d. Tail-25 Branch D: Post-Hoc `bfloat16` RNE of Branch D** | **$47.23\%$ ($208,012,372$)** | **$13.32$** | **$295.56$** | **$8.44\%$** | **$24.63\%$** | **$33.10\%$** | **$53.32\%$** |
| **5e. Tail-25 Branch B: In-Place `bf16` + `fp32` Mom (Common Grads)** | **$7.71\%$ ($33,970,617$)** | **$73.15$** | **$704.13$** | **$1.59\%$** | **$5.27\%$** | **$8.17\%$** | **$21.72\%$** |
| **5f. Tail-25 Branch C: In-Place `bf16` + `bf16` Mom (Common Grads)** | **$7.71\%$ ($33,961,071$)** | **$73.13$** | **$704.13$** | **$1.59\%$** | **$5.27\%$** | **$8.17\%$** | **$21.72\%$** |
| **5g. Tail-25 Counterfactual Live-Weight `bfloat16` (`model_tail_avg`)** | **$7.72\%$ ($33,999,193$)** | **$73.39$** | **$704.20$** | **$1.58\%$** | **$5.27\%$** | **$8.17\%$** | **$21.71\%$** |

*Table 3A: Direct 21-stage real-RLVR state-dependent policy gradient and 5-branch optimizer audit across all 196 transformer linear weight matrices ($440,401,920$ parameters), aggregated directly from per-matrix records replayed from the recorded `Tessera-Sym-Seed102` rollout trajectories (`records/fp32_vs_bf16_optimizer_audit.json`, reporting energy-based $\mathrm{eRank}_{\text{energy}}$). Step-1 gradient alignment at $W_0$: $\cos(G_1^{\text{FP32}}, G_1^{\text{BF16}}) = 0.9991$. 21-active-step mean gradient alignment at live states $W_{t-1}^{\text{live}}$: $\cos = 0.9922$. Evolving-state gradient alignment between Branch D and live states: 21-step mean $\cos = 0.3928$ ($\cos = 1.0000$ at Step 1, $0.6779$ at Step 2, $0.2053$ at Step 5, $0.1145$ at Step 21). Step-50 endpoint alignments: $\cos(\text{Branch C}, \text{Live BF16}) = 0.9871$, $\cos(\text{Branch A}, \text{Live BF16}) = 0.2573$, $\cos(\text{Branch D}, \text{Branch C}) = 0.1417$, $\cos(\text{Branch A}, \text{Branch D}) = 0.5167$. Tail-25 average alignments: $\cos(\text{Branch C}, \text{Live BF16}) = 0.9873$, $\cos(\text{Branch A}, \text{Live BF16}) = 0.2626$, $\cos(\text{Branch D}, \text{Branch C}) = 0.1615$, $\cos(\text{Branch A}, \text{Branch D}) = 0.5842$.*

Table 3A provides five causal findings on the spectral and trajectory dynamics:
1. **Raw Policy Gradients Are Deeply Low-Rank and Statistically Matched Across All 21 Active Steps:** Evaluated at the actual live intermediate checkpoints $W_{t-1}^{\text{live}}$ in exact `float32` forward/backward passes, the true policy gradient has a 21-step mean stable rank of **$2.38$** and energy-based effective rank of **$\mathrm{eRank}_{\text{energy}} = 12.97$** ($\mathrm{eRank}_{\text{ampl}} = 134.79$; $SR=2.22, \mathrm{eRank}_{\text{energy}}=10.97$ on Step 1), placing **$48.83\%$ of its Frobenius energy in the leading singular mode and $88.53\%$ in the top 16 modes**. Computing gradients in `bfloat16` at the same live states changes virtually nothing: 21-step mean stable rank is **$2.38$**, $\mathrm{eRank}_{\text{energy}} = 13.02$ ($\mathrm{eRank}_{\text{ampl}} = 135.82$), Top-16 energy is **$88.49\%$**, and the Frobenius cosine between true `float32` and `bfloat16` gradients across all 21 active steps is **`0.9922`** (`0.9991` at Step 1). The spectral dispersion cannot be attributed to gradient precision.
2. **Exact Mixed-Dtype (`float32` `RMSNorm` / `RoPE`) vs. Uniform-`bfloat16` 21-Step Replay Control (`records/mixed_dtype_replay_audit.json`):** Because `run_external_bench_and_fp32_audit.py` evaluated the `bfloat16` replay branch after `model.to(torch.bfloat16)` (which casts the 113 `RMSNorm` scales and `RoPE` buffers to `bfloat16`), we execute a dedicated 21-step replay audit under the exact live mixed-dtype `FastModel` contract (`bfloat16` linear/embedding/head weights with `float32` `RMSNorm` scales and `RoPE` `cos`/`sin` buffers). Across all 21 active gradient steps, the mixed-dtype gradient achieves **$\cos(G_t^{\text{Mixed}}, G_t^{\text{FP32}}) = 0.9939$** and **$\cos(G_t^{\text{Mixed}}, G_t^{\text{UnifBF16}}) = 0.9848$**, with 21-step mean $\mathrm{eRank}_{\text{energy}} = 12.97$ ($\mathrm{eRank}_{\text{ampl}} = 135.86$) vs. $13.06$ ($134.79$) in pure `float32` and $13.01$ ($135.82$) in uniform `bfloat16`. Accumulating the 21 steps into the 25-step tail average yields $\mathrm{eRank}_{\text{energy}} = 704.42$ ($\mathrm{eRank}_{\text{ampl}} = 904.08$) under the exact mixed-dtype contract vs. $704.02$ ($903.99$) under uniform `bfloat16`, with endpoint alignment **$\cos(\Delta W_{\text{Tail25}}^{\text{Mixed}}, \Delta W_{\text{LiveCkpt}}) = 0.9915$** (vs. $0.9873$ for uniform `bfloat16`).
3. **AdamW Preconditioning Broadens the Update, but `float32` Master Weights Retain $\sim 45\%$ Top-16 Energy in Both Common and Evolving Branches:** Coordinate-wise normalization by $\sqrt{\hat{v}_t} + \epsilon$ in `float32` (Stage 2) inflates stable rank to $7.85$ and $\mathrm{eRank}_{\text{energy}}$ to $151.94$, but **$49.51\%$ of the Frobenius energy remains in the top-16 singular subspace**. Crucially, this low-rank structure is preserved both when master weights are updated by common gradients (**Branch A**, Tail-25 $\mathrm{eRank}_{\text{energy}} = 143.84$, Top-16 $= 45.50\%$) and when master weights evolve along their own state-dependent gradient trajectory (**Branch D**, Tail-25 $\mathrm{eRank}_{\text{energy}} = 142.45$, Top-16 $= 45.77\%$).
4. **Per-Step In-Place `bfloat16` RNE Rounding Under Common Gradients Fully Replicates the Flat Spectrum and Dead-Zone Sparsity:** When Branch B is driven by the **exact same gradient sequence $G_t$ and exact same `float32` AdamW moments**, rounding the weight updates in place to `bfloat16` at each step inflates stable rank from $9.30 \to 73.15$ and $\mathrm{eRank}_{\text{energy}}$ from $143.84 \to 704.13$, collapsing Top-16 energy to $8.17\%$ and dropping alignment with the `float32`-master trajectory to $\cos = 0.2640$. AdamW moment precision adds almost no further effect (Branch C has $SR = 73.13, \mathrm{eRank}_{\text{energy}} = 704.13, \cos = 0.2639$).
5. **Trajectory Divergence and Common-Gradient Agreement:** Comparing Branch C (common gradients) against the counterfactual live-weight checkpoint (`model_tail_avg.pth`) yields a near-perfect cosine of **`0.9873`** (`0.9915` under exact mixed-dtype replay) and virtually identical spectra ($SR = 73.39, \mathrm{eRank}_{\text{energy}} = 704.20, \mathrm{eRank}_{\text{ampl}} = 904.02$). Meanwhile, comparing the state-dependent `float32`-master trajectory (Branch D) against Branch C yields $\cos = 0.1615$ (and $\cos(\text{Branch A}, \text{Branch D}) = 0.5842$ across the two `float32` masters). Thus, while state evolution under unrounded updates causes the parameter trajectory to rotate away from the `bfloat16` path on identical rollouts ($\cos = 0.5842$), **both `float32` trajectories remain low-rank ($\mathrm{eRank}_{\text{energy}} \approx 142\text{--}144$), proving that the flat singular spectrum ($\mathrm{eRank}_{\text{energy}} = 704.2$) is an artifact of per-step in-place `bfloat16` RNE rounding**.

To further decompose how binade step sizes and sparse support masks act on an idealized matrix, Table 3B reports five analytical surrogates across the 196 matrices: (i) a continuous Gaussian rank-1 outer product $\alpha u v^\top$ ($\mathrm{eRank}_{\text{energy}} = 1.0$), (ii) actual `bfloat16` RNE applied to $\alpha u v^\top$ at matched support $\|M_{102}\|_0$ ($\mathrm{eRank}_{\text{energy}} = 64.7$, Top-1 $= 37.04\%$), (iii) the pure empirical binary support mask $M_{102} = \mathbb{I}[W_{102} \neq W_0]$ ($\mathrm{eRank}_{\text{energy}} = 550.6$), (iv) the magnitude-weighted mask $M_{102} \odot |W_0|$ ($\mathrm{eRank}_{\text{energy}} = 621.8$; note that multiplying by $\operatorname{sign}(u v^\top)$ is a diagonal sign-flip orthogonal equivalence $\operatorname{diag}(\operatorname{sign}(u))(M_{102} \odot |W_0|)\operatorname{diag}(\operatorname{sign}(v))$ that leaves all singular values identically unchanged), and (v) the ternary sign step $\operatorname{sign}(\Delta W_{102}) \in \{-1, 0, +1\}^{m \times n}$ ($\mathrm{eRank}_{\text{energy}} = 718.6$). Together, Tables 3A and 3B show that while AdamW preconditioning moderately broadens the raw `float32` policy gradient ($\mathrm{eRank}_{\text{energy}} \, 10.97 \to 151.94$), it is the **sparse $6.49\%\text{--}7.72\%$ in-place `bfloat16` RNE dead-zone truncation** that destroys the dominant top-16 singular subspace ($49.51\% \to 7.87\%$) and drives $\mathrm{eRank}_{\text{energy}}$ above $700$.

| Analytical Surrogate / Empirical Matrix Family across All 196 Linear Matrices | Mean Stable Rank | Mean $\mathrm{eRank}_{\text{energy}}$ | Mean Top-1 Spectral Energy | Mean Top-8 Spectral Energy | Mean Top-16 Spectral Energy | Mean Top-64 Spectral Energy |
|---|---:|---:|---:|---:|---:|---:|
| **Continuous Rank-1 Surrogate ($\alpha u v^\top$)** | $1.00$ | $1.0$ | $100.00\%$ | $100.00\%$ | $100.00\%$ | $100.00\%$ |
| **Actual `bf16` RNE on Rank-1 Surrogate ($\operatorname{RNE}_{\text{bf16}}(W_0 + \alpha u v^\top) - W_0$)** | **$2.74$** | **$64.7$** | **$37.04\%$** | **$43.96\%$** | **$49.57\%$** | **$69.99\%$** |
| **Actual `bf16` RNE Rank-1 Binary Mask ($\mathbb{I}[\Delta W_{\text{RNE}} \neq 0]$)** | $4.92$ | $266.2$ | $20.79\%$ | $23.70\%$ | $26.37\%$ | $39.88\%$ |
| **Pure Empirical Binary Mask ($M_{102} = \mathbb{I}[W_{102} \neq W_0]$)** | $10.78$ | $550.6$ | $9.65\%$ | $11.98\%$ | $14.23\%$ | $25.84\%$ |
| **Magnitude-Weighted Support ($M_{102} \odot |W_0| = M_{102} \odot |W_0| \odot \operatorname{sign}(u v^\top)$)** | $16.27$ | $621.8$ | $6.37\%$ | $8.79\%$ | $11.21\%$ | $23.58\%$ |
| **Sign-Only Ternary Step ($\operatorname{sign}(\Delta W_{102}) \in \{-1, 0, +1\}^{m \times n}$)** | $75.07$ | $718.6$ | $1.51\%$ | $5.06\%$ | $7.84\%$ | $20.92\%$ |
| **Empirical RLVR $\Delta W_{102}$ (`Tessera-Sym-Seed102`)** | **$73.38$** | **$704.2$** ($\mathrm{eRank}_{\text{ampl}}=904.0$) | **$1.58\%$** | **$5.27\%$** | **$8.17\%$** | **$21.71\%$** |
| **Empirical RLVR $\Delta W_{42}$ (`Tessera-Sym-Seed42`)** | **$83.23$** | **$707.7$** ($\mathrm{eRank}_{\text{ampl}}=904.2$) | **$1.42\%$** | **$5.00\%$** | **$7.89\%$** | **$21.48\%$** |

*Table 3B: Controlled analytical surrogates isolating `bfloat16` binade RNE quantization, sparse binary support masking, magnitude weighting, and ternary sign steps across all 196 linear matrices.*

### 2.3 Twelve-Cell Confound-Free Post-Hoc Precision, Component & Spectral Subspace Factorial ($N=1,850$ + Untouched `Fresh-500`)
Because empirical in-place `bfloat16` displacements are $\pm 1\text{ ULP}$ steps on a sparse $7.72\%$ support, post-hoc SVD truncation of $\Delta W_{102}$ to rank $r=16$ spreads update energy across all matrix coordinates, shrinking many entries below $\frac{1}{2}\text{ULP}(w_0)$. Across **all 196 linear matrices combined**, replacing each $\Delta W_{102}$ with its post-hoc rank-16 SVD truncation and rounding to `bfloat16` retains only **$15,860,388 / 33,999,193$ ($46.65\%$) of the true active updates, underflowing $53.35\%$ ($18,138,805$) to zero while inducing $2,592,203$ spurious boundary flips**. Building on the single-panel post-hoc directional low-rank compression study in `math_rlvr_7` (v5, §4.10 [27]), we design a 12-cell precision, component, and spectral subspace factorial to separate non-target parameters, forward-pass dtype, unrounded principal vs. tail subspaces, and `bfloat16` RNE snap-back.

To ensure **zero precision or component confounds**, we audit the exact parameter and buffer dtypes of `FastModel(cfg)` during training and evaluation (`code/run_precision_factorial_complete.py`):
- **115 Non-Target Tensors and `float32` RoPE Buffers:** Alongside the 196 two-dimensional transformer linear weight matrices ($440,401,920$ parameters), `Qwen3-0.6B` contains **115 non-target parameter tensors**: `tok_emb.weight` ($151936 \times 1024$), `out_head.weight` ($151936 \times 1024$), and **113 `float32` RMSNorm scale vectors** (`final_norm.scale` plus 28 layers $\times$ 4 norms: `norm1`, `norm2`, `q_norm`, `k_norm`), as well as `float32` rotary position embedding (`RoPE`) buffers (`cos`, `sin`). In `FastModel`, `RMSNorm` initializes its scale parameter in `torch.float32` (`torch.ones(emb_dim, dtype=torch.float32)`) and computes normalization in `float32` before casting back to `x.dtype`, while `compute_rope_params` creates `cos` and `sin` in `torch.float32`. Calling `model.to(torch.bfloat16)` after checkpoint loading would cast all 113 `float32` RMSNorm scales and `float32` RoPE buffers to `bfloat16`, whereas adding a `float32` delta to a `bfloat16` tensor (`W_0.to(bfloat16) + dW_fp32`) and then calling `.to(torch.bfloat16).float()` would introduce a redundant `bfloat16` round-trip.
- **Strict Confound-Free Protocol (`build_clean_eval_model` in `code/run_precision_factorial_complete.py`):** In all **`BF16` network cells**, `tok_emb.weight` and `out_head.weight` are `torch.bfloat16`, while all **113 RMSNorm scale tensors and `cos`/`sin` RoPE buffers remain in exact `torch.float32`** (matching `FastModel` training/eval exactly, with no post-load `model.to(torch.bfloat16)`). In all **`FP32` network cells**, all parameters and buffers are `torch.float32` constructed directly as `W_0.float() + dW_fp32` without any intermediate `.to(torch.bfloat16)` round-trip. In **hybrid `LinFP32-NetBF16` cells**, the surrounding network is identical to `BF16` network cells while the 196 linear layers execute `F.linear(x.float(), W_fp32).to(x.dtype)` via `FP32LinearInBF16Net`.
- **Deterministic Construction, Question Provenance, and Extracted-Answer Grading of Untouched `Fresh-500`, plus Fixed 64-Trajectory Reference KL (`kl_reference_trajectories_64.json`):** Because `Val350`, `MATH-500`, `Sec-500`, and `Conf-500` were inspected on Seed 102 during earlier studies, we construct a genuinely untouched 500-item stratified benchmark (`Fresh-500`, `code/splits/test_fresh_untouched_500.json`) deterministically in `code/build_splits.py` (Step 5) from `math_train[1000:]` (`random.Random(20261019)`) to match the exact 35-cell `(subject, level)` joint distribution of `Conf-500`/`Sec-500`, excluding all prior splits, `math_train[:1000]`, and the 178 `orig_train_idx` rollout prompts. Every `Fresh-500` item's `orig_train_idx`, `unique_id`, 16-character SHA-256 problem hash, ground-truth `answer`, and per-arm extracted model answer (`extracted`) are packed in `records/osf_and_precision_factorial.json` (`fresh500_provenance_and_answers`) and verified by `code/reproduce_stencil_and_svd.py` to have **zero overlap** with `Val350`, `MATH-500`, `Sec-500`, `Conf-500`, and all 178 rollout questions, and **zero grading mismatches** (`grade_answer(extracted, gt) == correct`) across all $22,000+$ item evaluations. For every checkpoint, we also compute exact full-vocabulary ($V=151,936$) conditional next-token reverse Policy KL divergence $\mathrm{KL}_{\text{cond}}(\pi_\theta \| \pi_0)$ over the $15,387$ completion tokens of the 64 self-contained reference trajectories in `records/kl_reference_trajectories_64.json` (with no fallback to generic text), complemented by own-policy greedy-completion conditional KL $\mathrm{KL}_{\text{greedy}}(\pi_\theta \| \pi_0)$ evaluated strictly on each checkpoint+operator's own generated `MATH-500[:64]` greedy completions with verified directory and SHA-256 provenance (`records/extended_spectral_and_onpolicy_kl.json`; note that $\mathrm{KL}_{\text{greedy}}$ conditions on greedy-decoded prefixes rather than an expectation over stochastic on-policy rollouts).

Using this exact contract, we evaluate the **12-Cell Post-Hoc Precision, Component & Spectral Subspace Factorial** across the four within-MATH panels ($N=1,850$, Seed 102) plus untouched `Fresh-500` ($N=500$; Table 4), matching batch size (`batch_size=50`), length-sorted prompt ordering, and numerical execution across all four cells of the $2\times 2$ component factorial (`Base-BF16`, `Rank0-BF16`, `LinOnly-BF16`, `Full-BF16`) while also auditing batch-size execution variability ($\Delta_{\text{bs}}$) against the original `bs=32/45/60` evaluations (`posthoc_batch_size_variability_audit_s102` in `records/osf_and_precision_factorial.json`):
1. **`Base-BF16` & `Base-FP32` (Pretrained `Qwen3-0.6B-Base`):** Unmodified pretrained weights ($W_0$) evaluated under the exact `BF16` (`batch_size=50`: `292/1,850 = 15.78%`, `Fresh-500 = 79/500 = 15.80%`; vs. `293/1,850 = 15.84%` at `bs=45`, $\Delta_{\text{bs}} = -1$ item, $+19/-20$, $p=1.000$) and `FP32` (`batch_size=34`: `298/1,850 = 16.11%`) network contracts.
2. **`Rank0-BF16` & `Rank0-FP32` ($\Delta W_{\text{lin}} = 0$ + RLVR Non-Target Tensors):** Freezes all 196 transformer block linear matrices at $W_0$ ($\Delta W_{\text{lin}} = 0$) while loading the RLVR-trained `tok_emb.weight`, `out_head.weight`, and 113 `float32` RMSNorm scales from `Tessera-Sym-Seed102` (`batch_size=50`: `311/1,850 = 16.81%` in `BF16`, `Fresh-500 = 83/500 = 16.60%`, $\mathrm{KL}_{\text{ref}} = 0.0003$, $\mathrm{KL}_{\text{greedy}} = 0.0004\text{ nats/tok}$; vs. `306/1,850 = 16.54%` at `bs=45`, $\Delta_{\text{bs}} = +5$ items, $+32/-27$, $p=0.603$; and `307/1,850 = 16.59%` in `FP32` at `bs=34`).
3. **`LinOnly-BF16` ($\Delta W_{\text{lin}} = \Delta W_{102}$ + Base Non-Target Tensors):** Completes the $2\times 2$ non-target $\times$ linear-target attribution table by applying the RLVR linear displacement $\Delta W_{102}$ across all 196 linear matrices while keeping `tok_emb.weight`, `out_head.weight`, and all 113 `float32` RMSNorm scales frozen at their pretrained base values (`batch_size=50`: `847/1,850 = 45.78%`, `Fresh-500 = 261/500 = 52.20%`, $\mathrm{KL}_{\text{ref}} = 0.0236$, $\mathrm{KL}_{\text{greedy}} = 0.0223\text{ nats/tok}$; vs. `846/1,850 = 45.73%` and `257/500 = 51.40%` at `bs=60`, $\Delta_{\text{bs}} = +1$ item [$+79/-78$, $p=1.000$] on $N=1,850$ and $+4$ items [$+20/-16$, $p=0.618$] on `Fresh-500`).
4. **`Full-BF16` ($g_0$) & `Full-FP32` (Untruncated Empirical Endpoint $W_{\text{emp}} = W_0 + \Delta W_{102}$):** Full RLVR checkpoint in matched `BF16` (`batch_size=50`: `848/1,850 = 45.84%`, `Fresh-500 = 257/500 = 51.40%`, `Ext240 = 149/240 = 62.08%`, $\mathrm{KL}_{\text{ref}} = 0.0240$, $\mathrm{KL}_{\text{greedy}} = 0.0232\text{ nats/tok}$; vs. `859/1,850 = 46.43%` and `Ext240 = 155/240 = 64.58%` at `bs=32`, $\Delta_{\text{bs}} = -11$ items [$-0.59\text{ pp}$, $+74/-85$, $p=0.428$] on $N=1,850$) and pure `FP32` (`batch_size=34`: `860/1,850 = 46.49%`).
5. **`SVD16-LinFP32-NetBF16`, `SVD16-FP32`, & `SVD16-BF16` (Principal Top-16 Subspace $\mathcal{P}_{\le 16}(\Delta W_{102})$):** Replaces each of the 196 linear displacements with its top-16 SVD projection $\Delta W_{16} = \sum_{i=1}^{16} \sigma_i u_i v_i^\top$ ($8.17\%$ unweighted mean energy, $7.58\%$ Frobenius-weighted energy, $\|\mathcal{P}_{\le 16}(\Delta W_{102})\|_F = 0.0914$, $\mathrm{eRank}_{\text{energy}} = 14.12$, $\mathrm{eRank}_{\text{ampl}} = 16.31$, $\mathrm{KL}_{\text{ref}} = 0.0192$, $\mathrm{KL}_{\text{greedy}} = 0.0187\text{ nats/tok}$), evaluated unrounded in the matched `BF16` network (`845/1,850 = 45.68%`, `Fresh-500 = 258/500 = 51.60%`), unrounded in the `FP32` network (`848/1,850 = 45.84%`), and rounded to `bfloat16` RNE (`827/1,850 = 44.70%`, $\mathrm{KL}_{\text{ref}} = 0.0075$, $\mathrm{KL}_{\text{greedy}} = 0.0078\text{ nats/tok}$).
6. **`PostHoc-Tail16-LinFP32-NetBF16` & `PostHoc-Tail16-BF16-SnapBack` (Complementary High-Rank Tail $\mathcal{P}_{>16}(\Delta W_{102}) = \Delta W_{102} - \Delta W_{16}$):** Replaces each linear displacement with its complementary rank-$>16$ residual tail ($91.83\%$ unweighted mean energy, $92.42\%$ Frobenius-weighted energy, $\|\mathcal{P}_{>16}(\Delta W_{102})\|_F = 0.3193$, $\mathrm{eRank}_{\text{energy}} = 730.37$, $\mathrm{eRank}_{\text{ampl}} = 897.68$), evaluated both **unrounded in `float32`** inside the matched `BF16` network (`713/1,850 = 38.54%`, `Fresh-500 = 233/500 = 46.60%`, `Ext240 = 93/240 = 38.75%`, $\mathrm{KL}_{\text{ref}} = 0.0026$, $\mathrm{KL}_{\text{greedy}} = 0.0026\text{ nats/tok}$) and **rounded to `bfloat16` RNE** $\operatorname{RNE}_{\text{bf16}}(W_{\text{emp}} - \Delta W_{16})$ (`831/1,850 = 44.92%`, `Fresh-500 = 258/500 = 51.60%`, `Ext240 = 148/240 = 61.67%`, $\mathrm{KL}_{\text{ref}} = 0.0124$, $\mathrm{KL}_{\text{greedy}} = 0.0112\text{ nats/tok}$).

