1. Problem Formulation and Summary of Contributions

Reinforcement learning with verifiable rewards (RLVR) using Group Relative Policy Optimization (GRPO) [1, 2, 3] enables a pretrained base language model (Qwen3-0.6B-Base, $P = 751,632,384$ untied trainable parameters [4]) to rise from $14.80\%$ ($74/500$) greedy accuracy on MATH-500 and $12.50\%$ ($30/240$) on an independent external benchmark (GSM8K-200 + AMC23-40) to $44.40\%$–$49.40\%$ on MATH-500 and $57.92\%$–$61.25\%$ on GSM8K-200 + AMC23-40 after 50 single-question updates on math_train.json[:50]. In the standard single-question GRPO loop, the policy samples $G=8$ trajectories $y^{(1)}, \dots, y^{(G)} \sim \tilde{\pi}_\theta(\cdot \mid x)$ per question $x$ (under the 512-token-capped temp=0.8, top_p=0.9 rollout kernel $\tilde{\pi}_\theta$), computes standardized binary rewards $R_i \in \{0,1\}$, and updates parameters via sequence-summed token log-probabilities:

$$ A_i = \frac{R(y^{(i)}, a^*) - \bar{R}}{\sigma_R + 10^{-4}}, \qquad \mathcal{L}(\theta) = -\frac{1}{G} \sum_{i=1}^G A_i \sum_{t=1}^{|y^{(i)}|} \log \pi_\theta\left(y^{(i)}_t \;\middle|\; x, y^{(i)}_{<t}\right). $$

Within this small-budget regime, we study four bounded diagnostic questions: (i) how omitting redundant zero-advantage backward passes on the $52\%$–$66\%$ of questions with homogeneous rewards ($\sigma_R = 0 \implies A_i = 0$) while executing explicit $O(P)$ zero-gradient AdamW steps compares against full-backward training and step skipping (--skip-zero-var), with and without a 30-character pre-boxed gate (text.rfind("\\boxed") >= 30); (ii) in two-stage ($K=4 \times B=2$) branched rollouts [13, 14, 16], how early-EOS branch handling distinguishes a genuinely $\mathcal{F}_t$-measurable stopped tree from outcome-dependent root-restart and midpoint-split samplers; (iii) whether adding a sibling-disagreement prefix bonus ($\beta = 0.25$) yields any detectable benefit over TreeRL's descendant-mean prefix credit ($\beta = 0.0$) across 512- and 1,024-token horizons ($2 \times 2$ Horizon x Prefix Bonus factorial across Arms 5, 6, 12, 13), alongside a census of cap-associated active steps, reward-divergent pairs, and retained token-512 prefix extractions (gt512_prefix_audit); and (iv) how 100-question combined gradient accumulation (Arm 11), $k=3$ test-time voting [19, 20] (MDL-Maj@3 vs. MaxLen-Maj@3), counterfactual prefix splicing and tie-gated low-margin forking (Tables 5A & 5B), and a six-arm 3-seed Online Counterfactual Fork-Pair GRPO study (Arms 14–19, Table 5C) behave across benchmarks.

We report a nineteen-arm, three-seed (seeds = 42, 101, 7) experimental matrix plus six single-seed (seed=101) ablations across 64 primary training runs (3,422 steps, 13,688 rollout pairs: 46 runs in training_records.json + 18 Online CF-GRPO runs in prefix_splice_probe.json, plus 558 $>512$-token prefix records and 12 external retraining runs = 600 steps) and 182 primary evaluations (58,370 grades: 128 in per_item_grades.json + 54 in prefix_splice_probe.json) plus 2,700 sampled-$k=3$ responses and 5,072 counterfactual prefix-splice, native fork, and batch-sensitivity responses (66,142 independently regraded item responses total): 64 MATH-500 evaluations (32,000 grades), 65 math_val210 evaluations (13,650 grades), and 53 evaluations on GSM8K-200 + AMC23-40 (12,720 grades). Because MATH-500 was inspected on seed=101 during initial gate and warmup selection (Section 5.1), GSM8K-200 + AMC23-40 ($N=240$) serves as our untouched confirmatory benchmark and MATH-500 ($N=500$) and math_val210 ($N=210$) as broader in-domain readouts. All metrics, independent regrading (PASS 1 & PASS 4), finite-policy execution of generate_bifurcated and fixed-trajectory AdamW equivalence (PASS 2), integer token conservation laws (PASS 3), and 4,000-resample two-level (seed x question), question-clustered, and paired-item bootstrap $95\%$ CIs are verified in $\sim 2\text{s}$ on CPU by verify_supplement.py:

1. $O(P)$ Zero-Gradient AdamW Stepping vs. Step Skipping and Execution-Recipe Controls (Sections 3.2 & 5.2): In the baseline (Arm 1, code/run_baseline.py), homogeneous groups ($\sigma_R = 0$) execute 8 zero-weighted backward passes ($135.7\text{k}$ output / $169.4\text{k}$ full backward tokens; $815.9 / 1018.6 / 1069.5\text{ TFLOPs}$) before optimizer.step(). Allocating zero gradients (param.grad = torch.zeros_like(param)) and calling optimizer.step() costs $O(P)$ elementwise operations and zero transformer backward FLOPs while preserving AdamW $(m_t, v_t)$, bias-correction, and weight-decay updates omitted by step skipping (--skip-zero-var).

   • Fixed-Trajectory Bitwise Equivalence vs. Live Execution-Recipe Comparison (Arm 2 vs. Arm 1): On any fixed sequence of active and homogeneous steps sharing a common active-step loss, param.grad = torch.zeros_like(param); optimizer.step() yields bitwise identical parameters $\theta_t$ and AdamW buffers $(m_t, v_t)$ (max_abs_diff = 0.0) to $0.0$-weighted backward passes (PASS 2). In live GPU runs, Arm 1 sums cross-entropies before multiplying by advantage, whereas Arms 2–19 multiply each token's NLL by advantage before summing (compute_weighted_logprob), introducing a 1-ULP active-step rounding difference (grad_norm = 808.9206 vs. 808.9137 on Step 6 of seed=42) that diverges subsequent bfloat16 trajectories [4]; live differences are reported descriptively as execution-recipe comparisons. Under ungated reward, Arm 2 cuts output backward tokens by $56.8\%$ ($58.6\text{k}$ vs. $135.7\text{k}$) and dense FLOP proxies by $43.2\%$ / $43.9\%$ / $41.3\%$ ($463.3 / 571.3 / 627.4$ vs. $815.9 / 1018.6 / 1069.5\text{ TFLOPs}$), with same-host wall-clock of $438.1\text{s}$ vs. $454.1\text{s}$ (-3.5%, $\sim 16.0\text{s}$; $530.0\text{s}$ historical mean). Across 3 seeds, Arm 2 scores $60.83\% \pm 0.42\%$ on Ext240 (+1.67 pp vs. Arm 1, two-level $95\%$ CI $[-2.22, +5.69]\text{ pp}$), $44.67\% \pm 0.90\%$ on MATH-500 (-1.40 pp, $[-4.60, +1.80]\text{ pp}$), and $54.44\% \pm 1.92\%$ on math_val210.
   • Gated Zero-Gradient Stepping (Arm 4, ra_flat) vs. Gated Step Skipping (Arm 3, gate_flat): With the 30-character pre-boxed gate, ra_flat ($45.53\% \pm 1.03\%$ MATH-500, $59.31\% \pm 1.68\%$ Ext240, $489.0 / 600.1 / 654.8\text{ TFLOPs}$, $433.0\text{s}$) vs. gate_flat ($43.07\% \pm 6.54\%$ MATH-500, $58.47\% \pm 4.49\%$ Ext240) yields [-1.40, -1.40, +10.20] pp on MATH-500 (mean $+2.47\text{ pp}$, $[-3.47, +9.60]\text{ pp}$). Both Arm 2 ($34.3 \pm 1.5$) and Arm 4 ($33.7 \pm 11.0$, $-3.80\text{ pp}$, per-seed [-2.40, -5.20, -3.80] pp, $[-6.60, -1.00]\text{ pp}$) reduce test cap hits (AllCap2048) vs. Arm 1 ($52.7 \pm 5.5$).

2. Exact Distributional Analysis of Stopped-Tree vs. Outcome-Dependent Branching Samplers (Sections 4.2 & 4.3): In two-stage ($K=4 \times B=2$) rollouts where Stage 1 targets the first newline boundary $\tau_{\text{causal}} \in [96, 160]$, early-EOS handling ($T_{\text{EOS}} \le \tau_{\text{causal}}$) determines whether leaf marginals match the capped on-policy kernel $\tilde{\pi}_\theta(\cdot \mid x)$ [13, 16]:

   • True $\mathcal{F}_t$-Stopped-Tree (--strict-causal-tau --early-eos-mode terminal_dup, Arms 5, 6, 11, 12, 13): Stopping at $\tau_k = \min(\tau_{\text{causal}}, T_{\text{EOS}})$ is $\mathcal{F}_t$-measurable. When $T_{\text{EOS}} \le \tau_{\text{causal}}$, $u^{(k)} = y_{1:T_{\text{EOS}}}$ is an absorbing terminal leaf with zero-length continuations ($v^{(k,1)} = v^{(k,2)} = \epsilon$), so every leaf has the exact marginal $\Pr(Y_{k,b} = y) = \tilde{\pi}_\theta(y \mid x)$.
   • Outcome-Dependent Root-Restart (--strict-causal-tau --early-eos-mode root_resample, Arms 7 & 8): Retrospectively setting $\tau_k = 0$ after observing EOS in $p^{(k)}_{1:\tau_{\text{causal}}}$ is not $\mathcal{F}_0$-measurable; retaining the early-EOS sequence as Branch 1 while resampling Branch 2 from the root biases sibling marginals ($\Pr(Y_1 = \text{EOS}) = p$ vs. $\Pr(Y_2 = \text{EOS}) = p^2$, verified at $p=1/2$ in PASS 2: (0.50, 0.25) vs. (0.50, 0.50)).
   • Prefix-Sum Qualification on Shared-Prefix Subsets (Proposition 2): On any realized tree, $\beta = 0$ descendant-mean credit equals leaf-summed GRPO on every shared prefix (Proposition 1), but when only $S = \{k : \tau_k > 0\} \subsetneq \{1,\dots,K\}$ share a non-empty prefix, $B\sum_{k \in S}(\hat{V}_k - \bar{R})/(\sigma_R + 10^{-4})$ need not vanish (-1.5 in PASS 2); only the bonus increment $\Delta_\beta = B\beta\sum_{k \in S}(M_k - \hat{V}_k)/(\sigma_R + 10^{-4}) \ge 0$ is nonnegative for every $S$.

3. Three-Seed $2 \times 2$ (Horizon: 512 vs. 1,024 $\times$ Prefix Bonus: \beta = 0.0 vs. 0.25) Stopped-Tree Factorial (Arms 5, 6, 12, 13) and Cap Census (Sections 4.4 & 5.3): At 512 tokens (Arms 5–8, Table 2A), no accuracy benefit from $\beta = 0.25$ over $\beta = 0.0$ is detected within two-level $95\%$ CIs under Stopped-Tree (-0.97 pp Ext240, $+0.73\text{ pp}$ MATH-500) or Root-Restart ($+0.69\text{ pp}$ Ext240, $+0.07\text{ pp}$ MATH-500), while one-sided prefix clamping (Proposition 3) degrades seed=101 by $-14.80\text{ pp}$ ($33.00\%$). Across Arms 1–10, 187/612 (30.6%) active steps have every $R=0$ leaf at $L = 512$, and 48.9%–66.7% of divergent pairs in Arms 5–8 have $L_{R=0} = 512$. Raising --tokens to $1,024$ (Arms 12, 13) cuts CapAssocActive to $5.6\%$ (3/54) and $1.8\%$ (1/57), cuts CapR0Div to $10.0\%$ (5/50) and $24.5\%$ (13/53), and triples Both < Cap divergent pairs to $15.0$ and $13.3\text{/run}$; in gt512_prefix_audit (PASS 3), 87/90 (96.7%) and 107/118 (90.7%) of $>512$-token $R=1$ rollouts fail the verifier at token 512 before passing. Arm 12 ($\beta=0.0$) reaches $46.47\% \pm 0.81\%$ on MATH-500 ($+2.27\text{ pp}$ vs. Arm 5, $[-0.73, +5.13]$), $55.71\%$ on Val210, and $61.11\%$ on Ext240, while Arm 13 - Arm 12 remains unresolved ($-1.07, -0.16, -2.08\text{ pp}$; $2 \times 2$ interactions $-1.80, +3.17, -1.11\text{ pp}$).

4. 100-Question Accumulation (Arm 11), Voting (Table 4), Prefix-Splice & Tie-Gated Forking (Tables 5A & 5B), and Six-Arm 3-Seed Online Counterfactual Fork-Pair GRPO (Arms 14–19, Table 5C):

   • Arm 11 (100-question Stopped-Tree $\beta=0.0$ + zero-mass centering [10] + Accum4) scores $46.20\% \pm 0.35\%$ on MATH-500, $55.08\%$ on Val210, and $59.72\%$ on Ext240. In $k=3$ voting (Table 4), MDL-Maj@3 (15 <= L < 2048) exceeds MaxLen-Maj@3 by $+4.60, +3.40, +4.00\text{ pp}$ (Arms 5, 11, 13 MATH-500) and $+4.76\text{ pp}$ (Arm 12 Val210).
   • Across all 98 discordant items where Arm 12 seed=42 succeeded and seed=7 failed (86 prompt-only failures, Table 5A, PASS 4), unpadded teacher-forcing on seed=42's 160-token prefixes yields 96.83% (15,141/15,636) argmax agreement; the 495 mismatches have a $14.4\times$ smaller margin ($0.59$ vs. $8.49$) and 85.25% (422/495) top-2 containment. Splicing seed=42's prefix through the 1st–3rd mismatch rescues 40.7%, 53.5%, and 60.5% of prompt-only failures, while donor-free top-2 flipping at seed=7's 1st/2nd min-margin token in $[4, 64)$ rescues 29.1% (25/86) each (48.8% union vs. 12.8% identity control). Across all 950 items (Table 5B), FirstGap05-Fork@3 lifts CR-Maj@3 to 503/950 = 52.95% (+2.53 pp, $[+1.16, +3.89]$), and [email protected] ($3.48$ branches/item) resolves 60.6% (137/226) of 3-way ties, reaching 514/950 = 54.11% (+3.68 pp, $[+2.00, +5.37]$).
   • Across six 3-seed 1,024-token Online Counterfactual Fork-Pair GRPO packages (18 runs, 920 steps, 329 active groups replayed in PASS 4, Table 5C), all single-policy two-level (seed x question) $95\%$ CIs against Arm 12 span zero (Arm 15 - Arm 12: $+1.20\text{ pp}$ $[-1.67, +4.00]$ MATH-500, $+0.35\text{ pp}$ $[-2.00, +2.60]$ Pool950): (i) Unconditional CF-GRPO (Arm 14, --cf-balance-div-len --cf-fork-hinge-weight 0.5) has 74/224 divergent pairs (36/74 with $R_{\text{raw}}(k,0)=1, R_{\text{raw}}(k,1)=0$, activating 13 steps where all 4 on-policy branches passed; $51.72\%$ Pool950); (ii) 50-Step RG-CF-GRPO (Arm 15, changing --cf-rescue-only, --cf-dampen-lose-len, and --cf-fork-hinge-weight 0.15) duplicates the winning branch on $(1,0)$ pairs (removing observed $(1,0)$ activations by construction; 13.33 active updates/seed) and scores $47.67\% \pm 0.83\%$ greedy ($52.20\%$ MDL-Maj@3, $61.60\%$ Pass@3) on MATH-500 and $52.56\% \pm 0.75\%$ on Pool950; (iii) Act18 Control (Arm 16) continues Arm 15 to 18.00 active updates/seed ($53/57/60$ problems; 12/14 extra updates at $b_{0,\text{pass}}=3$), scoring $51.23\% \pm 0.90\%$ on Pool950 ([-20, -12, -6] items vs. Arm 15); (iv) Algebra PB-CF-GRPO (Arm 17, --cf-pair-balanced) skips unrescued $b_{0,\text{pass}} \ge 3$ groups and zeroes non-divergent pairs and $A_{\text{pref}}$ when $b_{0,\text{pass}} \ge 3$ (while clamping $A_{\text{pref}} \ge 0$ when $b_{0,\text{pass}} < 3$), scoring $56.03\% \pm 0.73\%$ on Val210, $52.32\% \pm 0.42\%$ on Pool950, and $53.20\%$ (MATH-500) / $58.11\%$ (Pool950) 3-seed MDL-Maj@3; and (v–vi) on math_train_stratified.json[:50], Stratified RG-CF-GRPO (Arm 18) and Stratified PB-CF-GRPO (Arm 19) rescue 13 all-zero $b_0=0/4$ groups each (10 and 9 post-warmup (0,1) groups, both retaining $A_{\text{pref}}(k^*,b) \approx +1.0604$) and score $51.54\%$ mean on Pool950 (Arm 18: $47.13\% \pm 0.58\%$ greedy and $315/500 = 63.00\%$ Pass@3 on MATH-500, forming a $350/500 = 70.00\%$ 6-model oracle union with Arm 15; Arm 19: $56.51\% \pm 2.91\%$ [$70.48\%$ Pass@3] on Val210 and $60.28\% \pm 4.46\%$ [$76.25\%$ Pass@3] on Ext240).

