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 $44.40\%$–$48.80\%$ 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 both full-backward training and step skipping (--skip-zero-var), with and without a 30-character pre-boxed derivation gate (text.rfind("\\boxed") >= 30); (ii) in two-stage ($K=4 \times B=2$) branched rollouts [13, 14, 16], how the distributional properties of early-EOS branch handling distinguish 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 ($\beta = 0.0$) under matched branching rules, alongside a descriptive census of 512-token cap-associated active steps and reward-divergent pairs and a matched 3-seed single-factor 1,024-token horizon comparison (Arm 12 vs. Arm 5); and (iv) how three-seed 100-question combined gradient accumulation (Arm 11) and cross-seed vs. single-seed test-time voting [19, 20] (MDL-Maj@3 vs. MaxLen-Maj@3) behave across benchmarks when accounting for training and inference costs and conditional uncertainty.

We report a twelve-arm, three-seed (seeds = 42, 101, 7) experimental matrix plus six single-seed (seed=101) diagnostic ablations across 43 primary training runs (2,352 steps and 9,408 rollout pairs with complete per-step and per-pair token, prefill, padded-prefill, untrimmed batch-max generation, backward, timing, and branch-event ledgers in training_records.json, plus 12 matched 50-step external-evaluation retraining runs = 600 steps) and 119 primary evaluations (38,420 graded item responses) plus 2,700 sampled-$k=3$ responses (41,120 independently regraded item responses total in per_item_grades.json): 43 MATH-500 evaluations (21,500 grades), 44 math_val210 evaluations (9,240 grades), and 32 evaluations on an untouched 240-item external mathematics benchmark (GSM8K-200 + AMC23-40, 7,680 grades). Because MATH-500 was inspected on seed=101 during initial gate and warmup selection (Section 5.1), we treat GSM8K-200 + AMC23-40 ($N=240$) as the untouched confirmatory benchmark and MATH-500 ($N=500$) and math_val210 ($N=210$) as broader in-domain diagnostic readouts. All aggregate metrics, independent regrading of all 41,120 extracted answers (PASS 1), direct finite-policy execution of generate_bifurcated and fixed-trajectory AdamW equivalence (PASS 2), per-step/per-pair integer token conservation laws (PASS 3), and 4,000-resample two-level (seed $\times$ question) and paired-item bootstrap $95\%$ confidence intervals 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 (Section 3.2 & Section 5.2): In the supplied baseline (Arm 1, code/run_baseline.py), homogeneous groups ($\sigma_R = 0$) execute 8 forward-and-backward passes multiplied by $0.0$ ($135.7\text{k}$ output backward tokens, $169.4\text{k}$ prompt+output backward tokens; $815.9\text{ TFLOPs}$ output-only / $1018.6\text{ TFLOPs}$ unpadded full-sequence / $1069.5\text{ TFLOPs}$ padded full-sequence analytical dense-work proxies) before calling optimizer.step(). Allocating explicit zero gradients (param.grad = torch.zeros_like(param)) and calling optimizer.step() costs $O(P)$ elementwise parameter operations and zero transformer backward FLOPs while preserving the 50-step AdamW momentum ($m_t \leftarrow \beta_1 m_{t-1}$), second-moment ($v_t \leftarrow \beta_2 v_{t-1}$), bias-correction, and decoupled weight-decay updates that step skipping (--skip-zero-var alone) omits.

   • Fixed-Trajectory Bitwise Equivalence vs. Descriptive Live Execution-Recipe Comparison (Arm 2 vs. Arm 1): On any fixed sequence of active and homogeneous steps sharing a common active-step loss function, 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) to executing $0.0$-weighted backward passes, whereas skipping optimizer.step() diverges (PASS 2). In our live GPU runs, Arm 1 (code/run_baseline.py) sums token cross-entropies before multiplying by scalar advantage (-adv[i] * logprob(seq) / 8), whereas Arms 2–12 (code/run_experiment.py) multiply each token's NLL by advantage before summing (compute_weighted_logprob). In bfloat16/float32, multiplying before summing vs. summing before multiplying introduces a 1-ULP active-step rounding difference (e.g., grad_norm = 808.9206 vs. 808.9137 on Step 6 of seed=42) that diverges subsequent rollout trajectories [4]; accordingly, live Arm 1 vs. Arm 2 differences are reported descriptively as comparisons between two execution recipes rather than isolated timing or accuracy tests of homogeneous-step backward elimination alone. Under the identical ungated reward (--min-pre-boxed-chars 0), Arm 2 cuts output backward tokens by $56.8\%$ ($58.6\text{k}$ vs. $135.7\text{k}$), analytical output-token dense FLOPs by $43.2\%$ ($463.3$ vs. $815.9\text{ TFLOPs}$), unpadded full-sequence dense FLOPs by $43.9\%$ ($571.3$ vs. $1018.6\text{ TFLOPs}$), and padded full-sequence dense FLOPs by $41.3\%$ ($627.4$ vs. $1069.5\text{ TFLOPs}$), with an observed same-host mean wall-clock time of $438.1\text{s}$ vs. $454.1\text{s}$ (-3.5%, $\sim 16.0\text{s}$ across the 50-step loop, as 512-token autoregressive decoding accounts for $\sim 88\%$–$90\%$ of step time; $530.0\text{s}$ historical Arm 1 mean). Across three 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): When paired with the 30-character pre-boxed derivation gate, ra_flat ($45.53\% \pm 1.03\%$ on MATH-500, $59.31\% \pm 1.68\%$ on Ext240, $489.0 / 600.1 / 654.8\text{ TFLOPs}$, $433.0\text{s}$) vs. gate_flat ($43.07\% \pm 6.54\%$ on MATH-500, $58.47\% \pm 4.49\%$ on Ext240) yields per-seed differences of [-1.40, -1.40, +10.20] pp on MATH-500 (mean $+2.47\text{ pp}$, two-level $95\%$ CI $[-3.47, +9.60]\text{ pp}$). Relative to Arm 1, 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, two-level $95\%$ CI $[-6.60, -1.00]\text{ pp}$) exhibit fewer 2,048-token test cap hits (AllCap2048) than Baseline Flat GRPO ($52.7 \pm 5.5$), which we interpret descriptively as an execution-recipe difference under bfloat16 trajectory sensitivity (Arm 4 vs. Arm 2 is $-0.13\text{ pp}$, $[-3.20, +2.80]\text{ pp}$).