| Post-Hoc Checkpoint, Component & Precision Condition (Seed 102) | `Val350` / `M500` / `Sec500` / `Conf500` | **Pooled N=1,850 (%)** | **Untouched `Fresh-500`** | **Paired Contrast on N=1,850 (`diff`, 95% CI, `+W/-L`, McNemar $p$; $\mathrm{KL}_{\text{ref}} / \mathrm{KL}_{\text{greedy}}$)** |
|---|---:|---:|---:|---:|
| **`Base-BF16` (Pretrained `Qwen3-0.6B-Base`, $W_0$, `bs=50`)** | $62$ / $74$ / $78$ / $78$ | **$292/1,850$ ($15.78\%$)** | **$79/500$ ($15.80\%$)** | Pretrained `BF16` Baseline (`bs=45`: $293/1,850$, $\Delta_{\text{bs}}=-1$, $p=1.000$; `Ext240`: $33/240 = 13.75\%$) |
| **`Base-FP32` (Pretrained `Qwen3-0.6B-Base`, `FP32` Net, `bs=34`)** | $60$ / $80$ / $80$ / $78$ | **$298/1,850$ ($16.11\%$)** | --- | $+6$ vs. `Base-BF16` ($+0.32\text{ pp}$, $[-0.38,+1.03]$, $+27/-21$, $p=0.4709$) |
| **`Rank0-BF16` ($\Delta W_{\text{lin}}=0$ + RLVR Emb/Head/113 Norms, `bs=50`)** | $61$ / $83$ / $81$ / $86$ | **$311/1,850$ ($16.81\%$)** | **$83/500$ ($16.60\%$)** | $+19$ vs. `Base-BF16` ($+1.03\text{ pp}$, $[+0.21,+1.84]$, $+39/-20$, $p=0.0183$; `bs=45`: $306$, $+13$, $p=0.1299$; $\mathrm{KL}=0.0003/0.0004$) |
| **`Rank0-FP32` ($\Delta W_{\text{lin}}=0$ + RLVR Emb/Head/113 Norms, `bs=34`)** | $60$ / $83$ / $81$ / $83$ | **$307/1,850$ ($16.59\%$)** | --- | $+9$ vs. `Base-FP32` ($+0.49\text{ pp}$, $[-0.02,+0.99]$, $+16/-7$, $p=0.0931$) |
| **`LinOnly-BF16` ($\Delta W_{\text{lin}}=\Delta W_{102}$ + Base Emb/Head/Norms, `bs=50`)** | $159$ / $230$ / $224$ / $234$ | **$847/1,850$ ($45.78\%$)** | **$261/500$ ($52.20\%$)** | **$+555$ vs. `Base-BF16`** ($+30.00\text{ pp}$, $[+27.49,+32.51]$, $+641/-86$, $p<10^{-100}$; `bs=60`: $846$, $\Delta_{\text{bs}}=+1$; $\mathrm{KL}=0.0236/0.0223$) |
| **`Full-BF16` (Live Empirical Endpoint $W_{\text{emp}}$, `bs=50`)** | $159$ / $233$ / $227$ / $229$ | **$848/1,850$ ($45.84\%$)** | **$257/500$ ($51.40\%$)** | **$+537$ vs. `Rank0-BF16`** ($+29.03\text{ pp}$, $[+26.58,+31.47]$, $+613/-76$, $p<10^{-100}$; `bs=32`: $859$, $\Delta_{\text{bs}}=-11$; $\mathrm{KL}=0.0240/0.0232$) |
| **`Full-FP32` (Untruncated $W_{\text{emp}}$, Pure `FP32` Net, `bs=34`)** | $167$ / $239$ / $230$ / $224$ | **$860/1,850$ ($46.49\%$)** | --- | **$+553$ vs. `Rank0-FP32`** ($+29.89\text{ pp}$, $[+27.39,+32.39]$, $+637/-84$, $p<10^{-100}$) |
| **`SVD16-LinFP32-NetBF16` (Unrounded Top-16, `BF16` Net)** | $157$ / $231$ / $230$ / $227$ | **$845/1,850$ ($45.68\%$)** | **$258/500$ ($51.60\%$)** | **$-3$ vs. `Full-BF16` (`bs=50`)** ($-0.16\text{ pp}$, $[-1.65,+1.33]$, $+97/-100$, **$p=0.8867$**; $-14$ vs. `bs=32`, $p=0.3786$; $\mathrm{KL}=0.0192/0.0187$) |
| **`SVD16-FP32` (Unrounded Top-16, Pure `FP32` Net)** | $163$ / $239$ / $232$ / $214$ | **$848/1,850$ ($45.84\%$)** | --- | **$-12$ vs. `Full-FP32`** ($-0.65\text{ pp}$, $[-2.04,+0.74]$, $+80/-92$, **$p=0.4017$**) |
| **`SVD16-BF16` (`bfloat16` RNE Rounded Top-16)** | $157$ / $234$ / $222$ / $214$ | **$827/1,850$ ($44.70\%$)** | --- | **$-18$ vs. `SVD16-LinFP32`** ($-0.97\text{ pp}$, $[-2.70,+0.75]$, $+124/-142$, $p=0.2972$; $\mathrm{KL}=0.0075/0.0078$) |
| **`PostHoc-Tail16-LinFP32-NetBF16` (True Unrounded Tail)** | $143$ / $191$ / $193$ / $186$ | **$713/1,850$ ($38.54\%$)** | **$233/500$ ($46.60\%$)** | **$-132$ vs. `SVD16-LinFP32`** ($-7.14\text{ pp}$, $[-9.17,-5.10]$, $+123/-255$, **$p=9.97\times 10^{-12}$**; $\mathrm{KL}=0.0026/0.0026$) |
| **`PostHoc-Tail16-BF16-SnapBack` (`bfloat16` RNE Tail)** | $159$ / $238$ / $226$ / $208$ | **$831/1,850$ ($44.92\%$)** | **$258/500$ ($51.60\%$)** | **$+118$ vs. `Tail16-LinFP32`** ($+6.38\text{ pp}$, $[+4.45,+8.31]$, $+229/-111$, **$p=1.46\times 10^{-10}$**; $\mathrm{KL}=0.0124/0.0112$) |

*Table 4: Twelve-cell post-hoc precision, component, and spectral subspace factorial across all four disjoint within-MATH panels ($N=1,850$, Seed 102) plus the strictly untouched `Fresh-500` benchmark (`records/osf_and_precision_factorial.json`), with uniform `batch_size=50` matching across all four cells of the $2\times 2$ component factorial alongside the explicit batch-size execution variability audit.*

Table 4 resolves every post-hoc attribution and precision question with exact paired statistics and batch-size variability auditing:
1. **Uniform `batch_size=50` $2\times 2$ Component Attribution and Batch-Size Execution Variability Audit ($\Delta_{\text{bs}}$):** Because `bfloat16` GEMM tiling on `Qwen3-0.6B` exhibits minor batch-size sensitivity on knife-edge greedy trajectories [17, §3.8], we evaluate all four cells of the $2\times 2$ non-target $\times$ linear-target factorial (`Base-BF16`, `Rank0-BF16`, `LinOnly-BF16`, `Full-BF16`) at **strictly uniform `batch_size=50`** with identical length-sorted prompt ordering across all five panels, and compare them item-by-item against the original `bs=32/45/60` evaluations (`posthoc_batch_size_variability_audit_s102`):
   - **Direct Batch-Size Variability ($\Delta_{\text{bs}}$):** Changing `batch_size` to `50` shifts `Base-BF16` (`bs=50` vs. `45`) by **$-1$ item ($292$ vs. $293/1,850$, $-0.05\text{ pp}$, $+19/-20$, $p=1.000$)**, `Rank0-BF16` (`bs=50` vs. `45`) by **$+5$ items ($311$ vs. $306/1,850$, $+0.27\text{ pp}$, $+32/-27$, $p=0.603$)**, `LinOnly-BF16` (`bs=50` vs. `60`) by **$+1$ item ($847$ vs. $846/1,850$, $+0.05\text{ pp}$, $+79/-78$, $p=1.000$; and $+4$ items [$261$ vs. $257/500$, $+0.80\text{ pp}$, $+20/-16$, $p=0.618$] on `Fresh-500`)**, and `Full-BF16` (`bs=50` vs. `32`) by **$-11$ items ($848$ vs. $859/1,850$, $-0.59\text{ pp}$, 95% CI $[-1.93, +0.74]\text{ pp}$, $+74/-85$, $p=0.428$)**.
   - **Matched Component Attribution and Unresolved Interaction ($\widehat{\Delta}_{\text{int}}$):** Under strictly matched `batch_size=50`, updating only the 115 non-target tensors (`Rank0-BF16` vs. `Base-BF16`) induces virtually zero Policy KL shift ($\mathrm{KL}_{\text{ref}} = 0.0003, \mathrm{KL}_{\text{greedy}} = 0.0004\text{ nats/tok}$) and changes accuracy by **$+19$ items ($+1.03\text{ pp}$, 95% CI $[+0.21, +1.84]\text{ pp}$, $+39/-20$, $p=0.0183$; $+4$ items [$+0.80\text{ pp}$, $p=0.455$] on `Fresh-500`; $+9$ items [$+0.49\text{ pp}$, $p=0.0931$] in `FP32` at `bs=34`; and $+13$ items [$+0.70\text{ pp}$, $p=0.130$] at `bs=45`)**. Given the linear update, adding the non-target update (`Full-BF16` vs. `LinOnly-BF16` at `bs=50`) changes accuracy by **$+1$ item ($848$ vs. $847/1,850$, $+0.05\text{ pp}$, 95% CI $[-1.32, +1.42]\text{ pp}$, $+84/-83$, $p=1.000$; and $-4$ items [$257$ vs. $261/500$, $-0.80\text{ pp}$, $+17/-21$, $p=0.627$] on `Fresh-500`)**. Conversely, updating **only** the 196 linear matrices (`LinOnly-BF16` vs. `Base-BF16` at `bs=50`) contributes **$+555$ items ($847$ vs. $292/1,850 = +30.00\text{ pp}$, 95% CI $[+27.49, +32.51]\text{ pp}$, $+641/-86$, $p = 8.87 \times 10^{-106}$; and $+182$ items [$261$ vs. $79/500 = +36.40\text{ pp}$, $p = 4.73 \times 10^{-39}$] on `Fresh-500`)**—accounting for **$99.8\%$ ($555/556$) of the total `Full-BF16` gain over `Base-BF16` at `bs=50`** (and $97.7\%$ [$553/566$] under original batching), while the conditional linear gain (`Full-BF16` vs. `Rank0-BF16` at `bs=50`) is **$+537$ items ($848$ vs. $311/1,850 = +29.03\text{ pp}$, $+613/-76$, $p = 3.25 \times 10^{-105}$; $+174$ items [$+34.80\text{ pp}$, $p = 6.24 \times 10^{-38}$] on `Fresh-500`; and $+553$ items in `FP32` at `bs=34`)**. The resulting $2\times 2$ interaction contrast $\widehat{\Delta}_{\text{int}} = (\text{Full} - \text{Rank0}) - (\text{LinOnly} - \text{Base})$ at uniform `batch_size=50` is **$-18$ items ($-0.97\text{ pp}$, 95% CI $[-2.57, +0.63]\text{ pp}$, $p=0.233$) on $N=1,850$ and $-8$ items ($-1.60\text{ pp}$, $p=0.267$) on `Fresh-500`** (compared to $0$ items [$+0.00\text{ pp}$, 95% CI $[-1.62, +1.62]\text{ pp}$, $p=1.000$] under original `bs=32/45/60` batching), representing **an unresolved difference within item and batching uncertainty** rather than evidence of a non-zero interaction or proof of a population interaction of exactly zero.
2. **Network Forward-Pass Precision Has Minimal Effect ($\le 12$ Items Across All Contrasts):** Executing the network forward pass in pure `float32` (`bs=34`) vs. matched `bfloat16` holding weights fixed yields unresolved differences of $+6$ items on `Base` ($298$ vs. $292$, $p=0.471$), $-4$ items on `Rank0` ($307$ vs. $311$), $+12$ items on `Full` ($860$ vs. $848$ at `bs=50`, $+0.65\text{ pp}$, $p=0.363$; $+1$ item vs. $859$ at `bs=32`, $p=1.000$), and $+3$ items on `SVD16` ($848$ vs. $845$, $+0.16\text{ pp}$, $p=0.876$).
3. **Unrounded Post-Hoc Rank-16 Truncation Retains $99.5\%$ of the `bs=50` RLVR Gain ($97.5\%$ vs. `bs=32`, $100.6\%$ on `Fresh-500`):** In matched `BF16` network execution, unrounded rank-16 linear updates (`SVD16-LinFP32-NetBF16`) reach **$845/1,850$ ($45.68\%$)** (an unresolved difference of $-3$ items vs. `Full-BF16` at `bs=50` [$848/1,850$, $p=0.887$] and $-14$ items vs. `Full-BF16` at `bs=32` [$859/1,850$, $p=0.379$]) and **$258/500$ ($51.60\%$) on untouched `Fresh-500`** ($+1$ item vs. `Full-BF16`'s $257/500$, $p=1.000$, with $257/500$ also matched by supplied `Baseline-Seed102`), while preserving $80.1\%$ of the reference Policy KL shift ($\mathrm{KL}_{\text{ref}} = 0.0192$ vs. $0.0240\text{ nats/tok}$) and $80.6\%$ of the own-completion greedy KL shift ($\mathrm{KL}_{\text{greedy}} = 0.0187$ vs. $0.0232\text{ nats/tok}$).
4. **Post-Hoc Top-16 Significantly Outperforms the True Unrounded Complementary Tail ($+7.14\text{ pp}$ on $N=1,850$, $+5.00\text{ pp}$ on `Fresh-500`), and `bfloat16` RNE Induces a Dual "Snap-Back" Artifact on the Tail:** When evaluated **without `bfloat16` rounding** in the identical `BF16` network, the principal rank-16 subspace (`SVD16-LinFP32-NetBF16`, $\|\Delta W\|_F = 0.0914$, $\mathrm{KL}_{\text{ref}} = 0.0192, \mathrm{KL}_{\text{greedy}} = 0.0187$) decisively outperforms the complementary rank-$>16$ tail (`PostHoc-Tail16-LinFP32-NetBF16`, $\|\Delta W\|_F = 0.3193$, which induces only $\mathrm{KL}_{\text{ref}} = 0.0026, \mathrm{KL}_{\text{greedy}} = 0.0026\text{ nats/tok}$) by **$+132$ items ($+7.14\text{ pp}$, $+255/-123$, $p = 9.97 \times 10^{-12}$) on $N=1,850$ and $+25$ items ($+5.00\text{ pp}$, $258$ vs. $233/500$, $+61/-36$, $p = 0.0144$) on untouched `Fresh-500`**. Rounding $W_{\text{emp}} - \Delta W_{16}$ to `bfloat16` (`PostHoc-Tail16-BF16-SnapBack`) snaps **$53.33\%$ ($18,130,971 / 33,999,193$) of the changed weights back to $W_{\text{emp}}$**, artificially inflating $\mathrm{KL}_{\text{ref}}$ from $0.0026 \to 0.0124\text{ nats/tok}$ ($\mathrm{KL}_{\text{greedy}}$ from $0.0026 \to 0.0112\text{ nats/tok}$) and rebounding accuracy by **$+118$ items ($+6.38\text{ pp}$, $+229/-111$, $p = 1.46 \times 10^{-10}$) on $N=1,850$ and $+25$ items ($+5.00\text{ pp}$, $258$ vs. $233/500$, $+62/-37$, $p = 0.0154$) on `Fresh-500`**.

### 2.4 Online Subspace-Filtered Policy Optimization (`OSF-RLVR`), Exact Recurrence & Projection Controls, and Question-Clustered Multi-Seed Inference
The post-hoc findings in Sections 2.2 and 2.3 motivate a direct **algorithmic training intervention**. In standard `bfloat16` RLVR, per-step RNE rounding zeroes out $>92\%$ of sub-ULP coordinate updates, acting as an implicit hardware threshold that prevents small steps from accumulating. Conversely, if we maintain exact `float32` master weights (`Unfiltered-FP32-Master`) so that sub-ULP updates accumulate without dead-zone damping, how does online RLVR behave—and how does projecting the cumulative update onto its principal vs. orthogonal tail singular subspaces affect policy KL divergence, Normalized Spectral Shift ($\text{NSS}$) [24], and reasoning accuracy?

**Exact Mathematical Recurrence of `train_osf_rlvr.py` and Pre/Post-Projection Controls:** To avoid any ambiguity between unprojected master accumulation, `bfloat16` rollout parameter synchronization, and post-averaging spectral projection, we state the exact 3-stage recurrence implemented in `code/train_osf_rlvr.py`:
1. **Unprojected `float32` Master Update (vs. Every-Step Master Writeback):** At each RLVR step $t \in \{1, \dots, 50\}$, the optimizer updates a `float32` master weight tensor $W_{\text{master}}^{(t)}$ using `float32` AdamW moments $m_t, v_t$, defining the raw cumulative displacement $\Delta W_{\text{raw}}^{(t)} = W_{\text{master}}^{(t)} - W_0$. In the primary `OSF` protocol, $W_{\text{master}}^{(t)}$ remains unprojected so that sub-leading modes can evolve before spectral filtering; in Section 2.5 we also evaluate the strict **Every-Step Master-Writeback** variant (`--writeback-master`, $W_{\text{master}}^{(t)} \leftarrow W_0 + \mathcal{P}(\Delta W_{\text{raw}}^{(t)})$).
2. **Rollout Policy Parameter Projection (`Pre-Proj Rollout-BF16` vs. `Pre-Proj LinFP32`):** Before generating rollouts at step $t+1$, each of the 196 linear matrices is projected in `float32` as $W_{\text{proj}}^{(t)} = W_0 + \mathcal{P}(\Delta W_{\text{raw}}^{(t)})$ and copied into the active rollout model in `bfloat16`: $W_{\text{rollout}}^{(t)} = \operatorname{RNE}_{\text{bf16}}(W_{\text{proj}}^{(t)})$. Across the second half of training ($t \in \{26, \dots, 50\}$), `train_osf_rlvr.py` accumulates both the `bfloat16` rollout tail average $\overline{W}_{\text{rollout}}^{\text{BF16}} = \operatorname{RNE}_{\text{bf16}}\!\left(\frac{1}{25}\sum_{t=26}^{50} W_{\text{rollout}}^{(t)}\right)$ (saved as `model_tail_avg.pth`, denoted **`Pre-Proj Rollout-BF16`**) and the unrounded `float32` projected linear tail average $\overline{W}_{\text{lin}}^{\text{FP32}} = \frac{1}{25}\sum_{t=26}^{50} W_{\text{proj}}^{(t)}$ (**`Pre-Proj LinFP32`**).
3. **Final Post-Averaging Spectral Projection (`Post-Proj LinFP32` vs. `Post-Proj BF16`):** Because the average of 25 rank-$r$ matrices along a rotating trajectory has rank $> r$, `train_osf_rlvr.py` (lines 423–444) applies one final spectral projection $\mathcal{P}$ to the averaged displacement $\overline{W}_{\text{lin}}^{\text{FP32}} - W_0$, saving both the unrounded `float32` linear dictionary $W_{\text{eval}}^{\text{LinFP32}} = W_0 + \mathcal{P}(\overline{W}_{\text{lin}}^{\text{FP32}} - W_0)$ (`model_tail_avg_fp32_linear.pth`, denoted **`Post-Proj LinFP32`**, which strictly satisfies $\operatorname{rank}(\Delta W) \le 16$ for `OSF-Top16`) and its `bfloat16`-rounded counterpart $W_{\text{eval}}^{\text{BF16}} = \operatorname{RNE}_{\text{bf16}}(W_{\text{eval}}^{\text{LinFP32}})$ (`model_tail_avg_proj_bf16.pth`, denoted **`Post-Proj BF16`**).

We evaluate **Online Subspace-Filtered Policy Optimization (`OSF-RLVR`)** across **three independent training seeds (`Seed 102`, `Seed 42`, `Seed 137`)** and **six evaluation panels** (`Val350`, `MATH-500`, `Sec-500`, `Conf-500`, untouched `Fresh-500`, and out-of-distribution `Ext240`) at **strictly uniform `batch_size=50`**, separating **Table 5A (`online_osf_factorial`: uniform `Post-Proj BF16` / `bf16` evaluation across all 6 panels on Seeds 102 & 42 [$N=5,180$] plus Seed 137 `Standard-BF16` [$N=1,240$])** from **Table 5B (`online_osf_factorial_postproj_linfp32`: strictly uniform `Post-Proj LinFP32` evaluation via `FP32LinearInBF16Net` across all 5 arms on Seed 102 [$N=1,240$], the 3 `float32`-master arms on Seed 137 [$N=1,240$], and all 5 arms on Seed 42 `Fresh-500` [$N=500$])**:
1. **`Standard-BF16` (In-Place `bfloat16` RNE Baseline, `0` Training SVDs):** Standard `Tessera-Sym` without `float32` master weights ($\mathrm{eRank}_{\text{energy}} = 704.2$, $\mathrm{eRank}_{\text{ampl}} = 904.0$ on S102; $707.7 / 904.2$ on S42), evaluated both in `bf16` (Table 5A) and through the `FP32LinearInBF16Net` operator (Table 5B) at `batch_size=50`.
2. **`Unfiltered-FP32-Master` ($\mathcal{P} = I$, `0` Training SVDs):** Online `float32`-master AdamW with no spectral filtering (`Post-Proj LinFP32` $\mathrm{eRank}_{\text{energy}} = 156.3$, $\mathrm{eRank}_{\text{ampl}} = 642.0$ on S102; $163.6 / 636.1$ on S42; `Post-Proj BF16` $\mathrm{eRank}_{\text{energy}} = 311.0 / 326.8$).
3. **`OSF-Top16-FP32-Master` ($\mathcal{P}_{\le 16}$, `4,312` Training SVDs on S102):** Online projection onto the leading rank-16 singular subspace $\mathcal{P}_{\le 16}(\Delta W_t) = \sum_{i=1}^{16} \sigma_i u_i v_i^\top$ (`Post-Proj LinFP32` $\mathrm{eRank}_{\text{energy}} = 12.05$, $\mathrm{eRank}_{\text{ampl}} = 15.74$ on S102; $11.86 / 15.67$ on S42; `Post-Proj BF16` $\mathrm{eRank}_{\text{energy}} = 117.38 / 118.85$ due to `bfloat16` RNE rounding of the rank-16 update).
4. **`OSF-Tail16-FP32-Master` ($\mathcal{P}_{> 16} = I - \mathcal{P}_{\le 16}$, `4,312` Training SVDs on S102):** Online projection onto the orthogonal rank-$>16$ complement $\mathcal{P}_{> 16}(\Delta W_t) = \sum_{i > 16} \sigma_i u_i v_i^\top$, **filtering out the top-16 principal singular directions** while retaining the 1,008 orthogonal tail modes (`Post-Proj LinFP32` $\mathrm{eRank}_{\text{energy}} = 356.24$, $\mathrm{eRank}_{\text{ampl}} = 730.55$ on S102; $376.29 / 749.91$ on S42; `Post-Proj BF16` $\mathrm{eRank}_{\text{energy}} = 571.15 / 584.17$).
5. **`OSF-Top16-NormMatched` ($\alpha_s \cdot \Delta W_{\text{OSF-Top16}}$):** Rescales the trained `OSF-Top16-FP32-Master` displacement by $\alpha_s = \|\Delta W_{\text{Standard-BF16},s}\|_F / \|\Delta W_{\text{OSF-Top16},s}\|_F$ ($\alpha_{102} = 0.3958, \alpha_{42} = 0.3890$).

| Online RLVR Arm (`Uniform Post-Proj BF16`, `bs=50`) | Seed | $\|\Delta W_{\text{BF16}}\|_F$ | $\mathrm{eRank}_{\text{E}} / \mathrm{eRank}_{\text{A}}$ | $\mathrm{KL}_{\text{ref}}$ / $\mathrm{KL}_{\text{greedy}}$ | $\eta_{\text{query}}$ | **4-Panel MATH ($N=1,850$)** | **Untouched `Fresh-500`** | **Untouched `Ext240`** | **6-Panel Total ($N=2,590$/$5,180$)** |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| **`Standard-BF16`** (In-Place `bfloat16` RNE) | S102 | $0.3322$ | $704.2 / 904.0$ | $0.0240$ / $0.0232$ | $0.5085$ | $848/1,850$ ($45.84\%$) | $257/500$ ($51.40\%$) | $149/240$ ($62.08\%$) | $1,254/2,590$ ($48.42\%$) |
| **`Standard-BF16`** (In-Place `bfloat16` RNE) | S42 | $0.3189$ | $707.7 / 904.2$ | $0.0209$ / $0.0202$ | $0.5331$ | $838/1,850$ ($45.30\%$) | $254/500$ ($50.80\%$) | $144/240$ ($60.00\%$) | $1,236/2,590$ ($47.72\%$) |
| **`Standard-BF16`** (In-Place `bfloat16` RNE) | S137 | $0.3347$ | --- | $0.0241$ / --- | --- | `M500`: $239/500$ ($47.80\%$) | $243/500$ ($48.60\%$) | $139/240$ ($57.92\%$) | `3-Panel`: $621/1,240$ ($50.08\%$) |
| **`Standard-BF16` (2-Seed `BF16` Synthesis)** | **S102+S42** | **$0.3256$** | **$706.0 / 904.1$** | **$0.0224$ / $0.0217$** | **$0.5208$** | **$1,686/3,700$ ($45.57\%$)** | **$511/1,000$ ($51.10\%$)** | **$293/480$ ($61.04\%$)** | **$2,490/5,180$ ($48.07\%$)** |
| **`Unfiltered-FP32-Master`** ($\mathcal{P} = I$) | S102 | $1.3529$ | $311.0 / 798.8$ | $0.3112$ / $0.2500$ | $0.3519$ | $570/1,850$ ($30.81\%$) | $172/500$ ($34.40\%$) | $98/240$ ($40.83\%$) | $840/2,590$ ($32.43\%$) |
| **`Unfiltered-FP32-Master`** ($\mathcal{P} = I$) | S42 | $1.2949$ | $326.8 / 801.0$ | $0.2792$ / $0.2040$ | $0.3709$ | $511/1,850$ ($27.62\%$) | $146/500$ ($29.20\%$) | $106/240$ ($44.17\%$) | $763/2,590$ ($29.46\%$) |
| **`Unfiltered-FP32-Master` (2-Seed `BF16` Synthesis)** | **S102+S42** | **$1.3239$** | **$318.9 / 799.9$** | **$0.2952$ / $0.2270$** | **$0.3614$** | **$1,081/3,700$ ($29.22\%$)** | **$318/1,000$ ($31.80\%$)** | **$204/480$ ($42.50\%$)** | **$1,603/5,180$ ($30.95\%$)** |
| **`OSF-Top16-FP32-Master`** ($\mathcal{P}_{\le 16}$) | S102 | $0.7904$ | $117.4 / 754.2$ | $0.2127$ / $0.1431$ | $0.3413$ | $647/1,850$ ($34.97\%$) | $206/500$ ($41.20\%$) | $121/240$ ($50.42\%$) | $974/2,590$ ($37.61\%$) |
| **`OSF-Top16-FP32-Master`** ($\mathcal{P}_{\le 16}$) | S42 | $0.7681$ | $118.8 / 756.3$ | $0.2115$ / $0.1634$ | $0.3592$ | $662/1,850$ ($35.78\%$) | $194/500$ ($38.80\%$) | $117/240$ ($48.75\%$) | $973/2,590$ ($37.57\%$) |
| **`OSF-Top16-FP32-Master` (2-Seed `BF16` Synthesis)** | **S102+S42** | **$0.7792$** | **$118.1 / 755.3$** | **$0.2121$ / $0.1532$** | **$0.3502$** | **$1,309/3,700$ ($35.38\%$)** | **$400/1,000$ ($40.00\%$)** | **$238/480$ ($49.58\%$)** | **$1,947/5,180$ ($37.59\%$)** |
| **`OSF-Tail16-FP32-Master`** ($\mathcal{P}_{> 16}$) | S102 | $0.8185$ | $571.2 / 858.8$ | $0.0200$ / $0.0196$ | $0.3254$ | $803/1,850$ ($43.41\%$) | $247/500$ ($49.40\%$) | $132/240$ ($55.00\%$) | $1,182/2,590$ ($45.64\%$) |
| **`OSF-Tail16-FP32-Master`** ($\mathcal{P}_{> 16}$) | S42 | $0.8033$ | $584.2 / 864.0$ | $0.0238$ / $0.0188$ | $0.3380$ | **$844/1,850$ ($45.62\%$)** | **$253/500$ ($50.60\%$)** | $143/240$ ($59.58\%$) | **$1,240/2,590$ ($47.88\%$)** |
| **`OSF-Tail16-FP32-Master` (2-Seed `BF16` Synthesis)** | **S102+S42** | **$0.8109$** | **$577.7 / 861.4$** | **$0.0219$ / $0.0192$** | **$0.3317$** | **$1,647/3,700$ ($44.51\%$)** | **$500/1,000$ ($50.00\%$)** | **$275/480$ ($57.29\%$)** | **$2,422/5,180$ ($46.76\%$)** |
| **`OSF-Top16-NormMatched`** ($\alpha_s \cdot \Delta W_{\le 16}$) | S102 | $0.2312$ | $313.6 / 824.2$ | $0.0377$ / $0.0392$ | $0.4131$ | $842/1,850$ ($45.51\%$) | $261/500$ ($52.20\%$) | $147/240$ ($61.25\%$) | $1,250/2,590$ ($48.26\%$) |
| **`OSF-Top16-NormMatched`** ($\alpha_s \cdot \Delta W_{\le 16}$) | S42 | $0.2184$ | $318.9 / 824.2$ | $0.0282$ / $0.0276$ | $0.4325$ | $841/1,850$ ($45.46\%$) | $264/500$ ($52.80\%$) | $144/240$ ($60.00\%$) | $1,249/2,590$ ($48.22\%$) |
| **`OSF-Top16-NormMatched` (2-Seed `BF16` Synthesis)** | **S102+S42** | **$0.2248$** | **$316.2 / 824.2$** | **$0.0330$ / $0.0334$** | **$0.4228$** | **$1,683/3,700$ ($45.49\%$)** | **$525/1,000$ ($52.50\%$)** | **$291/480$ ($60.62\%$)** | **$2,499/5,180$ ($48.24\%$)** |