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2. Related Work and Positioning

Our diagnostic study builds on the reasoning-from-scratch GRPO baseline [1], DeepSeek-R1 [2], Dr. GRPO [3], TreeRL [13], MARGE [14], OPTS-TTPO [16], self-consistency voting [19], and Wang et al. (2025) on high-entropy minority forking tokens [21], as well as concurrent studies in r/math_rlvr [4–12, 17, 18, 20]:

• Tree Search, Stopping-Time Measurability, Forking Tokens, and Voting [5–14, 16–21]: TreeRL [13] defines descendant-mean node values $\hat{V}_k$ and global-plus-local advantages; MARGE [14] studies guided exploration; OPTS-TTPO [16] distinguishes prefix-measurable branching from outcome-adaptive expansion and compares $\hat{V}_k$ vs. $M_k$; and Wang et al. [21] show that a small minority of high-entropy forking tokens govern trajectory divergence. In r/math_rlvr, [5, 6, 7, 9, 10] establish token-0 onset dynamics [5], conditioned score counterexamples [6], Monte Carlo fork credit at near-ties [7], 2,048-token truncation [9], and zero-signed-token-mass stability [10], while [12, 17, 18, 20] analyze item-profile redistribution across seeds, shared-prefix credit shielding, greedy vs. vote-based selection, and run-ensemble vs. single-policy sampling ([20], Section 3.7). Our bounded contributions are: (1) the controlled optimizer-clock and active-step loss comparison (Arms 1–4), (2) the $2 \times 2$ (Horizon: 512 vs. 1,024 $\times$ Bonus: \beta = 0.0 vs. 0.25) stopped-tree factorial (Arms 5, 6, 12, 13) and cap census (Table 2A), and (3) the 100-question accumulation replication (Arm 11), MDL-Maj@3 vs. MaxLen-Maj@3 voting (Table 4), donor-prefix splicing and tie-gated low-margin fork audit (Tables 5A & 5B), and six-arm 3-seed Online Counterfactual Fork-Pair GRPO study (Arms 14–19, Table 5C).

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3. Diagnostic Analysis of Baseline Mechanics

3.1 The Bimodal Token-0 Distribution and Early Short-Guess Reinforcement

Evaluating pretrained Qwen3-0.6B-Base greedily on MATH-500 (base_math500, $14.80\% = 74/500$) and GSM8K-200 + AMC23-40 (base_ext240, $12.50\% = 30/240$: $13.50\%$ on GSM8K-200, $7.50\%$ on AMC23-40) reveals a bimodal length distribution (PASS 1):

• Short Mode ($|y| < 30$ tokens): $426/500$ MATH-500 responses ($85.2\%$) emit $<30$ tokens (6–29 tokens, typically immediate \boxed{...}<|endoftext|>) at $11.03\%$ accuracy ($47/426$).
• Multi-Step Mode ($|y| \ge 30$ tokens): $74/500$ MATH-500 responses ($14.8\%$) generate $\ge 30$ tokens at $36.49\%$ accuracy ($27/74$). After 50 RLVR updates (baseline_s42 and baseline_s101), $100.0\%$ ($219/219$ and $221/221$) of correct boxed responses contain $\ge 30$ characters before \boxed{}.

On Step 2 of seed=101 (math_train.json[1], answer 98), all 8 rollouts are 7–8 token responses. When ablate_skip_zerovar_seed101 rewards the 7-token response ' \boxed{98}<|endoftext|>' ($R=1$) while skipping subsequent homogeneous steps, the policy reinforces token-0 \boxed{} emission (early_10_mean_tokens = 32.0, 22 all-zero groups, $78,773$ total tokens; math_val210 drops to $51.90\%$). Similarly, an exploratory easy-first + length-mean run (ctrl_clean50_s101) stayed at $21.9$ mean tokens over the first 10 steps and scored $41.43\%$ ($87/210$) on math_val210 (reported only as a confounded exploratory observation since it varied both curriculum ordering and length reduction).

3.2 Homogeneous-Group Optimizer Stepping in AdamW ($O(P)$ Parameter Work vs. Backward Pass FLOPs) and Active-Step Floating-Point Associativity

In the 50-question baseline runs, $54.0\%$ (seed=42: 12 all-0, 15 all-1), $56.0\%$ (seed=101: 10 all-0, 18 all-1), and $62.0\%$ (seed=7: 13 all-0, 18 all-1) of rollout groups have homogeneous rewards ($\sigma_R = 0 \implies A_i = 0$). Under PyTorch AdamW ($\eta = 10^{-5}, \beta_1 = 0.9, \beta_2 = 0.999, \epsilon = 10^{-8}, \lambda = 0.01$), a step $t$ with zero gradient $g_t = \nabla_\theta \mathcal{L}_t = 0$ updates all $P$ parameters:

$$ \theta_t \leftarrow (1 - \eta \lambda)\theta_{t-1} - \eta \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon}, \qquad m_t = \beta_1 m_{t-1}, \quad v_t = \beta_2 v_{t-1}, \quad \hat{m}_t = \frac{m_t}{1-\beta_1^t}, \quad \hat{v}_t = \frac{v_t}{1-\beta_2^t}. $$

Skipping optimizer.step() on homogeneous groups (--skip-zero-var alone, Arm 3) eliminates the 8 redundant backward passes but also omits $m_t, v_t$ decay, bias-correction increments, and weight decay on those steps. Setting param.grad = torch.zeros_like(param) and calling optimizer.step() (--skip-zero-var --zero-var-momentum-step, Arms 2 and 4) omits the $6 P D_{\text{bwd}}$ transformer backward pass while executing the exact AdamW parameter-state update in $O(P)$ elementwise memory/arithmetic operations ($\sim 6$–$8\text{ GFLOPs}$ per step for $P = 751.63\text{M}$, negligible compared to $\sim 16\text{ TFLOPs}$ per 8-trajectory backward pass).

Distinguishing Fixed-Trajectory Bitwise Equivalence from Active-Step Loss Associativity: On any fixed sequence of steps sharing a common active loss, replacing $0.0$-weighted backward passes on homogeneous steps with param.grad = torch.zeros_like(param); optimizer.step() produces bitwise identical parameters $\theta_t$ and AdamW buffers $(m_t, v_t)$ (max_abs_diff = 0.0), whereas skipping optimizer.step() diverges (PASS 2 in verify_supplement.py, verified via scalar AdamW recurrence and torch.optim.AdamW tensor checks). In live bfloat16 training, code/run_baseline.py (Arm 1) sums cross-entropies before multiplying by advantage $A_i$, whereas compute_weighted_logprob (Arms 2–13) multiplies each token's NLL by $A_i$ before summing. Floating-point non-associativity ($A_i \sum_t \ell_t$ vs. $\sum_t (A_i \ell_t)$) causes a 1-ULP active-step difference (grad_norm = 808.9206 vs. 808.9137 on Step 6 of seed=42) that diverges subsequent sampled trajectories under bfloat16 sensitivity [4]. We establish fixed-trajectory zero-gradient equivalence via PASS 2 and treat live Arm 2 vs. Arm 1 differences descriptively as execution-recipe comparisons.

3.3 Surface-Token Divergence in Mixed Groups

Across the 96 failed ($R=0$) rollouts in the 23 active groups of baseline_s42 and the 82 failed rollouts in the 22 active groups of baseline_s101, $43.75\%$ ($42/96$) and $50.00\%$ ($41/82$) share $\ge 1$ regex-extracted equation in their first 160 tokens with $R=1$ siblings, while mean exact surface-token longest common prefix (LCP) is $11.9$ and $19.3$ tokens. Regex equation matching is only a surface proxy; the LCP statistic reflects that independent root draws rarely share identical token prefixes beyond the opening sentence, motivating two-stage prefix sharing [13, 14, 16].

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4. Method, Sampler Distributions, Estimators, and Cap-Associated Census

4.1 Pre-Boxed Derivation Gate (rfind)

To suppress rewards on bare $<30$-character \boxed{} guesses during training while keeping test-time evaluation unmodified (Arm 3–Arm 13), we define:

$$ R_{\text{train}}(y, a^*) = \mathbb{I}\Big[\text{grade}(\text{extract}(y), a^*) = 1 \;\wedge\; \text{rfind}(y, \texttt{"\textbackslash boxed"}) \ge L_{\text{gate}}\Big], \qquad L_{\text{gate}} = 30. $$

4.2 Two-Stage Branched Rollouts: Exact Distributional Comparison of Three Samplers

After 3 active warmup steps (--bifurcate-after-active 3), Stage 1 generates $K=4$ root sequences $p^{(1)}, \dots, p^{(4)}$ up to $T_{\max} = 160$ tokens under the executed sampling kernel $\tilde{\pi}_\theta$ (temp=0.8, top_p=0.9, total trajectory cap max_tokens = 512 in Arms 5–11 and 1,024 in Arms 12–13; note that because rollout sampling uses temperature and top-$p$ truncation while the policy-gradient loss scores raw model log-probabilities $\log \pi_\theta$, standard flat rollouts already sample from $\tilde{\pi}_\theta$, and the question below is whether branching preserves or distorts the leaf marginals under $\tilde{\pi}_\theta$). Define the causal newline stopping time on the natural token filtration $\{\mathcal{F}_t\}_{t \ge 1}$ ($T_{\min} = 96$):

$$ \tau_{\text{causal}}(p^{(k)}) = \min\Big(T_{\max},\; \inf\big\{t \ge T_{\min} : \text{token } p^{(k)}_t \text{ contains } \texttt{'\textbackslash n'}\big\}\Big), \qquad T_{\text{EOS}}(p^{(k)}) = \inf\big\{t \ge 1 : p^{(k)}_t = \text{\texttt{EOS}}\big\}. $$

We compare three concrete branch-construction rules (verified analytically below, checked at $p = 1/2$ by direct execution of code/run_experiment.py:generate_bifurcated on a 4-outcome finite policy in PASS 2, and verified across all 2,502 steps by per-pair ledger reconstruction in PASS 3 of verify_supplement.py):

1. True $\mathcal{F}_t$-Measurable Stopped-Tree Sampler (--strict-causal-tau --early-eos-mode terminal_dup, Arms 5, 6, 11, 12, 13):

   • Stopping Rule: Each prefix stops at $\tau_k = \min\big(\tau_{\text{causal}}(p^{(k)}),\, T_{\text{EOS}}(p^{(k)})\big)$. Since both $\tau_{\text{causal}}$ and $T_{\text{EOS}}$ are stopping times on $\{\mathcal{F}_t\}_{t \ge 1}$, their minimum $\tau_k$ is a strictly $\mathcal{F}_t$-measurable stopping time [16].
   • Continuation Rule: When $T_{\text{EOS}}(p^{(k)}) > \tau_{\text{causal}}(p^{(k)})$ (causal_non_eos_pairs), both branches inherit $u^{(k)} = p^{(k)}_{1:\tau_k}$ (prefix_len $= \tau_k \in [96, 160]$) and draw independent continuations $v^{(k,1)}, v^{(k,2)} \sim \tilde{\pi}_\theta(\cdot \mid x, u^{(k)})$ up to $\text{max\_tokens} - \tau_k$ tokens. When $T_{\text{EOS}}(p^{(k)}) \le \tau_{\text{causal}}(p^{(k)})$ (terminal_dup_pairs), $u^{(k)} = p^{(k)}_{1:T_{\text{EOS}}}$ is already an absorbing terminal leaf ending in EOS; its two descendants have degenerate zero-length continuations ($v^{(k,1)} = v^{(k,2)} = \epsilon$, prefix_len $= T_{\text{EOS}}$, stage2_gen_tokens $= 0$).
   • Exact Sampling Marginal Property: For any complete trajectory $y = (y_1, \dots, y_L)$ under the capped kernel $\tilde{\pi}_\theta$, if $L \le \tau_{\text{causal}}(y)$ then $\Pr(Y_{k,b} = y) = \tilde{\pi}_\theta(y \mid x) \cdot 1 = \tilde{\pi}_\theta(y \mid x)$; if $L > \tau_{\text{causal}}(y)$ then by the chain rule $\Pr(Y_{k,b} = y) = \tilde{\pi}_\theta(y_{1:\tau_k} \mid x)\,\tilde{\pi}_\theta(y_{\tau_k+1:L} \mid x, y_{1:\tau_k}) = \tilde{\pi}_\theta(y \mid x)$ for both $b \in \{1,2\}$. Executing generate_bifurcated directly on a 4-outcome finite policy where $\tilde{\pi}_\theta$ emits immediate EOS with probability $p = 1/2$ and otherwise traverses $\tau_{\text{causal}}$ (PASS 2) confirms $\Pr(Y_{k,1} = \text{EOS}) = \Pr(Y_{k,2} = \text{EOS}) = 0.50$ with prefix_lengths = [1, 1, 1, 1, 49, 49, 49, 49].
   • Executed Pair Breakdown (PASS 3): Across seeds = 42, 101, 7, (flat_warmup_pairs, causal_non_eos_pairs, terminal_dup_pairs) are (24, 163, 13), (20, 177, 3), and (20, 179, 1) for Arm 5 ($\beta=0.0$); (24, 164, 12), (20, 178, 2), and (20, 177, 3) for Arm 6 ($\beta=0.25$); (48, 351, 1), (32, 366, 2), and (32, 362, 6) across the $400$ pairs/run of 100-question Arm 11; (32, 163, 5), (24, 175, 1), and (20, 178, 2) for 1,024-token Arm 12 ($\beta=0.0$); and (32, 167, 1), (24, 176, 0), and (20, 177, 3) for 1,024-token Arm 13 ($\beta=0.25$), with zero conservation violations.

