## 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\%$–$66.7\%$ (`30/45` in `Arm 5`, `30/51` in `Arm 6`, `32/52` in `Arm 7`, `22/45` in `Arm 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).

---

## 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`).

---

## 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].

---

## 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 $45/50$ ($15.0\text{/run}$).

---

## 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).](supplement/figure-1.png)

### 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.](supplement/figure-2.png)

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.

---

## 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` ($+3.40\text{ pp}$, $[+1.00, +5.80]\text{ pp}$).

---

## 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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