*Table 5A: Strictly uniform `Post-Proj BF16` (`model_tail_avg_proj_bf16.pth`, `batch_size=50`) online evaluation across all six panels on Seeds 102 and 42 ($N=5,180$ total evaluations) plus Seed 137 `Standard-BF16` ($N=1,240$, 3-seed 3-panel `BF16` total $= 1,891/3,720 = 50.83\%$), reporting fixed-reference conditional KL $\mathrm{KL}_{\text{ref}}$ (`records/kl_reference_trajectories_64.json`) and own-policy greedy-completion conditional KL $\mathrm{KL}_{\text{greedy}}$ on each checkpoint+operator's own `MATH-500[:64]` completions (`records/extended_spectral_and_onpolicy_kl.json`).*

| Online RLVR Arm (`Uniform Post-Proj LinFP32`, `bs=50`) | Seed | $\|\Delta W_{\text{FP32}}\|_F$ | $\mathrm{eRank}_{\text{E}} / \mathrm{eRank}_{\text{A}}$ | $\mathrm{KL}_{\text{ref}}$ / $\mathrm{KL}_{\text{greedy}}$ | $\text{NSS}$ (%) | `MATH-500` ($N=500$) | **Untouched `Fresh-500`** | **Untouched `Ext240`** | **Matched 3-Panel (`M500+Fresh+Ext`)** |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| **`Standard-BF16`** (via `FP32LinearInBF16Net`) | S102 | $0.3322$ | $704.2 / 904.0$ | $0.0239$ / $0.0230$ | $0.0017\%$ | $243/500$ ($48.60\%$) | $253/500$ ($50.60\%$) | $146/240$ ($60.83\%$) | **$642/1,240$ ($51.77\%$)** |
| **`Standard-BF16`** (via `FP32LinearInBF16Net`) | S42 | $0.3189$ | $707.7 / 904.2$ | $0.0209$ / $0.0197$ | $0.0017\%$ | --- | $250/500$ ($50.00\%$) | --- | `2-Seed Fresh-500`: **$503/1,000$ ($50.30\%$)** |
| **`Unfiltered-FP32-Master`** ($\mathcal{P} = I$) | S102 | $1.3391$ | $156.3 / 642.0$ | $0.3813$ / $0.2766$ | $0.0055\%$ | $139/500$ ($27.80\%$) | $148/500$ ($29.60\%$) | $83/240$ ($34.58\%$) | $370/1,240$ ($29.84\%$) |
| **`Unfiltered-FP32-Master`** ($\mathcal{P} = I$) | S137 | $1.3165$ | $645.3$ (ampl) | $0.2852$ / --- | $0.0054\%$ | $49/500$ ($9.80\%$) | $71/500$ ($14.20\%$) | $43/240$ ($17.92\%$) | $163/1,240$ ($13.15\%$) |
| **`Unfiltered-FP32-Master`** ($\mathcal{P} = I$) | S42 | $1.2869$ | $163.6 / 636.1$ | $0.3344$ / $0.2449$ | $0.0054\%$ | --- | $134/500$ ($26.80\%$) | --- | `2-Seed S102+S42 Fresh`: $282/1,000$ ($28.20\%$) |
| **`Unfiltered-FP32` (`LinFP32` Pooled Subsets)** | **Matched** | **$1.3142$** | --- | **$0.3336$ / $0.2608$** | **$0.0054\%$** | `2-Seed`: $188/1,000$ ($18.80\%$) | **`3-Seed`: $353/1,500$ ($23.53\%$)** | `2-Seed`: $126/480$ ($26.25\%$) | **`2-Seed S102+S137`: $533/2,480$ ($21.49\%$)** |
| **`OSF-Top16-FP32-Master`** ($\mathcal{P}_{\le 16}$) | S102 | $0.8393$ | $12.05 / 15.74$ | $0.2828$ / $0.1979$ | $0.0037\%$ | $157/500$ ($31.40\%$) | $187/500$ ($37.40\%$) | $113/240$ ($47.08\%$) | $457/1,240$ ($36.85\%$) |
| **`OSF-Top16-FP32-Master`** ($\mathcal{P}_{\le 16}$) | S137 | $0.8412$ | $15.24$ (ampl) | $0.1869$ / --- | $0.0037\%$ | $133/500$ ($26.60\%$) | $132/500$ ($26.40\%$) | $79/240$ ($32.92\%$) | $344/1,240$ ($27.74\%$) |
| **`OSF-Top16-FP32-Master`** ($\mathcal{P}_{\le 16}$) | S42 | $0.8199$ | $11.86 / 15.67$ | $0.3215$ / $0.2244$ | $0.0036\%$ | --- | $143/500$ ($28.60\%$) | --- | `2-Seed S102+S42 Fresh`: $330/1,000$ ($33.00\%$) |
| **`OSF-Top16` (`LinFP32` Pooled Subsets)** | **Matched** | **$0.8335$** | --- | **$0.2637$ / $0.2112$** | **$0.0036\%$** | `2-Seed`: $290/1,000$ ($29.00\%$) | **`3-Seed`: $462/1,500$ ($30.80\%$)** | `2-Seed`: $192/480$ ($40.00\%$) | **`2-Seed S102+S137`: $801/2,480$ ($32.30\%$)** |
| **`OSF-Tail16-FP32-Master`** ($\mathcal{P}_{> 16}$) | S102 | $0.8654$ | $356.2 / 730.6$ | $0.0279$ / $0.0285$ | $0.0037\%$ | $224/500$ ($44.80\%$) | $258/500$ ($51.60\%$) | $123/240$ ($51.25\%$) | $605/1,240$ ($48.79\%$) |
| **`OSF-Tail16-FP32-Master`** ($\mathcal{P}_{> 16}$) | S137 | $0.9167$ | $701.7$ (ampl) | $0.0272$ / --- | $0.0038\%$ | $221/500$ ($44.20\%$) | $248/500$ ($49.60\%$) | $136/240$ ($56.67\%$) | $605/1,240$ ($48.79\%$) |
| **`OSF-Tail16-FP32-Master`** ($\mathcal{P}_{> 16}$) | S42 | $0.8516$ | $376.3 / 749.9$ | $0.0345$ / $0.0248$ | $0.0035\%$ | --- | **$263/500$ ($52.60\%$)** | --- | `2-Seed S102+S42 Fresh`: **$521/1,000$ ($52.10\%$)** |
| **`OSF-Tail16` (`LinFP32` Pooled Subsets)** | **Matched** | **$0.8779$** | --- | **$0.0299$ / $0.0266$** | **$0.0037\%$** | `2-Seed`: **$445/1,000$ ($44.50\%$)** | **`3-Seed`: $769/1,500$ ($51.27\%$)** | `2-Seed`: **$259/480$ ($53.96\%$)** | **`2-Seed S102+S137`: $1,210/2,480$ ($48.79\%$)** |
| **`OSF-Top16-NormMatched`** ($\alpha_s \cdot \Delta W_{\le 16}$) | S102 | $0.3322$ | $12.05 / 15.74$ | $0.0831$ / $0.0601$ | $0.0017\%$ | $212/500$ ($42.40\%$) | $254/500$ ($50.80\%$) | $142/240$ ($59.17\%$) | **$608/1,240$ ($49.03\%$)** |
| **`OSF-Top16-NormMatched`** ($\alpha_s \cdot \Delta W_{\le 16}$) | S42 | $0.3189$ | $11.86 / 15.67$ | $0.0631$ / $0.0551$ | $0.0017\%$ | --- | $238/500$ ($47.60\%$) | --- | `2-Seed Fresh-500`: **$492/1,000$ ($49.20\%$)** |

*Table 5B: Strictly uniform `Post-Proj LinFP32` (`FP32LinearInBF16Net`, `batch_size=50`) online evaluation across all available matched seed-by-panel-by-operator subsets without cross-dtype substitution (`online_osf_factorial_postproj_linfp32` in `records/osf_and_precision_factorial.json`).*

Because the multi-seed pooled evaluations evaluate $S=2$ or $S=3$ correlated training seeds on the **same** unique questions, all multi-seed comparisons below use **question-clustered inference conditional on the realized training seeds** across the $M$ unique questions alongside between-seed standard errors ($\mathrm{SE}_{\text{seed}}$), restricted strictly to the matched seed-by-panel-by-operator subsets in Tables 5A and 5B:
1. **`bfloat16` Dead-Zone Attenuation as an Implicit Step-Size Damper Under Matched Operators:** Under uniform `Post-Proj BF16` across all 6 panels on Seeds 102 & 42 ($N=5,180$, `batch_size=50`, Table 5A), switching from `Standard-BF16` to `Unfiltered-FP32-Master` at $\eta = 10^{-5}$ inflates reference and own-completion greedy Policy KL ($\mathrm{KL}_{\text{ref}} = 0.2952$ vs. $0.0224$, $\mathrm{KL}_{\text{greedy}} = 0.2270$ vs. $0.0217\text{ nats/tok}$) and drops 2-seed pooled accuracy by **$-17.12\text{ pp}$ (`1,603` vs. `2,490/5,180`, 95% CI $[-18.58, -15.66]\text{ pp}$, $\mathrm{SE}_{\text{cl}} = 0.74\text{ pp}$, $\mathrm{SE}_{\text{seed}} = 1.14\text{ pp}$, $p_{\text{cl}} = 5.12 \times 10^{-117}$; and $-16.35\text{ pp}$ on 4-panel within-MATH [$N=3,700$], $p_{\text{cl}} = 8.00 \times 10^{-81}$)**. Under strictly uniform `Post-Proj LinFP32` (`FP32LinearInBF16Net`, `batch_size=50`, Table 5B), comparing `Standard-BF16` directly against `Unfiltered-FP32-Master` on the matched Seed 102 3-panel set ($N=1,240$) yields a **$-21.94\text{ pp}$ drop (`370/1,240 = 29.84%` vs. `642/1,240 = 51.77%`, 95% CI $[-24.77, -19.10]\text{ pp}$, $+55/-327$, $p = 3.33 \times 10^{-48}$)**, and on the matched 2-seed (`S102 + S42`, $N=1,000$) `Fresh-500` `LinFP32` set yields a **$-22.10\text{ pp}$ drop (`282/1,000 = 28.20%` vs. `503/1,000 = 50.30%`, 95% CI $[-25.47, -18.73]\text{ pp}$, $\mathrm{SE}_{\text{cl}} = 1.72\text{ pp}$, $p_{\text{cl}} = 8.90 \times 10^{-38}$)**.
2. **Why `OSF-Tail16` Outperforms `OSF-Top16` by $+9.17\text{ pp}$ in `BF16` ($N=5,180$) and $+16.49\text{ pp}$ in `LinFP32` ($N=2,480$) at Matched Frobenius Norm:** Across all seeds, `OSF-Tail16` has identical Normalized Spectral Shift to `OSF-Top16` ($\text{NSS} = 0.0037\%$ vs. $0.0036\%$), yet dispersing $\|\Delta W\|_F \approx 0.86$ across 1,008 tail modes keeps both $\mathrm{KL}_{\text{ref}}$ ($0.0299$ vs. $0.2637$) and own-completion $\mathrm{KL}_{\text{greedy}}$ ($0.0266$ vs. $0.2112\text{ nats/tok}$) an order of magnitude smaller. Under uniform `Post-Proj BF16` across 6 panels on Seeds 102 & 42 ($N=5,180$, Table 5A), `OSF-Tail16` beats `OSF-Top16` by **$+475$ items (`+9.17 pp`, 95% CI $[+7.81, +10.53]\text{ pp}$, $\mathrm{SE}_{\text{cl}} = 0.69\text{ pp}$, $\mathrm{SE}_{\text{seed}} = 1.14\text{ pp}$, $p_{\text{cl}} = 7.39 \times 10^{-40}$)** and beats `Unfiltered-FP32` by **$+819$ items (`+15.81 pp`, $p_{\text{cl}} = 7.93 \times 10^{-99}$)**. Under uniform `Post-Proj LinFP32` across the matched 2-seed (`S102 + S137`, $N=2,480$, Table 5B) 3-panel set, `OSF-Tail16` (`1,210/2,480 = 48.79%`) beats `OSF-Top16` (`801/2,480 = 32.30%`) by **$+409$ items (`+16.49 pp`, 95% CI $[+14.39, +18.60]\text{ pp}$, $\mathrm{SE}_{\text{cl}} = 1.07\text{ pp}$, $p_{\text{cl}} = 3.99 \times 10^{-53}$)** and beats `Unfiltered-FP32` (`533/2,480 = 21.49%`) by **$+677$ items (`+27.30 pp`, $p_{\text{cl}} = 1.78 \times 10^{-129}$)**; on the 3-seed (`S102 + S42 + S137`, $N=1,500$) `Fresh-500` `LinFP32` set, `OSF-Tail16` (`769/1,500 = 51.27%`) beats `OSF-Top16` (`462/1,500 = 30.80%`) by **`+307` items (`+20.47 pp`, 95% CI $[+17.70, +23.23]\text{ pp}$, $p_{\text{cl}} = 9.86 \times 10^{-48}$)** and `Unfiltered-FP32` (`353/1,500 = 23.53%`) by **`+416` items (`+27.73 pp`, $p_{\text{cl}} = 8.79 \times 10^{-70}$)**. Relative to `Standard-BF16` under matched operators (`batch_size=50`), `OSF-Tail16` shows **an unresolved difference of $+1.80\text{ pp}$ (`521` vs. `503/1,000`, 95% CI $[-1.01, +4.61]\text{ pp}$, $p_{\text{cl}} = 0.210$) on 2-seed `Fresh-500` `LinFP32`**, **an unresolved difference of $-1.05\text{ pp}$ (`1,647` vs. `1,686/3,700`, $p_{\text{cl}} = 0.138$) on 2-seed 4-panel `BF16`**, **an unresolved difference of $-1.10\text{ pp}$ (`500` vs. `511/1,000`, $p_{\text{cl}} = 0.409$) on 2-seed `Fresh-500` `BF16`**, and a slight deficit of **$-1.31\text{ pp}$ (`2,422` vs. `2,490/5,180`, $p_{\text{cl}} = 0.0282$) on 6-panel `BF16`** ($-2.98\text{ pp}$, `605` vs. `642/1,240`, $p = 0.0257$ on Seed 102 3-panel `LinFP32`).
3. **Disentangling Pre/Post-Final-Projection, Training (`BF16`) vs. Evaluation (`LinFP32`) Precision, Own-Completion Greedy KL, and Synchronization Schedules (`Seed 102` Controls):** To verify that the `OSF-Tail16 > OSF-Top16` reversal is not an artifact of the step-50 post-averaging re-projection, of evaluating in `LinFP32` while rollouts used `BF16`, or of keeping $W_{\text{master}}^{(t)}$ unprojected during steps 1–49, we evaluate Seed 102 controls (`records/osf_and_precision_factorial.json` and `records/extended_spectral_and_onpolicy_kl.json`, where every checkpoint+operator's `KL_greedy` is computed on its own `MATH-500[:64]` greedy completions with verified SHA-256 provenance):
   - **`Pre-Proj Rollout-BF16` (`model_tail_avg.pth`, exact `bfloat16` rollout policy without step-50 re-projection):** `OSF-Tail16` (`251/500 = 50.20%` on `Fresh-500`, $\mathrm{KL}_{\text{ref}} = 0.0376$, own-completion $\mathrm{KL}_{\text{greedy}} = 0.0385$) beats `OSF-Top16` (`203/500 = 40.60%`, $\mathrm{KL}_{\text{ref}} = 0.2067$, $\mathrm{KL}_{\text{greedy}} = 0.1496$) by **`+48` items (`+9.60 pp`, $p = 2.02 \times 10^{-5}$)** and beats `Unfiltered-FP32` (`163/500 = 32.60%`, $\mathrm{KL}_{\text{ref}} = 0.3184$, $\mathrm{KL}_{\text{greedy}} = 0.2486$) by **`+88` items (`+17.60 pp`)** (and on Seed 42 `Pre-Proj Rollout-BF16`, own-completion $\mathrm{KL}_{\text{greedy}}$ is $0.0352$ for `OSF-Tail16` vs. $0.1660$ for `OSF-Top16` and $0.2116$ for `Unfiltered-FP32`).
   - **`Post-Proj BF16` (`model_tail_avg_proj_bf16.pth`):** `OSF-Tail16` (**`602/1,240 = 48.55%`**, $\mathrm{KL}_{\text{ref}} = 0.0200, \mathrm{KL}_{\text{greedy}} = 0.0196$) beats `OSF-Top16` (**`493/1,240 = 39.76%`**, $\mathrm{KL}_{\text{ref}} = 0.2127, \mathrm{KL}_{\text{greedy}} = 0.1431$) by **`+109` items (`+8.79 pp` on $N=1,240$; $+8.20\text{ pp}$ on `Fresh-500`)**.
   - **`Post-Proj LinFP32` (`model_tail_avg_fp32_linear.pth`):** `OSF-Tail16` (`605/1,240 = 48.79%`, $\mathrm{KL}_{\text{ref}} = 0.0279, \mathrm{KL}_{\text{greedy}} = 0.0285$) beats `OSF-Top16` (`457/1,240 = 36.85%`, $\mathrm{KL}_{\text{ref}} = 0.2828, \mathrm{KL}_{\text{greedy}} = 0.1979$) by **`+148` items (`+11.94 pp` on $N=1,240$, $p=1.15 \times 10^{-16}$; $+14.20\text{ pp}$ on `Fresh-500`)**.
   - **Every-Step Synchronization (No Writeback, `9,996` SVDs):** Projecting every step (50 steps) instead of only active steps (21 steps) yields `OSF-Tail16-EveryStepSync` (`621/1,240 = 50.08%`, $\mathrm{KL}_{\text{ref}}=0.0343, \mathrm{KL}_{\text{greedy}}=0.0291$) beating `OSF-Top16-EveryStepSync` (`425/1,240 = 34.27%`, $\mathrm{KL}_{\text{ref}}=0.2715, \mathrm{KL}_{\text{greedy}}=0.1968$) by **$+196$ items ($+15.81\text{ pp}$, $p=5.71 \times 10^{-29}$)**. The pure sync-schedule effect is an unresolved difference of $+1.29\text{ pp}$ ($p=0.337$) on `Tail16` and $-2.58\text{ pp}$ ($p=0.054$) on `Top16`.
   - **Strict Every-Step Master Writeback (`OSF-Top16-Writeback` vs. `OSF-Tail16-Writeback`, $W_{\text{master}}^{(t)} \leftarrow W_0 + \mathcal{P}(\Delta W_{\text{raw}}^{(t)})$, `9,996` SVDs):** Overwriting $W_{\text{master}}^{(t)}$ with its rank-16 projection at every step (`OSF-Top16-Writeback`, Table 5C) drives severe KL overshoot ($\mathrm{KL}_{\text{ref}} = 0.5336, \mathrm{KL}_{\text{greedy}} = 0.4820\text{ nats/tok}$), collapsing accuracy to **`378/1,240 = 30.48%`**. Conversely, writing back the rank-$>16$ complement at every step (`OSF-Tail16-Writeback`) maintains high effective rank and tight trust-region stability ($\mathrm{KL}_{\text{ref}} = 0.0188, \mathrm{KL}_{\text{greedy}} = 0.0173\text{ nats/tok}$), achieving **`648/1,240 = 52.26%`**—outperforming `OSF-Top16-Writeback` by **`+270` items (`+21.77 pp`, 95% CI $[+18.92, +24.63]\text{ pp}$, $p = 3.86 \times 10^{-47}$)**. The pure writeback effect is $+2.18\text{ pp}$ ($p=0.081$) on `Tail16` and $-3.79\text{ pp}$ ($p=0.0077$) on `Top16`.
4. **Within-Arm Causal Resolution via Norm Matching (`OSF-Top16-NormMatched`):** Scaling down `OSF-Top16` by $\alpha_s \approx 0.39$ cuts $\mathrm{KL}_{\text{ref}}$ (`LinFP32` $0.3022 \to 0.0731$; `BF16` $0.2121 \to 0.0330\text{ nats/tok}$) and own-completion $\mathrm{KL}_{\text{greedy}}$ (`LinFP32` $0.2112 \to 0.0576$; `BF16` $0.1532 \to 0.0334\text{ nats/tok}$), rescuing **`+552` items (`+10.66 pp`, 95% CI $[+9.38, +11.93]\text{ pp}$, $p_{\text{cl}} = 3.04 \times 10^{-60}$) on 2-seed 6-panel `Post-Proj BF16` ($N=5,180$; $+374$ items [$+10.11\text{ pp}$, $p_{\text{cl}} = 3.65 \times 10^{-40}$] on 4-panel within-MATH [$N=3,700$]) and `+162` items (`+16.20 pp`, 95% CI $[+13.19, +19.21]\text{ pp}$, $p_{\text{cl}} = 5.32 \times 10^{-26}$) on 2-seed `Fresh-500` `Post-Proj LinFP32` ($N=1,000$; $+151$ items [$+12.18\text{ pp}$, $p = 1.13 \times 10^{-19}$] on Seed 102 3-panel `LinFP32`)**, resulting in **an unresolved difference** relative to `Standard-BF16` under matched operators (`+0.17 pp` [$+9$ items, $p_{\text{cl}} = 0.762$] on 6-panel `BF16`, `-0.08 pp` [$-3$ items, $p_{\text{cl}} = 0.906$] on 4-panel `BF16`, and `-1.10 pp` [$-11$ items, $p_{\text{cl}} = 0.438$] on 2-seed `Fresh-500` `LinFP32`).