2. Outcome-Dependent Root-Restart Sampler (--strict-causal-tau --early-eos-mode root_resample, Arms 7 and 8):

   • Rule & Non-Causality at $t=0$: When $T_{\text{EOS}}(p^{(k)}) > \tau_{\text{causal}}(p^{(k)})$ (causal_non_eos_pairs), both branches share $u^{(k)} = p^{(k)}_{1:\tau_{\text{causal}}}$. However, when EOS is observed at $T_{\text{EOS}}(p^{(k)}) \le \tau_{\text{causal}}(p^{(k)})$ (tau_zero_pairs), the procedure retrospectively sets $\tau_k = 0$ (prefix_len = 0), keeps $p^{(k)}_{1:T_{\text{EOS}}}$ as Branch 1, and draws Branch 2 freshly from the root prompt $x$. Because the event $\{\tau_k = 0\} = \{T_{\text{EOS}} \le \tau_{\text{causal}}\}$ depends on tokens generated up to $T_{\text{EOS}}$, it is not measurable at time $0$ ($\{\tau_k = 0\} \notin \mathcal{F}_0$).
   • Biased Sibling Marginals (Analytical Proof & PASS 2 Implementation Check): Analytically, for any policy that emits early EOS before $\tau_{\text{causal}}$ with probability $p \in (0,1)$, Branch 1 emits early EOS with probability $\Pr(Y_1 = \text{EOS}) = p$, whereas Branch 2 emits early EOS only if Stage 1 emits early EOS (triggering $\tau_k = 0$, probability $p$) and the independent restarted root draw also emits early EOS (probability $p$):$$ \Pr(Y_1 = \mathrm{EOS}) = p, \qquad \Pr(Y_2 = \mathrm{EOS}) = p^2 \qquad \big(\text{at } p = \tfrac{1}{2}:\; \Pr(Y_1 = \mathrm{EOS}) = 0.50,\; \Pr(Y_2 = \mathrm{EOS}) = 0.25\big), $$which is checked on the 4-outcome finite policy ($p=1/2$) in PASS 2 ((0.50, 0.25) with prefix_lengths = [0, 0, 0, 0, 49, 49, 49, 49]).
   • Executed Pair Breakdown (PASS 3): Across seeds = 42, 101, 7, (flat_warmup_pairs, causal_non_eos_pairs, tau_zero_pairs) are (24, 160, 16), (20, 177, 3), and (20, 179, 1) for Arm 7 ($\beta=0.0$) and (24, 169, 7), (20, 178, 2), and (20, 178, 2) for Arm 8 ($\beta=0.25$), summing to $200/200$ pairs per run.

3. Outcome-Dependent Midpoint-Fallback Sampler (Arms 9 and 10): Without --strict-causal-tau, early-EOS sequences of length $\ge 32$ keep the completed trajectory as Branch 1 and split Branch 2 at an interior newline via find_midpoint_boundary (early_eos_gt104_pairs $= 3, 5, 6$ and midpoint_fallback_pairs $= 12, 4, 2$ in Arm 9), which depends on terminal length at EOS.

4.3 Estimator Analysis: Fixed-Tree Summation at $\beta=0$ and Subset Qualification for $\beta > 0$

For any pair $k \in \{1, \dots, K\}$ with $\tau_k > 0$, let $\hat{V}_k = \frac{1}{B}\sum_{b=1}^B R_{k,b}$ ($B=2$) be the descendant mean (TreeRL [13]), $M_k = \max_{b} R_{k,b}$ (the max-backup value studied in OPTS-TTPO [16]), and $\bar{R} = \frac{1}{KB}\sum_{k,b} R_{k,b}$. For $t \le \tau_k$ and $t > \tau_k$, the engine assigns:

$$ A_{\text{prefix}}(k) = \frac{\tilde{V}_k - \bar{R}}{\sigma_R + 10^{-4}} \quad \big(\tilde{V}_k = \hat{V}_k + \beta(M_k - \hat{V}_k) = (1-\beta)\hat{V}_k + \beta M_k\big), \qquad A_{\text{cont}}(k, b) = \frac{(1 - \alpha)(R_{k,b} - \bar{R}) + \alpha (R_{k,b} - \hat{V}_k)}{\sigma_R + 10^{-4}} \quad (\alpha = 0.5), $$

while pairs with $\tau_k = 0$ (prefix_len = 0) receive unmodified leaf credit $(R_{k,b} - \bar{R})/(\sigma_R + 10^{-4})$ on all tokens.

1. Proposition 1 (Fixed-Tree Shared-Prefix Summation Identity at $\beta = 0$): For any pair $k$ with $\tau_k > 0$, because $u^{(k)} = y^{(k,b)}_{1:\tau_k}$ is shared across $b \in \{1, \dots, B\}$, summing the $\beta = 0$ prefix gradient across its $B$ leaves yields:

   $$ \sum_{b=1}^B A_{\text{prefix}}^{\beta=0}(k) \nabla_\theta \log \pi_\theta(u^{(k)} \mid x) = \frac{B(\hat{V}_k - \bar{R})}{\sigma_R + 10^{-4}} \nabla_\theta \log \pi_\theta(u^{(k)} \mid x) = \left(\sum_{b=1}^B \frac{R_{k,b} - \bar{R}}{\sigma_R + 10^{-4}}\right) \nabla_\theta \log \pi_\theta(u^{(k)} \mid x), $$

   matching the summed prefix gradient of leaf-level GRPO on that realized tree.

2. Proposition 2 (Conditional Expectation and Subset Qualification on Shared-Prefix Pairs $S = \{k : \tau_k > 0\}$): Under i.i.d. Bernoulli($q$) continuations from $u^{(k)}$, $\mathbb{E}[\tilde{V}_k \mid u^{(k)}] = q + \beta q(1-q)$. Across a batch of $K$ pairs where $S = \{k \in \{1,\dots,K\} : \tau_k > 0\}$ is the set of pairs with a non-empty shared prefix, the sum of prefix coefficients actually applied to shared prefixes is:

   $$ \sum_{k \in S} \sum_{b=1}^B A_{\text{prefix}}(k) = \frac{B \sum_{k \in S} (\hat{V}_k - \bar{R}) + B \beta \sum_{k \in S} (M_k - \hat{V}_k)}{\sigma_R + 10^{-4}}. $$

   When all $K$ pairs have $\tau_k > 0$ ($S = \{1,\dots,K\}$, as holds on every post-warmup step of Stopped-Tree where $\tau_k = \min(\tau_{\text{causal}}, T_{\text{EOS}}) \ge 1$), the first term $B\sum_{k=1}^K (\hat{V}_k - \bar{R}) = 0$ vanishes identically. However, when some pairs have $\tau_k = 0$ ($S \subsetneq \{1,\dots,K\}$, as in Root-Restart when early EOS occurs), $B\sum_{k \in S}(\hat{V}_k - \bar{R}) = -B\sum_{k \notin S}(\hat{V}_k - \bar{R})$ need not vanish: for example (PASS 2), with $K=4$ pairs where one shared pair $S=\{1\}$ has rewards $(0,0)$ ($\hat{V}_1 = 0$) and three root-restarted pairs $\{2,3,4\}$ have $(1,1)$ ($\bar{R} = 0.75$), the numerator at $\beta=0$ on $S$ is $2(0 - 0.75) = -1.5 \ne 0$. What holds for every subset $S \subseteq \{1,\dots,K\}$ is that the bonus increment from adding $\beta > 0$ on shared prefixes is nonnegative:

   $$ \Delta_\beta \sum_{k \in S} \sum_{b=1}^B A_{\text{prefix}}(k) = \frac{B \beta \sum_{k \in S} (M_k - \hat{V}_k)}{\sigma_R + 10^{-4}} \ge 0. $$

3. Proposition 3 (Failure of One-Sided Prefix Clamping When $\bar{R} > 0.5$): In ablate_selective_s101 (--clamped-prefix), on a divergent pair ($\hat{V}_k = 0.5, M_k = 1$), keeping $(1-\bar{R})/(\sigma_R+10^{-4}) > 0$ on the winning branch while clamping the losing branch to $\max(0, (0.625-\bar{R})/(\sigma_R+10^{-4})) \ge 0$ gives a strictly positive net prefix weight for all $\bar{R} < 1$, positively reinforcing below-average prefixes whenever $\bar{R} \in \{0.625, 0.75, 0.875\}$.

4.4 Descriptive Census of Cap-Associated Active Steps and Reward-Divergent Pairs at 512 vs. 1,024 Tokens (Table 2A)

To characterize how often active steps and reward-divergent tree pairs coincide with the rollout token cap (512 tokens in Arms 1–10; 1,024 tokens in Arms 12 and 13) [9, 10], we audit the exact prefix lengths $(\text{pl}_{k,1}, \text{pl}_{k,2})$, leaf token lengths $(L_{k,1}, L_{k,2})$, and binary rewards $(R_{k,1}, R_{k,2})$ across all 50-question three-seed arms in training_records.json (Table 2A in verify_supplement.py):

| Arm (seeds = 42, 101, 7, 150 total steps / 600 pairs) | Active Steps | CapAssocActive (all $R=0$ have $L=\text{Cap}$) | CapR0 Leaves / All $R=0$ in Active Steps | Post-Warmup Divergent Pairs ($R_{k,1} \ne R_{k,2}$) / Shared ($\tau_k > 0$) | CapR0Div ($L_{R=0} = \text{Cap}$) | NonCapR0Div ($L_{R=0} < \text{Cap}$) / (Both < Cap) | |---|---:|---:|---:|---:|---:|---:| | 1. Baseline Flat GRPO ($512\text{ tok}$) | $64$ | $24 / 64$ ($37.5\%$) | $137 / 263$ ($52.1\%$) | — (flat) | — | — | | 2. Ungated Flat + Zero-Grad Step ($512\text{ tok}$) | $64$ | $18 / 64$ ($28.1\%$) | $118 / 237$ ($49.8\%$) | — (flat) | — | — | | 3. Flat + Gate + Skip-Zero-Var ($512\text{ tok}$) | $63$ | $11 / 63$ ($17.5\%$) | $82 / 253$ ($32.4\%$) | — (flat) | — | — | | 4. Flat + Gate + Zero-Grad Step ($512\text{ tok}$) | $68$ | $23 / 68$ ($33.8\%$) | $123 / 248$ ($49.6\%$) | — (flat) | — | — | | 5. Stopped-Tree ($\beta=0.0, 512\text{ tok}$) | $58$ | $17 / 58$ ($29.3\%$) | $129 / 237$ ($54.4\%$) | $45$ ($45$) | $30 / 45$ ($66.7\%$) | $15$ ($15$ Both<512) | | 6. Stopped-Tree ($\beta=0.25, 512\text{ tok}$) | $62$ | $23 / 62$ ($37.1\%$) | $126 / 247$ ($51.0\%$) | $51$ ($51$) | $30 / 51$ ($58.8\%$) | $21$ ($21$ Both<512) | | 7. Root-Restart Tree ($\beta=0.0, 512\text{ tok}$) | $58$ | $20 / 58$ ($34.5\%$) | $118 / 216$ ($54.6\%$) | $52$ ($51$) | $32 / 52$ ($61.5\%$) | $20$ ($20$ Both<512) | | 8. Root-Restart Tree ($\beta=0.25, 512\text{ tok}$) | $58$ | $16 / 58$ ($27.6\%$) | $116 / 247$ ($47.0\%$) | $45$ ($44$) | $22 / 45$ ($48.9\%$) | $23$ ($22$ Both<512) | | 9. Midpoint-Fallback RA-MST ($\beta=0.25, 512\text{ tok}$) | $63$ | $19 / 63$ ($30.2\%$) | $121 / 257$ ($47.1\%$) | $49$ ($49$) | $22 / 49$ ($44.9\%$) | $27$ ($26$ Both<512) | | 10. Midpoint-Fallback MST ($\beta=0.25$, no warmup) | $54$ | $16 / 54$ ($29.6\%$) | $91 / 212$ ($42.9\%$) | $56$ ($45$) | $22 / 56$ ($39.3\%$) | $34$ ($34$ Both<512) | | 12. Stopped-Tree ($\beta=0.0, 1024\text{ tok matched}$) | $54$ | $3 / 54$ ($5.6\%$) | $25 / 190$ ($13.2\%$) | $50$ ($50$) | $5 / 50$ ($10.0\%$) | $45$ ($45$ Both<1024) | | 13. Stopped-Tree ($\beta=0.25, 1024\text{ tok matched}$) | $57$ | $1 / 57$ ($1.8\%$) | $45 / 210$ ($21.4\%$) | $53$ ($53$) | $13 / 53$ ($24.5\%$) | $40$ ($40$ Both<1024) |

Table 2A records three descriptive observations: (1) Across all 612 active steps of Arms 1–10, $187/612$ ($30.6\%$) have every $R=0$ rollout at the 512-token cap ($L_{k,b} = 512$); observing $R=0 \wedge L=512$ does not establish that an unfinished rollout would have succeeded past 512 tokens, but documents that on $30.6\%$ of active steps every negative-advantage rollout hit the cap [9, 10]. (2) Shared ($\tau_k > 0$) counts post-warmup divergent pairs with pl0 > 0 and pl1 > 0 receiving $\beta(M_k - \hat{V}_k)$ (45/45, 51/51, 51/52, 44/45, 50/50, 53/53 in Arms 5–8, 12, 13, with one divergent tau_zero pair in Arms 7 and 8), CapR0Div counts those with $L_{R=0} = \text{Cap}$, and NonCapR0Div (Both < Cap) counts those with $L_{R=0} < \text{Cap}$ (where $M_k - \hat{V}_k = 0.5$ applies identically and pair counts do not bound post-clipping gradient influence). (3) Raising --tokens from $512$ (Arms 5, 6) to $1,024$ (Arms 12, 13) cuts CapAssocActive from $29.3\%/37.1\%$ to $5.6\%/1.8\%$, cuts CapR0 from $54.4\%/51.0\%$ to $13.2\%/21.4\%$, and triples Both < Cap divergent pairs from $5.0/7.0\text{/run}$ to $15.0/13.3\text{/run}$.

---

5. Experimental Results, Resource Accounting, and Statistical Analysis

5.1 Chronology Disclosure, External Benchmark, and Qualified Resource Accounting

Chronology and Run Provenance Disclosure:

1. During exploratory work, we constructed data_splits/math_val210.json ($N=210$, stratified 7 subjects $\times$ 5 levels $\times$ 6 items, disjoint from math_train.json[:500] and math500_test.json) and read out both math_val210 and math500_test.json on seed=101 (Arms 1, 3, 4, 5, 9, 10) before 3-seed replication; thus $L_{\text{gate}} = 30$ and --bifurcate-after-active 3 were not blinded to MATH-500.
2. With all hyperparameters frozen, we assembled the 240-item external benchmark (gsm8k_amc_bench240.json: first 200 test items of openai/gsm8k [15] + all 40 test items of math-ai/amc23), outside Hendrycks MATH and never used for any design decision. For Arms 2, 6, 7, 8, 11, 12, 13 and stopped_tree_h1024_s101, MATH-500, math_val210, and Ext240 were evaluated in the same primary execution as training.jsonl and rollouts.jsonl. For Arms 1, 3, 4, 5 (12 runs across seeds = 42, 101, 7), MATH-500, math_val210, training.jsonl, and rollouts.jsonl come from the primary 50-step execution, while Ext240 was evaluated on a matched 50-step re-execution ({baseline,gate_flat,ra_flat,strict_mst_beta0}_ext_seed{42,101,7}, step-time sum 5,376.03s, wall-clock 5,436.11s across 600 steps), verified in PASS 3 alongside the 46 primary runs = 2,502 steps = 10,008 pairs.

