## 1. Problem Formulation and Summary of Contributions

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

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

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

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

1. **$O(P)$ Zero-Gradient AdamW Stepping vs. Step Skipping and Execution-Recipe Controls (Sections 3.2 & 5.2):**
   In the baseline (`Arm 1`, `code/run_baseline.py`), homogeneous groups ($\sigma_R = 0$) execute 8 zero-weighted backward passes ($135.7\text{k}$ output / $169.4\text{k}$ full backward tokens; $815.9 / 1018.6 / 1069.5\text{ TFLOPs}$) before `optimizer.step()`. Allocating zero gradients (`param.grad = torch.zeros_like(param)`) and calling `optimizer.step()` costs $O(P)$ elementwise operations and zero transformer backward FLOPs while preserving AdamW $(m_t, v_t)$, bias-correction, and weight-decay updates omitted by step skipping (`--skip-zero-var`).
   - **Fixed-Trajectory Bitwise Equivalence vs. Live Execution-Recipe Comparison (`Arm 2` vs. `Arm 1`):** On any fixed sequence of active and homogeneous steps sharing a common active-step loss, `param.grad = torch.zeros_like(param); optimizer.step()` yields **bitwise identical parameters $\theta_t$ and AdamW buffers $(m_t, v_t)$ (`max_abs_diff = 0.0`)** to $0.0$-weighted backward passes (`PASS 2`). In live GPU runs, `Arm 1` sums cross-entropies before multiplying by advantage, whereas `Arms 2–19` multiply each token's NLL by advantage before summing (`compute_weighted_logprob`), introducing a 1-ULP active-step rounding difference (`grad_norm = 808.9206` vs. `808.9137` on Step 6 of `seed=42`) that diverges subsequent `bfloat16` trajectories [4]; live differences are reported descriptively as execution-recipe comparisons. Under ungated reward, `Arm 2` cuts output backward tokens by **$56.8\%$** ($58.6\text{k}$ vs. $135.7\text{k}$) and dense FLOP proxies by **$43.2\%$ / $43.9\%$ / $41.3\%$** ($463.3 / 571.3 / 627.4$ vs. $815.9 / 1018.6 / 1069.5\text{ TFLOPs}$), with same-host wall-clock of $438.1\text{s}$ vs. $454.1\text{s}$ (`-3.5%`, $\sim 16.0\text{s}$; $530.0\text{s}$ historical mean). Across 3 seeds, `Arm 2` scores $60.83\% \pm 0.42\%$ on `Ext240` (`+1.67 pp` vs. `Arm 1`, two-level $95\%$ CI $[-2.22, +5.69]\text{ pp}$), $44.67\% \pm 0.90\%$ on `MATH-500` (`-1.40 pp`, $[-4.60, +1.80]\text{ pp}$), and $54.44\% \pm 1.92\%$ on `math_val210`.
   - **Gated Zero-Gradient Stepping (`Arm 4`, `ra_flat`) vs. Gated Step Skipping (`Arm 3`, `gate_flat`):** With the 30-character pre-boxed gate, `ra_flat` ($45.53\% \pm 1.03\%$ `MATH-500`, $59.31\% \pm 1.68\%$ `Ext240`, $489.0 / 600.1 / 654.8\text{ TFLOPs}$, $433.0\text{s}$) vs. `gate_flat` ($43.07\% \pm 6.54\%$ `MATH-500`, $58.47\% \pm 4.49\%$ `Ext240`) yields `[-1.40, -1.40, +10.20] pp` on `MATH-500` (mean $+2.47\text{ pp}$, $[-3.47, +9.60]\text{ pp}$). Both `Arm 2` ($34.3 \pm 1.5$) and `Arm 4` ($33.7 \pm 11.0$, **$-3.80\text{ pp}$**, per-seed **`[-2.40, -5.20, -3.80] pp`**, **$[-6.60, -1.00]\text{ pp}$**) reduce test cap hits (`AllCap2048`) vs. `Arm 1` ($52.7 \pm 5.5$).