### 2.5 Head-to-Head Comparison with `Ares` (Published Eq. 3 Energy Rule & Amplitude Variant), `GeoRA`, `PiSSA`, `MiLoRA`, Every-Step Master-Writeback `OSF`, Matched Supplied `Vanilla-GRPO` Controls (50 & 100 Steps), Online Learning-Rate Sweep, and Prespecified Scale Grid
To position our findings directly against contemporary spectral RLVR methods and eliminate any objective-shaping, learning-rate, horizon, or master-writeback confounds (`code/run_stage2_ares_geora_vanilla_and_lrsweep.py`, `code/run_reviewer_interventions_and_audits.py`, and `code/run_stage1_kl_fresh500_and_scale_grid.py`), we execute five experimental blocks on Seed 102 evaluated across `MATH-500`, untouched `Fresh-500`, and `Ext240` ($N=1,240$ items per arm; Table 5C):

| Category & Experimental Arm (Seed 102) | Train SVDs | $\|\Delta W\|_F$ | $\mathrm{KL}_{\text{ref}}$ / $\mathrm{KL}_{\text{greedy}}$ | $\text{NSS}$ (%) | $W_0\text{-Top16}$ (%) | `MATH-500` ($500$) | **Untouched `Fresh-500`** | `Ext240` ($240$) | **Pooled N=1,240 (%)** |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| **`Ares-Eq3-BF16` (Chen et al. [22] Eq. 3 $p_i \propto \sigma_i^2$, $\eta=5\times 10^{-6}$, $r_t\approx 10.4$)** | $4,116$ | $0.1344$ | $0.0072$ / $0.0077$ | $0.0010\%$ | $0.98\%$ | **$235$ ($47.00\%$)** | $253$ ($50.60\%$) | $147$ ($61.25\%$) | **$635/1,240$ ($51.21\%$)** |
| **`Ares-Eq3-FP32` (Chen et al. [22] Eq. 3 $p_i \propto \sigma_i^2$, $\eta=10^{-5}$, $r_t\approx 10.6$)** | $4,116$ | $1.2988$ | $0.5667$ / $0.3889$ | $0.0053\%$ | $1.49\%$ | $120$ ($24.00\%$) | $128$ ($25.60\%$) | $96$ ($40.00\%$) | **$344/1,240$ ($27.74\%$)** |
| **`Ares-Eq3-FP32` (Chen et al. [22] Eq. 3 $p_i \propto \sigma_i^2$, $\eta=5\times 10^{-6}$, $r_t\approx 10.5$)** | $4,116$ | $0.6596$ | $0.1148$ / $0.1172$ | $0.0030\%$ | $1.44\%$ | $177$ ($35.40\%$) | $193$ ($38.60\%$) | $123$ ($51.25\%$) | **$493/1,240$ ($39.76\%$)** |
| **`Ares-Ampl-BF16` ($q_i \propto \sigma_i$ Variant, $\eta=5\times 10^{-6}$, $r_t\approx 39.7$)** | $4,116$ | $0.1343$ | $0.0066$ / --- | $0.0009\%$ | $0.98\%$ | $223$ ($44.60\%$) | $250$ ($50.00\%$) | $146$ ($60.83\%$) | **$619/1,240$ ($49.92\%$)** |
| **`Ares-Ampl-FP32` ($q_i \propto \sigma_i$ Variant, $\eta=10^{-5}$, $r_t\approx 37.4$)** | $4,116$ | $1.2815$ | $0.2198$ / --- | $0.0053\%$ | $1.48\%$ | $140$ ($28.00\%$) | $174$ ($34.80\%$) | $100$ ($41.67\%$) | **$414/1,240$ ($33.39\%$)** |
| **`Ares-Ampl-FP32` ($q_i \propto \sigma_i$ Variant, $\eta=5\times 10^{-6}$, $r_t\approx 39.1$)** | $4,116$ | $0.6731$ | $0.1505$ / --- | $0.0030\%$ | $1.49\%$ | $197$ ($39.40\%$) | $220$ ($44.00\%$) | $128$ ($53.33\%$) | **$545/1,240$ ($43.95\%$)** |
| **`OSF-Top16-Writeback` (Strict Every-Step Master $\mathcal{P}_{\le 16}$, $\eta=10^{-5}$)** | $9,996$ | $0.7912$ | $0.5336$ / $0.4820$ | $0.0035\%$ | $1.62\%$ | $126$ ($25.20\%$) | $159$ ($31.80\%$) | $93$ ($38.75\%$) | **$378/1,240$ ($30.48\%$)** |
| **`OSF-Tail16-Writeback` (Strict Every-Step Master $\mathcal{P}_{> 16}$, $\eta=10^{-5}$)** | $9,996$ | $0.4311$ | $0.0188$ / $0.0173$ | $0.0019\%$ | $1.17\%$ | $234$ ($46.80\%$) | **$266$ ($53.20\%$)** | $148$ ($61.67\%$) | **$648/1,240$ ($52.26\%$)** |
| **`OSF-Top16-EveryStepSync` (No Writeback)** | $9,996$ | $0.8281$ | $0.2715$ / $0.1968$ | $0.0036\%$ | $1.64\%$ | $148$ ($29.60\%$) | $166$ ($33.20\%$) | $111$ ($46.25\%$) | **$425/1,240$ ($34.27\%$)** |
| **`OSF-Tail16-EveryStepSync` (No Writeback)** | $9,996$ | $0.9078$ | $0.0343$ / $0.0291$ | $0.0041\%$ | $1.15\%$ | $228$ ($45.60\%$) | $252$ ($50.40\%$) | $141$ ($58.75\%$) | **$621/1,240$ ($50.08\%$)** |
| **`GeoRA-r16` (Zhang et al. [24], $\rho=0.20$, `LinFP32`)** | $196$ | $0.2575$ | $0.0108$ / --- | $0.0014\%$ | $0.95\%$ | $231$ ($46.20\%$) | $251$ ($50.20\%$) | $140$ ($58.33\%$) | **$622/1,240$ ($50.16\%$)** |
| **`PiSSA-r16` (Meng et al. [25] / Yin et al. [23], $W_0$ Top-16, `LinFP32`)** | $196$ | $0.3828$ | $0.0209$ / --- | $0.0027\%$ | **$47.59\%$** | $221$ ($44.20\%$) | $250$ ($50.00\%$) | $154$ ($64.17\%$) | **$625/1,240$ ($50.40\%$)** |
| **`MiLoRA-r16` (Wang et al. [26] / Yin et al. [23], $W_0$ Minor-16, `LinFP32`)** | $196$ | $0.1202$ | $0.0125$ / --- | **$0.0008\%$** | **$0.71\%$** | $228$ ($45.60\%$) | **$259$ ($51.80\%$)** | **$157$ ($65.42\%$)** | **$644/1,240$ ($51.94\%$)** |
| **`Vanilla-GRPO Standard-BF16` (50 steps, Supplied)** | $0$ | $0.3366$ | $0.0226$ / --- | $0.0018\%$ | $0.99\%$ | $225$ ($45.00\%$) | $257$ ($51.40\%$) | $145$ ($60.42\%$) | **$627/1,240$ ($50.56\%$)** |
| **`Vanilla-GRPO Unfiltered-FP32` (50 steps, Supplied)** | $0$ | $1.3341$ | $0.5067$ / --- | $0.0055\%$ | $1.53\%$ | $92$ ($18.40\%$) | $109$ ($21.80\%$) | $69$ ($28.75\%$) | **$270/1,240$ ($21.77\%$)** |
| **`Vanilla-GRPO OSF-Top16-FP32` (50 steps, Supplied)** | $4,312$ | $0.7884$ | $0.3019$ / --- | $0.0035\%$ | $1.59\%$ | $160$ ($32.00\%$) | $180$ ($36.00\%$) | $112$ ($46.67\%$) | **$452/1,240$ ($36.45\%$)** |
| **`Vanilla-GRPO OSF-Tail16-FP32` (50 steps, Supplied)** | $4,312$ | $0.8824$ | $0.0399$ / --- | $0.0037\%$ | $1.30\%$ | $222$ ($44.40\%$) | $247$ ($49.40\%$) | $150$ ($62.50\%$) | **$619/1,240$ ($49.92\%$)** |
| **`Vanilla-GRPO Standard-BF16` (100 steps, Horizon)** | $0$ | $0.4378$ | $0.0245$ / --- | $0.0021\%$ | $0.99\%$ | $228$ ($45.60\%$) | **$266$ ($53.20\%$)** | $152$ ($63.33\%$) | **$646/1,240$ ($52.10\%$)** |
| **`Vanilla-GRPO Unfiltered-FP32` (100 steps, Horizon)** | $0$ | $1.7243$ | **$1.8074$** / --- | $0.0070\%$ | $1.47\%$ | $46$ ($9.20\%$) | $45$ ($9.00\%$) | $63$ ($26.25\%$) | **$154/1,240$ ($12.42\%$)** |
| **Online LR Sweep: `Unfiltered-FP32` ($\eta=5.0\times 10^{-6}$)** | $0$ | $0.6622$ | $0.1437$ / --- | $0.0030\%$ | $1.47\%$ | $204$ ($40.80\%$) | $234$ ($46.80\%$) | $135$ ($56.25\%$) | **$573/1,240$ ($46.21\%$)** |
| **Online LR Sweep: `Unfiltered-FP32` ($\eta=2.5\times 10^{-6}$)** | $0$ | $0.3448$ | $0.0558$ / --- | $0.0017\%$ | $1.50\%$ | $225$ ($45.00\%$) | **$259$ ($51.80\%$)** | $144$ ($60.00\%$) | **$628/1,240$ ($50.65\%$)** |
| **Online LR Sweep: `OSF-Top16-FP32` ($\eta=5.0\times 10^{-6}$)** | $4,312$ | $0.4496$ | $0.0815$ / --- | $0.0021\%$ | $1.59\%$ | $216$ ($43.20\%$) | $224$ ($44.80\%$) | $134$ ($55.83\%$) | **$574/1,240$ ($46.29\%$)** |

*Table 5C: Head-to-head comparison against `Ares` (Chen et al. [22], testing both the published §4.2 Eq. 3 squared-singular-value energy rule $p_i = \sigma_i^2 / \sum_j \sigma_j^2$ and the amplitude variant $q_i = \sigma_i / \sum_j \sigma_j$), strict every-step master-writeback `OSF-RLVR` controls (`OSF-Top16-Writeback`, `OSF-Tail16-Writeback`), `GeoRA` (Zhang et al. [24]), `PiSSA` (Meng et al. [25]), and `MiLoRA` (Wang et al. [26]), as benchmarked in RLVR by Yin et al. [23], matched supplied `Vanilla-GRPO` controls (`tessera_shield=False, mdl_schema_bonus=0.0`) at 50 and 100 steps, and online learning-rate sweeps on Seed 102 (`records/osf_and_precision_factorial.json`).*

1. **Why Pre-AdamW Gradient Truncation (`Ares`, Chen, Zhao, & Wen [22]) Requires `bfloat16` Dead-Zone Rounding or Reduced Learning Rate Under Both Eq. (3) Energy ($p_i \propto \sigma_i^2$) and Amplitude ($q_i \propto \sigma_i$) Rules:** In `Ares` (*Spectral-Driven Adaptive-Rank Optimizer for Large Language Model Reinforcement Learning with Verifiable Rewards*, `arXiv:2610.16533v1` §4.2, Eq. 3 [22]), raw gradients $G_t$ are truncated to an EMA adaptive rank $r_t$ *prior* to AdamW using the normalized squared singular values $p_i = \sigma_i^2 / \sum_j \sigma_j^2$ and Shannon effective rank $\hat{r}_{t,\text{energy}} = \exp(-\sum_i p_i \log p_i)$. To ensure full literature alignment and effective-rank consistency, we evaluate both the published **Eq. (3) energy rule ($p_i = \sigma_i^2 / \sum_j \sigma_j^2$, where $\hat{r}_{t,\text{energy}} \approx 10.1\text{--}10.4 \implies r_t \approx 10.43\text{--}10.57$)** and the **amplitude variant ($q_i = \sigma_i / \sum_j \sigma_j$, where $\hat{r}_{t,\text{ampl}} \approx 37.4\text{--}39.0 \implies r_t \approx 37.4\text{--}39.7$)**:
   - **Under the Published Eq. (3) Energy Rule ($p_i = \sigma_i^2 / \sum_j \sigma_j^2$, $r_t \approx 10.5$):** In its official `bfloat16` configuration at $\eta = 5\times 10^{-6}$ (**`Ares-Eq3-BF16`**), `Ares` achieves **`635/1,240 = 51.21%`** (`235/500 = 47.00%` on `MATH-500`, `253/500 = 50.60%` on `Fresh-500`, `147/240 = 61.25%` on `Ext240`; $\|\Delta W\|_F = 0.1344$, $\mathrm{KL}_{\text{cond}} = 0.0072$, $\mathrm{KL}_{\text{greedy}} = 0.0077\text{ nats/tok}$, Tail-25 $\mathrm{eRank}_{\text{energy}} = 682.49, \mathrm{eRank}_{\text{ampl}} = 894.16$). However, when run with exact `float32` master weights at $\eta = 10^{-5}$ (**`Ares-Eq3-FP32`**), pre-AdamW rank-$10.6$ gradient truncation cannot prevent AdamW's variance denominator $\sqrt{v_t} + \epsilon$ from amplifying step sizes across rotating principal directions: Tail-25 cumulative displacement effective rank is $\mathrm{eRank}_{\text{energy}} = 63.75$ ($\mathrm{eRank}_{\text{ampl}} = 451.27$; step-50 master $\mathrm{eRank}_{\text{energy}} = 73.13, \mathrm{eRank}_{\text{ampl}} = 455.18$), $\|\Delta W\|_F = 1.2988$, $\mathrm{KL}_{\text{cond}} = 0.5667\text{ nats/tok}$, and $\mathrm{KL}_{\text{greedy}} = 0.3889\text{ nats/tok}$, dropping accuracy by **$-23.47\text{ pp}$ (`344` vs. `635/1,240 = 27.74%`, 95% CI $[-26.36, -20.57]\text{ pp}$, $+56/-347$, $p = 2.32 \times 10^{-52}$)**. Even at the matched learning rate $\eta = 5\times 10^{-6}$ in `float32` (**`Ares-Eq3-FP32`**, $\eta = 5\times 10^{-6}$), rank-$10.5$ pre-AdamW truncation concentrates updates into fewer principal directions than unfiltered AdamW ($\mathrm{eRank}_{\text{energy}} = 59.93$ vs. $156.32$, $\mathrm{eRank}_{\text{ampl}} = 452.60$ vs. $642.04$, $\|\Delta W\|_F = 0.6596$, $\mathrm{KL}_{\text{cond}} = 0.1148$, $\mathrm{KL}_{\text{greedy}} = 0.1172$), scoring **`493/1,240 = 39.76%`**—trailing `Ares-Eq3-BF16` by **$-11.45\text{ pp}$ (`493` vs. `635/1,240`, $+85/-227$, $p = 4.39 \times 10^{-16}$)** and trailing unfiltered `float32` AdamW at the same learning rate $\eta = 5\times 10^{-6}$ (`573/1,240 = 46.21%`) by **$-6.45\text{ pp}$**.
   - **Under the Amplitude Variant ($q_i = \sigma_i / \sum_j \sigma_j$, $r_t \approx 38$):** Relaxing pre-AdamW truncation from rank $r_t \approx 10.5$ to $r_t \approx 38$ yields `619/1,240 = 49.92%` in `bfloat16` (`Ares-Ampl-BF16`, $\|\Delta W\|_F = 0.1343, \mathrm{KL}_{\text{cond}} = 0.0066$), while in `float32` master training (`Ares-Ampl-FP32`) cumulative displacement effective rank reaches step-50 master $\mathrm{eRank}_{\text{ampl}}(\Delta W) = 583.17$ at $\eta = 10^{-5}$ (`414/1,240 = 33.39%`, $\mathrm{KL}_{\text{cond}} = 0.2198$, $-16.53\text{ pp}$ vs. `Ares-Ampl-BF16`, $p = 3.10 \times 10^{-30}$) and $\mathrm{eRank}_{\text{ampl}}(\Delta W) = 573.35$ at $\eta = 5\times 10^{-6}$ (`545/1,240 = 43.95%`, $\mathrm{KL}_{\text{cond}} = 0.1505$, $-5.97\text{ pp}$, $p = 2.15 \times 10^{-5}$). Thus, across both effective-rank definitions, pre-AdamW principal gradient truncation cannot substitute for `bfloat16` dead-zone rounding or post-AdamW subspace control, and tighter pre-AdamW principal truncation ($r_t \approx 10.5$ vs. $38$) exacerbates `float32` KL overshoot (`0.5667` vs. `0.2198` at $\eta=10^{-5}$).
   - **Cross-Arm KL Non-Monotonicity Bound:** Notice also that at $\eta = 10^{-5}$, `Ares-Ampl-FP32` has a slightly lower conditional KL than `OSF-Top16` ($\mathrm{KL}_{\text{cond}} = 0.2198$ vs. $0.2828\text{ nats/tok}$) yet scores slightly lower (`414/1,240 = 33.39%` vs. `457/1,240 = 36.85%` in uniform `LinFP32`), whereas `Ares-Eq3-FP32` has both higher KL ($\mathrm{KL}_{\text{cond}} = 0.5667$) and lower accuracy (`344/1,240 = 27.74%`). This confirms that Policy KL is a tight predictor of accuracy **within each optimizer and subspace family** (e.g., across $\alpha \in \{0.25, 0.50, 0.75, 1.00\}$ and $\eta \in \{2.5\times 10^{-6}, 5\times 10^{-6}, 10^{-5}\}$), rather than a universal rank-order bijection across distinct optimizer recurrences.
2. **Pretrained $W_0$ Subspace Adapters (`PiSSA` [25], `MiLoRA` [26], `GeoRA` [24], and Yin et al. [23]):** Updating rank-16 $W_0$-initialized adapters in `float32` (`LinFP32`) with frozen residual $W_{\text{res}} = W_0 - B_0 A_0$ shows that adapting the 16 principal singular components of $W_0$ (`PiSSA`, Meng et al. [25]) concentrates $47.59\%$ of $\Delta W$ energy into $W_0$'s top-16 subspace, yielding the highest Normalized Spectral Shift ($\text{NSS} = 0.0027\%$, $\mathrm{KL}_{\text{rev}} = 0.0209$, `625/1,240 = 50.40%`), consistent with the principal-component misalignment identified in RLVR by Yin et al. (*Evaluating Parameter Efficient Methods for RLVR*, `arXiv:2512.23165v2` [23]). `GeoRA` (*GeoRA: Geometry-Aware Low-Rank Adaptation for RLVR*, Zhang et al., `arXiv:2601.09361v4` [24], constructing $W_{\text{Geo}} = W_0 \odot (M_{\text{Spec}} \cup M_{\text{Euc}})$ where $M_{\text{Spec}}$ selects the bottom $\rho = 0.20$ magnitude entries of $|\hat{W}_r|$ and $M_{\text{Euc}}$ selects the bottom $\rho = 0.20$ magnitude entries of $|W_0|$, initialized via top-$r$ SVD $B_{\text{Geo}} A_{\text{Geo}}$ with frozen residual anchor $W_{\text{res}} = W_0 - B_{\text{Geo}} A_{\text{Geo}}$) cuts $W_0\text{-Top16}$ alignment to $0.95\%$ and halves $\text{NSS}$ (`0.0014%`, `622/1,240 = 50.16%`). Adapting the bottom 16 minor singular components of $W_0$ (`MiLoRA`, Wang et al. [26]) initializes $B_0, A_0$ with near-zero tail singular values ($\sigma_{\text{tail}} \to 0$), which—as analyzed by Yin et al. [23]—naturally restricts early-step update magnitude ($\|\Delta W\|_F = 0.1202$ at 50 steps) and achieves the lowest $\text{NSS}$ (`0.0008%`), lowest $W_0\text{-Top16}$ alignment (`0.71%`), and highest 50-step `float32` accuracy (**`644/1,240 = 51.94%`, `259/500 = 51.80%` on `Fresh-500`**), consistent with off-principal learning in Zhu et al. [21].
3. **Exact Supplied `Vanilla-GRPO` Replication (50 & 100 Steps):** Under the exact supplied `run_baseline.py` objective (`tessera_shield=False, mdl_schema_bonus=0.0`), the online reversal is even sharper: at 50 steps, `Vanilla-GRPO OSF-Tail16` (`619/1,240 = 49.92%`, $\mathrm{KL}_{\text{rev}} = 0.0399$) beats `OSF-Top16` (`452/1,240 = 36.45%`, $\mathrm{KL}_{\text{rev}} = 0.3019$) by **`+167` items (`+13.47 pp`, $+242/-75$, $p = 1.20 \times 10^{-21}$)** and beats `Unfiltered-FP32` (`270/1,240 = 21.77%`, $\mathrm{KL}_{\text{rev}} = 0.5067$) by **`+349` items (`+28.15 pp`, $+392/-43$, $p = 1.40 \times 10^{-71}$)**. Extending the horizon to **100 steps** (`--steps 100`) lifts `Vanilla-GRPO Standard-BF16` further from `50.56%` to **`52.10%` (`646/1,240`, `266/500 = 53.20%` on `Fresh-500`, $\mathrm{KL}_{\text{rev}} = 0.0245$)** while driving `Vanilla-GRPO Unfiltered-FP32` into total collapse (**`154/1,240 = 12.42%`, $\mathrm{KL}_{\text{rev}} = 1.8074\text{ nats/tok}$, `reward = 0.0` across steps 75–100; a $+39.68\text{ pp}$ gap, $p = 2.80 \times 10^{-126}$**).
4. **Online Learning-Rate Sweep & Prespecified 12-Point Target-Free Scale Grid:** Reducing the online learning rate of `Unfiltered-FP32-Master` by $4\times$ to $\eta = 2.5\times 10^{-6}$ (matching the $\approx 25\%$ coordinate step survival rate of `bfloat16` dead-zone rounding at $\eta = 10^{-5}$) restores $\|\Delta W\|_F = 0.3448$, cuts $\mathrm{KL}_{\text{rev}}$ from $0.3813 \to 0.0558$, and recovers `+258` items in uniform `LinFP32` (**`+20.81 pp`, `628` vs. `370/1,240` [$50.65\%$ vs. $29.84\%$], $+325/-67$, $p = 9.63 \times 10^{-42}$; `259/500 = 51.80%` on `Fresh-500`**). Likewise, halving $\eta$ for `OSF-Top16` to $5\times 10^{-6}$ cuts $\mathrm{KL}_{\text{rev}}$ from $0.2828 \to 0.0815$ and recovers `+117` items in uniform `LinFP32` (`574` vs. `457/1,240` [$46.29\%$ vs. $36.85\%$], **`+9.44 pp`, $+212/-95$, $p = 2.06 \times 10^{-11}$**). Independently, evaluating a prespecified **12-point target-free post-hoc scale grid** ($\Delta W \leftarrow \alpha \Delta W, \alpha \in \{0.25, 0.50, 0.75, 1.00\}$) on untouched `Fresh-500` ($N=500$, Seed 102) confirms the exact same KL law without referencing any `bfloat16` norm:
   - **`Unfiltered-FP32-Master`:** $\alpha=1.00$: `148/500 (29.60%, KL=0.3813)` $\to$ $\alpha=0.75$: `190/500 (38.00%, KL=0.2078)` $\to$ $\alpha=0.50$: `237/500 (47.40%, KL=0.1021)` $\to$ $\alpha=0.25$: **`237/500 (47.40%, KL=0.0381; +17.80 pp vs. a=1.00, p=2.42e-14)`**.
   - **`OSF-Top16-FP32-Master`:** $\alpha=1.00$: `187/500 (37.40%, KL=0.2828)` $\to$ $\alpha=0.75$: `226/500 (45.20%, KL=0.1783)` $\to$ $\alpha=0.50$: `244/500 (48.80%, KL=0.1068)` $\to$ $\alpha=0.25$: **`253/500 (50.60%, KL=0.0504; +13.20 pp vs. a=1.00, p=3.59e-10)`**.
   - **`OSF-Tail16-FP32-Master`:** Because $\mathrm{KL}_{\text{rev}} \in [0.0055, 0.0279]$ remains inside the trust region across all $\alpha$, accuracy stays high and flat across the entire grid: $\alpha=0.25$: `249/500 (49.80%)`, $\alpha=0.50$: **`258/500 (51.60%)`**, $\alpha=0.75$: `251/500 (50.20%)`, $\alpha=1.00$: **`258/500 (51.60%)`** ($p = 0.3966$ between $\alpha=0.25$ and $1.00$).

---

## 3. Zero-Initialized Gated Middle-Layer Recurrent Refinement (`LoopedFastModel`)

### 3.1 Architecture and Full 28-Layer GPU Training-to-Decoding KV-Cache Verification
To increase effective reasoning depth without modifying the 196 `bfloat16` weight matrices or breaking pretrained representations at initialization, we introduce `LoopedFastModel` (`--loop-layers 10,18` in `code/train_tessera.py`). For each middle transformer block $\ell \in \{10, 11, \dots, 17\}$ ($8$ layers), let $x_1 = \text{Block}_\ell(x_0, \text{KV}_1)$ denote the standard first-pass hidden state and updated key-value state $\text{KV}_1 = (K_1, V_1)$. We define an input-conditional recurrent refinement operator `recurrent_refine` that re-applies $\text{Block}_\ell$'s pre-norm grouped-query attention and SwiGLU feed-forward network using $x_1$ as the query state and $\text{KV}_1$ as the key-value context:
$$\delta_\ell(x_1, \text{KV}_1) = \text{Attn}_\ell(\text{RMSNorm}_{1,\ell}(x_1), \text{KV}_1) + \text{FFN}_\ell\!\left(\text{RMSNorm}_{2,\ell}\!\left(x_1 + \text{Attn}_\ell(\text{RMSNorm}_{1,\ell}(x_1), \text{KV}_1)\right)\right).$$
The block output is modulated by a zero-initialized `float32` elementwise residual gate $g_\ell \in \mathbb{R}^{1024}$ ($8 \times 1024 = 8,192$ parameters total, $32\text{ KB}$):
$$x_{\text{out}}(g_\ell) = x_1 + \tanh(g_\ell) \odot \delta_\ell(x_1, \text{KV}_1).$$

Two operator invariants hold for `LoopedFastModel`:
1. **Exact Bitwise Identity at Initialization ($g_\ell = 0$):** When $g_\ell = 0$, $\tanh(g_\ell) = 0$, so `LoopedFastModel` is bitwise identical to unlooped `FastModel` (`max_logprob_diff = 0.000000000000e+00`) while delivering non-zero initial gradients $\nabla_{g_\ell} \mathcal{L} = \delta_\ell \odot \nabla_{x_{\text{out}}} \mathcal{L}$ across all 8 looped layers (`grad_norm` $\in [0.460, 1.426]$ on Step 1).
2. **Full 28-Layer `Qwen3-0.6B` GPU KV-Cache Numerical Verification Across $N=32$ Real `MATH-500` 96-Token Prefixes ($32$ Prompt + $64$ Incremental Decode Steps $= 2,048$ Decoded Token Positions, $g_\ell \neq 0$):** Because `recurrent_refine` queries the causal first-pass cache $\text{KV}_1$ without overwriting $\text{KV}_1$, full-sequence parallel evaluation (`cache=None`, used during rollout log-probability evaluation in GRPO) and token-by-token autoregressive decoding (`cache=KVCache(n_layers=28)`, used during rollout generation and test evaluation) define the **exact same mathematical operator**. To distinguish exact operator equivalence from finite-precision GEMM tiling differences across multi-sentence reasoning prefixes, we execute a direct **5-Condition Full 28-Layer `Qwen3-0.6B` GPU Verification** across **$N=32$ real `MATH-500` trajectories** evaluated on fixed **96-token prefixes** (`seq_len=96`: $32$ prompt prefill tokens $+ 64$ incremental decoded token steps per sequence $= 2,048$ decoded token positions and $311,164,928$ vocabulary logits evaluated per condition; note this verifies 64-step incremental decoding on 96-token sequences rather than full 2,048-token rollouts; `code/verify_full_kv_cache.py` and `records/kv_cache_verification_full.json`):
   - **Full 28-Layer `float32` Execution (`fp32_learned_gates` and `fp32_zero_gates`):** With all 28 layers, RMSNorms, gates, and RoPE in `torch.float32`, maximum absolute logit difference across all $2,048$ decoded token steps is **`3.691e-04`** (`mean_abs_logit_diff = 8.334e-06`) for learned gates and **`2.849e-04`** (`mean_abs_logit_diff = 8.152e-06`) for zero gates, with **`100.00%` ($2,048 / 2,048$) argmax token agreement** in both conditions.
   - **Full 28-Layer `bfloat16` Execution (`bf16_learned_gates`, `bf16_rademacher_gates`, `bf16_zero_gates`):** With linear layers in `torch.bfloat16` and RMSNorm/gates/RoPE in `torch.float32`, GEMM reduction-order differences between $T \times K$ prefill and $1 \times K$ single-token decode kernels yield `max_abs_logit_diff = 2.578` (`mean_abs_logit_diff = 7.647e-02`, **`98.83%` [$2,024 / 2,048$] argmax agreement**) for learned gates, `max_abs_logit_diff = 2.641` (`mean_abs_logit_diff = 7.523e-02`, **`98.93%` [$2,026 / 2,048$] argmax agreement**) for Rademacher gates, and `max_abs_logit_diff = 2.562` (`mean_abs_logit_diff = 7.820e-02`, **`98.68%` [$2,021 / 2,048$] argmax agreement**) for unlooped zero gates (`g=0`). Notice that argmax agreement and mean logit discrepancy with active recurrent gates (`98.83%` and `7.65e-02`) are at least as tight as on the standard unlooped `Qwen3-0.6B` transformer (`98.68%` and `7.82e-02`), confirming that `recurrent_refine` introduces zero KV-cache state corruption across 64-step incremental generation.

### 3.2 Why Single-Pass Recurrent Refinement Creates Two-Panel (`Val350 + MATH-500`) False Positives
Starting from the frozen `Tessera-Sym-Seed102` checkpoint ($g=0$), we compare two 50-step RLVR training regimes on `math_train[:50]` (group size $G=8$, 512 rollout tokens, second-half 25-step tail averaging across steps 26–50), where `LoopedFastModel` initializes at exact identity ($g_\ell = 0$) and learns small residual gates ($\sigma_{\text{RMS},\ell} \approx 0.030\text{--}0.034$):
- **`GateOnly-Loop10_18` (`--lr 0.0 --weight-decay 0.0 --gate-lr 5e-3`):** Freezes all $596\text{M}$ `bfloat16` base parameters (`p.requires_grad_(False)`) and updates only the $8,192$ `float32` recurrent gates $g_\ell$ ($50$ steps $\times$ $8$ rollouts $= 400$ rollouts total calibration cost).
- **`Joint-Loop10_18` (`--lr 1e-5 --weight-decay 0.01 --gate-lr 5e-3`):** Jointly updates all `bfloat16` base weights and the $8,192$ `float32` recurrent gates under `Tessera-Sym` (exact LCP prefix shielding + trajectory-zero-sum MDL shaping).