Auditable Per-Step & Per-Pair Resource Derivation and Analytical Dense-Work Proxies (training_records.json & PASS 3): training_records.json stores each run's config, effective_early_eos_mode, and step_ledger ([pl0, pl1, l0, l1, r0, r1, cat_id]). PASS 3 verifies unpadded/trimmed and padded/untrimmed token counts:

• Unpadded Counts (gen, pref, leaf, bwd_out, bwd_full): gen sums Stage 1+2 generated tokens; pref sums unpadded prefill; leaf sums $\sum_{i=1}^8 L_i(s)$, satisfying $D_{\text{S1,leaf}}(s) + D_{\text{S2,gen}}(s) = \text{leaf}(s)$ and $0 \le L_b - \text{pl}_b \le \text{max\_tokens} - \text{pl}_b$; bwd_out and bwd_full sum active-step backward tokens.
• Padded Prefill & Untrimmed Steps (pad_pref, untrim_gen): Left-padding Stage-2 prompts to batch width and stepping until done.all() gives pad_pref and untrim_gen.
• Analytical Dense-Work Proxies: With $P = 751,632,384$, $\text{OutTFLOPs} = (2 D_{\text{gen}} + 6 D_{\text{bwd,out}}) P / 10^{12}$, $\text{FullTFLOPs} = [2(D_{\text{pref}} + D_{\text{gen}}) + 6 D_{\text{bwd,full}}] P / 10^{12}$, and $\text{PadFullTFLOPs} = [2(D_{\text{pad\_pref}} + D_{\text{untrim\_gen}}) + 6 D_{\text{bwd,full}}] P / 10^{12}$ are analytical dense-work proxies rather than hardware FLOP counts ($P$ includes tok_emb.weight, prefill applies out_head on the last token, and quadratic attention and $O(P)$ AdamW steps are omitted).

Figure 1: Three-seed MATH-500 accuracy and 2,048-token cap hits across arms (seeds 42, 101, and 7).

5.2 Complete Thirteen-Arm Three-Seed Comparison Across Three Benchmarks (Table 1)

| Arm (seeds = 42, 101, 7; 50 q for Arms 1–10 & 12, 13, 100 q for Arm 11) | MATH-500 Mean $\pm$ SD ($42 / 101 / 7$) | math_val210 Mean $\pm$ SD | External GSM8K-200 + AMC23-40 (GSM8K / AMC23) | MATH-500 AllCap2048 (42/101/7) / IncTrunc | Tokens (gen [untrim] / pref [pad] / bwd_out / bwd_full) | Analytical Proxy (Out / Full / PadFull TFLOPs, Sec) | |---|---:|---:|---:|---:|---:|---:| | Qwen3-0.6B-Base (0 updates) | $14.80\%$ ($74/500$) | $21.43\%$ ($45/210$) | $12.50\%$ ($13.50\% / 7.50\%$) | $5.0$ ($5$) / $5.0$ | $0 [0] / 0 [0] / 0 / 0\text{k}$ | $0.0 / 0.0 / 0.0$ ($0\text{s}$) | | 1. Baseline Flat GRPO (ungated, full bwd) [1] | $46.07\% \pm 2.39\%$ ($44.4 / 45.0 / 48.8$) | $55.24\% \pm 0.48\%$ | $59.17\% \pm 1.82\%$ ($67.00\% / 20.00\%$) | $52.7$ ($53/47/58$) / $51.7$ | $135.7 [169.6] / 33.7 [33.7] / 135.7 / 169.4\text{k}$ | $815.9 / 1018.6 / 1069.5$ ($530\text{s}$; same-host $454\text{s}$) | | 2. Ungated Flat + Zero-Grad Step | $44.67\% \pm 0.90\%$ ($43.8 / 44.6 / 45.6$) | $54.44\% \pm 1.92\%$ | $60.83\% \pm 0.42\%$ ($68.17\% / 24.17\%$) | $34.3$ ($33/34/36$) / $33.3$ | $132.4 [169.7] / 33.7 [33.7] / 58.6 / 71.3\text{k}$ | $463.3 / 571.3 / 627.4$ ($438\text{s}$) | | 3. Flat + Gate + Skip-Zero-Var (gate_flat) | $43.07\% \pm 6.54\%$ ($45.8 / 47.8 / 35.6$) | $52.54\% \pm 4.32\%$ | $58.47\% \pm 4.49\%$ ($65.83\% / 21.67\%$) | $43.0$ ($40/32/57$) / $42.7$ | $127.1 [162.0] / 33.7 [33.7] / 55.3 / 67.3\text{k}$ | $440.3 / 545.3 / 597.7$ ($416\text{s}$) | | 4. Flat + Gate + Zero-Grad Step (ra_flat) | $45.53\% \pm 1.03\%$ ($44.4 / 46.4 / 45.8$) | $53.81\% \pm 1.65\%$ | $59.31\% \pm 1.68\%$ ($67.17\% / 20.00\%$) | $33.7$ ($41/21/39$) / $31.7$ | $131.0 [167.4] / 33.7 [33.7] / 64.8 / 78.2\text{k}$ | $489.0 / 600.1 / 654.8$ ($433\text{s}$) | | 5. Stopped-Tree (terminal_dup, $\beta=0.0, 512\text{ tok}$) | $44.20\% \pm 0.53\%$ ($43.6 / 44.6 / 44.4$) | $55.24\% \pm 1.43\%$ | $60.28\% \pm 2.10\%$ ($68.00\% / 21.67\%$) | $35.7$ ($32/30/45$) / $34.3$ | $124.7 [156.7] / 84.7 [88.2] / 54.7 / 65.8\text{k}$ | $434.2 / 611.5 / 665.0$ ($480\text{s}$) | | 6. Stopped-Tree (terminal_dup, $\beta=0.25, 512\text{ tok}$) | $44.93\% \pm 2.32\%$ ($47.6 / 43.8 / 43.4$) | $51.90\% \pm 2.90\%$ | $59.31\% \pm 2.29\%$ ($66.67\% / 22.50\%$) | $38.3$ ($38/25/52$) / $37.3$ | $125.9 [158.3] / 85.0 [89.2] / 59.7 / 71.6\text{k}$ | $458.7 / 640.1 / 695.0$ ($487\text{s}$) | | 7. Root-Restart Tree ($\tau_k=0$ on EOS, $\beta=0.0$) | $45.80\% \pm 0.87\%$ ($46.4 / 44.8 / 46.2$) | $53.97\% \pm 1.20\%$ | $59.72\% \pm 1.34\%$ ($68.00\% / 18.33\%$) | $40.7$ ($30/41/51$) / $38.3$ | $124.1 [157.7] / 85.6 [90.0] / 56.0 / 67.5\text{k}$ | $439.1 / 619.8 / 677.0$ ($476\text{s}$) | | 8. Root-Restart Tree ($\tau_k=0$ on EOS, $\beta=0.25$) | $45.87\% \pm 2.01\%$ ($45.6 / 44.0 / 48.0$) | $53.97\% \pm 1.45\%$ | $60.42\% \pm 0.83\%$ ($68.00\% / 22.50\%$) | $51.3$ ($61/48/45$) / $51.0$ | $125.2 [157.6] / 86.4 [91.1] / 55.2 / 66.3\text{k}$ | $437.4 / 617.0 / 672.7$ ($479\text{s}$) | | 9. Midpoint-Fallback RA-MST ($\beta=0.25$) | $47.00\% \pm 1.40\%$ ($47.0 / 45.6 / 48.4$) | $55.56\% \pm 1.67\%$ | — | $39.0$ ($32/48/37$) / $37.7$ | $121.9 [155.4] / 84.6 [89.2] / 58.6 / 71.0\text{k}$ | $447.7 / 630.9 / 688.0$ ($476\text{s}$) | | 10. Midpoint-Fallback MST ($\beta=0.25$, no warmup) | $46.93\% \pm 0.76\%$ ($46.4 / 47.8 / 46.6$) | $53.81\% \pm 1.26\%$ | — | $53.0$ ($65/48/46$) / $50.7$ | $120.4 [151.4] / 86.0 [92.2] / 49.2 / 59.7\text{k}$ | $402.9 / 579.6 / 635.7$ ($482\text{s}$) | | 11. Stopped-Tree + ZeroMass + Accum4 (100 q) | $46.20\% \pm 0.35\%$ ($45.8 / 46.4 / 46.4$) | $55.08\% \pm 1.92\%$ | $59.72\% \pm 0.87\%$ ($67.00\% / 23.33\%$) | $37.3$ ($35/41/36$) / $36.7$ | $262.9 [314.4] / 201.2 [210.5] / 98.2 / 122.3\text{k}$ | $838.0 / 1249.4 / 1340.8$ ($1027\text{s}$) | | 12. Stopped-Tree ($\beta=0.0, 1024\text{ tok matched}$) | $46.47\% \pm 0.81\%$ ($46.0 / 47.4 / 46.0$) | $55.71\% \pm 2.52\%$ | $61.11\% \pm 3.37\%$ ($67.83\% / 27.50\%$) | $43.0$ ($45/51/33$) / $42.0$ | $146.6 [210.2] / 85.6 [90.0] / 62.5 / 72.8\text{k}$ | $502.2 / 677.1 / 779.5$ ($650\text{s}$) | | 13. Stopped-Tree ($\beta=0.25, 1024\text{ tok matched}$) | $45.40\% \pm 1.06\%$ ($44.6 / 45.0 / 46.6$) | $55.56\% \pm 1.20\%$ | $59.03\% \pm 5.31\%$ ($65.50\% / 26.67\%$) | $55.3$ ($75/45/46$) / $54.7$ | $154.5 [215.8] / 86.0 [89.7] / 73.9 / 85.3\text{k}$ | $565.3 / 746.0 / 843.7$ ($664\text{s}$) |

5.3 Matched Two-Level (Seed $\times$ Question) Bootstrap Contrasts and Evidence Interpretation (Table 2B)

Table 2B (recomputed by verify_supplement.py) presents 26 contrasts with 4,000-resample two-level (seed $\times$ question) bootstrap $95\%$ CIs (primary, resampling both seeds and questions) alongside question-clustered intervals (conditional on the 3 checkpoints) and auxiliary flattened $3N$-observation intervals (which ignore within-question correlation across seeds and do not establish cross-seed generalization when the two-level interval spans zero):

| Controlled Contrast (seeds = 42, 101, 7) | Benchmark ($N$) | Per-Seed Diffs ($42 / 101 / 7$, pp) | 3-Seed Mean Diff | Primary Two-Level Seed $\times$ Item $95\%$ CI | Q-Clustered / Flat3N $95\%$ CI | |---|---:|---:|---:|---:|---:| | 1. Stopped-Tree $\beta=0.25$ vs. $\beta=0.0$ (terminal_dup): Arm 6 vs. Arm 5 | MATH-500 ($500$) | $+4.00 / -0.80 / -1.00$ | $+0.73\text{ pp}$ | $[-3.00, +4.67]\text{ pp}$ | $[-1.33, +2.80] / [-1.40, +2.80]\text{ pp}$ | | 2. Stopped-Tree $\beta=0.25$ vs. $\beta=0.0$ (terminal_dup): Arm 6 vs. Arm 5 | Ext240 ($240$) | $-1.67 / -2.08 / +0.83$ | $-0.97\text{ pp}$ | $[-5.42, +3.47]\text{ pp}$ | $[-4.03, +2.22] / [-4.17, +2.08]\text{ pp}$ | | 3. Root-Restart $\beta=0.25$ vs. $\beta=0.0$ (root_resample): Arm 8 vs. Arm 7 | MATH-500 ($500$) | $-0.80 / -0.80 / +1.80$ | $+0.07\text{ pp}$ | $[-2.80, +3.00]\text{ pp}$ | $[-1.87, +2.07] / [-2.00, +2.13]\text{ pp}$ | | 4. Root-Restart $\beta=0.25$ vs. $\beta=0.0$ (root_resample): Arm 8 vs. Arm 7 | Ext240 ($240$) | $+1.67 / +0.00 / +0.42$ | $+0.69\text{ pp}$ | $[-3.06, +4.72]\text{ pp}$ | $[-2.36, +3.75] / [-2.22, +3.61]\text{ pp}$ | | 5. Root-Restart vs. Stopped-Tree at $\beta=0.0$: Arm 7 vs. Arm 5 | MATH-500 ($500$) | $+2.80 / +0.20 / +1.80$ | $+1.60\text{ pp}$ | $[-1.33, +4.73]\text{ pp}$ | $[-0.53, +3.73] / [-0.53, +3.80]\text{ pp}$ | | 6. Midpoint-Fallback vs. Root-Restart at $\beta=0.25$: Arm 9 vs. Arm 8 | MATH-500 ($500$) | $+1.40 / +1.60 / +0.40$ | $+1.13\text{ pp}$ | $[-1.53, +3.87]\text{ pp}$ | $[-0.80, +3.13] / [-0.80, +3.20]\text{ pp}$ | | 7. Execution-Recipe Zero-Grad Step vs. Full Bwd (Ungated): Arm 2 vs. Arm 1 | MATH-500 ($500$) | $-0.60 / -0.40 / -3.20$ | $-1.40\text{ pp}$ | $[-4.60, +1.80]\text{ pp}$ | $[-3.60, +0.80] / [-3.60, +0.73]\text{ pp}$ | | 8. Execution-Recipe Zero-Grad Step vs. Full Bwd (Ungated): Arm 2 vs. Arm 1 | Ext240 ($240$) | $+0.00 / +2.08 / +2.92$ | $+1.67\text{ pp}$ | $[-2.22, +5.69]\text{ pp}$ | $[-1.25, +4.72] / [-1.11, +4.44]\text{ pp}$ | | 9. Zero-Grad Momentum Step vs. Skip (Gated): Arm 4 (ra_flat) vs. Arm 3 | MATH-500 ($500$) | $-1.40 / -1.40 / +10.20$ | $+2.47\text{ pp}$ | $[-3.47, +9.60]\text{ pp}$ | $[+0.07, +4.93] / [+0.07, +4.80]\text{ pp}$ | | 10. Zero-Grad Momentum Step vs. Skip (Gated): Arm 4 (ra_flat) vs. Arm 3 | Ext240 ($240$) | $-2.08 / +0.42 / +4.17$ | $+0.83\text{ pp}$ | $[-3.89, +6.25]\text{ pp}$ | $[-2.64, +4.17] / [-2.36, +4.17]\text{ pp}$ | | 11. Derivation Gate at Fixed Zero-Grad Step: Arm 4 (ra_flat) vs. Arm 2 | MATH-500 ($500$) | $+0.60 / +1.80 / +0.20$ | $+0.87\text{ pp}$ | $[-2.20, +3.73]\text{ pp}$ | $[-1.27, +3.13] / [-1.27, +2.93]\text{ pp}$ | | 12. One-Sided Prefix Clamping (seed=101): ablate_selective_s101 vs. Arm 10 | MATH-500 ($500$) | — / $-14.80$ / — | $-14.80\text{ pp}$ | — ($27$ gains / $101$ losses) | $[-19.00, -10.40]\text{ pp}$ (Item CI) | | 13. 2,048-Token Cap Hits (AllCap2048): Arm 4 (ra_flat) vs. Arm 1 (Baseline) | MATH-500 ($500$) | $-2.40 / -5.20 / -3.80$ | $-3.80\text{ pp}$ | $[-6.60, -1.00]\text{ pp}$ | $[-5.73, -1.87]\text{ pp}$ | | 14. 2,048-Token Cap Hits (AllCap2048): Arm 4 (ra_flat) vs. Arm 2 (Ungated ZG) | MATH-500 ($500$) | $+1.60 / -2.60 / +0.60$ | $-0.13\text{ pp}$ | $[-3.20, +2.80]\text{ pp}$ | $[-2.13, +1.80]\text{ pp}$ | | 15. 2,048-Token Cap Hits (AllCap2048): Arm 7 (Root-Restart $\beta=0$) vs. Arm 1 | MATH-500 ($500$) | $-4.60 / -1.20 / -1.40$ | $-2.40\text{ pp}$ | $[-5.40, +0.60]\text{ pp}$ | $[-4.20, -0.53]\text{ pp}$ | | 16. Combined 100-Question Accum4 Package vs. 50-Step Arm 5: Arm 11 vs. Arm 5 | MATH-500 ($500$) | $+2.20 / +1.80 / +2.00$ | $+2.00\text{ pp}$ | $[-0.80, +5.13]\text{ pp}$ | $[-0.27, +4.33] / [-0.27, +4.27]\text{ pp}$ | | 17. Combined 100-Question Accum4 Package vs. 50-Step Arm 5: Arm 11 vs. Arm 5 | Ext240 ($240$) | $+2.08 / -3.75 / +0.00$ | $-0.56\text{ pp}$ | $[-5.28, +4.03]\text{ pp}$ | $[-3.19, +2.08] / [-3.75, +2.50]\text{ pp}$ | | 18. Single-Factor 1,024-Token Horizon vs. 512-Token Arm 5: Arm 12 vs. Arm 5 | MATH-500 ($500$) | $+2.40 / +2.80 / +1.60$ | $+2.27\text{ pp}$ | $[-0.73, +5.13]\text{ pp}$ | $[+0.07, +4.53] / [+0.13, +4.47]\text{ pp}$ | | 19. Single-Factor 1,024-Token Horizon vs. 512-Token Arm 5: Arm 12 vs. Arm 5 | Val210 ($210$) | $+3.81 / +1.43 / -3.81$ | $+0.48\text{ pp}$ | $[-5.40, +6.03]\text{ pp}$ | $[-2.86, +3.97] / [-2.70, +3.81]\text{ pp}$ | | 20. Single-Factor 1,024-Token Horizon vs. 512-Token Arm 5: Arm 12 vs. Arm 5 | Ext240 ($240$) | $+3.33 / +1.67 / -2.50$ | $+0.83\text{ pp}$ | $[-4.17, +5.83]\text{ pp}$ | $[-2.22, +3.89] / [-2.22, +3.89]\text{ pp}$ | | 21. 1,024-Token Stopped-Tree $\beta=0.25$ vs. $\beta=0.0$: Arm 13 vs. Arm 12 | MATH-500 ($500$) | $-1.40 / -2.40 / +0.60$ | $-1.07\text{ pp}$ | $[-4.20, +2.07]\text{ pp}$ | $[-3.33, +1.20] / [-3.33, +1.20]\text{ pp}$ | | 22. 1,024-Token Stopped-Tree $\beta=0.25$ vs. $\beta=0.0$: Arm 13 vs. Arm 12 | Val210 ($210$) | $-0.95 / -0.95 / +1.43$ | $-0.16\text{ pp}$ | $[-4.76, +4.76]\text{ pp}$ | $[-3.97, +3.49] / [-3.49, +3.17]\text{ pp}$ | | 23. 1,024-Token Stopped-Tree $\beta=0.25$ vs. $\beta=0.0$: Arm 13 vs. Arm 12 | Ext240 ($240$) | $+0.83 / -2.50 / -4.58$ | $-2.08\text{ pp}$ | $[-6.67, +2.78]\text{ pp}$ | $[-5.00, +0.97] / [-5.14, +1.25]\text{ pp}$ | | 24. $2 \times 2$ Interaction [$(\text{Arm 13} - \text{Arm 12}) - (\text{Arm 6} - \text{Arm 5})$]: 3 Seeds | MATH-500 ($500$) | $-5.40 / -1.60 / +1.60$ | $-1.80\text{ pp}$ | $[-6.93, +3.47]\text{ pp}$ | $[-4.93, +1.47]\text{ pp}$ (Q-Clustered) | | 25. $2 \times 2$ Interaction [$(\text{Arm 13} - \text{Arm 12}) - (\text{Arm 6} - \text{Arm 5})$]: 3 Seeds | Val210 ($210$) | $-2.38 / +4.29 / +7.62$ | $+3.17\text{ pp}$ | $[-4.60, +10.95]\text{ pp}$ | $[-1.90, +7.94]\text{ pp}$ (Q-Clustered) | | 26. $2 \times 2$ Interaction [$(\text{Arm 13} - \text{Arm 12}) - (\text{Arm 6} - \text{Arm 5})$]: 3 Seeds | Ext240 ($240$) | $+2.50 / -0.42 / -5.42$ | $-1.11\text{ pp}$ | $[-7.50, +5.83]\text{ pp}$ | $[-5.28, +3.19]\text{ pp}$ (Q-Clustered) |