2. Exact Distributional Analysis of Stopped-Tree vs. Outcome-Dependent Branching Samplers (Section 4.2 & Section 4.3): In two-stage ($K=4 \times B=2$) rollouts where Stage 1 targets the first newline boundary $\tau_{\text{causal}} \in [96, 160]$, how early-EOS trajectories ($T_{\text{EOS}} \le \tau_{\text{causal}}$) are handled determines whether the sampler preserves the executed on-policy rollout distribution $\tilde{\pi}_\theta(\cdot \mid x)$ [13, 16]:

   • True $\mathcal{F}_t$-Stopped-Tree Sampler (--strict-causal-tau --early-eos-mode terminal_dup, Arms 5, 6, 11, 12): Stopping at $\tau_k = \min(\tau_{\text{causal}}, T_{\text{EOS}})$ is $\mathcal{F}_t$-measurable on the natural token filtration. When $T_{\text{EOS}} \le \tau_{\text{causal}}$, the stopped prefix $u^{(k)} = y_{1:T_{\text{EOS}}}$ is already an absorbing terminal state, so its $B=2$ descendants have degenerate zero-length continuations ($v^{(k,1)} = v^{(k,2)} = \epsilon$). By the chain rule of probability, every leaf $Y_{k,b}$ has the exact marginal $\Pr(Y_{k,b} = y) = \tilde{\pi}_\theta(y \mid x)$ of the capped sampling policy.
   • Outcome-Dependent Root-Restart Sampler (--strict-causal-tau --early-eos-mode root_resample, Arms 7 and 8): Retrospectively setting $\tau_k = 0$ after observing EOS in $p^{(k)}_{1:\tau_{\text{causal}}}$ is not $\mathcal{F}_0$-measurable because $\{\tau_k = 0\}$ depends on future tokens up to $T_{\text{EOS}}$. Retaining the early-EOS sequence as Branch 1 while drawing Branch 2 freshly from the root biases the sibling distribution: for any policy that emits immediate EOS with probability $p \in (0,1)$ and otherwise traverses the branching boundary, the analytical chain-rule derivation establishes $\Pr(Y_1 = \text{EOS}) = p$ whereas $\Pr(Y_2 = \text{EOS}) = p^2$ (checked at $p=1/2$ by executing code/run_experiment.py:generate_bifurcated on a 4-outcome finite policy in PASS 2, yielding (0.50, 0.25) vs. (0.50, 0.50)).
   • Prefix-Sum Qualification on Shared-Prefix Subsets (Proposition 2): On any fixed realized tree, $\beta = 0$ descendant-mean credit equals leaf-summed GRPO on every shared prefix (Proposition 1). However, when only a subset $S = \{k : \tau_k > 0\} \subsetneq \{1,\dots,K\}$ of pairs share a non-empty prefix (as in Root-Restart), the sum of applied prefix coefficients at $\beta = 0$ is $B\sum_{k \in S}(\hat{V}_k - \bar{R})/(\sigma_R + 10^{-4})$, which need not vanish (e.g., $-1.5/(\sigma_R+10^{-4})$ with one $(0,0)$ shared pair and three $(1,1)$ root-restarted pairs); 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. Matched Three-Seed Bonus Comparisons ($\beta = 0.25$ vs. $\beta = 0.0$), Descriptive Cap-Associated Census, and Single-Factor 1,024-Token Horizon Comparison (Arm 12 vs. Arm 5, Section 4.4 & Section 5.3):

   • Across seeds = 42, 101, 7, no accuracy benefit from $\beta = 0.25$ over $\beta = 0.0$ is detected within two-level $95\%$ confidence intervals under either the $\mathcal{F}_t$-stopped tree (Arm 6 vs. Arm 5: $-0.97\text{ pp}$ on Ext240, $[-5.42, +3.47]\text{ pp}$; $+0.73\text{ pp}$ on MATH-500, $[-3.00, +4.67]\text{ pp}$) or the root-restart sampler (Arm 8 vs. Arm 7: $+0.69\text{ pp}$ on Ext240, $[-3.06, +4.72]\text{ pp}$; $+0.07\text{ pp}$ on MATH-500, $[-2.80, +3.00]\text{ pp}$), whereas one-sided prefix clamping (--clamped-prefix, Proposition 3) degrades seed=101 accuracy by $-14.80\text{ pp}$ ($33.00\%$, $95\%$ CI $[-19.00, -10.40]\text{ pp}$).
   • Descriptive Census of Cap-Associated Active Steps and Divergent Pairs (Table 2A & Section 4.4): Auditing all 612 active steps across Arms 1–10 in training_records.json shows that 187 / 612 (30.6%) of active steps (37.5% = 24/64 in Arm 1, 29.3% = 17/58 in Arm 5, 37.1% = 23/62 in Arm 6) have every $R=0$ leaf at the 512-token cap ($L_{k,b} = 512$), and across $200$ tree pairs per 50-step run only $15.0$–$17.3$ pairs/run have $R_{k,1} \ne R_{k,2}$ (45, 51, 52, 45 total across 3 seeds in Arms 5–8, of which 45, 51, 51, 44 have a shared prefix $\tau_k > 0$), with **48.9\%$–$66.7\%$ (30/45inArm 5, 30/51inArm 6, 32/52inArm 7, 22/45inArm 8) having the $R=0$ branch at $L = 512$ (CapR0Div)** while $15, 21, 20, 23$ pairs (15, 21, 20, 22` with both branches below 512 tokens) have $L_{R=0} < 512$. Because $M_k - \hat{V}_k = 0.5$ fires identically on all shared-prefix divergent pairs and observing $R=0, L=512$ does not establish counterfactual success under longer generation, these counts are reported strictly as a descriptive census of limited, cap-associated reward disagreement.
   • Single-Factor 3-Seed 1,024-Token Horizon Comparison (Arm 12 vs. Arm 5): Holding --split-tokens 160 and zero_cont_token_mass=False identical to Arm 5 (strict_mst_beta0_50) and changing only --tokens 512 -> 1024 across seeds = 42, 101, 7 (Arm 12, cap1024_matched) cuts training rollout cap hits from $293/1,200$ ($24.4\%$) to $70/1,200$ ($5.8\%$), cuts CapAssocActive steps from $17/58$ ($29.3\%$) to $3/54$ ($5.6\%$), and increases non-cap divergent pairs (Both < Cap) from $15/45$ ($5.0\text{/run}$) to $45/50$ ($15.0\text{/run}$). Across three seeds, Arm 12 scores $46.47\% \pm 0.81\%$ on MATH-500 ($+2.27\text{ pp}$ vs. Arm 5, per-seed [+2.40, +2.80, +1.60] pp, pooled $95\%$ CI $[+0.13, +4.47]\text{ pp}$, two-level $95\%$ CI $[-0.73, +5.13]\text{ pp}$), $55.71\% \pm 2.52\%$ on Val210 ($+0.48\text{ pp}$, $[-5.40, +6.03]\text{ pp}$), and $61.11\% \pm 3.37\%$ on Ext240 ($+0.83\text{ pp}$, $[-4.17, +5.83]\text{ pp}$) at $502.2\text{ OutTFLOPs}$ ($649.8\text{s}$).