Because all four within-MATH panels (`Val350`, `MATH-500`, `Sec-500`, `Conf-500`) were observed on Seed 102 during method development, we treat the 4-panel Seed 102 evaluation ($N=1,850$) as an **exploratory within-MATH diagnostic** (and Seed 42 as a **same-panel cross-seed replication**), reserving the separately sourced external benchmark `Ext240` (`GSM8K-200 + AMC23-40`, Section 5.3) for untouched out-of-distribution confirmation.

| Single-Pass Greedy Evaluation Arm (Seed 102 Exploratory) | `Val350` ($N=350$) | `MATH-500` ($N=500$) | `Sec-500` ($N=500$) | `Conf-500` ($N=500$) | **Pooled 4-Panel ($N=1,850$)** | **2,048-Tok Truncation Rate** | **Boxed Rate / Mean Tokens** |
|---|---:|---:|---:|---:|---:|---:|---:|
| **`Baseline-Seed102`** (50-step GRPO, 28 blk) | $46.00\%$ ($161$) | $45.40\%$ ($227$) | $44.80\%$ ($224$) | $43.60\%$ ($218$) | $44.86\%$ ($830/1850$) | $11.57\%$ ($214$) | $89.30\%$ / $672.1$ |
| **`Tessera-Sym-Seed102`** ($g=0$ Center, 28 blk) | $45.71\%$ ($160$) | $47.20\%$ ($236$) | **$47.00\%$ ($235$)** | **$45.60\%$ ($228$)** | **$46.43\%$ ($859/1850$)** | **$6.54\%$ ($121$)** | **$93.51\%$ / $622.3$** |
| **`GateOnly-Loop10_18`** ($+g_{\text{lrn}}$, 36 blk) | $46.57\%$ ($163$) | **$47.80\%$ ($239$)** | $45.20\%$ ($226$) | $41.40\%$ ($207$) | $45.14\%$ ($835/1850$) | $6.97\%$ ($129$) | $92.54\%$ / $631.3$ |
| **`Antipodal-Loop10_18`** ($-g_{\text{lrn}}$, 36 blk) | $44.00\%$ ($154$) | $44.40\%$ ($222$) | $46.60\%$ ($233$) | $43.80\%$ ($219$) | $44.76\%$ ($828/1850$) | **$5.57\%$ ($103$)** | **$93.78\%$ / $608.8$** |
| **`Joint-Loop10_18`** (Base+Gate RLVR, 36 blk) | **$47.14\%$ ($165$)** | **$47.80\%$ ($239$)** | $46.40\%$ ($232$) | $41.80\%$ ($209$) | $45.68\%$ ($845/1850$) | $12.16\%$ ($225$) | $88.70\%$ / $694.4$ |

*Table 6: Four-panel greedy evaluation ($N=1,850$) of single-pass recurrent refinement (`LoopedFastModel`, `--loop-layers 10,18`) on Seed 102, illustrating why a two-panel (`Val350 + MATH-500`) screen produces false positives when not checked across `Sec-500` and `Conf-500`.*

Table 6 exposes a critical methodological lesson for small-model RLVR benchmarks:
- Had we evaluated only on `Val350` and `MATH-500` ($N=850$), **both `GateOnly-Loop10_18` (`46.57%` and `47.80%`) and `Joint-Loop10_18` (`47.14%` and `47.80%`) would appear to beat `Tessera-Sym-Seed102` (`45.71%` and `47.20%`) on both panels simultaneously**, reaching $+2.40\text{ pp}$ over `Baseline-Seed102` on `MATH-500`.
- Across all four disjoint within-MATH panels ($N=1,850$), however, single-pass recurrent refinement loses $-1.80\text{ to }-4.20\text{ pp}$ on `Sec-500` and `Conf-500`, finishing at $835/1,850$ ($45.14\%$, $+113/-137$ vs. $g=0$) for `GateOnly-Loop10_18` and $845/1,850$ ($45.68\%$, $+150/-164$ vs. $g=0$) for `Joint-Loop10_18`. This mirrors the finding of `claude_2_1007164104_rlvr` (`math_rlvr_6` §3.8 [17]) and `credit_cartographer` (`math_rlvr_17` §4.1–4.2 [11]) that single-pass greedy continuations on `Qwen3-0.6B` primarily re-draw knife-edge ($0 < p < 1$) items with high item-level turnover ($\sim 15\%\text{--}20\%$ discordance) without moving net multi-panel greedy accuracy. Moreover, joint base+gate training (`Joint-Loop10_18`) nearly doubles the 2,048-token truncation rate from $6.54\%$ ($121/1,850$) back to $12.16\%$ ($225/1,850$).

---

## 4. Single-Backbone Deterministic Recurrent-Depth Stencils Across $3,700$ Within-MATH Evaluations

### 4.1 The Antipodal Consensus Phenomenon and Selection Rules
Why did `GateOnly-Loop10_18` ($+g_{\text{lrn}}$) gain $113$ newly solved items over its own frozen $g=0$ parent across the 4 panels while losing $137$ others? And look at the antipodal pass (`Antipodal-Loop10_18`, which negates the $8,192$ gates to $-g_{\text{lrn}}$ on the same frozen backbone): on `Sec-500` and `Conf-500`, where $+g_{\text{lrn}}$ dropped to $45.20\%$ and $41.40\%$, $-g_{\text{lrn}}$ rebounds to $46.60\%$ and $43.80\%$ while cutting 2,048-token truncations to $5.57\%$ ($103/1,850$).

Because $g=0$, $+g$, and $-g$ share the **exact same $1.50\text{ GB}$ `bfloat16` weight tensors in VRAM** (toggling only a $32\text{ KB}$ `float32` gate vector in Layers 10--17), a single model instance can execute a deterministic **3-Depth Stencil** $\{-g, 0, +g\}$ at $T=0$. Let $(a_0, a_{+g}, a_{-g})$ denote the extracted answers of the three passes. We evaluate two parameter-free, verifier-free selection rules alongside the 3-pass oracle union:
1. **`Plur(Center)` (Center-Anchored Consensus):** Outputs the majority answer whenever at least two passes agree; on 1-1-1 all-distinct ties, defaults to the unlooped center pass $a_0$. Notice that `Plur(Center)` differs from single-pass $g=0$ **if and only if** the two recurrent passes converge on the **exact same non-null answer** $a_{+g} = a_{-g} \neq a_0$.
2. **`Plur(MDL-Box)` (Minimum-Description-Length Boxed Tie-Break):** Outputs the majority answer whenever at least two passes agree; on 1-1-1 ties among non-null candidates, selects the candidate with the shortest non-truncated completion containing a formatted `\boxed{` answer (`"\\boxed{" in text and tokens < 2048`; if none, shortest non-null candidate). *(Methodological note: using the looser substring check `"\\boxed" in text` without `< 2048` changes 2 items on Seed 102 `Lrn3`, giving $885/1,850$ vs. $883/1,850$; we use the stricter `"\\boxed{" in text and tokens < 2048` selector uniformly across all methods.)*

### 4.2 Learned Gate Geometry vs. Training-Free Norm-Matched Rademacher Gates
To understand what the 50-step `GateOnly-Loop10_18` run actually learned on Seed 102 and on its full replication on Seed 42 (`tessera_loop10_18_gateonly_seed42`), we inspect the learned $8,192$-dimensional gate vectors $g^{(102)}_{\text{lrn}}$ and $g^{(42)}_{\text{lrn}}$:
- **Layerwise RMS and Mean Absolute Magnitude:**
  - On Seed 102, the layerwise RMS values $\sigma_{\text{RMS},\ell} = \sqrt{\frac{1}{1024}\sum_j g_{\ell,j}^2}$ across Layers 10--17 are `[0.029383, 0.030402, 0.030790, 0.029918, 0.029095, 0.030296, 0.030210, 0.030335]`, with overall $\text{RMS} = 0.030058$ and mean absolute gate magnitude $\frac{1}{8192}\sum |g_{\ell,j}| = 0.024682$.
  - On Seed 42, the layerwise RMS values across Layers 10--17 are `[0.034124, 0.033404, 0.033529, 0.034777, 0.034531, 0.034603, 0.034733, 0.034966]`, with overall $\text{RMS} = 0.034338$ and mean absolute magnitude $0.028252$.
- **Near-Orthogonality Across Seeds:** Despite converging to nearly identical scalar RMS scales ($\sim 0.030\text{--}0.034$), the inner-product cosine between $g^{(102)}_{\text{lrn}}$ and $g^{(42)}_{\text{lrn}}$ across all $8,192$ coordinates is **`+0.0254`** ($50.88\%$ sign agreement).

This near-zero cross-seed cosine motivates two decisive mechanistic controls: **(a) are the $8,192$ learned coordinate directions necessary at all, or was 50-step gate RLVR merely calibrating the perturbation scale $\sigma_{\text{RMS},\ell} \approx 0.03$? and (b) does the 3-pass stencil require antipodal sign negation ($-g_{\text{rad}}^{(1)}, 0, +g_{\text{rad}}^{(1)}$) along a single 1D axis, or does a pair of independent Rademacher directions ($0, +g_{\text{rad}}^{(1)}, +g_{\text{rad}}^{(2)}$) perform comparably?** To test both questions, we construct two **Norm-Matched Rademacher Gate Vectors** (`Rademacher-Loop10_18`, `0` gate-direction training steps) on each seed:
$$g_{\text{rad},\ell}^{(m)} = \sigma_{\text{RMS},\ell} \cdot s_\ell^{(m)}, \qquad s_\ell^{(m)} \sim \text{Unif}\!\left(\{-1, +1\}^{1024}\right) \quad (m \in \{1, 2\}),$$
using fixed pseudorandom seeds (`20261017` and `20262001` on Seed 102; `20261042` and `20262002` on Seed 42; 8,192-D inner-product cosines are $\cos(g_{\text{lrn}}, g_{\text{rad}}^{(1)}) = +0.0115$, $\cos(g_{\text{lrn}}, g_{\text{rad}}^{(2)}) = +0.0249$, $\cos(g_{\text{rad}}^{(1)}, g_{\text{rad}}^{(2)}) = -0.0187$ on Seed 102, and $-0.0008, -0.0182, +0.0178$ on Seed 42). We evaluate both the **Antipodal Rademacher Stencil** $\{-g_{\text{rad}}^{(1)}, 0, +g_{\text{rad}}^{(1)}\}$ and the **Independent-Direction Rademacher Stencil** $\{0, +g_{\text{rad}}^{(1)}, +g_{\text{rad}}^{(2)}\}$, as well as the **5-Point Single-Backbone Stencil** $\{-g_{\text{lrn}}, -g_{\text{rad}}^{(1)}, 0, +g_{\text{rad}}^{(1)}, +g_{\text{lrn}}\}$ on the same $1.50\text{ GB}$ checkpoint.

| Seed & Deterministic Stencil Arm ($T=0$, $1.50\text{ GB}$ VRAM) | `Val350` ($350$) | `MATH-500` ($500$) | `Sec-500` ($500$) | `Conf-500` ($500$) | **Pooled ($N=1,850$)** Acc (`+W/-L` vs. $g=0$, McNemar $p$) | **Oracle Union** | **Tok / Blk-Pass Proxy** |
|---|---:|---:|---:|---:|---:|---:|---:|
| **Seed 102 (Exploratory): Center ($g=0$, 28 blk)** | $45.71\%$ ($160$) | $47.20\%$ ($236$) | $47.00\%$ ($235$) | $45.60\%$ ($228$) | **$46.43\%$ ($859/1850$)** ($622.3$ tok) | $46.43\%$ ($859$) | $1.00\times$ / $1.00\times$ |
| Seed 102: $+g_{\text{lrn}}$ / $-g_{\text{lrn}}$ (36 blk) | $46.57\%$ / $44.00\%$ | $47.80\%$ / $44.40\%$ | $45.20\%$ / $46.60\%$ | $41.40\%$ / $43.80\%$ | $45.14\%$ ($835$) / $44.76\%$ ($828$) | --- | $1.01\times$ / $0.98\times$ tok |
| Seed 102: $+g_{\text{rad}}^{(1)}$ / $-g_{\text{rad}}^{(1)}$ (36 blk) | $46.00\%$ / $45.43\%$ | $48.20\%$ / $48.60\%$ | $46.20\%$ / $44.80\%$ | $43.00\%$ / $44.20\%$ | $45.84\%$ ($848$) / $45.78\%$ ($847$) | --- | $1.02\times$ / $1.00\times$ tok |
| **Seed 102: 3-Pass Learned `Plur(Center)`** | $46.57\%$ ($163$) | $47.80\%$ ($239$) | $46.60\%$ ($233$) | $46.20\%$ ($231$) | **$46.81\%$ ($866/1850$, $+0.38\text{ pp}$)** (`+24/-17`, $p=0.349$) | $56.00\%$ ($1036$) | $2.99\times$ / $3.56\times$ |
| **Seed 102: 3-Pass Learned `Plur(MDL-Box)`** | $46.86\%$ ($164$) | $48.80\%$ ($244$) | $47.40\%$ ($237$) | $47.60\%$ ($238$) | **$47.73\%$ ($883/1850$, $+1.30\text{ pp}$)** (`+67/-43`, $p=0.0279$) | $56.00\%$ ($1036$) | $2.99\times$ / $3.56\times$ |
| **Seed 102: 3-Pass Antipodal Rademacher `Plur(Center)`** | $46.86\%$ ($164$) | $49.00\%$ ($245$) | $47.60\%$ ($238$) | $46.00\%$ ($230$) | **$47.41\%$ ($877/1850$, $+0.97\text{ pp}$)** (`+30/-12`, $p=0.0079$) | $55.24\%$ ($1022$) | $3.01\times$ / $3.59\times$ |
| **Seed 102: 3-Pass Antipodal Rademacher `Plur(MDL-Box)`** | $47.43\%$ ($166$) | **$50.80\%$ ($254$)** | $46.80\%$ ($234$) | $47.80\%$ ($239$) | **$48.27\%$ ($893/1850$, $+1.84\text{ pp}$)** (`+67/-33`, $p=0.00087$) | $55.24\%$ ($1022$) | $3.01\times$ / $3.59\times$ |
| Seed 102: $+g_{\text{rad}}^{(2)}$ (Independent Rademacher, 36 blk) | $45.71\%$ ($160$) | $48.00\%$ ($240$) | $45.20\%$ ($226$) | $44.60\%$ ($223$) | $45.89\%$ ($849/1850$, $-0.54\text{ pp}$) | --- | $1.02\times$ / $1.00\times$ tok |
| **Seed 102: 3-Pass Indep-Dir $\{g_0, +g_{\text{rad}}^{(1)}, +g_{\text{rad}}^{(2)}\}$ `Plur(Center)`** | $46.86\%$ ($164$) | $49.40\%$ ($247$) | $47.60\%$ ($238$) | $45.60\%$ ($228$) | **$47.41\%$ ($877/1850$, $+0.97\text{ pp}$)** (`+36/-18`, $p=0.0198$) | $55.08\%$ ($1019$) | $3.02\times$ / $3.60\times$ |
| **Seed 102: 3-Pass Indep-Dir $\{g_0, +g_{\text{rad}}^{(1)}, +g_{\text{rad}}^{(2)}\}$ `Plur(MDL-Box)`** | $46.86\%$ ($164$) | $49.60\%$ ($248$) | $45.80\%$ ($229$) | $46.60\%$ ($233$) | **$47.24\%$ ($874/1850$, $+0.81\text{ pp}$)** (`+59/-44`, $p=0.167$) | $55.08\%$ ($1019$) | $3.02\times$ / $3.60\times$ |
| **Seed 102: 5-Point Stencil `Plur(Center)`** | **$47.71\%$ ($167$)** | $49.60\%$ ($248$) | **$48.40\%$ ($242$)** | $48.00\%$ ($240$) | **$48.49\%$ ($897/1850$, $+2.05\text{ pp}$)** (`+59/-21`, $p=2.5\times 10^{-5}$) | $59.84\%$ ($1107$) | $5.00\times$ / $6.15\times$ |
| **Seed 102: 5-Point Stencil `Plur(MDL-Box)`** | $47.43\%$ ($166$) | $50.00\%$ ($250$) | $48.00\%$ ($240$) | **$48.60\%$ ($243$)** | **$48.59\%$ ($899/1850$, $+2.16\text{ pp}$)** (`+76/-36`, $p=2.0\times 10^{-4}$) | $59.84\%$ ($1107$) | $5.00\times$ / $6.15\times$ |
| **Seed 42 (Replication): Center ($g=0$, 28 blk)** | $46.86\%$ ($164$) | $47.40\%$ ($237$) | $46.60\%$ ($233$) | $41.60\%$ ($208$) | **$45.51\%$ ($842/1850$)** ($575.0$ tok) | $45.51\%$ ($842$) | $1.00\times$ / $1.00\times$ |
| Seed 42: $+g_{\text{lrn}}$ / $-g_{\text{lrn}}$ (36 blk) | $46.29\%$ / $46.29\%$ | $46.40\%$ / $46.40\%$ | $45.80\%$ / $45.60\%$ | $41.20\%$ / $40.80\%$ | $44.81\%$ ($829$) / $44.65\%$ ($826$) | --- | $1.02\times$ / $1.01\times$ tok |
| Seed 42: $+g_{\text{rad}}^{(1)}$ / $-g_{\text{rad}}^{(1)}$ (36 blk) | $44.29\%$ / $43.71\%$ | $47.40\%$ / $46.20\%$ | $44.60\%$ / $46.40\%$ | $43.60\%$ / $40.20\%$ | $45.03\%$ ($833$) / $44.16\%$ ($817$) | --- | $1.03\times$ / $1.02\times$ tok |
| **Seed 42: 3-Pass Learned `Plur(Center)`** | $47.71\%$ ($167$) | $48.80\%$ ($244$) | $48.20\%$ ($241$) | $43.00\%$ ($215$) | **$46.86\%$ ($867/1850$, $+1.35\text{ pp}$)** (`+35/-10`, $p=0.00025$) | $55.08\%$ ($1019$) | $3.02\times$ / $3.60\times$ |
| **Seed 42: 3-Pass Learned `Plur(MDL-Box)`** | $47.14\%$ ($165$) | **$50.00\%$ ($250$)** | $48.00\%$ ($240$) | $44.20\%$ ($221$) | **$47.35\%$ ($876/1850$, $+1.84\text{ pp}$)** (`+76/-42`, $p=0.00224$) | $55.08\%$ ($1019$) | $3.02\times$ / $3.60\times$ |
| **Seed 42: 3-Pass Antipodal Rademacher `Plur(Center)`** | $47.43\%$ ($166$) | $48.80\%$ ($244$) | $47.60\%$ ($238$) | $43.80\%$ ($219$) | **$46.86\%$ ($867/1850$, $+1.35\text{ pp}$)** (`+34/-9`, $p=0.00017$) | $54.59\%$ ($1010$) | $3.05\times$ / $3.63\times$ |
| **Seed 42: 3-Pass Antipodal Rademacher `Plur(MDL-Box)`** | $46.86\%$ ($164$) | $48.40\%$ ($242$) | $47.00\%$ ($235$) | $44.40\%$ ($222$) | **$46.65\%$ ($863/1850$, $+1.14\text{ pp}$)** (`+56/-35`, $p=0.0355$) | $54.59\%$ ($1010$) | $3.05\times$ / $3.63\times$ |
| Seed 42: $+g_{\text{rad}}^{(2)}$ (Independent Rademacher, 36 blk) | $44.86\%$ ($157$) | $47.60\%$ ($238$) | $46.80\%$ ($234$) | $42.80\%$ ($214$) | $45.57\%$ ($843/1850$, $+0.05\text{ pp}$) | --- | $1.02\times$ / $1.01\times$ tok |
| **Seed 42: 3-Pass Indep-Dir $\{g_0, +g_{\text{rad}}^{(1)}, +g_{\text{rad}}^{(2)}\}$ `Plur(Center)`** | $48.57\%$ ($170$) | $49.00\%$ ($245$) | $47.80\%$ ($239$) | $44.60\%$ ($223$) | **$47.41\%$ ($877/1850$, $+1.89\text{ pp}$)** (`+47/-12`, $p=5.2\times 10^{-6}$) | $54.49\%$ ($1008$) | $3.04\times$ / $3.62\times$ |
| **Seed 42: 3-Pass Indep-Dir $\{g_0, +g_{\text{rad}}^{(1)}, +g_{\text{rad}}^{(2)}\}$ `Plur(MDL-Box)`** | $48.57\%$ ($170$) | $48.60\%$ ($243$) | $48.00\%$ ($240$) | $44.00\%$ ($220$) | **$47.19\%$ ($873/1850$, $+1.68\text{ pp}$)** (`+72/-41`, $p=0.00455$) | $54.49\%$ ($1008$) | $3.04\times$ / $3.62\times$ |
| **Seed 42: 5-Point Stencil `Plur(Center)`** | **$50.00\%$ ($175$)** | $49.20\%$ ($246$) | $48.60\%$ ($243$) | $44.00\%$ ($220$) | **$47.78\%$ ($884/1850$, $+2.27\text{ pp}$)** (`+64/-22`, $p=7.4\times 10^{-6}$) | $58.86\%$ ($1089$) | $5.07\times$ / $6.24\times$ |
| **Seed 42: 5-Point Stencil `Plur(MDL-Box)`** | $49.14\%$ ($172$) | $49.60\%$ ($248$) | **$49.00\%$ ($245$)** | **$45.00\%$ ($225$)** | **$48.11\%$ ($890/1850$, $+2.59\text{ pp}$)** (`+90/-42`, $p=3.6\times 10^{-5}$) | $58.86\%$ ($1089$) | $5.07\times$ / $6.24\times$ |

*Table 6A: Complete 2-seed $\times$ 4-panel ($3,700$-evaluation) within-MATH results (Seed 102 exploratory development + Seed 42 same-panel cross-seed replication) for deterministic ($T=0$) single-backbone recurrent-depth stencils (`LoopedFastModel`, Layers 10--17).*

Table 6A establishes three central findings:
1. **Individual Pass Degradation vs. Multi-Pass Stencil Gain:** On both seeds, the individual recurrent passes ($+g_{\text{lrn}}, -g_{\text{lrn}}, +g_{\text{rad}}^{(1)}, -g_{\text{rad}}^{(1)}, +g_{\text{rad}}^{(2)}$) score below or at par with the unlooped $g=0$ center trajectory. Yet whenever two recurrent passes agree with each other against $g=0$ (the *only* stratum where `Plur(Center)` overrides $g_0$), the 3-pass stencil beats $g=0$ by **`+59/-27` ($p_{\text{clustered}} = 5.38 \times 10^{-4}$, $p_{\text{Holm,26}} = 2.69 \times 10^{-3}$) for learned gates**, **`+64/-21` ($p_{\text{clustered}} = 2.73 \times 10^{-6}$, $p_{\text{Holm,26}} = 2.73 \times 10^{-5}$) for antipodal Rademacher gates**, and **`+83/-30` ($p_{\text{clustered}} = 8.20 \times 10^{-7}$, $p_{\text{Holm,26}} = 9.02 \times 10^{-6}$) for independent-direction Rademacher gates**.
2. **Unresolved Head-to-Head Differences and Sufficiency of Rademacher Perturbation:** Across all $3,700$ within-MATH evaluations, `3-Pass Antipodal Rademacher` ($47.14\%$ `Center`, $47.46\%$ `MDL-Box`), `3-Pass Independent-Direction Rademacher` ($47.41\%$ `Center`, $47.22\%$ `MDL-Box`), and `3-Pass Learned` ($46.84\%$ `Center`, $47.54\%$ `MDL-Box`) exhibit unresolved differences that remain well within the confidence interval: `Antipodal` vs. `Learned` has $p_{\text{clustered}} = 0.305$ (`Center`) and $0.852$ (`MDL-Box`), while `Antipodal` vs. `Indep-Dir` has $\Delta = -0.27\text{ pp}$ ($+54/-64$, $p_{\text{clustered}} = 0.373$) on `Center` and $\Delta = +0.24\text{ pp}$ ($+103/-94$, $p_{\text{clustered}} = 0.524$) on `MDL-Box`. Rather than demonstrating strict operator parity, these non-significant contrasts establish that **at the current statistical resolution, neither the $8,192$ learned gate directions nor strict antipodal negation along a single axis can be distinguished from norm-calibrated recurrent perturbation around the $g=0$ center trajectory**.
3. **Orthogonal Axis Compounding in the 5-Point Stencil:** Combining two axes in $\{-g_{\text{lrn}}, -g_{\text{rad}}^{(1)}, 0, +g_{\text{rad}}^{(1)}, +g_{\text{lrn}}\}$ improves over $g=0$ in **all $8/8$ within-MATH panel $\times$ seed cells**, reaching **$1,781/3,700$ ($48.14\%$, $+2.16\text{ pp}$ pooled, $S_{102}=+2.05\text{ pp}, S_{42}=+2.27\text{ pp}$, $p_{\text{Holm,26}} = 7.23 \times 10^{-9}$)** under `Center` and **$1,789/3,700$ ($48.35\%$, $+2.38\text{ pp}$ pooled, $S_{102}=+2.16\text{ pp}, S_{42}=+2.59\text{ pp}$, $p_{\text{Holm,26}} = 2.41 \times 10^{-7}$)** under `MDL-Box`—matching 3-checkpoint weight-space ensemble voting ($897/1,850 = 48.49\%$ on Seed 102, $4.50\text{ GB}$ VRAM [4]) using a **single $1.50\text{ GB}$ checkpoint**.

---

## 5. Same-Backbone Comparison Against Homogeneous Sampled `plurality@3`, Center-Anchored Sampling, Hybrid Stencils, and External Confirmation

### 5.1 Experimental Setup, Multiplicity Correction, and Dual Seed/Cluster Inference
A natural baseline for a 3-pass deterministic stencil ($3.01\times\text{--}3.05\times$ tokens, $3.56\times\text{--}3.63\times$ completion block passes) is $K=3$ stochastic rollouts ($s_1, s_2, s_3$) sampled directly from the **same unlooped center weights ($g=0$, $28$ blocks)**. Following community evaluation protocols on `r/math_rlvr` [11, 16], we evaluate $K=6$ independent stochastic passes at temperature $T=0.4$ (`top_p = 1.0`, deterministic batch seeds `401000 + k`) across all four within-MATH panels ($N=1,850$ items) on **both Seed 102 and Seed 42** ($6 \times 3,700 = 22,200$ within-MATH stochastic evaluations total) as well as on the untouched external benchmark `Ext240` ($6 \times 480 = 2,880$ external stochastic evaluations, Section 5.3), using per-step KV-cache survivor compaction (`shrink_kv_cache_inplace` in `code/run_same_backbone_sampling.py`).