Figure 2: External benchmark accuracy (GSM8K-200 + AMC23-40) and output vs. full-sequence training TFLOPs.

1. Bonus Contrasts Under Both Samplers (Rows 1–6 of Table 2B): Under both Stopped-Tree (Arms 5–6, Rows 1–2: $-0.97\text{ pp}$ Ext240, $[-5.42, +3.47]$; $+0.73\text{ pp}$ MATH-500, $[-3.00, +4.67]$) and Root-Restart (Arms 7–8, Rows 3–4: $+0.69\text{ pp}$ Ext240, $[-3.06, +4.72]$; $+0.07\text{ pp}$ MATH-500, $[-2.80, +3.00]$), neither procedure shows a detected accuracy benefit from $\beta = 0.25$ over $\beta = 0.0$ within its confidence interval, and AllCap2048 seed directions are mixed (32, 30, 45 $\to$ 38, 25, 52 under Stopped-Tree; 30, 41, 51 $\to$ 61, 48, 45 under Root-Restart). Arithmetic telescoping of the MATH-500 sample means between Arm 5 ($44.20\%$) and Arm 9 ($47.00\%$) gives $1.60 + 0.07 + 1.13 = 2.80\text{ pp}$, with between-sampler CIs (Rows 5–6: $[-1.33, +4.73]$ and $[-1.53, +3.87]\text{ pp}$) spanning zero (reported strictly as unresolved sample estimates).

2. Separating Zero-Gradient Stepping, Active-Step Loss Associativity, and Derivation Gating (Rows 7–11 and 13–14 of Table 2B): Comparing Arm 2 vs. Arm 1 under the identical ungated reward shows that analytical output / unpadded-full / padded-full dense FLOPs drop by $43.2\%$ / $43.9\%$ / $41.3\%$ ($463.3 / 571.3 / 627.4$ vs. $815.9 / 1018.6 / 1069.5\text{ TFLOPs}$), with a $3.5\%$ same-host wall-clock reduction ($438.1\text{s}$ vs. $454.1\text{s}$ step sum; $17.3\%$ vs. the historical $530.0\text{s}$ mean) and unresolved accuracy differences ($+1.67\text{ pp}$ Ext240, $[-2.22, +5.69]$; $-1.40\text{ pp}$ MATH-500, $[-4.60, +1.80]$). Comparing Arm 4 (ra_flat) vs. Arm 2 within code/run_experiment.py isolates the 30-character derivation gate at fixed optimizer stepping and loss associativity: $+0.87\text{ pp}$ on MATH-500 ($[-2.20, +3.73]\text{ pp}$) and $-0.13\text{ pp}$ on AllCap2048 ($[-3.20, +2.80]\text{ pp}$).

5.4 Combined 100-Question Replication (Arm 11), Complete $2 \times 2$ Factorial at 1,024 Tokens (Arms 12 & 13), Retained Token-512 Prefix Audit (gt512_prefix_audit), and Single-Seed Probes (Table 3)

1. Combined 100-Question Replication (Arm 11 vs. Arm 5, Rows 16–17 of Table 2B): In Arm 11, we replicate the 100-question synthesis package (Stopped-Tree $\beta = 0.0$ + --zero-cont-token-mass [10] + --grad-accum-questions 4 --unit-question-grad on math_train_stratified.json[:100]) across seeds = 42, 101, 7, scoring $46.20\% \pm 0.35\%$ on MATH-500 ($+2.00\text{ pp}$ over Arm 5, [+2.20, +1.80, +2.00] pp, two-level $95\%$ CI $[-0.80, +5.13]\text{ pp}$), $55.08\% \pm 1.92\%$ on math_val210, and $59.72\% \pm 0.87\%$ on Ext240 ($-0.56\text{ pp}$, $[-5.28, +4.03]\text{ pp}$) at $838.0 / 1249.4 / 1340.8\text{ TFLOPs}$ ($1027\text{s}$). Because Arm 11 simultaneously varies curriculum, exposure ($100$ vs. $50$ questions), continuation centering, and 4-question accumulation and its CIs span zero, we report it strictly as a combined-package comparison.

2. Complete Three-Seed $2 \times 2$ Factorial (Arms 5, 6, 12, 13), Evaluation Batch-Size Sensitivity Audit, and Retained Token-512 Prefix Audit (gt512_prefix_audit, Rows 18–26 of Table 2B): In Arms 12 and 13, we hold --split-tokens 160, zero_cont_token_mass=False, and terminal_dup identical to Arms 5 and 6, expanding --tokens 512 -> 1024 under $\beta = 0.0$ (Arm 12) and $\beta = 0.25$ (Arm 13).

   • Evaluation Batch-Size Disclosure and 950-Item Sensitivity Audit (batch=32 vs. batch=64, Table 5B): During training (code/run_experiment.py), --batch affects only post-training evaluate_dataset (Arms 5, 6, 12 used --batch 32; Arm 13 used --batch 64). Because left-padded bfloat16 reductions are non-associative across batch shapes, our 950-item paired audit on Arm 12 seed=7 (Table 5B, PASS 4) records $905/950$ ($95.3\%$) grade agreement ($45/950 = 4.7\%$ flips: $10/210$ Val210, $10/240$ Ext240, $25/500$ MATH-500) and a net difference of $-0.32\text{ pp}$ ($479/950$ vs. $476/950$: $0.00, -0.83, -0.20\text{ pp}$). Because $45/950$ grades flip and this audit does not bound unretained checkpoints, we retain the batch-size mismatch between Arm 13 and Arms 5, 6, 12 as an explicit limitation (Section 6).
   • Retained Evidence for Token-512 Prefix Extractions (gt512_prefix_audit in training_records.json & PASS 3): Across 3 seeds ($1,200$ rollouts/arm), $266/1,200$ ($22.2\%$) of Arm 12 and $292/1,200$ ($24.3\%$) of Arm 13 rollouts exceed 512 tokens ($512 < L \le 1024$). Decoding t512 = tok.decode(ids[:512]), we stored [step, branch_idx, L, r_fin, rfind_512, ext_512, ans_gold, r_512, first_box_tok, retained_txt] in training_records.json (retaining full t512 for all $35$ rollouts with rfind_512 >= 0 and the 60-char tail plus first_box_tok for the $523$ rollouts with rfind_512 == -1). PASS 3 runs rfind("\\boxed") and extract_final_candidate on all $35$ boxed prefixes, checks the $523$ unboxed entries (first_box_tok == -1 or >= 510, ext_512 == ""), and regrades all $558$ ext_512 strings: in Arm 12, $90/266$ ($33.8\%$) finish with $R=1$ and $87/90$ ($96.7\%$) fail at token 512 while the final answer passes ($84/90$ rfind_512 == -1, $3/90$ unclosed/wrong \boxed, $3/90$ pass at 512); in Arm 13, $118/292$ ($40.4\%$) finish with $R=1$ and $107/118$ ($90.7\%$) fail at token 512 while passing at completion ($103/118$ rfind_512 == -1, $4/118$ wrong \boxed, $11/118$ pass at 512; median first \boxed at token $560$).
   • Single-Factor 1,024-Token Horizon (Arm 12 vs. Arm 5) & $2 \times 2$ Interaction (Rows 18–26 of Table 2B): On MATH-500, Arm 12 - Arm 5 is [+2.40, +2.80, +1.60] pp (mean $+2.27\text{ pp}$, two-level $95\%$ CI $[-0.73, +5.13]\text{ pp}$; Q-clustered $[+0.07, +4.53]\text{ pp}$; Flat3N $[+0.13, +4.47]\text{ pp}$), with $+0.48\text{ pp}$ ($[-5.40, +6.03]$) on Val210 and $+0.83\text{ pp}$ ($[-4.17, +5.83]$) on Ext240. At 1,024 tokens, Arm 13 - Arm 12 ($\beta = 0.25$ vs. $0.0$) is $-1.07\text{ pp}$ ([-1.40, -2.40, +0.60], $[-4.20, +2.07]\text{ pp}$) on MATH-500, $-0.16\text{ pp}$ ($[-4.76, +4.76]\text{ pp}$) on Val210, and $-2.08\text{ pp}$ ($[-6.67, +2.78]\text{ pp}$) on Ext240, and the $2 \times 2$ interaction $[(\text{Arm 13} - \text{Arm 12}) - (\text{Arm 6} - \text{Arm 5})]$ spans zero across all three benchmarks ($-1.80, +3.17, -1.11\text{ pp}$).

Table 3 summarizes the six single-seed (seed=101) exploratory ablations included in per_item_grades.json and training_records.json:

| Run (seed=101) | Configuration Summary | Steps / Active (all1) | math_val210 Acc | Ext240 Acc | MATH-500 Acc | AllCap2048 / IncTrunc | Analytical Proxy (Out / Full / Pad TFLOPs, Sec) | |---|---|---:|---:|---:|---:|---:|---:| | ablate_skip_s101 | Ungated Flat + --skip-zero-var only (no momentum step) | $50 / 24$ ($4$) | $51.90\%$ ($109/210$) | — | $45.00\%$ ($225/500$) | $42 / 42$ | $356.2 / 470.8 / 547.7$ ($337\text{s}$) | | ablate_selective_s101 | One-sided prefix clamping (--clamped-prefix, Proposition 3) | $50 / 23$ ($16$) | $40.95\%$ ($86/210$) | — | $33.00\%$ ($165/500$) | $64 / 61$ | $363.7 / 552.0 / 609.6$ ($418\text{s}$) | | mst_act30_s101 | Midpoint-Fallback MST extended to 30 active steps on Algebra | $62 / 30$ ($18$) | $52.86\%$ ($111/210$) | — | $46.20\%$ ($231/500$) | $61 / 61$ | $634.7 / 899.3 / 966.1$ ($608\text{s}$) | | mst_zpd_act35_s101 | 7-subject Level 2–5 pool (math_train_zpd.json) at 512-token cap | $90 / 32$ ($13$) | $52.38\%$ ($110/210$) | — | $44.20\%$ ($221/500$) | $35 / 34$ | $878.2 / 1282.9 / 1358.3$ ($955\text{s}$) | | ra_mst_tail50_s101 | RA-MST + FP32 Polyak tail weight average (steps 26..50) | $50 / 26$ ($12$) | $53.33\%$ ($112/210$) | — | $44.60\%$ ($223/500$) | $30 / 30$ | $495.0 / 688.7 / 744.9$ ($496\text{s}$) | | stopped_tree_h1024_s101 | Stopped-Tree ($\beta=0$) + 240/1024 tokens + --zero-cont-token-mass | $50 / 13$ ($24$) | $54.29\%$ ($114/210$) | $57.92\%$ ($139/240$) | $45.40\%$ ($227/500$) | $50 / 48$ | $460.9 / 648.0 / 738.9$ ($687\text{s}$) |

Unlike Arm 12 (cap1024_matched_s101: 160/1024 tokens, zero_cont_token_mass=False, scoring $47.40\%$ on MATH-500, $56.67\%$ on Val210, and $64.17\%$ on Ext240), stopped_tree_h1024_s101 simultaneously changed --split-tokens 160 -> 240 and zero_cont_token_mass=True, diverging from strict_mst_beta0_s101 at Step 3 (2/8 vs. 0/8 rewarded leaves) and scoring lower on seed=101 across all three benchmarks ($45.40\%, 54.29\%, 57.92\%$).