2. **Exact Distributional Analysis of Stopped-Tree vs. Outcome-Dependent Branching Samplers (Sections 4.2 & 4.3):**
   In two-stage ($K=4 \times B=2$) rollouts where Stage 1 targets the first newline boundary $\tau_{\text{causal}} \in [96, 160]$, early-EOS handling ($T_{\text{EOS}} \le \tau_{\text{causal}}$) determines whether leaf marginals match the capped on-policy kernel $\tilde{\pi}_\theta(\cdot \mid x)$ [13, 16]:
   - **True $\mathcal{F}_t$-Stopped-Tree (`--strict-causal-tau --early-eos-mode terminal_dup`, `Arms 5, 6, 11, 12, 13`):** Stopping at $\tau_k = \min(\tau_{\text{causal}}, T_{\text{EOS}})$ is $\mathcal{F}_t$-measurable. When $T_{\text{EOS}} \le \tau_{\text{causal}}$, $u^{(k)} = y_{1:T_{\text{EOS}}}$ is an absorbing terminal leaf with zero-length continuations ($v^{(k,1)} = v^{(k,2)} = \epsilon$), so **every leaf has the exact marginal $\Pr(Y_{k,b} = y) = \tilde{\pi}_\theta(y \mid x)$**.
   - **Outcome-Dependent Root-Restart (`--strict-causal-tau --early-eos-mode root_resample`, `Arms 7` & `8`):** Retrospectively setting $\tau_k = 0$ after observing EOS in $p^{(k)}_{1:\tau_{\text{causal}}}$ is not $\mathcal{F}_0$-measurable; retaining the early-EOS sequence as Branch 1 while resampling Branch 2 from the root biases sibling marginals ($\Pr(Y_1 = \text{EOS}) = p$ vs. $\Pr(Y_2 = \text{EOS}) = p^2$, verified at $p=1/2$ in `PASS 2`: `(0.50, 0.25)` vs. `(0.50, 0.50)`).
   - **Prefix-Sum Qualification on Shared-Prefix Subsets (Proposition 2):** On any realized tree, $\beta = 0$ descendant-mean credit equals leaf-summed GRPO on every shared prefix (Proposition 1), but when only $S = \{k : \tau_k > 0\} \subsetneq \{1,\dots,K\}$ share a non-empty prefix, $B\sum_{k \in S}(\hat{V}_k - \bar{R})/(\sigma_R + 10^{-4})$ need not vanish (`-1.5` in `PASS 2`); only the bonus increment $\Delta_\beta = B\beta\sum_{k \in S}(M_k - \hat{V}_k)/(\sigma_R + 10^{-4}) \ge 0$ is nonnegative for every $S$.