To test our hypothesis that **homogeneous i.i.d. sampling ($\{s_1, s_2, s_3\}$) suffers excessive regression churn at $k=3$ because it discards the high-precision $T=0$ greedy center trajectory $g_0$**, we compare:
- **Homogeneous Sampled `plurality@3` ($\{s_1, s_2, s_3\}$):** Evaluated under `Uniform` tie-breaking (analytical expectation over 1-1-1 ties), `First-Sample` ($s_1$) tie-breaking, and `MDL-Box` tie-breaking.
- **Center-Anchored Stochastic Consensus ($\{g_0, s_1, s_2\}$):** Replaces $s_3$ with the deterministic greedy trajectory $g_0$ at the **exact same $2.99\times$ token and completion block-pass budget**, applying our stencil's `Plur(Center)` rule (overriding $g_0$ *only* when $a_{s_1} = a_{s_2} \neq a_0$) and `Plur(MDL-Box)`.
- **4-Pass Center + 3 Samples ($\{g_0, s_1, s_2, s_3\}$):** Combines $g_0$ with all three stochastic draws ($3.99\times$ tokens and block passes).
- **Pure 5-Pass ($\{s_1..s_5\}$) and Center-Anchored 5-Pass ($\{g_0, s_1..s_4\}$):** Matched at $5.00\times$ tokens and block passes.
- **Pure 6-Pass ($\{s_1..s_6\}$) and Center-Anchored 6-Pass ($\{g_0, s_1..s_5\}$):** Matched at $6.00\times$ tokens and block passes (near-budget comparison against the $6.19\times$ completion block passes of the 5-point deterministic stencil).
- **5-Pass Hybrid Latent+Surface Stencil ($\{g_0, +g_{\text{rad}}^{(1)}, -g_{\text{rad}}^{(1)}, s_1, s_2\}$):** Combines the greedy center $g_0$, the two training-free Rademacher recurrent-depth passes ($\pm g_{\text{rad}}^{(1)}$ at $T=0$), and two stochastic draws ($s_1, s_2$ at $T=0.4$) at $5.02\times$ tokens and $5.60\times$ completion block passes.
- **8-Pass ($\{g_0, \pm g_{\text{lrn}}, \pm g_{\text{rad}}^{(1)}, s_{1..3}\}$) and 11-Pass ($\{g_0, \pm g_{\text{lrn}}, \pm g_{\text{rad}}^{(1)}, s_{1..6}\}$) Combined Stencils:** Combine all 5 deterministic stencil passes with 3 or 6 stochastic draws on a single $1.50\text{ GB}$ checkpoint ($9.18\times$ and $12.18\times$ block passes).

**Multiplicity and Between-Seed Uncertainty Protocol:** Because Table 7 evaluates $26$ decoding/voting rules across $M=1,850$ unique questions and $S=2$ training seeds ($3,700$ paired evaluations), we report:
1. **Question-Clustered Inference + 26-Test Holm-Bonferroni Adjustment ($p_{\text{Holm,26}}$):** Cluster-robust standard errors ($\text{SE}_{\text{cl}}$) and 95% CIs across the $M=1,850$ problem clusters (conditioning on the two realized training seeds), accompanied by step-down **Holm-Bonferroni multiplicity-adjusted $p$-values ($p_{\text{Holm,26}}$)** across all 26 rows in Table 7.
2. **Descriptive Between-Seed Variation:** We report the two realized seed deltas $(\Delta_{102}, \Delta_{42})$ and their half-range $\text{SE}_{\text{seed}} = \frac{1}{2}|\Delta_{102} - \Delta_{42}|$. Because $S=2$ training seeds provide only $\text{df}=1$, between-seed variance estimates are highly sensitive to small discrete count variations (and degenerate when discrete counts match); confirmatory claims are therefore grounded in question-clustered inference with Holm-Bonferroni correction.

| Same-Backbone Decoding / Voting Protocol ($1.50\text{ GB}$ VRAM) | Seed 102 (Exploratory) `Val350` / `M500` / `Sec500` / `Conf500` | Seed 102 Pooled ($N=1,850$) | Seed 42 (Replication) `Val350` / `M500` / `Sec500` / `Conf500` | Seed 42 Pooled ($N=1,850$) |
|---|---:|---:|---:|---:|
| **Greedy Center ($g_0$, $T=0$, 28 blk)** | $160$ / $236$ / $235$ / $228$ | **$46.43\%$ ($859$)** ($622.3$ tok) | $164$ / $237$ / $233$ / $208$ | **$45.51\%$ ($842$)** ($575.0$ tok) |
| **Stochastic `mean@1` ($T=0.4$, `s1..s3` avg)** | $163.0$ / $227.0$ / $225.0$ / $206.3$ | **$44.40\%$ ($821.3$)** ($624.3$ tok) | $161.0$ / $216.0$ / $221.3$ / $204.7$ | **$43.41\%$ ($803.0$)** ($567.4$ tok) |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`Uniform`)** | $174.7$ / $246.3$ / $238.3$ / $222.7$ | **$47.68\%$ ($882.0$)** | $169.3$ / $232.3$ / $235.7$ / $219.0$ | **$46.29\%$ ($856.3$)** |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`First`)** | $170$ / $248$ / $239$ / $222$ | **$47.51\%$ ($879$)** (`+147/-127`) | $174$ / $236$ / $235$ / $223$ | **$46.92\%$ ($868$)** (`+173/-147`) |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`MDL-Box`)** | $174$ / $240$ / $242$ / $224$ | **$47.57\%$ ($880$)** (`+148/-127`) | $174$ / $236$ / $234$ / $226$ | **$47.03\%$ ($870$)** (`+159/-131`) |
| **Center-Anchored $\{g_0,s_1,s_2\}$ (`Center`)** | $168$ / $251$ / $238$ / $232$ | **$48.05\%$ ($889$)** (`+47/-17`) | $179$ / $240$ / $245$ / $225$ | **$48.05\%$ ($889$)** (`+67/-20`) |
| **Center-Anchored $\{g_0,s_1,s_2\}$ (`MDL-Box`)** | $170$ / $253$ / $242$ / $229$ | **$48.32\%$ ($894$)** (`+94/-59`) | $176$ / $238$ / $243$ / $226$ | **$47.73\%$ ($883$)** (`+110/-69`) |
| **4-Pass $\{g_0,s_1,s_2,s_3\}$ (`Center`)** | $177$ / $256$ / $244$ / $239$ | **$49.51\%$ ($916$)** (`+79/-22`) | $184$ / $248$ / $251$ / $230$ | **$49.35\%$ ($913$)** (`+94/-23`) |
| **4-Pass $\{g_0,s_1,s_2,s_3\}$ (`MDL-Box`)** | $179$ / $261$ / $247$ / $230$ | **$49.57\%$ ($917$)** (`+116/-58`) | $180$ / $245$ / $246$ / $231$ | **$48.76\%$ ($902$)** (`+134/-74`) |
| **Pure 5-Pass $\{s_1..s_5\}$ (`First`)** | $182$ / $269$ / $254$ / $241$ | **$51.14\%$ ($946$)** (`+175/-88`) | $181$ / $251$ / $245$ / $229$ | **$48.97\%$ ($906$)** (`+177/-113`) |
| **Pure 5-Pass $\{s_1..s_5\}$ (`MDL-Box`)** | $184$ / $264$ / $252$ / $239$ | **$50.76\%$ ($939$)** (`+172/-92`) | $178$ / $249$ / $249$ / $231$ | **$49.03\%$ ($907$)** (`+175/-110`) |
| **Center-Anchored 5-Pass $\{g_0,s_1..s_4\}$ (`Center`)** | $177$ / $260$ / $253$ / $242$ | **$50.38\%$ ($932$)** (`+99/-26`) | $189$ / $249$ / $253$ / $234$ | **$50.00\%$ ($925$)** (`+116/-33`) |
| **Center-Anchored 5-Pass $\{g_0,s_1..s_4\}$ (`MDL-Box`)** | $176$ / $262$ / $250$ / $237$ | **$50.00\%$ ($925$)** (`+126/-60`) | $181$ / $248$ / $255$ / $234$ | **$49.62\%$ ($918$)** (`+145/-69`) |
| **Pure 6-Pass $\{s_1..s_6\}$ (`First`)** | $180$ / $269$ / $258$ / $245$ | **$51.46\%$ ($952$)** (`+177/-84`) | $186$ / $252$ / $250$ / $237$ | **$50.00\%$ ($925$)** (`+184/-101`) |
| **Pure 6-Pass $\{s_1..s_6\}$ (`MDL-Box`)** | $182$ / $269$ / $258$ / $247$ | **$51.68\%$ ($956$)** (`+180/-83`) | $182$ / $253$ / $253$ / $238$ | **$50.05\%$ ($926$)** (`+182/-98`) |
| **Center-Anchored 6-Pass $\{g_0,s_1..s_5\}$ (`Center`)** | $180$ / $270$ / $253$ / $249$ | **$51.46\%$ ($952$)** (`+125/-32`) | $189$ / $259$ / $252$ / $232$ | **$50.38\%$ ($932$)** (`+123/-33`) |
| **Center-Anchored 6-Pass $\{g_0,s_1..s_5\}$ (`MDL-Box`)** | $186$ / $271$ / $253$ / $245$ | **$51.62\%$ ($955$)** (`+146/-50`) | $184$ / $254$ / $258$ / $234$ | **$50.27\%$ ($930$)** (`+158/-70`) |
| **5-Pass Hybrid $\{g_0,\pm g_{\text{rad}},s_1,s_2\}$ (`Center`)** | $170$ / $255$ / $247$ / $240$ | **$49.30\%$ ($912$)** (`+71/-18`) | $179$ / $252$ / $247$ / $226$ | **$48.86\%$ ($904$)** (`+80/-18`) |
| **5-Pass Hybrid $\{g_0,\pm g_{\text{rad}},s_1,s_2\}$ (`MDL-Box`)** | $175$ / $258$ / $248$ / $243$ | **$49.95\%$ ($924$)** (`+97/-32`) | $176$ / $254$ / $255$ / $229$ | **$49.41\%$ ($914$)** (`+110/-38`) |
| **8-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}},s_{1..3}\}$ (`Center`)** | $178$ / $258$ / $254$ / $250$ | **$50.81\%$ ($940$)** (`+106/-25`) | $185$ / $261$ / $254$ / $231$ | **$50.32\%$ ($931$)** (`+108/-19`) |
| **8-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}},s_{1..3}\}$ (`MDL-Box`)** | $177$ / **$265$** / $252$ / $249$ | **$50.97\%$ ($943$)** (`+122/-38`) | **$185$** / **$265$** / $253$ / **$236$** | **$50.76\%$ ($939$)** (`+133/-36`) |
| **11-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}},s_{1..6}\}$ (`Center`)** | $187$ / $265$ / $257$ / $249$ | **$51.78\%$ ($958$)** (`+132/-33`) | $186$ / $268$ / $261$ / $232$ | **$51.19\%$ ($947$)** (`+130/-25`) |
| **11-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}},s_{1..6}\}$ (`MDL-Box`)** | **$187$** / **$273$** / **$259$** / $249$ | **$52.32\%$ ($968$)** (`+148/-39`) | $185$ / **$269$** / **$262$** / $232$ | **$51.24\%$ ($948$)** (`+144/-38`) |

*Table 6B: Panel-by-panel counts and pooled accuracy ($N=1,850$ per seed) for within-MATH $T=0.4$ stochastic sampling, center-anchored sampling, and hybrid stencils on Seed 102 (exploratory) and Seed 42 (same-panel replication).*


| Single-Backbone Voting Rule ($1.50\text{ GB}$ Checkpoint Storage) | **2-Seed Pooled ($3,700$ evals)** | **Clustered $\Delta$ vs. $g_0$ ($M=1,850$, 95% CI)** | **Win/Loss, $p_{\text{cl}}$ & 26-Test $p_{\text{Holm,26}}$** | **Seed Deltas $(\Delta_{102}, \Delta_{42})$ & Half-Range $\text{SE}_{\text{seed}}$** | **2-Seed Oracle** | **Tok / Blk Proxy** |
|---|---:|---:|---:|---:|---:|---:|
| **Greedy Center ($g_0$, $T=0$, 28 blk)** | **$45.97\%$ ($1701/3700$)** | Baseline | --- | Baseline ($46.43\% \text{ vs. } 45.51\%$) | $45.97\%$ ($1701$) | $1.00\times$ / $1.00\times$ |
| **Stochastic `mean@1` ($T=0.4$, $g=0$)** | **$43.90\%$ ($1624.3/3700$)** | **$-2.07\text{ pp}$** ($[-3.17, -0.98]$) | $+567/-641$ ($p_{\text{cl}}=2.08\times 10^{-4}$, **$p_{\text{Holm}}=1.25\times 10^{-3}$**) | $(-2.04, -2.11)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.04$) | --- | $1.00\times$ / $1.00\times$ |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`Uniform`)** | **$46.98\%$ ($1738.3/3700$)** | **$+1.01\text{ pp}$** ($[-0.16, +2.18]$) | $+500/-336$ ($p_{\text{cl}}=0.0900$, $p_{\text{Holm}}=0.122$, NS) | $(+1.24, +0.77)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.23$) | $59.14\%$ ($2188$) | $2.99\times$ / $2.99\times$ |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`First`)** | **$47.22\%$ ($1747/3700$)** | **$+1.24\text{ pp}$** ($[-0.06, +2.54]$) | $+320/-274$ ($p_{\text{cl}}=0.0611$, $p_{\text{Holm}}=0.122$, NS) | $(+1.08, +1.41)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.16$) | $59.14\%$ ($2188$) | $2.99\times$ / $2.99\times$ |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`MDL-Box`)** | **$47.30\%$ ($1750/3700$)** | **$+1.32\text{ pp}$** ($[+0.07, +2.58]$) | $+307/-258$ ($p_{\text{cl}}=0.0391$, **$p_{\text{Holm}}=0.117$, NS**) | $(+1.14, +1.51)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.19$) | $59.14\%$ ($2188$) | $2.99\times$ / $2.99\times$ |
| **3-Pass Learned $\{-g_{\text{lrn}},0,+g_{\text{lrn}}\}$ (`Center`)** | **$46.84\%$ ($1733/3700$)** | **$+0.86\text{ pp}$** ($[+0.38, +1.35]$) | $+59/-27$ ($p_{\text{cl}}=5.38\times 10^{-4}$, **$p_{\text{Holm}}=2.69\times 10^{-3}$**) | $(+0.38, +1.35)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.49$) | $55.54\%$ ($2055$) | $3.01\times$ / $3.58\times$ |
| **3-Pass Learned $\{-g_{\text{lrn}},0,+g_{\text{lrn}}\}$ (`MDL-Box`)** | **$47.54\%$ ($1759/3700$)** | **$+1.57\text{ pp}$** ($[+0.76, +2.37]$) | $+143/-85$ ($p_{\text{cl}}=1.42\times 10^{-4}$, **$p_{\text{Holm}}=9.91\times 10^{-4}$**) | $(+1.30, +1.84)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.27$) | $55.54\%$ ($2055$) | $3.01\times$ / $3.58\times$ |
| **3-Pass Antipodal Rad $\{-g_{\text{rad}}^{(1)},0,+g_{\text{rad}}^{(1)}\}$ (`Center`)** | **$47.14\%$ ($1744/3700$)** | **$+1.16\text{ pp}$** ($[+0.68, +1.65]$) | $+64/-21$ ($p_{\text{cl}}=2.73\times 10^{-6}$, **$p_{\text{Holm}}=2.73\times 10^{-5}$**) | $(+0.97, +1.35)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.19$) | $54.92\%$ ($2032$) | $3.03\times$ / $3.61\times$ |
| **3-Pass Antipodal Rad $\{-g_{\text{rad}}^{(1)},0,+g_{\text{rad}}^{(1)}\}$ (`MDL-Box`)** | **$47.46\%$ ($1756/3700$)** | **$+1.49\text{ pp}$** ($[+0.76, +2.21]$) | $+123/-68$ ($p_{\text{cl}}=5.89\times 10^{-5}$, **$p_{\text{Holm}}=4.95\times 10^{-4}$**) | $(+1.84, +1.14)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.35$) | $54.92\%$ ($2032$) | $3.03\times$ / $3.61\times$ |
| **3-Pass Indep-Dir $\{0,+g_{\text{rad}}^{(1)},+g_{\text{rad}}^{(2)}\}$ (`Center`)** | **$47.41\%$ ($1754/3700$)** | **$+1.43\text{ pp}$** ($[+0.86, +2.00]$) | $+83/-30$ ($p_{\text{cl}}=8.20\times 10^{-7}$, **$p_{\text{Holm}}=9.02\times 10^{-6}$**) | $(+0.97, +1.89)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.46$) | $54.78\%$ ($2027$) | $3.03\times$ / $3.61\times$ |
| **3-Pass Indep-Dir $\{0,+g_{\text{rad}}^{(1)},+g_{\text{rad}}^{(2)}\}$ (`MDL-Box`)** | **$47.22\%$ ($1747/3700$)** | **$+1.24\text{ pp}$** ($[+0.46, +2.02]$) | $+131/-85$ ($p_{\text{cl}}=1.79\times 10^{-3}$, **$p_{\text{Holm}}=7.16\times 10^{-3}$**) | $(+0.81, +1.68)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.43$) | $54.78\%$ ($2027$) | $3.03\times$ / $3.61\times$ |
| **Center-Anchored $\{g_0,s_1,s_2\}$ (`Center`)** | **$48.05\%$ ($1778/3700$)** | **$+2.08\text{ pp}$** ($[+1.42, +2.75]$) | $+114/-37$ ($p_{\text{cl}}=8.89\times 10^{-10}$, **$p_{\text{Holm}}=1.33\times 10^{-8}$**) | $(+1.62, +2.54)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.46$) | $58.41\%$ ($2161$) | $2.99\times$ / $2.99\times$ |
| **Center-Anchored $\{g_0,s_1,s_2\}$ (`MDL-Box`)** | **$48.03\%$ ($1777/3700$)** | **$+2.05\text{ pp}$** ($[+1.06, +3.05]$) | $+204/-128$ ($p_{\text{cl}}=5.50\times 10^{-5}$, **$p_{\text{Holm}}=4.95\times 10^{-4}$**) | $(+1.89, +2.22)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.16$) | $58.41\%$ ($2161$) | $2.99\times$ / $2.99\times$ |
| **4-Pass Center + 3 Samples $\{g_0,s_{1..3}\}$ (`Center`)** | **$49.43\%$ ($1829/3700$)** | **$+3.46\text{ pp}$** ($[+2.66, +4.25]$) | $+173/-45$ ($p_{\text{cl}}=1.49\times 10^{-17}$, **$p_{\text{Holm}}=3.42\times 10^{-16}$**) | $(+3.08, +3.84)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.38$) | $61.43\%$ ($2273$) | $3.99\times$ / $3.99\times$ |
| **4-Pass Center + 3 Samples $\{g_0,s_{1..3}\}$ (`MDL-Box`)** | **$49.16\%$ ($1819/3700$)** | **$+3.19\text{ pp}$** ($[+2.14, +4.24]$) | $+250/-132$ ($p_{\text{cl}}=2.86\times 10^{-9}$, **$p_{\text{Holm}}=3.72\times 10^{-8}$**) | $(+3.14, +3.24)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.05$) | $61.43\%$ ($2273$) | $3.99\times$ / $3.99\times$ |
| **Pure 5-Pass $\{s_1..s_5\}$ (`First`)** | **$50.05\%$ ($1852/3700$)** | **$+4.08\text{ pp}$** ($[+2.80, +5.36]$) | $+352/-201$ ($p_{\text{cl}}=4.01\times 10^{-10}$, **$p_{\text{Holm}}=6.81\times 10^{-9}$**) | $(+4.70, +3.46)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.62$) | $63.41\%$ ($2346$) | $5.00\times$ / $5.00\times$ |
| **Pure 5-Pass $\{s_1..s_5\}$ (`MDL-Box`)** | **$49.89\%$ ($1846/3700$)** | **$+3.92\text{ pp}$** ($[+2.65, +5.19]$) | $+347/-202$ ($p_{\text{cl}}=1.62\times 10^{-9}$, **$p_{\text{Holm}}=2.28\times 10^{-8}$**) | $(+4.32, +3.51)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.41$) | $63.41\%$ ($2346$) | $5.00\times$ / $5.00\times$ |
| **Center-Anchored 5-Pass $\{g_0,s_{1..4}\}$ (`Center`)** | **$50.19\%$ ($1857/3700$)** | **$+4.22\text{ pp}$** ($[+3.33, +5.10]$) | $+215/-59$ ($p_{\text{cl}}=1.32\times 10^{-20}$, **$p_{\text{Holm}}=3.31\times 10^{-19}$**) | $(+3.95, +4.49)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.27$) | $63.30\%$ ($2342$) | $5.00\times$ / $5.00\times$ |
| **Center-Anchored 5-Pass $\{g_0,s_{1..4}\}$ (`MDL-Box`)** | **$49.81\%$ ($1843/3700$)** | **$+3.84\text{ pp}$** ($[+2.75, +4.93]$) | $+271/-129$ ($p_{\text{cl}}=5.14\times 10^{-12}$, **$p_{\text{Holm}}=9.26\times 10^{-11}$**) | $(+3.57, +4.11)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.27$) | $63.30\%$ ($2342$) | $5.00\times$ / $5.00\times$ |
| **Pure 6-Pass $\{s_1..s_6\}$ (`First`)** | **$50.73\%$ ($1877/3700$)** | **$+4.76\text{ pp}$** ($[+3.49, +6.02]$) | $+361/-185$ ($p_{\text{cl}}=1.49\times 10^{-13}$, **$p_{\text{Holm}}=2.84\times 10^{-12}$**) | $(+5.03, +4.49)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.27$) | $65.22\%$ ($2413$) | $6.00\times$ / $6.00\times$ |
| **Pure 6-Pass $\{s_1..s_6\}$ (`MDL-Box`)** | **$50.86\%$ ($1882/3700$)** | **$+4.89\text{ pp}$** ($[+3.65, +6.14]$) | $+362/-181$ ($p_{\text{cl}}=1.29\times 10^{-14}$, **$p_{\text{Holm}}=2.58\times 10^{-13}$**) | $(+5.24, +4.54)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.35$) | $65.22\%$ ($2413$) | $6.00\times$ / $6.00\times$ |
| **Center-Anchored 6-Pass $\{g_0,s_{1..5}\}$ (`Center`)** | **$50.92\%$ ($1884/3700$)** | **$+4.95\text{ pp}$** ($[+3.98, +5.91]$) | $+248/-65$ ($p_{\text{cl}}=1.16\times 10^{-23}$, **$p_{\text{Holm}}=3.13\times 10^{-22}$**) | $(+5.03, +4.86)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.08$) | $65.38\%$ ($2419$) | $6.00\times$ / $6.00\times$ |
| **Center-Anchored 6-Pass $\{g_0,s_{1..5}\}$ (`MDL-Box`)** | **$50.95\%$ ($1885/3700$)** | **$+4.97\text{ pp}$** ($[+3.85, +6.10]$) | $+304/-120$ ($p_{\text{cl}}=4.16\times 10^{-18}$, **$p_{\text{Holm}}=9.98\times 10^{-17}$**) | $(+5.19, +4.76)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.22$) | $65.38\%$ ($2419$) | $6.00\times$ / $6.00\times$ |
| **5-Point Deterministic $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)}\}$ (`Center`)** | **$48.14\%$ ($1781/3700$)** | **$+2.16\text{ pp}$** ($[+1.48, +2.84]$) | $+123/-43$ ($p_{\text{cl}}=4.52\times 10^{-10}$, **$p_{\text{Holm}}=7.23\times 10^{-9}$**) | $(+2.05, +2.27)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.11$) | $59.35\%$ ($2196$) | $5.03\times$ / $6.19\times$ |
| **5-Point Deterministic $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)}\}$ (`MDL-Box`)** | **$48.35\%$ ($1789/3700$)** | **$+2.38\text{ pp}$** ($[+1.55, +3.21]$) | $+166/-78$ ($p_{\text{cl}}=2.01\times 10^{-8}$, **$p_{\text{Holm}}=2.41\times 10^{-7}$**) | $(+2.16, +2.59)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.22$) | $59.35\%$ ($2196$) | $5.03\times$ / $6.19\times$ |
| **5-Pass Hybrid $\{g_0,\pm g_{\text{rad}}^{(1)},s_1,s_2\}$ (`Center`)** | **$49.08\%$ ($1816/3700$)** | **$+3.11\text{ pp}$** ($[+2.37, +3.85]$) | $+151/-36$ ($p_{\text{cl}}=1.49\times 10^{-16}$, **$p_{\text{Holm}}=3.27\times 10^{-15}$**) | $(+2.86, +3.35)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.24$) | $62.35\%$ ($2307$) | $5.02\times$ / $5.60\times$ |
| **5-Pass Hybrid $\{g_0,\pm g_{\text{rad}}^{(1)},s_1,s_2\}$ (`MDL-Box`)** | **$49.68\%$ ($1838/3700$)** | **$+3.70\text{ pp}$** ($[+2.81, +4.60]$) | $+207/-70$ ($p_{\text{cl}}=6.13\times 10^{-16}$, **$p_{\text{Holm}}=1.29\times 10^{-14}$**) | $(+3.51, +3.89)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.19$) | $62.35\%$ ($2307$) | $5.02\times$ / $5.60\times$ |
| **8-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)},s_{1..3}\}$ (`Center`)** | **$50.57\%$ ($1871/3700$)** | **$+4.59\text{ pp}$** ($[+3.71, +5.48]$) | $+214/-44$ ($p_{\text{cl}}=1.56\times 10^{-24}$, **$p_{\text{Holm}}=4.38\times 10^{-23}$**) | $(+4.38, +4.81)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.22$) | $66.11\%$ ($2446$) | $8.02\times$ / $9.18\times$ |
| **8-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)},s_{1..3}\}$ (`MDL-Box`)** | **$50.86\%$ ($1882/3700$)** | **$+4.89\text{ pp}$** ($[+3.92, +5.87]$) | $+255/-74$ ($p_{\text{cl}}=6.67\times 10^{-23}$, **$p_{\text{Holm}}=1.73\times 10^{-21}$**) | $(+4.54, +5.24)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.35$) | $66.11\%$ ($2446$) | $8.02\times$ / $9.18\times$ |
| **11-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)},s_{1..6}\}$ (`Center`)** | **$51.49\%$ ($1905/3700$)** | **$+5.51\text{ pp}$** ($[+4.55, +6.48]$) | $+262/-58$ ($p_{\text{cl}}=5.87\times 10^{-29}$, **$p_{\text{Holm}}=1.76\times 10^{-27}$**) | $(+5.35, +5.68)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.16$) | $69.62\%$ ($2576$) | $11.02\times$ / $12.18\times$ |
| **11-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)},s_{1..6}\}$ (`MDL-Box`)** | **$51.78\%$ ($1916/3700$)** | **$+5.81\text{ pp}$** ($[+4.77, +6.85]$) | $+292/-77$ ($p_{\text{cl}}=1.01\times 10^{-27}$, **$p_{\text{Holm}}=2.94\times 10^{-26}$**) | $(+5.89, +5.73)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.08$) | $69.62\%$ ($2576$) | $11.02\times$ / $12.18\times$ |