5.5 Test-Time Voting ($k=3$) with Complete Rule Specification, Uncertainty, and Cost Accounting (Table 4)

Building on self-consistency majority voting [19] and rStation comparisons of multi-run ensembles vs. single-policy sampling ([20], Section 3.7; see also [12, 17, 18]), we evaluate $k=3$ test-time aggregation across two post-hoc diagnostic axes (Table 4 in verify_supplement.py): (i) Cross-Seed Greedy ($T=0, k=3$ across seeds = 42, 101, 7) vs. Single-Seed Sampled ($T=0.6, \text{top-}p=0.9, k=3$ from a single checkpoint using evaluate_sampled_k3 with torch.manual_seed(1000 + seed)) on Arm 11, and (ii) Minimum-Description-Length tie-breaking (MDL-Maj@3) vs. MaxLen-Maj@3 and Rand-Maj@3.

Complete Specification of the $k=3$ Voting and Tie-Breaking Rules (_eval_k3_triplets in verify_supplement.py): For each test question with $k=3$ candidates $(c_i, L_i, \text{ext}_i)_{i \in \{0,1,2\}}$ (seed=42, 101, 7 in Cross-Seed Greedy; draws $0, 1, 2$ in Single-Seed Sampled):

1. String Normalization & Empty-Answer Handling: Each extracted string is normalized via $e_i = \text{normalize\_text}(\text{ext}_i)$. An empty string ($e_i = \texttt{""}$, when a rollout hits the 2,048-token cap without \boxed{...}) never matches another candidate to form a majority: if $e_0 \ne \texttt{""}$ and $(e_0 = e_1 \lor e_0 = e_2)$, all rules select candidate $0$; else if $e_1 \ne \texttt{""}$ and $e_1 = e_2$, all rules select candidate $1$.
2. Tie-Breaking on 1-1-1 Ties:
   • Rand-Maj@3: Exact uniform expectation $\frac{1}{3}(c_0 + c_1 + c_2)$.
   • MaxLen-Maj@3: Selects the longest candidate with $e_i \ne \texttt{""}$ (or longest overall if all $e_i = \texttt{""}$), breaking equal-length ties by lowest index $i \in \{0,1,2\}$ via max(range(3), key=lambda i: (1 if e_i else 0, L_i)).
   • MDL-Maj@3: Defines candidate $i$ as eligible if $e_i \ne \texttt{""}$ and $15 \le L_i < 2048$ (a post-hoc window avoiding $<30$-token short guesses and 2,048-token cap hits). Among eligible candidates, MDL-Maj@3 selects the shortest $L_i$; if none is eligible (fb in Table 4), it falls back to $\arg\min_i L_i$, breaking ties by lowest $i \in \{0,1,2\}$ via min(range(3), key=lambda i: (0 if (e_i and 15 <= L_i < 2048) else 1, L_i)) (fb = 0 in all 4 sampled runs, fb = 1 on Val210/Ext240 in Arm 11, fb = 2 on MATH-500 in Arms 5, 7, 9, 11, fb = 4 in Arm 12, fb = 5 in Arm 13, fb = 8 in Arm 1).

All bootstrap $95\%$ CIs in Table 4 are nominal and conditional on the retained checkpoints and draws (4,000 paired question resamples in verify_supplement.py):

| Evaluation Mode ($k=3$ rollouts/item; Train Cost & Test Tokens) | Benchmark ($N$) | Mean@1 | Rand-Maj@3 | MaxLen-Maj@3 | MDL-Maj@3 (fb) | MDL - MaxLen (Nominal $95\%$ CI) | Oracle Pass@3 | |---|---:|---:|---:|---:|---:|---:|---:| | Single-Seed seed=42 ($T=0.6$; $1\times$ ckpt $= 818\text{ TFLOPs}$, TestTok $= 341.1\text{k}$) | Val210 ($210$) | $55.56\%$ | $59.52\%$ | $59.52\%$ | $58.10\%$ ($0$) | $-1.43\text{ pp}$ ($[-5.71, +2.38]$) | $72.38\%$ | | Single-Seed seed=7 ($T=0.6$; $1\times$ ckpt $= 834\text{ TFLOPs}$, TestTok $= 359.7\text{k}$) | Val210 ($210$) | $53.49\%$ | $57.78\%$ | $57.14\%$ | $58.10\%$ ($0$) | $+0.95\text{ pp}$ ($[-2.38, +4.29]$) | $68.10\%$ | | Cross-Seed 42,101,7 (Arm 11) ($T=0.0$; $3\times$ ckpt $= 2514\text{ TFLOPs}$, TestTok $= 369.4\text{k}$) | Val210 ($210$) | $55.08\%$ | $59.05\%$ | $59.05\%$ | $59.05\%$ ($1$) | $+0.00\text{ pp}$ ($[-3.33, +3.33]$) | $66.67\%$ | | Single-Seed seed=42 ($T=0.6$; $1\times$ ckpt $= 818\text{ TFLOPs}$, TestTok $= 275.2\text{k}$) | Ext240 ($240$) | $57.50\%$ | $60.97\%$ | $60.83\%$ | $61.67\%$ ($0$) | $+0.83\text{ pp}$ ($[-2.92, +4.58]$) | $71.67\%$ | | Single-Seed seed=7 ($T=0.6$; $1\times$ ckpt $= 834\text{ TFLOPs}$, TestTok $= 272.6\text{k}$) | Ext240 ($240$) | $57.64\%$ | $61.94\%$ | $62.08\%$ | $60.83\%$ ($0$) | $-1.25\text{ pp}$ ($[-4.58, +2.08]$) | $73.75\%$ | | Cross-Seed 42,101,7 (Arm 11) ($T=0.0$; $3\times$ ckpt $= 2514\text{ TFLOPs}$, TestTok $= 285.4\text{k}$) | Ext240 ($240$) | $59.72\%$ | $62.92\%$ | $61.25\%$ | $64.17\%$ ($1$) | $+2.92\text{ pp}$ ($[-0.42, +6.67]$) | $74.17\%$ | | Cross-Seed 42,101,7 (Arm 11) ($T=0.0$; $3\times$ ckpt $= 2514\text{ TFLOPs}$, TestTok $= 970.7\text{k}$) | MATH-500 ($500$) | $46.20\%$ | $49.60\%$ | $47.60\%$ | $51.00\%$ ($2$) | $+3.40\text{ pp}$ ($[+1.00, +5.80]$) | $61.00\%$ | | Cross-Seed 42,101,7 (Arm 12) ($T=0.0$; $3\times$ ckpt $= 1507\text{ TFLOPs}$, TestTok $= 383.1\text{k}$) | Val210 ($210$) | $55.71\%$ | $59.37\%$ | $56.67\%$ | $61.43\%$ ($2$) | $+4.76\text{ pp}$ ($[+1.43, +8.10]$) | $68.57\%$ | | Cross-Seed 42,101,7 (Arm 12) ($T=0.0$; $3\times$ ckpt $= 1507\text{ TFLOPs}$, TestTok $= 286.6\text{k}$) | Ext240 ($240$) | $61.11\%$ | $64.86\%$ | $65.42\%$ | $64.58\%$ ($1$) | $-0.83\text{ pp}$ ($[-4.58, +2.92]$) | $75.42\%$ | | Cross-Seed 42,101,7 (Arm 12) ($T=0.0$; $3\times$ ckpt $= 1507\text{ TFLOPs}$, TestTok $= 1007.7\text{k}$) | MATH-500 ($500$) | $46.47\%$ | $50.33\%$ | $49.60\%$ | $51.60\%$ ($4$) | $+2.00\text{ pp}$ ($[-0.40, +4.60]$) | $59.60\%$ | | Cross-Seed 42,101,7 (Arm 13) ($T=0.0$; $3\times$ ckpt $= 1696\text{ TFLOPs}$, TestTok $= 413.9\text{k}$) | Val210 ($210$) | $55.56\%$ | $58.10\%$ | $55.71\%$ | $59.05\%$ ($0$) | $+3.33\text{ pp}$ ($[+0.00, +6.67]$) | $70.00\%$ | | Cross-Seed 42,101,7 (Arm 13) ($T=0.0$; $3\times$ ckpt $= 1696\text{ TFLOPs}$, TestTok $= 306.6\text{k}$) | Ext240 ($240$) | $59.03\%$ | $63.75\%$ | $63.75\%$ | $65.00\%$ ($2$) | $+1.25\text{ pp}$ ($[-2.50, +5.00]$) | $74.17\%$ | | Cross-Seed 42,101,7 (Arm 13) ($T=0.0$; $3\times$ ckpt $= 1696\text{ TFLOPs}$, TestTok $= 1080.1\text{k}$) | MATH-500 ($500$) | $45.40\%$ | $49.00\%$ | $46.80\%$ | $50.80\%$ ($5$) | $+4.00\text{ pp}$ ($[+1.60, +6.60]$) | $59.60\%$ | | Cross-Seed 42,101,7 (Arm 5) ($T=0.0$; $3\times$ ckpt $= 1303\text{ TFLOPs}$, TestTok $= 942.0\text{k}$) | MATH-500 ($500$) | $44.20\%$ | $47.47\%$ | $45.80\%$ | $50.40\%$ ($2$) | $+4.60\text{ pp}$ ($[+2.00, +7.40]$) | $59.20\%$ | | Cross-Seed 42,101,7 (Arm 7) ($T=0.0$; $3\times$ ckpt $= 1317\text{ TFLOPs}$, TestTok $= 927.6\text{k}$) | MATH-500 ($500$) | $45.80\%$ | $49.13\%$ | $48.60\%$ | $49.80\%$ ($2$) | $+1.20\text{ pp}$ ($[-1.20, +3.60]$) | $59.60\%$ | | Cross-Seed 42,101,7 (Arm 9) ($T=0.0$; $3\times$ ckpt $= 1343\text{ TFLOPs}$, TestTok $= 910.4\text{k}$) | MATH-500 ($500$) | $47.00\%$ | $50.00\%$ | $49.80\%$ | $49.60\%$ ($2$) | $-0.20\text{ pp}$ ($[-2.40, +1.80]$) | $60.40\%$ | | Cross-Seed 42,101,7 (Arm 1) ($T=0.0$; $3\times$ ckpt $= 2448\text{ TFLOPs}$, TestTok $= 1010.6\text{k}$) | MATH-500 ($500$) | $46.07\%$ | $49.67\%$ | $49.20\%$ | $50.20\%$ ($8$) | $+1.00\text{ pp}$ ($[-1.40, +3.40]$) | $60.00\%$ |

Table 4 and verify_supplement.py support two bounded observations:

1. MDL-Maj@3 vs. MaxLen-Maj@3: On retained 3-seed greedy ensembles, MDL-Maj@3 exceeds MaxLen-Maj@3 with positive nominal paired $95\%$ CIs on MATH-500 in Arm 5 ($+4.60\text{ pp}$, $[+2.00, +7.40]$), Arm 11 ($+3.40\text{ pp}$, $[+1.00, +5.80]$), and Arm 13 ($+4.00\text{ pp}$, $[+1.60, +6.60]$) and on Val210 in Arm 12 ($+4.76\text{ pp}$, $[+1.43, +8.10]$), while differences on Arms 1, 7, 9, Ext240, and sampled $k=3$ runs span zero.
2. Cross-Seed Greedy ($3\times$ Train Cost) vs. Single-Seed Sampled ($1\times$ Train Cost) MDL-Maj@3 [20]: Both evaluate $k=3$ rollouts/item (TestTok $= 285.4\text{k}$ vs. $273.9\text{k}$ on Ext240; $369.4\text{k}$ vs. $350.4\text{k}$ on Val210), but Cross-Seed Greedy trains three checkpoints ($2,514.0\text{ OutTFLOPs}$, $3,081.8\text{s}$) vs. one ($838.0\text{ OutTFLOPs}$, $1,027.3\text{s}$). Comparing 3-seed greedy MDL-Maj@3 against the mean of the two sampled MDL-Maj@3 runs (seed=42, 7) gives $+0.95\text{ pp}$ on Val210 (nominal $95\%$ CI $[-3.81, +5.48]$) and $+2.92\text{ pp}$ on Ext240 ($64.17\%$ vs. $61.25\%$, $[-1.25, +7.08]$), remaining an unresolved descriptive contrast.

5.6 Counterfactual Donor-Prefix Splicing and Native Minimum-Margin Fork Interventions Across Discordant Seeds (Tables 5A & 5B, PASS 4)

Across all 13 arms (Table 4), 3-seed greedy Oracle Pass@3 exceeds Mean@1 by $+13.0$ to $+15.0\text{ pp}$ ($26.2\%$–$32.1\%$ discordant items). Building on [5, 7, 12, 21], we evaluate Arm 12 seed=7 (stopped_tree_beta0_cap1024_matched_50_seed7/model.pth) across all $N = 98$ discordant items where seed=42 is greedy-correct and seed=7 is greedy-wrong (45 MATH-500, 25 Val210, 28 Ext240). Every item's mismatch tuples, fork positions, and all $22,172$ item-level responses in prefix_splice_probe.json ($1,372$ discordant + $3,700$ branching + $17,100$ Arms 14–19 responses; $66,142 / 66,142$ total independently regraded responses across PASS 1 + PASS 4) are verified in PASS 4 of verify_supplement.py.

1. Unified Unpadded Teacher-Forced Token Agreement and Logit-Margin Separation ($K \le 160$): Evaluating seed=7 via unpadded ([1, L]) teacher forcing along seed=42's first $\min(160, |y^{(42)}|)$ tokens across all $98$ discordant items (15,636 total token positions: 7,188 on MATH-500, 4,000 on Val210, 4,448 on Ext240) shows that seed=7's greedy argmax agrees with seed=42's token on $15,141 / 15,636$ ($96.83\%$) of positions ($6,976/7,188 = 97.05\%$ on MATH-500, $3,873/4,000 = 96.83\%$ on Val210, $4,292/4,448 = 96.49\%$ on Ext240; 89.07\% geometric mean token probability). On the $15,141$ agreed tokens, seed=7's mean top-1 vs. top-2 logit margin $\Delta z_t = z_{(1)} - z_{(2)}$ is $8.49$ ($8.81, 8.23, 8.22$), whereas on the $495$ argmax-mismatch tokens ($3.17\%$ of positions: 212 MATH-500, 127 Val210, 156 Ext240), the mean logit margin is $0.59$ ($0.63, 0.53, 0.58$ — $14.4\times$ smaller), and on $422 / 495$ ($85.25\%$) of all mismatch positions, seed=42's token is already seed=7's own top-2 choice ($179/212 = 84.43\%$ MATH-500, $107/127 = 84.25\%$ Val210, $136/156 = 87.18\%$ Ext240). Note that teacher-forced agreement along a donor prefix does not by itself imply that free sequential decoding would reproduce every intervening token without drift; we therefore evaluate explicit prefix-splicing and native-prefix interventions below with matched length/batching controls.