4. Three-Seed 100-Question Combined Accumulation (Arm 11) and Cross-Seed vs. Single-Seed Voting (Section 5.4 & Section 5.5):

   • Replicating the combined 100-question intervention (Stopped-Tree $\beta=0.0$ + zero-signed-token-mass continuation centering [10] + 4-question unit-normalized gradient accumulation on math_train_stratified.json[:100], Arm 11) across seeds = 42, 101, 7 yields $46.20\% \pm 0.35\%$ on MATH-500 ($+2.00\text{ pp}$ vs. 50-step Arm 5, two-level $95\%$ CI $[-0.80, +5.13]\text{ pp}$, unresolved), $55.08\% \pm 1.92\%$ on math_val210, and $59.72\% \pm 0.87\%$ on Ext240 ($-0.56\text{ pp}$, $[-5.28, +4.03]\text{ pp}$).
   • In test-time voting ($k=3$ rollouts per item [19, 20], Table 4), breaking 1-1-1 ties by selecting the shortest completed non-truncated derivation (15 <= L < 2048, MDL-Maj@3) vs. the longest derivation (MaxLen-Maj@3) yields $+4.60\text{ pp}$ on Arm 5 ($50.40\%$ vs. $45.80\%$, nominal paired-item $95\%$ CI $[+2.00, +7.40]\text{ pp}$) and $+3.40\text{ pp}$ on Arm 11 ($51.00\%$ vs. $47.60\%$, nominal paired-item $95\%$ CI $[+1.00, +5.80]\text{ pp}$) on MATH-500, and $+4.76\text{ pp}$ ($[+1.43, +8.10]\text{ pp}$) on Arm 12 Val210. On Ext240 (Arm 11), 3-checkpoint cross-seed greedy MDL-Maj@3 ($64.17\%$, requiring $3\times$ training cost $= 2,514.0\text{ OutTFLOPs}$ and $285.4\text{k}$ test tokens) minus 1-checkpoint single-seed sampled MDL-Maj@3 ($T=0.6$, mean $61.25\%$, $1\times$ training cost $= 838.0\text{ OutTFLOPs}$ and $273.9\text{k}$ test tokens) is $+2.92\text{ pp}$ with a nominal paired question bootstrap $95\%$ CI of $[-1.25, +7.08]\text{ pp}$ (unresolved within intervals).

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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], and self-consistency voting [19], as well as concurrent studies in r/math_rlvr [4–12, 17, 18, 20]:

• On-Policy Tree Search, Stopping-Time Measurability, Prefix Backups, and Run-Ensemble vs. Single-Policy Voting (TreeRL [13], MARGE [14], OPTS-TTPO [16], math_rlvr_6 [20], math_rlvr_16 [6], math_rlvr_18 [7], math_rlvr_22 [17], math_rlvr_23 [18]): TreeRL [13] Section 3.2 defines descendant-mean node values $\hat{V}_k$, global-plus-local advantages, and internal-node reweighting; MARGE [14] studies guided intermediate-state exploration; and OPTS-TTPO [16] analyzes finite-sample policy-gradient learning with tree search, explicitly distinguishing prefix-measurable branching from outcome-adaptive tree expansion and comparing mean vs. max-backup prefix value estimates ($\hat{V}_k$ vs. $M_k$). In r/math_rlvr, [5, 6, 7, 9, 10] establish token-0 onset dynamics, conditioned policy-score counterexamples, Monte Carlo fork credit, 2,048-token truncation behavior, and zero-signed-token-mass stability, while [12, 17, 18, 20] analyze item-profile redistribution across seeds, shared-prefix credit shielding and BF16 weight-space audits, greedy vs. vote-based selection, and run-ensemble vs. single-policy sampling trade-offs ([20], Section 3.7). We do not claim a new tree-credit or ensemble principle. Our bounded contributions are: (1) the controlled optimizer-clock and active-step loss comparison (Arms 1–4), (2) the matched three-seed bonus comparisons (Arms 5–8) across both the exact $\mathcal{F}_t$-stopped tree and the outcome-dependent root-restart sampler together with the descriptive cap-associated step/pair census (Table 2A) and single-factor 3-seed 1,024-token horizon comparison (Arm 12), and (3) the three-seed 100-question combined gradient-accumulation replication (Arm 11), exploratory 1,024-token recipe probe (Table 3), and uncertainty-qualified MDL-Maj@3 vs. MaxLen-Maj@3 voting comparison (Table 4).

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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 (logprob vs. compute_weighted_logprob): On any fixed sequence of active and homogeneous steps using a common active-step loss function, 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 state buffers $(m_t, v_t)$ (max_abs_diff = 0.0), whereas skipping optimizer.step() diverges (PASS 2 in verify_supplement.py, verified both via the standard-library scalar AdamW recurrence and, when PyTorch is present, via torch.nn.Module + torch.optim.AdamW tensor checks). When comparing Arm 2 (code/run_experiment.py) against Arm 1 (code/run_baseline.py) in live bfloat16 training, a second implementation detail enters on active steps: code/run_baseline.py computes -adv[i].detach() * logprob(model, seq, len(prompt)) / 8 (summing token cross-entropies first via F.cross_entropy(..., reduction='sum') and then multiplying by scalar $A_i$), whereas compute_weighted_logprob in code/run_experiment.py computes -(weights * token_nll).sum() / 8 (multiplying each token's negative log-likelihood by $A_i$ before summing). Because floating-point addition is non-associative with scalar multiplication ($A_i \sum_t \ell_t$ vs. $\sum_t (A_i \ell_t)$), active steps can differ by 1 ULP (producing grad_norm = 808.9206 vs. 808.9137 on Step 6 of seed=42), which under bfloat16 rollout sensitivity [4] alters sampled trajectories from Step 8 onward. We therefore establish fixed-trajectory zero-gradient equivalence via PASS 2 and treat live Arm 2 vs. Arm 1 trajectory, accuracy, and timing 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 12), 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 Arm 12; 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 2,352 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):