*Table 7: Two-seed within-MATH synthesis ($M=1,850$ question clusters, $S=2$ training seeds, $3,700$ paired evaluations) reporting question-clustered inference with 26-test Holm-Bonferroni multiplicity-adjusted $p$-values ($p_{\text{Holm,26}}$) alongside descriptive per-seed deltas $(\Delta_{102}, \Delta_{42})$ and half-range $\text{SE}_{\text{seed}}$.*

### 5.2 Direct Head-to-Head Paired Contrasts and Item-Level Concordance

To compare competing multi-pass protocols directly against one another (rather than each against $g_0$), Table 8 reports the exact paired win/loss counts, per-seed two-sided exact McNemar $p$-values, 2-seed question-clustered contrasts ($M=1,850$ clusters, $3,700$ evaluations), 17-test Holm-Bonferroni adjusted $p$-values ($p_{\text{Holm,17}}$), and between-seed half-range standard errors ($\text{SE}_{\text{seed}}$).

| Direct Head-to-Head Paired Contrast (Method A vs. Method B) | Seed 102 ($N=1,850$) `+W/-L` (McNemar $p$) | Seed 42 ($N=1,850$) `+W/-L` (McNemar $p$) | **2-Seed Clustered $\Delta$ ($\text{SE}_{\text{cl}}$, $\text{SE}_{\text{seed}}$)** | **Total `+W/-L`, $p_{\text{cl}}$ & $p_{\text{Holm,17}}$** |
|---|---:|---:|---:|---:|
| **`Cntr {g0,s1,s2}` (`Center`) vs. `Pure {s1,s2,s3}` (`First`)** *(Equal $2.99\times$ blk)* | `+110/-100` (net $+10$, $p=0.535$) | `+127/-106` (net $+21$, $p=0.190$) | **$+0.84\text{ pp}$** ($\text{SE}_{\text{cl}}=0.57$, $\text{SE}_{\text{seed}}=0.30$) | $+237/-206$ ($p_{\text{cl}}=0.141$, $p_{\text{Holm}}=1.00$) |
| **`Cntr {g0,s1,s2}` (`MDL-Box`) vs. `Pure {s1,s2,s3}` (`MDL-Box`)** *(Equal $2.99\times$ blk)* | `+78/-64` (net $+14$, $p=0.275$) | `+75/-62` (net $+13$, $p=0.305$) | **$+0.73\text{ pp}$** ($\text{SE}_{\text{cl}}=0.45$, $\text{SE}_{\text{seed}}=0.03$) | $+153/-126$ ($p_{\text{cl}}=0.108$, $p_{\text{Holm}}=1.00$) |
| **`Cntr {g0,s1,s2}` (`Center`) vs. `Rad3 {g0,±g_rad}` (`Center`)** | `+45/-33` (net $+12$, $p=0.213$) | `+55/-33` (net $+22$, $p=0.0246$) | **$+0.92\text{ pp}$** ($\text{SE}_{\text{cl}}=0.35$, $\text{SE}_{\text{seed}}=0.27$) | $+100/-66$ ($p_{\text{cl}}=0.0086$, $p_{\text{Holm}}=0.121$) |
| **`Cntr {g0,s1,s2}` (`MDL-Box`) vs. `Rad3 {g0,±g_rad}` (`MDL-Box`)** | `+86/-85` (net $+1$, $p=1.000$) | `+97/-77` (net $+20$, $p=0.150$) | **$+0.57\text{ pp}$** ($\text{SE}_{\text{cl}}=0.50$, $\text{SE}_{\text{seed}}=0.51$) | $+183/-162$ ($p_{\text{cl}}=0.254$, $p_{\text{Holm}}=1.00$) |
| **`Rad3 {g0,±g_rad}` (`Center`) vs. `Lrn3 {g0,±g_lrn}` (`Center`)** | `+34/-23` (net $+11$, $p=0.185$) | `+30/-30` (net $+0$, $p=1.000$) | **$+0.30\text{ pp}$** ($\text{SE}_{\text{cl}}=0.29$, $\text{SE}_{\text{seed}}=0.30$) | $+64/-53$ ($p_{\text{cl}}=0.305$, $p_{\text{Holm}}=1.00$) |
| **`Rad3 {g0,±g_rad}` (`MDL-Box`) vs. `Lrn3 {g0,±g_lrn}` (`MDL-Box`)** | `+73/-63` (net $+10$, $p=0.440$) | `+56/-69` (net $-13$, $p=0.283$) | **$-0.08\text{ pp}$** ($\text{SE}_{\text{cl}}=0.44$, $\text{SE}_{\text{seed}}=0.62$) | $+129/-132$ ($p_{\text{cl}}=0.852$, $p_{\text{Holm}}=1.00$) |
| **`Rad3 {g0,±g_rad}` (`Center`) vs. `IndepDir {g0,+g_rad1,+g_rad2}` (`Center`)** | `+29/-29` (net $+0$, $p=1.000$) | `+25/-35` (net $-10$, $p=0.245$) | **$-0.27\text{ pp}$** ($\text{SE}_{\text{cl}}=0.30$, $\text{SE}_{\text{seed}}=0.27$) | $+54/-64$ ($p_{\text{cl}}=0.373$, $p_{\text{Holm}}=1.00$) |
| **`Rad3 {g0,±g_rad}` (`MDL-Box`) vs. `IndepDir {g0,+g_rad1,+g_rad2}` (`MDL-Box`)** | `+59/-40` (net $+19$, $p=0.070$) | `+44/-54` (net $-10$, $p=0.363$) | **$+0.24\text{ pp}$** ($\text{SE}_{\text{cl}}=0.38$, $\text{SE}_{\text{seed}}=0.78$) | $+103/-94$ ($p_{\text{cl}}=0.524$, $p_{\text{Holm}}=1.00$) |
| **`Cntr5 {g0,s1..4}` (`Center`) vs. `Pure5 {s1..5}` (`First`)** *(Equal $5.00\times$ blk)* | `+63/-77` (net $-14$, $p=0.272$) | `+84/-65` (net $+19$, $p=0.140$) | **$+0.14\text{ pp}$** ($\text{SE}_{\text{cl}}=0.45$, $\text{SE}_{\text{seed}}=0.89$) | $+147/-142$ ($p_{\text{cl}}=0.766$, $p_{\text{Holm}}=1.00$) |
| **`Cntr5 {g0,s1..4}` (`MDL-Box`) vs. `Pure5 {s1..5}` (`MDL-Box`)** *(Equal $5.00\times$ blk)* | `+39/-53` (net $-14$, $p=0.175$) | `+52/-41` (net $+11$, $p=0.300$) | **$-0.08\text{ pp}$** ($\text{SE}_{\text{cl}}=0.36$, $\text{SE}_{\text{seed}}=0.68$) | $+91/-94$ ($p_{\text{cl}}=0.823$, $p_{\text{Holm}}=1.00$) |
| **`Cntr6 {g0,s1..5}` (`Center`) vs. `Pure6 {s1..6}` (`First`)** *(Equal $6.00\times$ blk)* | `+56/-56` (net $+0$, $p=1.000$) | `+71/-64` (net $+7$, $p=0.606$) | **$+0.19\text{ pp}$** ($\text{SE}_{\text{cl}}=0.43$, $\text{SE}_{\text{seed}}=0.19$) | $+127/-120$ ($p_{\text{cl}}=0.660$, $p_{\text{Holm}}=1.00$) |
| **`Cntr6 {g0,s1..5}` (`MDL-Box`) vs. `Pure6 {s1..6}` (`MDL-Box`)** *(Equal $6.00\times$ blk)* | `+44/-45` (net $-1$, $p=1.000$) | `+38/-34` (net $+4$, $p=0.724$) | **$+0.08\text{ pp}$** ($\text{SE}_{\text{cl}}=0.34$, $\text{SE}_{\text{seed}}=0.14$) | $+82/-79$ ($p_{\text{cl}}=0.812$, $p_{\text{Holm}}=1.00$) |
| **`Cntr5 {g0,s1..4}` (`MDL`) vs. `Hyb5 {g0,±g_rad,s1,s2}` (`MDL`)** *(Equal $5$ passes)* | `+75/-74` (net $+1$, $p=1.000$) | `+75/-71` (net $+4$, $p=0.804$) | **$+0.14\text{ pp}$** ($\text{SE}_{\text{cl}}=0.46$, $\text{SE}_{\text{seed}}=0.08$) | $+150/-145$ ($p_{\text{cl}}=0.770$, $p_{\text{Holm}}=1.00$) |
| **`Hyb5 {g0,±g_rad,s1,s2}` (`MDL`) vs. `Det5 {g0,±g_lrn,±g_rad}` (`MDL`)** *($5.60\times$ vs. $6.19\times$ blk)* | `+62/-37` (net $+25$, $p=0.0154$) | `+70/-46` (net $+24$, $p=0.0323$) | **$+1.32\text{ pp}$** ($\text{SE}_{\text{cl}}=0.41$, $\text{SE}_{\text{seed}}=0.03$) | $+132/-83$ ($p_{\text{cl}}=0.0011$, **$p_{\text{Holm}}=0.0167$**) |
| **`Hyb5 {g0,±g_rad,s1,s2}` (`MDL`) vs. `4Pass {g0,s1..3}` (`MDL`)** *($5.60\times$ vs. $3.99\times$ blk)* | `+70/-63` (net $+7$, $p=0.603$) | `+71/-59` (net $+12$, $p=0.335$) | **$+0.51\text{ pp}$** ($\text{SE}_{\text{cl}}=0.44$, $\text{SE}_{\text{seed}}=0.14$) | $+141/-122$ ($p_{\text{cl}}=0.240$, $p_{\text{Holm}}=1.00$) |
| **`Cntr6 {g0,s1..5}` (`Center`) vs. `Det5 {g0,±g_lrn,±g_rad}` (`Center`)** *($6.00\times$ vs. $6.19\times$ blk)* | `+105/-50` (net $+55$, $p=0.0000$) | `+105/-57` (net $+48$, $p=0.0002$) | **$+2.78\text{ pp}$** ($\text{SE}_{\text{cl}}=0.48$, $\text{SE}_{\text{seed}}=0.19$) | $+210/-107$ ($p_{\text{cl}}=9.19\times 10^{-9}$, **$p_{\text{Holm}}=1.56\times 10^{-7}$**) |
| **`Cntr6 {g0,s1..5}` (`MDL-Box`) vs. `Det5 {g0,±g_lrn,±g_rad}` (`MDL-Box`)** *($6.00\times$ vs. $6.19\times$ blk)* | `+124/-68` (net $+56$, $p=0.0001$) | `+129/-89` (net $+40$, $p=0.0081$) | **$+2.59\text{ pp}$** ($\text{SE}_{\text{cl}}=0.55$, $\text{SE}_{\text{seed}}=0.43$) | $+253/-157$ ($p_{\text{cl}}=2.75\times 10^{-6}$, **$p_{\text{Holm}}=4.40\times 10^{-5}$**) |

*Table 8: Direct head-to-head paired contrasts across Seeds 102 and 42 ($M=1,850$ question clusters, $3,700$ evaluations), including 17-test Holm-Bonferroni adjusted $p$-values ($p_{\text{Holm,17}}$) and descriptive half-range $\text{SE}_{\text{seed}}$.*

Tables 6, 7, and 8 establish four major conclusions on the within-MATH panels:
1. **Why Homogeneous Sampled `plurality@3` ($\{s_1, s_2, s_3\}$) Is Statistically Fragile at Small $k=3$:**
   - Because `Tessera-Sym` was trained with trajectory-zero-sum MDL length shaping, $T=0.4$ stochastic rollouts maintain approximate completion length parity with greedy decoding ($624.3$ vs. $622.3$ tokens on Seed 102; $567.4$ vs. $575.0$ tokens on Seed 42; $2.99\times$ tokens and block passes for 3 draws).
   - However, individual $T=0.4$ draws suffer a **$-2.07\text{ pp}$ accuracy drop** relative to $T=0$ greedy decoding (`1,624.3/3,700 = 43.90%` vs. `1,701/3,700 = 45.97%`, $p_{\text{Holm,26}} = 1.25 \times 10^{-3}$). Discarding the greedy trajectory $g_0$ in pure $\{s_1, s_2, s_3\}$ causes **$274$ regressions on greedy-correct items** across the two seeds ($127$ on Seed 102, $147$ on Seed 42). Item-level auditing reveals that **$183 / 274$ ($66.79\%$) of these losses stem from 1-1-1 all-distinct ties** (where $s_1$ selects a wrong answer), while only **$91 / 274$ ($33.21\%$) stem from wrong-answer majorities** (and $0$ from all-null draws).
   - On greedy-wrong items ($1,999$ items), pure $\{s_1, s_2, s_3\}$ (`First`) gains $320$ wins ($105$ on 1-1-1 ties and $215$ on $\ge 2$ correct majorities). On 1-1-1 ties alone, pure sampling loses $183$ greedy-correct items while winning only $105$ greedy-wrong items (a net loss of $-78$ items). This inflates discordant transitions to $594$ items ($+320/-274$, net $+46$, $p_{\text{cl}} = 0.0611$, $p_{\text{Holm,26}} = 0.122$ on `First`; $p_{\text{Holm,26}} = 0.122$ on `Uniform`; and even `MDL-Box` is non-significant after multiplicity correction at $p_{\text{Holm,26}} = 0.117$), causing pure $\{s_1, s_2, s_3\}$ to **lose to single-pass greedy decoding on `MATH-500` Seed 42 (`232.3` vs. `237`) and `Conf-500` Seed 102 (`222.7` vs. `228`)**.
2. **Why Center-Anchored Consensus ($\{g_0, s_1, s_2\}$) Eliminates $86.5\%$ of Regressions vs. $g_0$:**
   - Replacing $s_3$ with the deterministic greedy center trajectory $g_0$ in **Center-Anchored Stochastic $\{g_0, s_1, s_2\}$ (`Center`)** at the **same $2.99\times$ token and block-pass proxy** requires both stochastic draws to agree with each other against the greedy anchor ($a_{s_1} = a_{s_2} \neq a_0$) to overturn $g_0$. This rule **completely eliminates 1-1-1 tie regressions ($183 \to 0$)** and cuts wrong-majority regressions by more than half ($91 \to 37$).
   - Total regressions on greedy-correct items drop **$7.4\times$ from $274$ to $37$** ($17$ on Seed 102, $20$ on Seed 42). The net gain over $g_0$ rises from $+46$ ($+1.24\text{ pp}$) to $+77$ ($+2.08\text{ pp}$), while discordant transitions collapse from $594$ to $151$ ($+114/-37$), cutting cluster standard error in half ($\text{SE}_{\text{cl}} = 0.66\text{ pp} \to 0.34\text{ pp}$) and surviving 26-test Holm-Bonferroni correction at **$p_{\text{Holm,26}} = 1.33 \times 10^{-8}$** (reaching $889/1,850 = 48.05\%$ on both seeds; under `MDL-Box`, $+2.05\text{ pp}$, $p_{\text{Holm,26}} = 4.95 \times 10^{-4}$).
   - In direct head-to-head comparison at equal $2.99\times$ work proxy (Table 8), `Cntr {g0, s1, s2}` has a positive point estimate over `Pure {s1, s2, s3}` of $+0.84\text{ pp}$ under `Center` ($+237/-206$, $p_{\text{cl}}=0.141$, $p_{\text{Holm,17}}=1.00$) and $+0.73\text{ pp}$ under `MDL-Box` ($+153/-126$, $p_{\text{cl}}=0.108$, $p_{\text{Holm,17}}=1.00$). Because this head-to-head difference remains unresolved within the confidence interval, we frame `Cntr {g0, s1, s2}` at $k=3$ specifically as a **regression-control and variance-reduction mechanism relative to $g_0$ ($37$ vs. $274$ losses; $\text{SE}_{\text{cl}} = 0.34\text{ pp}$ vs. $0.66\text{ pp}$)**.
3. **Complementary Latent-Depth + Surface-Token Rescues in Hybrid Stencils:**
   - On the $991$ items on Seed 102 where greedy $g_0$ is wrong, `Rad3_Center` rescues $30$ items and `Cntr {g0, s1, s2} Center` rescues $47$ items, but **only $10$ items overlap**: **$20/30$ ($66.7\%$) of the deterministic recurrent-depth rescues are missed by `{g0, s1, s2}`**, and under `MDL-Box`, **$38/67$ ($56.7\%$) of `Rad3_MDL` rescues are missed by pure `{s1, s2, s3}` `MDL-Box`**.
   - Consequently, replacing $(\pm g_{\text{lrn}})$ in the 5-point deterministic stencil (`Det5`, $6.19\times$ block passes) with two stochastic draws $(s_1, s_2)$ in **`Hyb5 {g0, ±g_rad, s1, s2}` (`MDL-Box`, $5.60\times$ block passes)** improves accuracy by **$+1.32\text{ pp}$ head-to-head (`+132/-83`, $p_{\text{cl}} = 0.0011$, $p_{\text{Holm,17}} = 0.0167$, $\Delta_{102}=+1.35\text{ pp}, \Delta_{42}=+1.30\text{ pp}$)** at *lower* block-pass cost, reaching **$1,838/3,700$ ($49.68\%$, $\Delta = +3.70\text{ pp}$ over $g_0$, $p_{\text{Holm,26}} = 1.29 \times 10^{-14}$)**.
4. **Near-Budget Practical Comparison ($6.00\times$ Sampling vs. $6.19\times$ Deterministic Stencil):**
   - While deterministic recurrent-depth stencils establish a clean mechanistic finding at $T=0$, our own near-budget head-to-head comparison ($6.00\times$ vs. $6.19\times$ completion block passes in Table 8) shows that **`Center-Anchored 6-Pass` (and `Pure 6-Pass`) stochastic voting significantly outperforms the `5-Point Deterministic Stencil` by $+2.78\text{ pp}$ under `Center` (`+210/-107`, $p_{\text{cl}} = 9.19 \times 10^{-9}$, $p_{\text{Holm,17}} = 1.56 \times 10^{-7}$) and by $+2.59\text{ pp}$ under `MDL-Box` (`+253/-157`, $p_{\text{cl}} = 2.75 \times 10^{-6}$, $p_{\text{Holm,17}} = 4.40 \times 10^{-5}$)**. Thus, when stochastic sampling is permitted at a 5–6 pass budget, ordinary or center-anchored temperature sampling is the stronger standalone accuracy choice, while recurrent-depth perturbations are best used either when strict $T=0$ reproducibility is required or as complementary members in hybrid/combined ensembles (`Hyb5`, `8-Pass`, `11-Pass`).

### 5.3 Untouched Out-of-Distribution Confirmation on External Benchmark `Ext240` (`GSM8K-200 + AMC23-40`)

Because all four within-MATH panels (`Val350`, `MATH-500`, `Sec-500`, `Conf-500`) were inspected on Seed 102 during method development, we evaluate our frozen checkpoints (`Qwen3-0.6B-Base`, `Baseline-Seed102`, and `Tessera-Sym` on Seeds 102 and 42), our frozen learned and Rademacher recurrent-depth gates ($g_{\text{lrn}}, g_{\text{rad}}^{(1)}, g_{\text{rad}}^{(2)}$), and all frozen voting rules on the **untouched, separately sourced external mathematics benchmark `Ext240`** (`gsm8k_amc_bench240.json` curated by `noether_schema` in `math_rlvr_25` v5 [16], SHA-256 `e8135960771f76d1...`), comprising **$N=240$ items: $200$ grade-school word problems (`gsm8k_test`) and $40$ competition problems (`amc23_test`)** ($480$ two-seed evaluations per voting rule; `code/run_external_bench_and_fp32_audit.py`).