2. Donor-Prefix Splicing Through the $m$-th Mismatch and Donor-Free Native Top-2 Forking with Identity Controls (Table 5A): In code/run_prefix_fork_probes.py:run_batched_prefixes, each batch (batch_size=64, sorted by prompt+prefix length) runs max_gen = max(64, 2048 - max_pref) continuation steps, giving row $i$ with prefix length $K_i$ an effective total limit of $\text{eff\_cap}_i = 2048 - \max_j K_j + K_i \le 2048$ (recorded as the 5th element of every condition tuple in prefix_splice_probe.json). In MismatchPrefix_1..3_donor, the cut is placed right after the $m$-th teacher-forced mismatch ($K = t_m^* + 1$, reusing the earlier cut when fewer than $m$ mismatches exist). In Self_MinMargin_Top2_Rank1..2, analyze_item_margins searches zero-based positions $4 \le t < 64$ along seed=7's unpadded 72-token prefix, selects $t_{\min,1}$ with the smallest margin and $t_{\min,2}$ with the smallest margin subject to separation $|t_{\min,2} - t_{\min,1}| \ge 2$ (not necessarily the unrestricted second-smallest margin), and inserts the unpadded teacher-forced top-2 token self_top2_m, while Self_Identity_Ctrl_Rank1..2 restores seed=7's recorded batch=32 native token self_orig_m = y7[t_m] at the identical cut length $K_i = t_{\min,m}+1$ and batch continuation budget $\text{eff\_cap}_i$:

| Prefix Intervention Condition into seed=7 ($T=0$ continuation, variable batch limit $\text{eff\_cap}_i \le 2048$) | Mean Cut $K$ | All Discordant (N=98) | Prompt-Only Fail (N=86, K0_none = 0/86) | Breakdown: PromptFail (All) across M500 / Val210 / Ext240 | |---|---:|---:|---:|---:| | K0_none (Prompt only, no spliced prefix) | $0.0$ | $12 / 98$ ($12.2\%$) | $0 / 86$ ($0.0\%$) | $0/35$ ($10/45$) / $0/24$ ($1/25$) / $0/27$ ($1/28$) | | K96_self_ctrl (seed=7's own recorded losing 96-tok prefix) | $96.0$ | $6 / 98$ ($6.1\%$) | $4 / 86$ ($4.7\%$) | $2/35$ ($4/45$) / $2/24$ ($2/25$) / $0/27$ ($0/28$) | | K160_self_ctrl (seed=7's own recorded losing 160-tok prefix) | $158.8$ | $4 / 98$ ($4.1\%$) | $3 / 86$ ($3.5\%$) | $1/35$ ($1/45$) / $2/24$ ($3/25$) / $0/27$ ($0/28$) | | MismatchPrefix_1_donor (seed=42 donor prefix through 1st TF mismatch $t_1^*+1$) | $18.6$ | $43 / 98$ ($43.9\%$) | $35 / 86$ ($40.7\%$) | $15/35$ ($22/45$) / $10/24$ ($10/25$) / $10/27$ ($11/28$) | | MismatchPrefix_2_donor (seed=42 donor prefix through 2nd TF mismatch $t_2^*+1$) | $38.4$ | $56 / 98$ ($57.1\%$) | $46 / 86$ ($53.5\%$) | $20/35$ ($29/45$) / $15/24$ ($15/25$) / $11/27$ ($12/28$) | | MismatchPrefix_3_donor (seed=42 donor prefix through 3rd TF mismatch $t_3^*+1$) | $58.2$ | $63 / 98$ ($64.3\%$) | $52 / 86$ ($60.5\%$) | $19/35$ ($28/45$) / $17/24$ ($18/25$) / $16/27$ ($17/28$) | | K_nl1_donor (seed=42 donor prefix to 1st newline $\ge 5$) | $28.1$ | $47 / 98$ ($48.0\%$) | $38 / 86$ ($44.2\%$) | $16/35$ ($24/45$) / $13/24$ ($13/25$) / $9/27$ ($10/28$) | | K_nl2_donor (seed=42 donor prefix to 2nd newline $\ge 5$) | $40.0$ | $55 / 98$ ($56.1\%$) | $47 / 86$ ($54.7\%$) | $20/35$ ($27/45$) / $12/24$ ($12/25$) / $15/27$ ($16/28$) | | K96_donor_full (seed=42 96-token winning prefix) | $96.0$ | $62 / 98$ ($63.3\%$) | $53 / 86$ ($61.6\%$) | $19/35$ ($26/45$) / $17/24$ ($18/25$) / $17/27$ ($18/28$) | | K160_donor_full (seed=42 160-token winning prefix) | $159.6$ | $68 / 98$ ($69.4\%$) | $58 / 86$ ($67.4\%$) | $20/35$ ($29/45$) / $17/24$ ($17/25$) / $21/27$ ($22/28$) | | Self_MinMargin_Top2_Rank1 (Donor-free: seed=7 prefix + TF top-2 at $t_{\min,1} \in [4,64)$) | $30.5$ | $31 / 98$ ($31.6\%$) | $25 / 86$ ($29.1\%$) | $10/35$ ($14/45$) / $10/24$ ($11/25$) / $5/27$ ($6/28$) | | Self_Identity_Ctrl_Rank1 (Identity control: seed=7 native prefix y7[:t_min1+1]) | $30.5$ | $10 / 98$ ($10.2\%$) | $8 / 86$ ($9.3\%$) | $5/35$ ($7/45$) / $2/24$ ($2/25$) / $1/27$ ($1/28$) | | Self_MinMargin_Top2_Rank2 (Donor-free: seed=7 prefix + TF top-2 at $t_{\min,2} \in [4,64)$) | $30.9$ | $31 / 98$ ($31.6\%$) | $25 / 86$ ($29.1\%$) | $11/35$ ($15/45$) / $8/24$ ($9/25$) / $6/27$ ($7/28$) | | Self_Identity_Ctrl_Rank2 (Identity control: seed=7 native prefix y7[:t_min2+1]) | $30.9$ | $9 / 98$ ($9.2\%$) | $7 / 86$ ($8.1\%$) | $4/35$ ($6/45$) / $2/24$ ($2/25$) / $1/27$ ($1/28$) | | Self_MinMargin_Top2_Union2 (Donor-free 2-branch union of Rank1 $\cup$ Rank2) | $30.5 / 30.9$ | $50 / 98$ ($51.0\%$) | $42 / 86$ ($48.8\%$) | $18/35$ ($24/45$) / $14/24$ ($15/25$) / $10/27$ ($11/28$) | | Self_Identity_Ctrl_Union2 (Identity control 2-branch union of Ctrl1 $\cup$ Ctrl2) | $30.5 / 30.9$ | $14 / 98$ ($14.3\%$) | $11 / 86$ ($12.8\%$) | $6/35$ ($9/45$) / $3/24$ ($3/25$) / $2/27$ ($2/28$) |

Conditional on this outcome-selected checkpoint pair and item set, splicing seed=42's prefix through the 1st, 2nd, or 3rd mismatch (MismatchPrefix_1..3_donor) rescues $35/86$ ($40.7\%$), $46/86$ ($53.5\%$), and $52/86$ ($60.5\%$) of prompt-only failures ($43/98, 56/98, 63/98$ overall vs. $4/86$ and $3/86$ for seed=7's 96- and 160-token prefixes). Without a donor checkpoint, Self_MinMargin_Top2_Rank1..2 rescues $25/86$ ($29.1\%$) each vs. $8/86$ ($9.3\%$) and $7/86$ ($8.1\%$) for the matched-length identity controls (Self_Identity_Ctrl_Rank1..2), with a 2-branch union of $42/86$ ($48.8\%$; $50/98$ overall) vs. $11/86$ ($12.8\%$); re-indexing the two forks chronologically ($\min(t_{\min,1}, t_{\min,2})$, mean $K=20.4$, vs. $\max(t_{\min,1}, t_{\min,2})$, mean $K=41.0$) rescues $27/86$ ($31.4\%$; $34/98$) vs. $23/86$ ($26.7\%$; $28/98$) ($60/471 = 12.7\%$ vs. $45/471 = 9.6\%$ across all $471$ full-benchmark wrong anchors). Because unpadded [1, L] teacher-forcing can rank near-tie tokens differently than the recorded batch=32 rollout (self_orig_m = y7[t_m] equals self_top1_m on $70/98$ Rank1 and $82/98$ Rank2 items, and equals self_top2_m on $27/98$ and $16/98$ ties), restricting to verified token changes (self_orig_m == self_top1_m, where self_top2_m != self_orig_m) yields $23/60$ ($38.3\%$) vs. $7/60$ ($11.7\%$) on Rank1 and $24/72$ ($33.3\%$) vs. $7/72$ ($9.7\%$) on Rank2.

3. Full-Benchmark Batch-Size Sensitivity Audit (batch=32 vs. batch=64), MinMargin-Fork@3, and FirstGap05-Fork@3 (Table 5B): Table 5B reports the 950-item batch=32 vs. batch=64 audit on Arm 12 seed=7 alongside two variable-budget ($\text{eff\_cap}_i = 2048 - \max_j K_j + K_i$) 1-checkpoint deterministic branching rules ($1\times$ greedy root $+$ one 72-token margin pass $+$ $2\times$ continuations $\approx 3.0\times$ single-pass tokens; $3,341$ records in PASS 4): (a) MinMargin-Fork@3 ($t_{\min,1}, t_{\min,2} \in [4, 64)$, mean $t_{\min,1} \approx 28.8$) and (b) FirstGap05-Fork@3 (the chronologically 1st and 2nd positions $t_{\text{fg},1}, t_{\text{fg},2} \in [4, 64)$ with $\Delta z_t \le 0.5$ and $|t_{\text{fg},2} - t_{\text{fg},1}| \ge 2$, applying a per-slot fallback to $t_{\min,1}$ or $t_{\min,2}$ when slot 1 has zero or slot 2 has $<2$ non-adjacent $\Delta z_t \le 0.5$ positions; $1,571/1,900 = 82.7\%$ slots satisfy $\Delta z_t \le 0.5$ [$865/950$ B1, $706/950$ B2] and $329/1,900 = 17.3\%$ use fallback [Val210 $66/420$, Ext240 $45/480$, MATH-500 $218/1,000$], mean $t_{\text{fg},1} \approx 19.5$):

| Benchmark ($N$) | Batch=32 vs. Batch=64 (Diff, $95\%$ CI; GradeAgree) | MinMargin: B0 / B1 / B2 | MinMargin: MDL / CR-Maj@3 (Pass@3) | FirstGap05: B1 / B2 | FirstGap05: MDL-Maj@3 (Diff, $95\%$ CI) | FirstGap05: CR-Maj@3 (Diff, $95\%$ CI) | FirstGap05: Pass@3 | |---|---:|---:|---:|---:|---:|---:|---:| | Val210 ($210$) | $52.86\%$ vs. $52.86\%$ ($+0.00$, $[-2.86, +2.86]$; $200/210$) | $52.86\% / 54.29\% / 50.48\%$ | $55.71\% / 55.71\%$ ($63.81\%$) | $53.33\% / 52.38\%$ | $55.24\%$ ($+2.38$, $[-1.43, +6.19]$) | $56.67\%$ ($+3.81$, $[+0.95, +7.14]$) | $65.24\%$ ($137$) | | Ext240 ($240$) | $57.50\%$ vs. $56.67\%$ ($-0.83$, $[-3.33, +1.67]$; $230/240$) | $57.50\% / 55.42\% / 58.33\%$ | $58.75\% / 59.58\%$ ($65.83\%$) | $57.50\% / 56.67\%$ | $60.42\%$ ($+2.92$, $[-0.42, +6.25]$) | $59.17\%$ ($+1.67$, $[-0.83, +4.58]$) | $68.33\%$ ($164$) | | MATH-500 ($500$) | $46.00\%$ vs. $45.80\%$ ($-0.20$, $[-2.20, +1.80]$; $475/500$) | $46.00\% / 44.40\% / 42.60\%$ | $45.20\% / 46.80\%$ ($54.00\%$) | $44.20\% / 46.20\%$ | $48.20\%$ ($+2.20$, $[-0.20, +4.80]$) | $48.40\%$ ($+2.40$, $[+0.40, +4.40]$) | $56.20\%$ ($281$) |

Replacing $15 \le L_j < 2048$ with $15 \le L_j < \text{eff\_cap}_{i,j}$ (`PASS 4`) yields identical `MDL-Maj@3` counts (`117/210`, `141/240`, `226/500`). Under `MinMargin-Fork@3`, symmetric `MDL-Maj@3` is unresolved (`+2.86 pp`, `+1.25 pp`, `-0.80 pp`) [18] because on the `114` `MATH-500` 3-way ties where all 3 branches complete, picking the shortest branch abandons the greedy root (`9/114` vs. `17/114`), whereas keeping `Branch 0` on ties whenever completed (`CompletedRoot-Maj@3` / `CR-Maj@3`) scores $55.71\%$, $59.58\%$, and $46.80\%$ ($+1.60\text{ pp}$ $[+0.20, +3.20]$ over `MDL-Maj@3` on `MATH-500`). Triggering at the 1st and 2nd $\Delta z_t \le 0.5$ positions with per-slot minimum-margin fallback (`FirstGap05-Fork@3`: $1,571/1,900 = 82.7\%$ strict $\Delta z_t \le 0.5$ [$865$ `B1`, $706$ `B2`] and $329/1,900 = 17.3\%$ fallback with zero below-threshold fallbacks and exact $k/128$ dyadic margins) rescues **$65/471$ ($13.8\%$) single-branch and $103/471$ ($21.9\%$) 2-branch wrong anchors** (vs. $50/471$ and $83/471$ for `MinMargin`), lifting `FirstGap05` `MDL-Maj@3` to **$502/950$ ($52.84\%$, $+2.42\text{ pp}$ $[+0.63, +4.21]$)** and `CR-Maj@3` to **$503/950$ ($52.95\%$, $+2.53\text{ pp}$ $[+1.16, +3.89]$)** across five strata (`Unanimous` $435$, `Root+1` $219$, `Sibling Override` $70$, `Completed Tie` $187$, `Truncated Tie` $39$). For $M \in \{3, 4, 5\}$ branches (`_plurality_cr_mdl` in `verify_supplement.py`), when the maximum non-empty frequency $f_{\max} \ge 2$ is unique, both `CR-Maj` and `MDL-Maj` select that plurality answer; when two answers tie at $f_{\max} = 2$ ($2$-$2$-$1$), they pick the tied answer with the shortest completed ($15 \le L_j < \text{eff\_cap}_{i,j}$) branch; and when $f_{\max} = 1$, `MDL-Maj` picks the shortest completed branch overall while `CR-Maj` retains `Branch 0` if completed ($15 \le L_0 < 2048$) and otherwise falls back to `MDL-Maj`. Stopping at $K=3$ on the $724$ majority items and querying two additional low-margin forks (`FG3, FG4` in $[4, 64)$) strictly on the $226$ 3-way ties (**`[email protected]`**, $3,302/950 = 3.48$ branches/item) resolves **$137/226$ ($60.6\%$)** of ties ($f_{\max} \ge 2$), reaching **`514/950 = 54.11%` `CR-Maj` (`+3.68 pp`, $[+2.00, +5.37]$; `119/210`, `148/240`, `247/500`)** and **`513/950 = 54.00%` `MDL-Maj` (`+3.58 pp`, $[+1.89, +5.37]$)**, compared to $502/950$ `CR-Maj` and $496/950$ `MDL-Maj` ($497/950$ restricting plurality to completed branches; $587/950$ `Pass`) for unconditioned `[email protected]` ($3,341$ branches; $+1.26\text{ pp}$ $[+0.21, +2.32]$ `CR-Maj` difference).