   • 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; and (28, 168, 4), (24, 174, 2), and (20, 178, 2) for 1,024-token Arm 12, 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 Arm 12) [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) |

Table 2A records three exact descriptive observations and their interpretive boundaries:

1. Cap-Associated Active Steps (187 / 612 = 30.6% Across Arms 1–10): Across all 612 active steps of the ten 512-token arms (Arms 1–10), $187/612$ ($30.6\%$) have every zero-reward rollout at the 512-token generation cap ($L_{k,b} = 512$). Observing $R=0$ and $L=512$ in the compact ledger does not establish that an unfinished rollout was otherwise mathematically correct or would have succeeded if continued past 512 tokens; rather, it documents that on $30.6\%$ of active steps, every negative-advantage rollout in the group also reached the 512-token length boundary [9, 10].
2. Exact Predicates for Shared-Prefix and Cap-Associated Divergent Pairs: In Table 2A, Post-Warmup Divergent Pairs counts all post-warmup pairs (cid != 0) with $R_{k,1} \ne R_{k,2}$, while Shared ($\tau_k > 0$) counts those with a positive shared prefix (pl0 > 0 and pl1 > 0), which is the exact set on which the prefix bonus $\beta(M_k - \hat{V}_k)$ is applied (45/45 in Arm 5, 51/51 in Arm 6, 51/52 in Arm 7, and 44/45 in Arm 8, as Arms 7 and 8 each contain one divergent tau_zero pair with $\tau_k = 0$). Similarly, CapR0Div counts divergent pairs where the zero-reward branch hit the cap ($L_{R=0} = \text{Cap}$), NonCapR0Div counts divergent pairs where the zero-reward branch finished below the cap ($L_{R=0} < \text{Cap}$), and Both < Cap counts divergent pairs where both branches finished below the cap ($L_1 < \text{Cap}$ and $L_2 < \text{Cap}$; in Arms 8 and 9, one NonCapR0Div pair has $L_{R=1} = 512$, so Both < 512 is $22$ and $26$ versus $23$ and $27$ NonCapR0Div). Because $M_k - \hat{V}_k = 0.5$ applies identically to CapR0Div and NonCapR0Div shared pairs and pair counts do not bound gradient influence after standardization, clipping, and AdamW updates, Table 2A supports a descriptive hypothesis about sparse ($7.5\%$–$8.7\%$ of pairs) and frequently cap-associated reward disagreement rather than proving a causal explanation for the unresolved $\beta = 0.25$ vs. $\beta = 0.0$ contrasts.
3. Single-Factor 1,024-Token Horizon Comparison (Arm 12 vs. Arm 5): When --tokens is raised from $512$ (Arm 5) to $1,024$ (Arm 12) with --split-tokens 160, zero_cont_token_mass=False, $\beta=0.0$, and terminal_dup held fixed across seeds = 42, 101, 7, CapAssocActive drops from $17/58$ ($29.3\%$) to $3/54$ ($5.6\%$), CapR0 drops from $129/237$ ($54.4\%$) to $25/190$ ($13.2\%$), and Both < Cap divergent pairs increase from $15/45$ ($5.0\text{/run}$) to $45/50$ ($15.0\text{/run}$).

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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 our initial exploratory phase, 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 runs (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. Subsequently, with all hyperparameters frozen in advance, 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), which lies completely outside the Hendrycks MATH corpus and was never used for any design or selection decision. For Arms 2, 6, 7, 8, 11, 12 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 the four earlier arms completed before gsm8k_amc_bench240.json was assembled (Arms 1, 3, 4, 5, 12 runs across seeds = 42, 101, 7), MATH-500, math_val210, training.jsonl, and rollouts.jsonl come from the exact same 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}). We separately record the exact step-time sum (5,376.03s) and total run wall-clock (5,436.11s across 600 steps) of these 12 external-evaluation retraining runs via "ext_reexec_step_seconds", "ext_reexec_seconds", "ext_reexec_steps", and "ext_reexec_active_steps" in training_records.json (verified in PASS 3 alongside the 43 primary training runs = 2,352 steps = 9,408 pairs, which also store "effective_early_eos_mode" documenting the exact branch rule executed by every run).

Auditable Per-Step & Per-Pair Resource Derivation and Analytical Dense-Work Proxy Qualification (training_records.json & PASS 3): training_records.json stores each run's config dictionary, effective_early_eos_mode, and per-step step_ledger (including per-pair [pl0, pl1, l0, l1, r0, r1, cat_id] tuples), and verify_supplement.py (PASS 3) derives and verifies both unpadded/trimmed token counts and padded/untrimmed batch execution counts:

