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 $1.00$).

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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 $12.16\%$ ($225/1,850$).

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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.

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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) ($321/480 = 66.88\%$, $+4.38\text{ pp}$, $+30/-9$, $p_{\text{Holm,30}} = 0.0455$).

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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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