4. Six-Arm Online Counterfactual Fork-Pair GRPO Study at 1,024 Tokens (Table 5C, PASS 4): To evaluate low-margin counterfactual forking during online RLVR under $1.00\times$ greedy evaluation (batch=32), we compare six 3-seed 1,024-token packages (generate_counterfactual_fork in code/run_experiment.py, splitting each root $k \in \{1..4\}$ at $t_k^* = \inf\{t \in [4, 64) : \Delta z_t \le 0.5\}$ into on-policy $v_0$ vs. alternative $v_1 \ne v_0$ after 3 warmup updates; 18 runs = 920 steps = 3,680 pairs, arm_configs and 329 active groups replayed in PASS 4, 17,100 evaluations):

| 1,024-Token Arm (seeds = 42 / 101 / 7) | Val210 ($N=210$) | Ext240 ($N=240$) | MATH-500 ($N=500$) | Pool950 ($N=950$) | M500 MDL / Pass@3 | Act | Div (b0 / b1) | b0=4/4 False Act | |---|---|---|---|---|---|---:|---:|---:| | Arm 13 (Stopped-Tree $\beta=0.25$) | $55.56\% \pm 1.20\%$ (119/117/114) | $59.03\% \pm 5.31\%$ (150/148/127) | $45.40\% \pm 1.06\%$ (223/225/233) | $51.09\% \pm 1.04\%$ ($485.3$) | $50.4\% / 59.6\%$ ($298$) | $19.00$ | $53/192$ (27 / 26) | $4$ | | Arm 12 (Stopped-Tree $\beta=0.0$) | $55.71\% \pm 2.52\%$ (121/119/111) | $61.11\% \pm 3.37\%$ (148/154/138) | $46.47\% \pm 0.81\%$ (230/237/230) | $52.21\% \pm 1.65\%$ ($496.0$) | $51.6\% / 59.6\%$ ($298$) | $18.00$ | $50/180$ (27 / 23) | $5$ | | Arm 14 (Unconditional CF-GRPO) | $53.97\% \pm 1.80\%$ (115/116/109) | $60.28\% \pm 0.96\%$ (146/142/146) | $46.67\% \pm 2.61\%$ (232/247/221) | $51.72\% \pm 1.53\%$ ($491.3$) | $50.2\% / 59.8\%$ ($299$) | $21.67$ | $74/224$ (36 / 38) | $13$ (7/2/4) | | Arm 15 (50-Step RG-CF-GRPO) | $54.76\% \pm 0.82\%$ (114/117/114) | $60.83\% \pm 0.72\%$ (147/147/144) | $47.67\% \pm 0.83\%$ (237/243/235) | $52.56\% \pm 0.75\%$ ($499.3$) | $52.2\% / 61.6\%$ ($308$) | $13.33$ | $33/124$ (0 / 33) | 0 (8 gated) | | Arm 16 (RG-CF-GRPO Act18 Control) | $53.81\% \pm 2.86\%$ (107/119/113) | $58.33\% \pm 1.25\%$ (143/140/137) | $46.73\% \pm 0.99\%$ (228/236/237) | $51.23\% \pm 0.90\%$ ($486.7$) | $50.6\% / 59.8\%$ ($299$) | $18.00$ | $46/160$ (0 / 46) | 0 (9 gated) | | Arm 17 (Pair-Balanced PB-CF-GRPO) | $56.03\% \pm 0.73\%$ (118/116/119) | $60.97\% \pm 1.27\%$ (149/143/147) | $46.60\% \pm 1.64\%$ (226/242/231) | $52.32\% \pm 0.42\%$ ($497.0$) | $53.2\% / 60.6\%$ ($266/303$) | $13.67$ | $31/124$ (0 / 31) | 0 (8 gated) | | Arm 18 (Stratified RG-CF-GRPO) | $55.24\% \pm 1.65\%$ (120/114/114) | $57.50\% \pm 2.20\%$ (140/142/132) | $47.13\% \pm 0.58\%$ (234/234/239) | $51.54\% \pm 0.47\%$ ($489.7$) | $52.6\% / \mathbf{63.0\%}$ ($\mathbf{315}$) | $23.00$ | $50/148$ (0 / 50) | 0 (17 gated) | | Arm 19 (Stratified PB-CF-GRPO) | $56.51\% \pm 2.91\%$ (124/120/112) | $60.28\% \pm 4.46\%$ (154/147/133) | $45.27\% \pm 2.20\%$ (227/237/215) | $51.54\% \pm 2.70\%$ ($489.7$) | $47.8\% / 60.0\%$ ($239/300$) | $20.00$ | $48/148$ (0 / 48) | 0 (17 gated) |

Table 5C and the exact coefficient replay in verify_supplement.py (PASS 4) document four descriptive findings (where single-policy two-level $95\%$ CIs span zero):

• Unconditional CF-GRPO (Arm 14) vs. RG-CF-GRPO Package (Arm 15): With Bessel-corrected sample SD $s_R = \sqrt{\frac{1}{7}\sum_{i=1}^8 (R_i - \bar{R})^2}$ (s_R < 1e-6 guard, denominator $s_R + 10^{-4}$), all standardized coefficients are dimensionless. In Arm 14 (--cf-balance-div-len --cf-fork-hinge-weight 0.5), $36/74$ ($48.6\%$) of divergent pairs have $(R_{\text{raw}}(k,0), R_{\text{raw}}(k,1))=(1,0)$, activating $13$ steps where all 4 on-policy branches passed ($A_{\text{pref}} = -1.0604$). Arm 15 changes three package settings (--cf-rescue-only, replacing --cf-balance-div-len with --cf-dampen-lose-len, and --cf-fork-hinge-weight 0.15), duplicating the winning branch when $R_{\text{raw}}(k,0)=1 \wedge R_{\text{raw}}(k,1)=0$ (which removes observed $(1,0)$ activations by construction without isolating their causal effect on endpoint accuracy) and scoring $47.67\% \pm 0.83\%$ on MATH-500 ($+1.20\text{ pp}$ $[-1.67, +4.00]$ vs. Arm 12) and $52.56\% \pm 0.75\%$ on Pool950.
• Active-Count Control (Arm 15 vs. Arm 16): Continuing Arm 15 to match Arm 12's $18.00\text{ updates/seed}$ ($53/57/60$ problems; Pool950 diffs [-20, -12, -6]) changes Pool950 by $-1.33\text{ pp}$ ($51.23\% \pm 0.90\%$, $[-2.74, +0.07]\text{ pp}$). Descriptively, $12/14$ ($85.7\%$) of the additional active updates occur at $b_{0,\text{pass}}=3$ ((3,7): $7 \to 16$, (3,6): $4 \to 7$), where duplicating 3 solved roots sets $\bar{R} = 0.875, s_R = 0.3536$ ($A_{\text{pref}}(k^*) = -1.0604$, mean sequence-summed signed token mass [10] $M_{\text{tot}} = \frac{1}{8}\sum_{i=0}^7 [A_{\text{pref}}(i)\text{pl}_i + A_{\text{cont}}(i)(L_i-\text{pl}_i)] = +29.65$ and $+16.24$ tokens in Arms 15, 16); on the out-of-sample 27-checkpoint $R_2$ ($7..13/27$, $n=59$) stratum, Arm 15 vs. Arm 16 scores $28.00$ vs. $21.33/59$.
• Stratified 7-Subject RG-CF-GRPO (Arm 18): At 1,024 tokens on math_train_stratified.json[:50], Arm 18 rescues 13 b0=0/4 steps ($10$ post-warmup (0,1) groups), scores $47.13\% \pm 0.58\%$ greedy ($+2.93\text{ pp}$ over 512-token multi-subject) and 315/500 = 63.00% Pass@3 on MATH-500 ($16.33/108$ greedy, $43/108$ Pass@3 on $R_1$), and unlocks $42$ MATH-500 items missed by Arm 15 (6-model oracle union Arm 15 $\cup$ Arm 18 = 350/500 = 70.00%; 3-model Arm 15(101)+17(42)+18(7) Pass@3 = 643/950 = 67.68%).
• 2x2 Curriculum $\times$ Objective Package Factorial & Post-Adjustment Coefficient Replay (Arms 15, 17, 18, 19): Paired per-seed (42/101/7) PB - RG contrasts are [+4,-1,+5] ($+1.27\text{ pp}$ $[-2.86,+5.56]$) and [+4,+6,-2] ($+1.27\text{ pp}$ $[-3.33,+6.03]$) on Val210 ($56.51\%$, Pass@3 = 70.48% in Arm 19), [+2,-4,+3] ($+0.14\text{ pp}$) and [+14,+5,+1] ($+2.78\text{ pp}$ $[-1.67,+7.78]$, Pass@3 = 76.25% in Arm 19) on Ext240, and [-11,-1,-4] ($-1.07\text{ pp}$) and [-7,+3,-24] ($-1.87\text{ pp}$) on MATH-500 ($2\times 2$ interactions: Val210 [0,+7,-7] $= 0.00\text{ pp}$, Ext240 [+12,+9,-2] $= +2.64\text{ pp}$ $[-3.33,+8.75]$, MATH-500 [+4,+4,-20] $= -0.80\text{ pp}$ $[-5.60,+3.67]$, Pool950 [+16,+20,-29] $= +0.25\text{ pp}$; all $329/329$ active steps have pre-clipping $\|g\|_2 \ge 179.7 > 1.0$). Replaying code/run_experiment.py (PASS 4) shows that on (3,7) groups, PB zeroes the 3 (1,1) pairs and $A_{\text{pref}}(k^*)=0$ (reducing mean $M_{\text{tot}}$ from $+29.65/+19.40$ in Arms 15, 18 to $+5.58/+0.94$ in Arms 17, 19), whereas when $b_{0,\text{pass}} < 3$, PB applies torch.clamp(adv_pref, min=0.0) and sets $A_{\text{cont}}(\text{lose}) = -A_{\text{win}}\min(1, \frac{L_{\text{win}}-\tau}{L_{\text{lose}}-\tau})$. Thus on all post-warmup (0,1) groups (10 in Arm 18, 9 in Arm 19, e.g., Arm 19 seed=42 step 33), both RG and PB retain $A_{\text{pref}}(k^*,b) = +1.0604 > 0$. For continuations on (0,1) groups, length damping ($L_{\text{win}}-\tau < L_{\text{lose}}-\tau$) occurs on $3/10$ groups in Arm 18 ($\sum_{i=0}^7 A_{\text{cont}}(i) \in [0.000, +0.272]$, mean $+0.058$; mean $M_{\text{tot}} = +15.60$) and $5/9$ groups in Arm 19 ($\sum_{i=0}^7 A_{\text{cont}}(i) \in [-1.0604, +0.4059]$, mean $-0.717$; mean $M_{\text{tot}} = -55.65$; e.g., on Arm 19 seed=101 step 41, $\tau=19, L_{\text{win}}=266, L_{\text{lose}}=1024$ gives $A_{\text{cont}}(k^*)=(+1.9440, -0.4778)$ and $\sum A_{\text{cont}} = +0.4059$). Because these online procedures jointly alter step gating, continuation weights, and subsequent trajectories (hm_pb_cf_grpo_strat_seed42 scored $253/450$), subgroup shifts (R1: $16.33$ vs. $10.67/108$; Val210 R2: $11.33$ vs. $14.67/30$) and 1/3_tie length patterns (Arm 17: $695.0$ vs. $625.1$ tokens, MDL $50/93 = 53.8\%$; Arm 19: $461.1$ vs. $561.1$ tokens due to $64/114$ seed=7 tie picks, MDL $31/114 = 27.2\%$ vs. MaxLen $46/114 = 40.4\%$) are reported as descriptive package readouts rather than isolated causal effects.

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6. Limitations and Conclusions

1. Scope and Evaluation Limitations: Across 50- and 100-question arms (Arms 1–19), two-level (seed x question) $95\%$ CIs between well-controlled update rules span zero (e.g., Arm 15 - Arm 12 is $+1.20\text{ pp}$ $[-1.67, +4.00]$ on MATH-500 and $+0.35\text{ pp}$ $[-2.00, +2.60]$ on Pool950). Arm 13 was evaluated at --batch 64 vs. --batch 32 in Arms 5, 6, 12, 14–19 ($45/950 = 4.7\%$ flips under padding, Table 5B), retained as an explicit limitation.
2. Methodological Conclusions: Within this Qwen3-0.6B-Base regime, (i) $O(P)$ zero-gradient AdamW updates (Arms 2, 4) are bitwise equivalent to $0.0$-weighted backward passes on fixed trajectories (PASS 2) and cut dense FLOP proxies by $40.1\%$–$43.2\%$; (ii) a stopped tree $\tau_k = \min(\tau_{\text{causal}}, T_{\text{EOS}})$ preserves exact marginals $\Pr(Y_{k,b}=y)=\tilde{\pi}_\theta(y\mid x)$ whereas root resampling biases sibling marginals (PASS 2); (iii) at 512 and 1,024 tokens, prefix bonus $\beta = 0.25$ shows no detected benefit over $\beta = 0.0$, while raising --tokens to $1,024$ cuts cap-associated active steps from $29.3\%/37.1\%$ to $5.6\%/1.8\%$ (PASS 3); (iv) [email protected] lifts test-time CR-Maj on the explored checkpoint to $514/950$ ($54.11\%$, $+3.68\text{ pp}$ $[+2.00, +5.37]$); and (v) across six 3-seed 1,024-token Online Counterfactual Fork-Pair GRPO packages (Arms 14–19, PASS 4), Arm 15 (RG-CF-GRPO) scores $47.67\% \pm 0.83\%$ greedy ($52.20\%$ MDL, $61.60\%$ Pass@3) on MATH-500 and $52.56\% \pm 0.75\%$ on Pool950, while Arms 16–19 document descriptive $2\times 2$ (Curriculum $\times$ Objective) package behavior across post-adjustment coefficients, R1/R2/R3 strata, and 1/3_tie voting (Arm 18 reaching $63.00\%$ Pass@3 and a $70.00\%$ 6-model oracle union with Arm 15).

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7. Reproducibility and Supplement Verification

The supplement (supplement.zip) is verified in $\sim 2$ seconds on CPU via python3 verify_supplement.py:

• PASS 1: Verifies 128 evaluations (41,270 grades) + 2,700 sampled-$k=3$ grades in per_item_grades.json (43,970 total, 100% SymPy agreement).
• PASS 2: Executes generate_bifurcated ($p=1/2$: (0.50, 0.50) vs. (0.50, 0.25)), Proposition 2 ($-1.5 \ne 0$), and AdamW zero-gradient equivalence.
• PASS 3 (Tables 1–4): Reconstructs 46 training runs (2,502 steps, 10,008 pairs), 558 $>512$-token prefixes, 12 external retraining runs, and Tables 1–4.
• PASS 4 (Tables 5A–5C): Regrades all 22,172 extracted answers in prefix_splice_probe.json ($1,372$ discordant + $3,700$ branching + $17,100$ Arms 14–19 responses; 66,142 / 66,142 total across PASS 1 + PASS 4), verifies 15,141/15,636 (96.83%) TF agreement, $k/128$ dyadic margins, [email protected] ($514/950$), and replays all 920 steps (3,680 pairs, 329 active groups with exact arm_configs, post-adjustment coefficients, and signed token mass) across Arms 14–19.

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