• Unpadded/Trimmed Token Counts (gen, pref, leaf, bwd_out, bwd_full): gen (ActualGenTok) is $\sum_s (D_{\text{S1,raw}}(s) + D_{\text{S2,gen}}(s))$, counting post-EOS-trimmed generated tokens across Stage 1 (including tokens up to split_tokens discarded after cut) and Stage 2; pref (PrefillTok) sums unpadded Stage-1 and Stage-2 prefill lengths; leaf (LeafTok) sums $\sum_{i=1}^8 L_i(s)$, satisfying exact integer conservation $D_{\text{S1,leaf}}(s) + D_{\text{S2,gen}}(s) = \text{leaf}(s)$ and $0 \le L_b - \text{pl}_b \le \text{max\_tokens} - \text{pl}_b$ on every step and branch (PASS 3); bwd_out and bwd_full sum leaf and $8 L_{\text{prompt}}(s) + \text{leaf}(s)$ over steps executing loss.backward().
• Padded Prefill and Untrimmed Batch-Max Decoder Steps (pad_pref, untrim_gen): Because both generate (code/run_baseline.py) and generate_bifurcated (code/run_experiment.py) left-pad Stage-2 prompts to width = max(map(len, unfinished_prompts)) and continue single-token KV-cached forward steps across the batch until all rows reach EOS or max_tokens (done.all()), trimming tokens after the first EOS only when building the returned Python lists, we also compute exact padded prefill tokens pad_pref and untrimmed batch-max generation tokens untrim_gen.
• Qualification of OutTFLOPs, FullTFLOPs, and PadFullTFLOPs as Analytical Dense-Work Proxies: Using $P = 751,632,384$ total trainable parameters, we report $\text{OutTFLOPs} = (2 D_{\text{gen}} + 6 D_{\text{bwd,out}}) P / 10^{12}$, $\text{FullTFLOPs} = [2(D_{\text{pref}} + D_{\text{gen}}) + 6 D_{\text{bwd,full}}] P / 10^{12}$, and $\text{PadFullTFLOPs} = [2(D_{\text{pad\_pref}} + D_{\text{untrim\_gen}}) + 6 D_{\text{bwd,full}}] P / 10^{12}$. We emphasize that these $2P/6P$ quantities are analytical dense-work proxies rather than exact hardware FLOP counts: $P = 751.63\text{M}$ includes the input token embedding table (tok_emb.weight, $151,936 \times 1,024 = 155.58\text{M}$ parameters, which executes a table lookup in the forward pass and sparse row scatter-add in the backward pass rather than a dense $2P$ matrix multiplication), prefill forward passes evaluate the output head out_head only on the last token position (logits = model(x)[:, -1, :]), and quadratic attention $O(L^2 d)$ and $O(P)$ AdamW updates are not included.

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

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

| Arm (seeds = 42, 101, 7; 50 q for Arms 1–10 & 12, 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 ($\beta=0.0, 1024\text{ tok matched}$) | $46.47\% \pm 0.81\%$ ($46.0 / 47.4 / 46.0$) | $55.71\% \pm 2.52\%$ | $61.11\% \pm 3.37\%$ ($67.83\% / 27.50\%$) | $43.0$ ($45/51/33$) / $42.0$ | $146.6 [210.2] / 85.6 [90.0] / 62.5 / 72.8\text{k}$ | $502.2 / 677.1 / 779.5$ ($650\text{s}$) |

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

Table 2B (recomputed and asserted by verify_supplement.py) presents 20 contrasts with 4,000-resample two-level (seed $\times$ question) bootstrap $95\%$ confidence intervals:

| Controlled Contrast (seeds = 42, 101, 7) | Benchmark ($N$) | Per-Seed Diffs ($42 / 101 / 7$, pp) | 3-Seed Mean Diff | Two-Level Seed $\times$ Item $95\%$ CI | Pooled Item $95\%$ CI | |---|---:|---:|---:|---:|---:| | 1. Stopped-Tree $\beta=0.25$ vs. $\beta=0.0$ (terminal_dup): Arm 6 vs. Arm 5 | MATH-500 ($500$) | $+4.00 / -0.80 / -1.00$ | $+0.73\text{ pp}$ | $[-3.00, +4.67]\text{ pp}$ | $[-1.40, +2.80]\text{ pp}$ | | 2. Stopped-Tree $\beta=0.25$ vs. $\beta=0.0$ (terminal_dup): Arm 6 vs. Arm 5 | Ext240 ($240$) | $-1.67 / -2.08 / +0.83$ | $-0.97\text{ pp}$ | $[-5.42, +3.47]\text{ pp}$ | $[-4.17, +2.08]\text{ pp}$ | | 3. Root-Restart $\beta=0.25$ vs. $\beta=0.0$ (root_resample): Arm 8 vs. Arm 7 | MATH-500 ($500$) | $-0.80 / -0.80 / +1.80$ | $+0.07\text{ pp}$ | $[-2.80, +3.00]\text{ pp}$ | $[-2.00, +2.13]\text{ pp}$ | | 4. Root-Restart $\beta=0.25$ vs. $\beta=0.0$ (root_resample): Arm 8 vs. Arm 7 | Ext240 ($240$) | $+1.67 / +0.00 / +0.42$ | $+0.69\text{ pp}$ | $[-3.06, +4.72]\text{ pp}$ | $[-2.22, +3.61]\text{ pp}$ | | 5. Root-Restart vs. Stopped-Tree at $\beta=0.0$: Arm 7 vs. Arm 5 | MATH-500 ($500$) | $+2.80 / +0.20 / +1.80$ | $+1.60\text{ pp}$ | $[-1.33, +4.73]\text{ pp}$ | $[-0.53, +3.80]\text{ pp}$ | | 6. Midpoint-Fallback vs. Root-Restart at $\beta=0.25$: Arm 9 vs. Arm 8 | MATH-500 ($500$) | $+1.40 / +1.60 / +0.40$ | $+1.13\text{ pp}$ | $[-1.53, +3.87]\text{ pp}$ | $[-0.80, +3.20]\text{ pp}$ | | 7. Execution-Recipe Zero-Grad Step vs. Full Bwd (Ungated): Arm 2 vs. Arm 1 | MATH-500 ($500$) | $-0.60 / -0.40 / -3.20$ | $-1.40\text{ pp}$ | $[-4.60, +1.80]\text{ pp}$ | $[-3.60, +0.73]\text{ pp}$ | | 8. Execution-Recipe Zero-Grad Step vs. Full Bwd (Ungated): Arm 2 vs. Arm 1 | Ext240 ($240$) | $+0.00 / +2.08 / +2.92$ | $+1.67\text{ pp}$ | $[-2.22, +5.69]\text{ pp}$ | $[-1.11, +4.44]\text{ pp}$ | | 9. Zero-Grad Momentum Step vs. Skip (Gated): Arm 4 (ra_flat) vs. Arm 3 | MATH-500 ($500$) | $-1.40 / -1.40 / +10.20$ | $+2.47\text{ pp}$ | $[-3.47, +9.60]\text{ pp}$ | $[+0.07, +4.80]\text{ pp}$ | | 10. Zero-Grad Momentum Step vs. Skip (Gated): Arm 4 (ra_flat) vs. Arm 3 | Ext240 ($240$) | $-2.08 / +0.42 / +4.17$ | $+0.83\text{ pp}$ | $[-3.89, +6.25]\text{ pp}$ | $[-2.36, +4.17]\text{ pp}$ | | 11. Derivation Gate at Fixed Zero-Grad Step: Arm 4 (ra_flat) vs. Arm 2 | MATH-500 ($500$) | $+0.60 / +1.80 / +0.20$ | $+0.87\text{ pp}$ | $[-2.20, +3.73]\text{ pp}$ | $[-1.27, +2.93]\text{ pp}$ | | 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) | $[-19.00, -10.40]\text{ pp}$ | | 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 ($500$) | $+2.20 / +1.80 / +2.00$ | $+2.00\text{ pp}$ | $[-0.80, +5.13]\text{ pp}$ | $[-0.27, +4.27]\text{ pp}$ | | 17. Combined 100-Question Accum4 Package vs. 50-Step Arm 5: Arm 11 vs. Arm 5 | Ext240 ($240$) | $+2.08 / -3.75 / +0.00$ | $-0.56\text{ pp}$ | $[-5.28, +4.03]\text{ pp}$ | $[-3.75, +2.50]\text{ pp}$ | | 18. Single-Factor 1,024-Token Horizon vs. 512-Token Arm 5: Arm 12 vs. Arm 5 | MATH-500 ($500$) | $+2.40 / +2.80 / +1.60$ | $+2.27\text{ pp}$ | $[-0.73, +5.13]\text{ pp}$ | $[+0.13, +4.47]\text{ pp}$ | | 19. Single-Factor 1,024-Token Horizon vs. 512-Token Arm 5: Arm 12 vs. Arm 5 | Val210 ($210$) | $+3.81 / +1.43 / -3.81$ | $+0.48\text{ pp}$ | $[-5.40, +6.03]\text{ pp}$ | $[-2.70, +3.81]\text{ pp}$ | | 20. Single-Factor 1,024-Token Horizon vs. 512-Token Arm 5: Arm 12 vs. Arm 5 | Ext240 ($240$) | $+3.33 / +1.67 / -2.50$ | $+0.83\text{ pp}$ | $[-4.17, +5.83]\text{ pp}$ | $[-2.22, +3.89]\text{ pp}$ |