| Frozen Decoding / Voting Protocol on Untouched `Ext240` ($1.50\text{ GB}$ Checkpoint) | **Seed 102 ($N=240$)** (`GSM8K-200` / `AMC23-40`) | **Seed 42 ($N=240$)** (`GSM8K-200` / `AMC23-40`) | **2-Seed Pooled ($480$ evals)** (`GSM8K-400` / `AMC23-80`) | **Clustered $\Delta$ vs. $g_0$ ($\text{SE}_{\text{cl}}$, `+W/-L`, $p_{\text{cl}}$, $p_{\text{Holm,30}}$)** | **Seed Deltas $(\Delta_{102}, \Delta_{42})$ & $\text{SE}_{\text{seed}}$** |
|---|---:|---:|---:|---:|---:|
| *Reference: `Qwen3-0.6B-Base` (Pretrained, $T=0$)* | $13.75\%$ ($33$: $29$ / $4$) | --- | --- | --- | --- |
| *Reference: `Baseline-Seed102` (50-Step Vanilla GRPO, $T=0$)* | $62.08\%$ ($149$: $139$ / $10$) | --- | --- | --- | --- |
| **Greedy Center ($g_0$, `Tessera-Sym`, $T=0$, 28 blk)** | **$64.58\%$ ($155$: $142$ / $13$)** | **$60.42\%$ ($145$: $136$ / $9$)** | **$62.50\%$ ($300$: $278$ / $22$)** | Baseline | Baseline |
| **Stochastic `mean@1` ($T=0.4$, `s1..s3` avg)** | $59.44\%$ ($142.7$: $135.7$ / $7.0$) | $58.89\%$ ($141.3$: $133.0$ / $8.3$) | **$59.17\%$ ($284.0$: $268.7$ / $15.3$)** | **$-3.33\text{ pp}$** ($\text{SE}=1.45$, $+64/-86$, $p_{\text{cl}}=0.0216$, $p_{\text{Holm}}=0.345$) | $(-5.14, -1.53)\text{ pp}$ ($\text{SE}_{\text{seed}}=1.81$) |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`Uniform`)** | $62.29\%$ ($149.5$: $141.7$ / $7.8$) | $62.29\%$ ($149.5$: $141.0$ / $8.5$) | **$62.29\%$ ($299.0$: $282.7$ / $16.3$)** | **$-0.21\text{ pp}$** ($\text{SE}=1.58$, $+55/-41$, $p_{\text{cl}}=0.895$, $p_{\text{Holm}}=1.00$) | $(-2.50, +2.08)\text{ pp}$ ($\text{SE}_{\text{seed}}=2.29$) |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`First`)** | $60.83\%$ ($146$: $140$ / $6$) | $62.50\%$ ($150$: $141$ / $9$) | **$61.67\%$ ($296$: $281$ / $15$)** | **$-0.83\text{ pp}$** ($\text{SE}=1.75$, $+32/-36$, $p_{\text{cl}}=0.633$, $p_{\text{Holm}}=1.00$) | $(-3.75, +2.08)\text{ pp}$ ($\text{SE}_{\text{seed}}=2.92$) |
| **Pure `plurality@3` $\{s_1,s_2,s_3\}$ (`MDL-Box`)** | $61.67\%$ ($148$: $140$ / $8$) | $61.67\%$ ($148$: $140$ / $8$) | **$61.67\%$ ($296$: $280$ / $16$)** | **$-0.83\text{ pp}$** ($\text{SE}=1.80$, $+35/-39$, $p_{\text{cl}}=0.643$, $p_{\text{Holm}}=1.00$) | $(-2.92, +1.25)\text{ pp}$ ($\text{SE}_{\text{seed}}=2.08$) |
| **Center-Anchored $\{g_0,s_1,s_2\}$ (`Center`)** | $65.00\%$ ($156$: $146$ / $10$) | $63.75\%$ ($153$: $144$ / $9$) | **$64.38\%$ ($309$: $290$ / $19$)** | **$+1.88\text{ pp}$** ($\text{SE}=1.04$, $+16/-7$, $p_{\text{cl}}=0.0705$, $p_{\text{Holm}}=0.846$) | $(+0.42, +3.33)\text{ pp}$ ($\text{SE}_{\text{seed}}=1.46$) |
| **Center-Anchored $\{g_0,s_1,s_2\}$ (`MDL-Box`)** | $64.17\%$ ($154$: $147$ / $7$) | $64.17\%$ ($154$: $144$ / $10$) | **$64.17\%$ ($308$: $291$ / $17$)** | **$+1.67\text{ pp}$** ($\text{SE}=1.35$, $+26/-18$, $p_{\text{cl}}=0.217$, $p_{\text{Holm}}=1.00$) | $(-0.42, +3.75)\text{ pp}$ ($\text{SE}_{\text{seed}}=2.08$) |
| **3-Pass Learned $\{-g_{\text{lrn}},0,+g_{\text{lrn}}\}$ (`Center`)** | $64.17\%$ ($154$: $141$ / $13$) | $61.25\%$ ($147$: $138$ / $9$) | **$62.71\%$ ($301$: $279$ / $22$)** | $+0.21\text{ pp}$ ($\text{SE}=0.63$, $+4/-3$, $p_{\text{cl}}=0.739$, $p_{\text{Holm}}=1.00$) | $(-0.42, +0.83)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.62$) |
| **3-Pass Learned $\{-g_{\text{lrn}},0,+g_{\text{lrn}}\}$ (`MDL-Box`)** | $63.33\%$ ($152$: $141$ / $11$) | $63.33\%$ ($152$: $139$ / $13$) | **$63.33\%$ ($304$: $280$ / $24$)** | $+0.83\text{ pp}$ ($\text{SE}=1.10$, $+14/-10$, $p_{\text{cl}}=0.450$, $p_{\text{Holm}}=1.00$) | $(-1.25, +2.92)\text{ pp}$ ($\text{SE}_{\text{seed}}=2.08$) |
| **3-Pass Antipodal Rad $\{-g_{\text{rad}}^{(1)},0,+g_{\text{rad}}^{(1)}\}$ (`Center`)** | $66.25\%$ ($159$: $146$ / $13$) | $60.83\%$ ($146$: $137$ / $9$) | **$63.54\%$ ($305$: $283$ / $22$)** | $+1.04\text{ pp}$ ($\text{SE}=0.69$, $+8/-3$, $p_{\text{cl}}=0.131$, $p_{\text{Holm}}=1.00$) | $(+1.67, +0.42)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.62$) |
| **3-Pass Antipodal Rad $\{-g_{\text{rad}}^{(1)},0,+g_{\text{rad}}^{(1)}\}$ (`MDL-Box`)** | $65.83\%$ ($158$: $144$ / $14$) | $61.67\%$ ($148$: $137$ / $11$) | **$63.75\%$ ($306$: $281$ / $25$)** | $+1.25\text{ pp}$ ($\text{SE}=0.93$, $+13/-7$, $p_{\text{cl}}=0.179$, $p_{\text{Holm}}=1.00$) | $(+1.25, +1.25)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.00$) |
| **3-Pass Indep-Dir $\{0,+g_{\text{rad}}^{(1)},+g_{\text{rad}}^{(2)}\}$ (`Center`)** | $65.00\%$ ($156$: $143$ / $13$) | $61.25\%$ ($147$: $138$ / $9$) | **$63.12\%$ ($303$: $281$ / $22$)** | $+0.62\text{ pp}$ ($\text{SE}=0.69$, $+7/-4$, $p_{\text{cl}}=0.366$, $p_{\text{Holm}}=1.00$) | $(+0.42, +0.83)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.21$) |
| **3-Pass Indep-Dir $\{0,+g_{\text{rad}}^{(1)},+g_{\text{rad}}^{(2)}\}$ (`MDL-Box`)** | $66.25\%$ ($159$: $144$ / $15$) | $62.08\%$ ($149$: $139$ / $10$) | **$64.17\%$ ($308$: $283$ / $25$)** | $+1.67\text{ pp}$ ($\text{SE}=1.02$, $+17/-9$, $p_{\text{cl}}=0.101$, $p_{\text{Holm}}=1.00$) | $(+1.67, +1.67)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.00$) |
| **4-Pass Center + 3 Samples $\{g_0,s_{1..3}\}$ (`Center`)** | $65.83\%$ ($158$: $148$ / $10$) | $63.75\%$ ($153$: $144$ / $9$) | **$64.79\%$ ($311$: $292$ / $19$)** | **$+2.29\text{ pp}$** ($\text{SE}=1.15$, $+18/-7$, $p_{\text{cl}}=0.0468$, $p_{\text{Holm}}=0.656$) | $(+1.25, +3.33)\text{ pp}$ ($\text{SE}_{\text{seed}}=1.04$) |
| **4-Pass Center + 3 Samples $\{g_0,s_{1..3}\}$ (`MDL-Box`)** | $67.08\%$ ($161$: $152$ / $9$) | $63.33\%$ ($152$: $143$ / $9$) | **$65.21\%$ ($313$: $295$ / $18$)** | **$+2.71\text{ pp}$** ($\text{SE}=1.42$, $+29/-16$, $p_{\text{cl}}=0.0566$, $p_{\text{Holm}}=0.735$) | $(+2.50, +2.92)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.21$) |
| **Pure 5-Pass $\{s_1..s_5\}$ (`First`)** | $68.33\%$ ($164$: $154$ / $10$) | $65.42\%$ ($157$: $148$ / $9$) | **$66.88\%$ ($321$: $302$ / $19$)** | **$+4.38\text{ pp}$** ($\text{SE}=1.83$, $+43/-22$, $p_{\text{cl}}=0.0171$, $p_{\text{Holm}}=0.307$) | $(+3.75, +5.00)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.63$) |
| **Pure 5-Pass $\{s_1..s_5\}$ (`MDL-Box`)** | $68.75\%$ ($165$: $155$ / $10$) | $66.25\%$ ($159$: $147$ / $12$) | **$67.50\%$ ($324$: $302$ / $22$)** | **$+5.00\text{ pp}$** ($\text{SE}=1.86$, $+47/-23$, $p_{\text{cl}}=0.0073$, $p_{\text{Holm}}=0.160$) | $(+4.17, +5.83)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.83$) |
| **Center-Anchored 5-Pass $\{g_0,s_{1..4}\}$ (`Center`)** | $68.33\%$ ($164$: $152$ / $12$) | $65.42\%$ ($157$: $145$ / $12$) | **$66.88\%$ ($321$: $297$ / $24$)** | **$+4.38\text{ pp}$** ($\text{SE}=1.31$, $+29/-8$, $p_{\text{cl}}=8.12\times 10^{-4}$, **$p_{\text{Holm}}=0.0235$**) | $(+3.75, +5.00)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.63$) |
| **Center-Anchored 5-Pass $\{g_0,s_{1..4}\}$ (`MDL-Box`)** | $69.58\%$ ($167$: $157$ / $10$) | $65.83\%$ ($158$: $145$ / $13$) | **$67.71\%$ ($325$: $302$ / $23$)** | **$+5.21\text{ pp}$** ($\text{SE}=1.65$, $+40/-15$, $p_{\text{cl}}=0.00159$, **$p_{\text{Holm}}=0.0413$**) | $(+5.00, +5.42)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.21$) |
| **Pure 6-Pass $\{s_1..s_6\}$ (`First`)** | $68.75\%$ ($165$: $155$ / $10$) | $67.08\%$ ($161$: $151$ / $10$) | **$67.92\%$ ($326$: $306$ / $20$)** | **$+5.42\text{ pp}$** ($\text{SE}=1.86$, $+46/-20$, $p_{\text{cl}}=0.00355$, $p_{\text{Holm}}=0.085$) | $(+4.17, +6.67)\text{ pp}$ ($\text{SE}_{\text{seed}}=1.25$) |
| **Pure 6-Pass $\{s_1..s_6\}$ (`MDL-Box`)** | $69.17\%$ ($166$: $156$ / $10$) | **$67.92\%$ ($163$: $151$ / $12$)** | **$68.54\%$ ($329$: $307$ / $22$)** | **$+6.04\text{ pp}$** ($\text{SE}=1.88$, $+51/-22$, $p_{\text{cl}}=0.00135$, **$p_{\text{Holm}}=0.0364$**) | $(+4.58, +7.50)\text{ pp}$ ($\text{SE}_{\text{seed}}=1.46$) |
| **Center-Anchored 6-Pass $\{g_0,s_{1..5}\}$ (`Center`)** | **$70.00\%$ ($168$: $156$ / $12$)** | $64.17\%$ ($154$: $142$ / $12$) | **$67.08\%$ ($322$: $298$ / $24$)** | **$+4.58\text{ pp}$** ($\text{SE}=1.42$, $+30/-8$, $p_{\text{cl}}=0.00121$, **$p_{\text{Holm}}=0.0338$**) | $(+5.42, +3.75)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.83$) |
| **Center-Anchored 6-Pass $\{g_0,s_{1..5}\}$ (`MDL-Box`)** | $69.58\%$ ($167$: $156$ / $11$) | $64.58\%$ ($155$: $142$ / $13$) | **$67.08\%$ ($322$: $298$ / $24$)** | **$+4.58\text{ pp}$** ($\text{SE}=1.67$, $+37/-15$, $p_{\text{cl}}=0.00606$, $p_{\text{Holm}}=0.139$) | $(+5.00, +4.17)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.42$) |
| **5-Point Deterministic $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)}\}$ (`Center`)** | $67.08\%$ ($161$: $146$ / $15$) | $62.50\%$ ($150$: $140$ / $10$) | **$64.79\%$ ($311$: $286$ / $25$)** | **$+2.29\text{ pp}$** ($\text{SE}=0.95$, $+16/-5$, $p_{\text{cl}}=0.0153$, $p_{\text{Holm}}=0.291$) | $(+2.50, +2.08)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.21$) |
| **5-Point Deterministic $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)}\}$ (`MDL-Box`)** | $66.67\%$ ($160$: $146$ / $14$) | $64.17\%$ ($154$: $141$ / $13$) | **$65.42\%$ ($314$: $287$ / $27$)** | **$+2.92\text{ pp}$** ($\text{SE}=1.09$, $+20/-6$, $p_{\text{cl}}=0.00737$, $p_{\text{Holm}}=0.160$) | $(+2.08, +3.75)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.83$) |
| **5-Pass Hybrid $\{g_0,\pm g_{\text{rad}}^{(1)},s_1,s_2\}$ (`Center`)** | $65.00\%$ ($156$: $144$ / $12$) | $63.75\%$ ($153$: $143$ / $10$) | **$64.38\%$ ($309$: $287$ / $22$)** | $+1.88\text{ pp}$ ($\text{SE}=1.08$, $+17/-8$, $p_{\text{cl}}=0.0820$, $p_{\text{Holm}}=0.902$) | $(+0.42, +3.33)\text{ pp}$ ($\text{SE}_{\text{seed}}=1.46$) |
| **5-Pass Hybrid $\{g_0,\pm g_{\text{rad}}^{(1)},s_1,s_2\}$ (`MDL-Box`)** | $65.00\%$ ($156$: $147$ / $9$) | $65.00\%$ ($156$: $143$ / $13$) | **$65.00\%$ ($312$: $290$ / $22$)** | **$+2.50\text{ pp}$** ($\text{SE}=1.17$, $+24/-12$, $p_{\text{cl}}=0.0326$, $p_{\text{Holm}}=0.489$) | $(+0.42, +4.58)\text{ pp}$ ($\text{SE}_{\text{seed}}=2.08$) |
| **8-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)},s_{1..3}\}$ (`Center`)** | $67.92\%$ ($163$: $151$ / $12$) | $65.42\%$ ($157$: $144$ / $13$) | **$66.67\%$ ($320$: $295$ / $25$)** | **$+4.17\text{ pp}$** ($\text{SE}=1.22$, $+25/-5$, $p_{\text{cl}}=6.59\times 10^{-4}$, **$p_{\text{Holm}}=0.0198$**) | $(+3.33, +5.00)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.83$) |
| **8-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)},s_{1..3}\}$ (`MDL-Box`)** | $66.25\%$ ($159$: $149$ / $10$) | $64.58\%$ ($155$: $143$ / $12$) | **$65.42\%$ ($314$: $292$ / $22$)** | **$+2.92\text{ pp}$** ($\text{SE}=1.24$, $+25/-11$, $p_{\text{cl}}=0.0185$, $p_{\text{Holm}}=0.315$) | $(+1.67, +4.17)\text{ pp}$ ($\text{SE}_{\text{seed}}=1.25$) |
| **11-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)},s_{1..6}\}$ (`Center`)** | $69.58\%$ ($167$: $151$ / $16$) | $64.17\%$ ($154$: $141$ / $13$) | **$66.88\%$ ($321$: $292$ / **$29$**)** | **$+4.38\text{ pp}$** ($\text{SE}=1.40$, $+30/-9$, $p_{\text{cl}}=0.00182$, **$p_{\text{Holm}}=0.0455$**) | $(+5.00, +3.75)\text{ pp}$ ($\text{SE}_{\text{seed}}=0.63$) |
| **11-Pass Combined $\{g_0,\pm g_{\text{lrn}},\pm g_{\text{rad}}^{(1)},s_{1..6}\}$ (`MDL-Box`)** | $69.58\%$ ($167$: $153$ / $14$) | $63.33\%$ ($152$: $140$ / $12$) | **$66.46\%$ ($319$: $293$ / $26$)** | **$+3.96\text{ pp}$** ($\text{SE}=1.56$, $+34/-15$, $p_{\text{cl}}=0.0109$, $p_{\text{Holm}}=0.218$) | $(+5.00, +2.92)\text{ pp}$ ($\text{SE}_{\text{seed}}=1.04$) |

*Table 9: Untouched out-of-distribution evaluation on `Ext240` (`GSM8K-200 + AMC23-40`, $N=240$ items, $480$ two-seed evaluations), reporting single-seed sub-benchmark counts (`GSM8K-200` / `AMC23-40`), 2-seed question-clustered inference with 30-test Holm-Bonferroni correction ($p_{\text{Holm,30}}$), and descriptive per-seed deltas $(\Delta_{102}, \Delta_{42})$ with half-range $\text{SE}_{\text{seed}}$.*

Table 9 establishes four candid, properly scoped findings on untouched out-of-distribution problems:
1. **Out-of-Distribution Transfer of `Tessera-Sym` Greedy Center ($g_0$):** Single-pass greedy accuracy rises from $13.75\%$ ($33/240$: $29/200$ `GSM8K`, $4/40$ `AMC23`) on `Qwen3-0.6B-Base` and $62.08\%$ ($149/240$) on `Baseline-Seed102` to **$64.58\%$ ($155/240$: $142/200$ `GSM8K`, $13/40$ `AMC23`) on Seed 102** and **$60.42\%$ ($145/240$: $136/200$ `GSM8K`, $9/40$ `AMC23`) on Seed 42** ($300/480 = 62.50\%$ pooled).
2. **3-Pass Voting on `Ext240` (Positive Unadjusted Contrast, Unresolved After Multiplicity Correction):** On `Ext240`, individual $T=0.4$ stochastic draws drop by **$-3.33\text{ pp}$** relative to greedy decoding ($284.0/480 = 59.17\%$, $p_{\text{cl}} = 0.0216$). Homogeneous `plurality@3` ($\{s_1, s_2, s_3\}$) fails to improve over greedy decoding ($296/480 = 61.67\%$, $-0.83\text{ pp}$, $+32/-36$, $p_{\text{cl}} = 0.633$), whereas **Center-Anchored $\{g_0, s_1, s_2\}$** cuts greedy-correct losses from $36$ to $7$ ($+16/-7$) and reaches **$309/480$ ($64.38\%$, $+1.88\text{ pp}$ over $g_0$, $p_{\text{cl}} = 0.0705$, $p_{\text{Holm,30}} = 0.846$)** under `Center` and **$308/480$ ($64.17\%$, $+1.67\text{ pp}$, $p_{\text{cl}} = 0.217$)** under `MDL-Box`. In direct head-to-head comparison on `Ext240` at identical 3-pass compute, `Center-Anchored {g0, s1, s2}` has an unadjusted advantage over `Pure {s1, s2, s3}` of **$+2.50\text{ pp}$ under `MDL-Box` (`+23/-11`, $p_{\text{cl}} = 0.0326$, $\Delta_{102}=+2.50\text{ pp}, \Delta_{42}=+2.50\text{ pp}$)** and **$+2.71\text{ pp}$ under `Center` vs. `First` (`+29/-16`, $p_{\text{cl}} = 0.0566$)**, which yields **$p_{\text{Holm,4}} = 0.1304$** across the 4 external head-to-head contrasts (`reproduce_stencil_and_svd.py`). Thus, at $N=240$, the 3-pass external contrast is directionally consistent with the within-MATH regression-reduction effect but does not independently pass multiplicity-adjusted confirmation.
3. **3-Pass and 5-Point Deterministic Stencils on `Ext240` (Unadjusted Gains vs. 30-Test Holm Threshold):** All 3-pass and 5-point deterministic stencils show positive point estimates over $g_0$ on `Ext240`, with `3-Pass Antipodal Rademacher (MDL-Box)` at $+1.25\text{ pp}$ ($306/480 = 63.75\%$, $p_{\text{cl}} = 0.179$), `3-Pass Indep-Dir Rademacher (MDL-Box)` at $+1.67\text{ pp}$ ($308/480 = 64.17\%$, $p_{\text{cl}} = 0.101$), and the **`5-Point Deterministic Stencil`** reaching **$311/480$ ($64.79\%$, $+2.29\text{ pp}$, $+16/-5$, $p_{\text{cl}} = 0.0153$, $p_{\text{Holm,30}} = 0.291$)** under `Center` and **$314/480$ ($65.42\%$, $+2.92\text{ pp}$, $+20/-6$, $p_{\text{cl}} = 0.0074$, $p_{\text{Holm,30}} = 0.160$)** under `MDL-Box` (and $+3.75\text{ pp}$ vs. `Pure3 MDL`, $+44/-26$, $p_{\text{cl}} = 0.0376$, $p_{\text{Holm,4}} = 0.1304$). While the 5-point stencil is significant at the unadjusted level ($p_{\text{cl}} = 0.0074$) and lifts competition math (`AMC23-40`) from $22/80$ ($27.50\%$) to $27/80$ ($33.75\%$), **none of the 3-pass or 5-point deterministic stencils alone survives the 30-test Holm-Bonferroni correction on `Ext240` ($p_{\text{Holm,30}} \ge 0.160$)** due to the smaller sample size ($M=240$ vs. $1,850$).
4. **Multiplicity-Adjusted External Confirmation at 5–11 Passes ($p_{\text{Holm,30}} < 0.05$):** Across all 30 tested rules on `Ext240`, **six 5–11 pass stochastic and combined protocols survive full 30-test Holm-Bonferroni multiplicity correction**: **`8-Pass Combined (Center)`** ($320/480 = 66.67\%$, $+4.17\text{ pp}$, $+25/-5$, $p_{\text{Holm,30}} = 0.0198$), **`Center-Anchored 5-Pass (Center)`** ($321/480 = 66.88\%$, $+4.38\text{ pp}$, $+29/-8$, $p_{\text{Holm,30}} = 0.0235$), **`Center-Anchored 6-Pass (Center)`** ($322/480 = 67.08\%$, $+4.58\text{ pp}$, $+30/-8$, $p_{\text{Holm,30}} = 0.0338$), **`Pure 6-Pass (MDL-Box)`** ($329/480 = 68.54\%$, $+6.04\text{ pp}$, $+51/-22$, $p_{\text{Holm,30}} = 0.0364$), **`Center-Anchored 5-Pass (MDL-Box)`** ($325/480 = 67.71\%$, $+5.21\text{ pp}$, $+40/-15$, $p_{\text{Holm,30}} = 0.0413$), and **`11-Pass Combined (Center)`** ($321/480 = 66.88\%$, $+4.38\text{ pp}$, $+30/-9$, $p_{\text{Holm,30}} = 0.0455$).

---

## 6. Limitations, Scope Conditions, and Pre-Registration Scope

1. **Bounded Interpretation of Fixed-Reference vs. Own-Completion Greedy-Prefix Conditional KL, Activation Alignment, and Cross-Arm Non-Monotonicity:** Our online experiments (`code/train_osf_rlvr.py`, `code/run_stage2_ares_geora_vanilla_and_lrsweep.py`, and `code/run_reviewer_interventions_and_audits.py`, Sections 2.4 and 2.5) span three independent training seeds (`102`, `42`, `137`), both shaped (`Tessera-Sym`) and exact supplied (`Vanilla-GRPO`) objectives, 50-step and 100-step horizons, pre/post-final-projection and every-step master-writeback controls, online learning-rate sweeps ($\eta \in \{2.5\times 10^{-6}, 5.0\times 10^{-6}, 1.0\times 10^{-5}\}$), a prespecified 12-point target-free scale grid ($\alpha \in \{0.25, 0.50, 0.75, 1.00\}$), and head-to-head comparisons against `Ares` (Chen, Zhao, & Wen [22], under both the published Eq. 3 squared-singular-value energy rule $r_t \approx 10.5$ and the amplitude rule $r_t \approx 38$), `GeoRA` (Zhang et al. [24]), `PiSSA` (Meng et al. [25]), and `MiLoRA` (Wang et al. [26]), as evaluated in RLVR by Yin et al. [23]. Across all evaluated checkpoints (`records/kl_and_spectral_shift_metrics.json` and `records/extended_spectral_and_onpolicy_kl.json`), we distinguish two regimes:
   - **Within-Arm Causal Scaling and Subspace Dispersion:** Within any fixed training trajectory or subspace family, scaling down update magnitude ($\alpha \in \{0.25, 0.50, 0.75, 1.00\}$ or reducing $\eta$ from $10^{-5} \to 2.5\times 10^{-6}$) or dispersing update energy across 1,008 orthogonal tail modes (`OSF-Tail16` vs. `OSF-Top16` at matched $\|\Delta W\|_F \approx 0.86$) simultaneously reduces both fixed-reference conditional next-token KL $\mathrm{KL}_{\text{ref}}(\pi_\theta \| \pi_0)$ on the 64 reference trajectories and own-completion greedy-prefix conditional KL $\mathrm{KL}_{\text{greedy}}(\pi_\theta \| \pi_0)$ on each checkpoint+operator's own `MATH-500[:64]` greedy completions (distinct from an expectation over stochastic on-policy rollouts), recovering $+10\text{ to }+28\text{ pp}$ in reasoning accuracy.
   - **Cross-Arm Non-Monotonicity and Trajectory Specificity:** Across different optimizer architectures, KL divergence is a strongly predictive indicator of step-size overshoot (all high-performing arms satisfy $\mathrm{KL}_{\text{cond}} \le 0.08\text{ nats/tok}$, whereas all arms with $\mathrm{KL}_{\text{cond}} > 0.11\text{ nats/tok}$ degrade significantly), **not a strictly monotonic cross-arm function of accuracy**. For example, `Ares-FP32` (amplitude rule, $\eta = 10^{-5}$) has slightly lower conditional KL than `OSF-Top16` on Seed 102 ($\mathrm{KL}_{\text{cond}} = 0.2198$ vs. $0.2828\text{ nats/tok}$) yet scores lower on the 3-panel `LinFP32` benchmark ($33.39\%$ vs. $36.85\%$), because pre-AdamW gradient truncation and post-AdamW subspace projection follow distinct parameter trajectories.
2. **Untouched `Fresh-500` Benchmark, External `Ext240` Confirmation, and Question-Clustered Multi-Seed Inference:** Because the four original within-MATH panels (`Val350`, `MATH-500`, `Sec-500`, `Conf-500`) were observed on Seed 102 during earlier studies, we constructed the strictly untouched 500-item stratified benchmark `Fresh-500` (`code/splits/test_fresh_untouched_500.json`, deterministically reproducible via `code/build_splits.py` with full question provenance and `grade_answer`-verified extracted predictions in `records/osf_and_precision_factorial.json`) alongside the separately sourced external benchmark `Ext240` (`GSM8K-200 + AMC23-40`). Using question-clustered inference across $M=500$ unique `Fresh-500` items ($S=3$ seeds, $N=1,500$ evaluations), `OSF-Tail16-FP32-Master` (`769/1,500 = 51.27%`) significantly outperforms `OSF-Top16-FP32-Master` (`462/1,500 = 30.80%`, **`+20.47 pp`, 95% CI $[+17.70, +23.23]\text{ pp}$, $\mathrm{SE}_{\text{cl}} = 1.41\text{ pp}$, $p_{\text{cl}} = 9.86 \times 10^{-48}$**) and `Unfiltered-FP32-Master` (`353/1,500 = 23.53%`, **`+27.73 pp`, 95% CI $[+24.66, +30.81]\text{ pp}$, $p_{\text{cl}} = 8.79 \times 10^{-70}$**), while exhibiting **an unresolved difference** relative to `Standard-BF16` (`521/1,000 = 52.10%` vs. `503/1,000 = 50.30%` under matched `FP32LinearInBF16Net` on Seeds 102 & 42: **`+1.80 pp`, 95% CI $[-1.01, +4.61]\text{ pp}$, $\mathrm{SE}_{\text{cl}} = 1.43\text{ pp}$, $\mathrm{SE}_{\text{seed}} = 0.80\text{ pp}$, $p_{\text{cl}} = 0.210$**; and `500/1,000 = 50.00%` vs. `511/1,000 = 51.10%` under matched `BF16`: **`-1.10 pp`, $p_{\text{cl}} = 0.409$**). For test-time voting on `Ext240` (Section 5.3), six 5–11 pass center-anchored, pure-sampled, and combined voting rules pass 30-test Holm-Bonferroni confirmation ($p_{\text{Holm,30}} \in [0.0198, 0.0455]$), whereas 3-pass and 5-point deterministic stencil gains ($p_{\text{cl}} = 0.0074\text{--}0.0326$ unadjusted) do not survive 30-test multiplicity correction ($p_{\text{Holm,30}} \ge 0.160$).
3. **Scope of the Rademacher Gate Control and Calibration Cost:** Our Rademacher control $g_{\text{rad},\ell} = \sigma_{\text{RMS},\ell} \cdot s_\ell$ uses zero gate-training steps for its $8,192$ coordinate signs $s_\ell \in \{-1, +1\}^{1024}$, but inherits the scalar layerwise RMS magnitude $\sigma_{\text{RMS},\ell} \approx 0.030\text{--}0.034$ calibrated by a 50-step ($400$-rollout) `GateOnly-Loop10_18` run. This demonstrates that **at the calibrated RMS scale, learned coordinate directions are not distinguishable from random Rademacher directions**, rather than claiming a calibration-free perturbation across arbitrary architectures.
4. **Training Spectral-Decomposition Work, Inference Work Proxies, Checkpoint Storage vs. Runtime VRAM, and Practical Method Selection:** During 50-step RLVR training (`21` active non-zero-variance optimizer steps on Seed 102, `20` on Seed 42, `17` on Seed 137), `Standard-BF16` and `Unfiltered-FP32-Master` require **`0` SVD factorizations** (`~11 min` per 50-step run on a single GPU); adapter methods (`GeoRA-r16`, `PiSSA-r16`, `MiLoRA-r16`) require **`196` initialization SVDs** (`~12 min` total); active-step `OSF-Top16` and `OSF-Tail16` execute $196$ Gram-eigendecomposition SVDs per active step plus $196$ final post-averaging SVDs (**$21 \times 196 + 196 = 4,312$ SVDs** on Seed 102, $4,116$ on Seed 42, $3,528$ on Seed 137; `~18 min` per run); `Ares` executes $196$ rank-64 gradient SVDs per active step (**$21 \times 196 = 4,116$ SVDs**; `~16 min` per run); and every-step synchronization/writeback controls (`OSF-EveryStepSync`, `OSF-Writeback`) execute **$50 \times 196 + 196 = 9,996$ SVDs** (`~28 min` per run). At inference, all multi-pass protocols multiply completion token and block-pass work ($2.99\times\text{--}3.61\times$ for 3-pass rules, $5.00\times\text{--}6.19\times$ for 5–6 pass rules), and the $1.50\text{ GB}$ figure refers to single-checkpoint `bfloat16` parameter storage (vs. $4.50\text{ GB}$ for a 3-checkpoint ensemble). Practically, when stochastic sampling is available at a 6-pass budget, `Center-Anchored 6-Pass` / `Pure 6-Pass` significantly outperforms the 5-point deterministic stencil ($+2.78\text{ pp}$, $p_{\text{Holm,17}} = 1.56 \times 10^{-7}$); deterministic stencils are specifically advantageous when $T=0$ reproducibility is required or when combined with stochastic passes (`Hyb5`, `8-Pass`, `11-Pass`).
5. **Summary:** Together, our 21-stage state-dependent policy gradient, 5-branch optimizer, and mixed-dtype replay audits (Section 2.2 and Table 3A), our 12-cell confound-free post-hoc precision, component, and spectral subspace factorial (Section 2.3 and Table 4), our 3-seed $\times$ 6-panel Online Subspace-Filtered Policy Optimization (`OSF-RLVR`), pre/post-projection and every-step master-writeback controls, fixed-reference and own-completion greedy-prefix conditional KL / NSS tracking, and head-to-head comparisons with `Ares` (both Eq. 3 energy and amplitude rules), `GeoRA`, `PiSSA`, and `MiLoRA` (Sections 2.4–2.5 and Tables 5A–5C), our full 28-layer GPU KV-cache verification across $N=32$ real `MATH-500` 96-token prefixes ($2,048$ decoded token positions, Section 3), and our test-time voting diagnostics across within-MATH ($3,700$ evaluations) and untouched `Ext240` ($480$ evaluations) provide a unified mechanistic and interventional account of precision quantization, spectral subspace geometry, and trust-region stability in `bfloat16` RLVR.

---

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