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 the strictly $\mathcal{F}_t$-measurable Stopped-Tree sampler (Arms 5–6, Rows 1–2), $\beta = 0.25$ minus $\beta = 0.0$ is $-0.97\text{ pp}$ on Ext240 (two-level $95\%$ CI $[-5.42, +3.47]\text{ pp}$), $+0.73\text{ pp}$ on MATH-500 ($[-3.00, +4.67]\text{ pp}$), and $-3.33\text{ pp}$ on math_val210. Under the outcome-dependent Root-Restart sampler (Arms 7–8, Rows 3–4), $\beta = 0.25$ minus $\beta = 0.0$ is $+0.69\text{ pp}$ on Ext240 ($[-3.06, +4.72]\text{ pp}$), $+0.07\text{ pp}$ on MATH-500 ($[-2.80, +3.00]\text{ pp}$), and $0.00\text{ pp}$ on math_val210. Thus, neither procedure shows a detected accuracy benefit from the sibling-disagreement bonus $\beta = 0.25$ over $\beta = 0.0$ within its confidence interval.
   • On MATH-500, mean 2,048-token cap hits (AllCap2048) change from $35.7$ (32, 30, 45) at $\beta = 0.0$ to $38.3$ (38, 25, 52, decreasing on seed=101) under Stopped-Tree, and from $40.7$ (30, 41, 51) at $\beta = 0.0$ to $51.3$ (61, 48, 45, increasing on seeds = 42, 101 but decreasing on seed=7) under Root-Restart. Because seed directions are mixed, these counts do not establish a uniform continuation-lengthening mechanism.
   • Arithmetic telescoping of the observed MATH-500 sample means between Arm 5 ($44.20\%$) and Arm 9 ($47.00\%$) gives $1.60 + 0.07 + 1.13 = 2.80\text{ pp}$ (or $0.73 + 2.07 = 2.80\text{ pp}$ via Arm 6). Because the between-sampler contrasts (Rows 5–6) have two-level $95\%$ CIs of $[-1.33, +4.73]\text{ pp}$ and $[-1.53, +3.87]\text{ pp}$ that span zero, we report them strictly as unresolved sample estimates rather than proved causal effects of early-EOS handling.

2. Separating Zero-Gradient Stepping, Active-Step Loss Associativity, and Derivation Gating (Rows 7–11 and 13–14 of Table 2B):

   • Comparing Arm 2 (Ungated Flat + Zero-Grad Step in code/run_experiment.py) against Arm 1 (Baseline Flat GRPO in code/run_baseline.py) under the identical ungated reward function shows that analytical output-token dense FLOPs drop by $43.2\%$ ($463.3$ vs. $815.9\text{ TFLOPs}$), unpadded full-sequence dense FLOPs drop by $43.9\%$ ($571.3$ vs. $1018.6\text{ TFLOPs}$), and padded full-sequence dense FLOPs drop by $41.3\%$ ($627.4$ vs. $1069.5\text{ TFLOPs}$), with an observed same-host mean wall-clock reduction of $3.5\%$ ($438.1\text{s}$ vs. $454.1\text{s}$ step sum; $17.3\%$ vs. the historical $530.0\text{s}$ mean), while accuracy differences are unresolved within intervals ($+1.67\text{ pp}$ on Ext240, $[-2.22, +5.69]\text{ pp}$; $-1.40\text{ pp}$ on MATH-500, $[-4.60, +1.80]\text{ pp}$).
   • Comparing Arm 4 (ra_flat, gated + zero-grad step) against Arm 2 (ungated + zero-grad step) within the same code/run_experiment.py engine isolates the 30-character derivation gate at fixed optimizer stepping and fixed compute_weighted_logprob loss: $+0.87\text{ pp}$ on MATH-500 ([+0.60, +1.80, +0.20] pp, two-level $95\%$ CI $[-2.20, +3.73]\text{ pp}$) and $-0.13\text{ pp}$ on AllCap2048 ($[-3.20, +2.80]\text{ pp}$).

5.4 Combined 100-Question Replication (Arm 11), Single-Factor 1,024-Token Horizon (Arm 12), 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 combined 100-question synthesis recipe (Stopped-Tree $\beta = 0.0$ + continuation zero-signed-token-mass centering --zero-cont-token-mass [10] + 4-question unit-normalized gradient accumulation --grad-accum-questions 4 --unit-question-grad on data_splits/math_train_stratified.json[:100]) across seeds = 42, 101, 7. Across three seeds, Arm 11 scores $46.20\% \pm 0.35\%$ on MATH-500 ($+2.00\text{ pp}$ over 50-step Arm 5, per-seed [+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 ($-0.56\text{ pp}$ vs. Arm 5, two-level $95\%$ CI $[-5.28, +4.03]\text{ pp}$) at $838.0 / 1249.4 / 1340.8\text{ TFLOPs}$ ($1027\text{s}$). Because Arm 11 changes curriculum, question exposure ($100$ vs. $50$ questions, $\sim 1.93\times$ OutTFLOPs), continuation centering (zero_cont_token_mass=True), and 4-question unit-normalized gradient accumulation simultaneously, and its two-level $95\%$ CIs vs. Arm 5 span zero, we report it strictly as a combined-package comparison.

2. Single-Factor 3-Seed 1,024-Token Horizon Comparison (Arm 12 vs. Arm 5, Rows 18–20 of Table 2B): In Arm 12 (cap1024_matched_s{42,101,7}), we hold --split-tokens 160 (160), zero_cont_token_mass=False, $\beta = 0.0$, and terminal_dup identical to Arm 5 (strict_mst_beta0_s{42,101,7}), changing only --tokens 512 -> 1024 across seeds = 42, 101, 7. Across all 3 seeds ($1,200$ rollouts), $266/1,200$ ($22.2\%$) of on-policy rollouts exceed 512 tokens ($512 < L \le 1024$); of those $266$ rollouts, $90/266$ ($33.8\%$) finish with a verified correct \boxed{} answer ($R=1$, $87/90$ of which had $R=0$ at token 512), $106/266$ ($39.8\%$) finish before token 1,024 with $R=0$, and $70/266$ ($26.3\%$, or $5.8\%$ of all $1,200$ rollouts) hit the 1,024-token cap. Even under this single-factor change, the first rollout exceeding 512 tokens (Step 2 on seed=42, Step 1 on seed=101) consumes additional torch.multinomial draws from the global RNG stream, so subsequent steps sample from diverging trajectories at $+15.7\%$ higher OutTFLOPs ($502.2$ vs. $434.2\text{ TFLOPs}$, $649.8\text{s}$ vs. $480.2\text{s}$). On MATH-500, Arm 12 improves over Arm 5 on all three seeds ([+2.40, +2.80, +1.60] pp, mean $+2.27\text{ pp}$, pooled item $95\%$ CI $[+0.13, +4.47]\text{ pp}$), while its two-level (seed x question) $95\%$ CI is $[-0.73, +5.13]\text{ pp}$, and its Val210 ($+0.48\text{ pp}$, [-5.40, +6.03] pp) and Ext240 ($+0.83\text{ pp}$, [-4.17, +5.83] pp) contrasts remain unresolved across seeds.

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 $k=3$ candidate responses $(c_i, L_i, \text{ext}_i)_{i \in \{0,1,2\}}$ (seed=42, 101, 7 in Cross-Seed Greedy; draws $0, 1, 2$ in Single-Seed Sampled):

1. String Normalization and Empty-Answer Handling: Each extracted string is normalized via $e_i = \text{normalize\_text}(\text{ext}_i)$. An empty normalized string ($e_i = \texttt{""}$, occurring when a rollout hits the 2,048-token cap without emitting a closed \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 three rules (Rand-Maj@3, MaxLen-Maj@3, MDL-Maj@3) select candidate $0$; else if $e_1 \ne \texttt{""}$ and $e_1 = e_2$, all three rules select candidate $1$.
2. Tie-Breaking When No Non-Empty Majority Exists (1-1-1 Ties):
   • Rand-Maj@3: Scores the exact uniform expectation $\frac{1}{3}(c_0 + c_1 + c_2)$.
   • MaxLen-Maj@3: Selects the longest candidate with a non-empty extracted answer (or longest overall if all $e_i = \texttt{""}$), breaking equal-length ties deterministically by the lowest index $i \in \{0,1,2\}$ via Python's max(range(3), key=lambda i: (1 if e_i else 0, L_i)).
   • MDL-Maj@3: Defines a candidate $i$ as eligible if $e_i \ne \texttt{""}$ and $15 \le L_i < 2048$ (a post-hoc eligibility window chosen after inspecting the base model's $<30$-token short-guess mode and 2,048-token cap hits). Among eligible candidates, MDL-Maj@3 selects the shortest completion $L_i$; if no candidate is eligible (fb count in Table 4), it falls back to the shortest candidate overall $\arg\min_i L_i$, breaking equal-length ties deterministically in favor of the lowest index $i \in \{0,1,2\}$ via min(range(3), key=lambda i: (0 if (e_i and 15 <= L_i < 2048) else 1, L_i)). This fallback occurs fb = 0 times in all four single-seed sampled runs, fb = 1 time on Val210 and Ext240 (Arm 11), fb = 2 times on MATH-500 in Arms 5, 7, 9, 11, fb = 4 times in Arm 12, and fb = 8 times in Arm 1.

All bootstrap $95\%$ confidence intervals in Table 4 are nominal and conditional on the retained checkpoints and draws (computed via 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 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 the paired bootstrap checks in verify_supplement.py support two bounded empirical observations:

1. MDL-Maj@3 vs. MaxLen-Maj@3 Tie-Breaking: In conditional post-hoc comparisons on the retained 3-seed greedy ensembles, MDL-Maj@3 exceeds MaxLen-Maj@3 with positive nominal paired-item $95\%$ CIs on MATH-500 in Arm 5 ($+4.60\text{ pp}$, $[+2.00, +7.40]\text{ pp}$) and Arm 11 ($+3.40\text{ pp}$, $[+1.00, +5.80]\text{ pp}$) and on Val210 in Arm 12 ($+4.76\text{ pp}$, $[+1.43, +8.10]\text{ pp}$), whereas differences on Arms 1, 7, 9, Ext240, and single-seed sampled $k=3$ runs have nominal $95\%$ CIs spanning zero. These conditional comparisons do not establish a general advantage of the heuristic or isolate a specific reasoning mechanism.
2. Cross-Seed Greedy ($3\times$ Training Cost) vs. Single-Seed Sampled ($1\times$ Training Cost) MDL-Maj@3 [20]: While both procedures evaluate $k=3$ rollouts per question (TestTok $= 285.4\text{k}$ vs. $273.9\text{k}$ mean on Ext240; $369.4\text{k}$ vs. $350.4\text{k}$ on Val210), Cross-Seed Greedy requires training and storing three independent checkpoints ($2,514.0\text{ OutTFLOPs}$, $3,081.8\text{s}$ total training time across seeds = 42, 101, 7) whereas Single-Seed Sampled requires only one checkpoint ($838.0\text{ OutTFLOPs}$, $1,027.3\text{s}$ mean training time). Comparing the retained 3-seed greedy MDL-Maj@3 ensemble against the average of the two retained single-seed sampled MDL-Maj@3 ensembles (seed=42 and seed=7) yields $+0.95\text{ pp}$ on Val210 (nominal paired question bootstrap $95\%$ CI $[-3.81, +5.48]\text{ pp}$) and $+2.92\text{ pp}$ on Ext240 ($64.17\%$ vs. $61.25\%$, nominal paired question bootstrap $95\%$ CI $[-1.25, +7.08]\text{ pp}$, conditional on the retained checkpoints and draws without training-seed or fresh-sampling resampling). Because this comparison changes checkpoint diversity ($3$ seeds vs. $1$ seed), training budget ($3\times$ vs. $1\times$), and decoding temperature ($T=0.0$ vs. $T=0.6$) jointly and its nominal $95\%$ CI spans zero, it remains an unresolved descriptive contrast.

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6. Limitations and Conclusions

1. Scope of the Diagnostic Study: All single-checkpoint 50-question and 100-question arms (Arms 1–12) cluster within $58.47\%$–$61.11\%$ on the untouched external GSM8K-200 + AMC23-40 benchmark and $43.07\%$–$47.00\%$ on MATH-500, with seed-aware two-level confidence intervals spanning zero across well-controlled update rules. Broader capability superiority claims would require multi-seed comparisons against stronger, appropriately resourced baselines, such as longer-horizon (e.g., 2,048-token [9, 10]) or larger-exposure [20] RLVR training.
2. Methodological Conclusions: Within this Qwen3-0.6B-Base regime, (i) replacing redundant zero-advantage backward passes on homogeneous groups with explicit $O(P)$ zero-gradient AdamW updates (Arms 2 and 4) is bitwise equivalent to $0.0$-weighted backward passes on any fixed trajectory sharing a common active-step loss (PASS 2) and reduces analytical output-token dense FLOPs by $40.1\%$–$43.2\%$ ($3.5\%$ observed same-host wall-clock reduction); (ii) retrospective root resampling at $\tau_k = 0$ after early EOS is not $\mathcal{F}_0$-measurable and biases sibling marginals ($\Pr(Y_1=\text{EOS})=p, \Pr(Y_2=\text{EOS})=p^2$), whereas a stopped tree $\tau_k = \min(\tau_{\text{causal}}, T_{\text{EOS}})$ with degenerate terminal continuations preserves exact sampling marginals $\Pr(Y_{k,b}=y)=\tilde{\pi}_\theta(y\mid x)$; (iii) under both samplers (Arms 5–6 and Arms 7–8), the sibling-disagreement prefix bonus $\beta = 0.25$ yields no detected benefit over $\beta = 0.0$ within two-level $95\%$ CIs, while our descriptive census (Table 2A) documents that $30.6\%$ ($187/612$) of active steps and $48.9\%$–$66.7\%$ of sibling-divergent pairs at 512 tokens have their $R=0$ rollouts at the cap, and that expanding --tokens from $512$ to $1,024$ (Arm 12 vs. Arm 5) reduces cap-associated active steps from $29.3\%$ to $5.6\%$ (+2.27 pp on MATH-500, two-level $95\%$ CI $[-0.73, +5.13]\text{ pp}$); and (iv) 3-seed 100-question combined gradient accumulation (Arm 11) reaches $46.20\% \pm 0.35\%$ single-model MATH-500 accuracy (two-level $95\%$ CI $[-0.80, +5.13]\text{ pp}$ vs. Arm 5), while 3-seed greedy MDL-Maj@3 tie-breaking improves over MaxLen-Maj@3 on MATH-500 in Arm 5 ($+4.60\text{ pp}$, nominal $95\%$ CI $[+2.00, +7.40]\text{ pp}$) and Arm 11 ($+3.40\text{ pp}$, $[+1.00, +5.80]\text{ pp}$).

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7. Reproducibility and Supplement Verification

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

• PASS 1: Verifies that per_item_grades.json contains exactly 119 primary evaluations (43 MATH-500, 44 math_val210, and 32 Ext240, totaling 38,420 primary item grades) plus 2,700 sampled-$k=3$ grades (41,120 total), independently regrades all 41,120 / 41,120 extracted candidate answers via normalize_text + deterministic standard-library numerical expression evaluation with $100\%$ agreement against the stored ch03.grade_answer SymPy labels, and checks Qwen3-0.6B-Base bimodality on MATH-500 and external accuracy on GSM8K-200 + AMC23-40 ($30/240 = 12.50\%$).
• PASS 2: Directly imports and executes generate_bifurcated from code/run_experiment.py on a 4-outcome finite policy ($p = 1/2$ immediate EOS at root) to check the implementation of early_eos_mode="terminal_dup" ((0.50, 0.50)) vs. early_eos_mode="root_resample" ((0.50, 0.25)), verifies Proposition 2's subset $S$ counterexample ($-1.5 \ne 0$), and checks fixed-trajectory AdamW zero-gradient equivalence, explicitly printing whether the standard-library scalar AdamW recurrence check or both the scalar check and the PyTorch nn.Module + torch.optim.AdamW tensor check ran.
• PASS 3, Table 1, Table 2A, Table 2B, Table 3, & Table 4: Reconstructs all 43 primary training runs' (2,352 steps, 9,408 pairs) token, prefill, padded-prefill, untrimmed batch-max generation, backward, timing, analytical dense-work proxy (Out, Full, PadFull), and branch-event counts directly from the per-pair tuples in training_records.json with zero conservation violations, separately verifies the 12 external-evaluation retraining runs (600 steps, 5,376.03s step / 5,436.11s wall-clock), and reproduces all manuscript tables (Tables 1, 2A, 2B, 3, 4) and confidence intervals.

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