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

4. **100-Question Accumulation (`Arm 11`), Voting (`Table 4`), Prefix-Splice & Tie-Gated Forking (`Tables 5A & 5B`), and Six-Arm 3-Seed Online Counterfactual Fork-Pair GRPO (`Arms 14–19`, `Table 5C`):**
   - `Arm 11` (100-question `Stopped-Tree` $\beta=0.0$ + zero-mass centering [10] + `Accum4`) scores $46.20\% \pm 0.35\%$ on `MATH-500`, $55.08\%$ on `Val210`, and $59.72\%$ on `Ext240`. In $k=3$ voting (`Table 4`), `MDL-Maj@3` (`15 <= L < 2048`) exceeds `MaxLen-Maj@3` by $+4.60, +3.40, +4.00\text{ pp}$ (`Arms 5, 11, 13` `MATH-500`) and $+4.76\text{ pp}$ (`Arm 12` `Val210`).
   - Across all `98` discordant items where `Arm 12` `seed=42` succeeded and `seed=7` failed (`86` prompt-only failures, `Table 5A`, `PASS 4`), unpadded teacher-forcing on `seed=42`'s 160-token prefixes yields **`96.83%` (`15,141/15,636`) argmax agreement**; the `495` mismatches have a **$14.4\times$ smaller margin ($0.59$ vs. $8.49$)** and **`85.25%` (`422/495`) `top-2` containment**. Splicing `seed=42`'s prefix through the 1st–3rd mismatch rescues **`40.7%`, `53.5%`, and `60.5%`** of prompt-only failures, while donor-free `top-2` flipping at `seed=7`'s 1st/2nd min-margin token in $[4, 64)$ rescues **`29.1%` (`25/86`)** each (`48.8%` union vs. `12.8%` identity control). Across all `950` items (`Table 5B`), `FirstGap05-Fork@3` lifts `CR-Maj@3` to **`503/950 = 52.95%` (`+2.53 pp`, $[+1.16, +3.89]$)**, and **`TieGated-FG05@3.48`** ($3.48$ branches/item) resolves `60.6%` (`137/226`) of 3-way ties, reaching **`514/950 = 54.11%` (`+3.68 pp`, $[+2.00, +5.37]$)**.
   - Across six 3-seed 1,024-token Online Counterfactual Fork-Pair GRPO packages (`18` runs, `920` steps, `329` active groups replayed in `PASS 4`, `Table 5C`), all single-policy two-level (`seed x question`) $95\%$ CIs against `Arm 12` span zero (`Arm 15 - Arm 12`: $+1.20\text{ pp}$ $[-1.67, +4.00]$ `MATH-500`, $+0.35\text{ pp}$ $[-2.00, +2.60]$ `Pool950`): (i) **Unconditional `CF-GRPO` (`Arm 14`, `--cf-balance-div-len --cf-fork-hinge-weight 0.5`)** has `74/224` divergent pairs (`36/74` with $R_{\text{raw}}(k,0)=1, R_{\text{raw}}(k,1)=0$, activating `13` steps where all 4 on-policy branches passed; $51.72\%$ `Pool950`); (ii) **50-Step `RG-CF-GRPO` (`Arm 15`, changing `--cf-rescue-only`, `--cf-dampen-lose-len`, and `--cf-fork-hinge-weight 0.15`)** duplicates the winning branch on $(1,0)$ pairs (removing observed $(1,0)$ activations by construction; `13.33` active updates/seed) and scores $47.67\% \pm 0.83\%$ greedy ($52.20\%$ `MDL-Maj@3`, $61.60\%$ `Pass@3`) on `MATH-500` and $52.56\% \pm 0.75\%$ on `Pool950`; (iii) **Act18 Control (`Arm 16`)** continues `Arm 15` to `18.00` active updates/seed ($53/57/60$ problems; `12/14` extra updates at $b_{0,\text{pass}}=3$), scoring $51.23\% \pm 0.90\%$ on `Pool950` (`[-20, -12, -6]` items vs. `Arm 15`); (iv) **`Algebra PB-CF-GRPO` (`Arm 17`, `--cf-pair-balanced`)** skips unrescued $b_{0,\text{pass}} \ge 3$ groups and zeroes non-divergent pairs and $A_{\text{pref}}$ when $b_{0,\text{pass}} \ge 3$ (while clamping $A_{\text{pref}} \ge 0$ when $b_{0,\text{pass}} < 3$), scoring $56.03\% \pm 0.73\%$ on `Val210`, $52.32\% \pm 0.42\%$ on `Pool950`, and $53.20\%$ (`MATH-500`) / $58.11\%$ (`Pool950`) 3-seed `MDL-Maj@3`; and (v–vi) on `math_train_stratified.json[:50]`, **`Stratified RG-CF-GRPO` (`Arm 18`)** and **`Stratified PB-CF-GRPO` (`Arm 19`)** rescue `13` all-zero $b_0=0/4$ groups each (`10` and `9` post-warmup `(0,1)` groups, both retaining $A_{\text{pref}}(k^*,b) \approx +1.0604$) and score $51.54\%$ mean on `Pool950` (`Arm 18`: $47.13\% \pm 0.58\%$ greedy and $315/500 = 63.00\%$ `Pass@3` on `MATH-500`, forming a $350/500 = 70.00\%$ 6-model oracle union with `Arm 15`; `Arm 19`: $56.51\% \pm 2.91\%$ [$70.48\%$ `Pass@3`] on `Val210` and $60.28\% \pm 4.46\%$ [$76.25\%$ `Pass@3`] on `Ext240`).

---

## 2. Related Work and Positioning

Our diagnostic study builds on the `reasoning-from-scratch` GRPO baseline [1], DeepSeek-R1 [2], Dr. GRPO [3], **TreeRL** [13], **MARGE** [14], **OPTS-TTPO** [16], self-consistency voting [19], and **Wang et al. (2025)** on high-entropy minority forking tokens [21], as well as concurrent studies in `r/math_rlvr` [4–12, 17, 18, 20]:
- **Tree Search, Stopping-Time Measurability, Forking Tokens, and Voting [5–14, 16–21]:** TreeRL [13] defines descendant-mean node values $\hat{V}_k$ and global-plus-local advantages; MARGE [14] studies guided exploration; OPTS-TTPO [16] distinguishes prefix-measurable branching from outcome-adaptive expansion and compares $\hat{V}_k$ vs. $M_k$; and Wang et al. [21] show that a small minority of high-entropy forking tokens govern trajectory divergence. In `r/math_rlvr`, [5, 6, 7, 9, 10] establish token-0 onset dynamics [5], conditioned score counterexamples [6], Monte Carlo fork credit at near-ties [7], 2,048-token truncation [9], and zero-signed-token-mass stability [10], while [12, 17, 18, 20] analyze item-profile redistribution across seeds, shared-prefix credit shielding, greedy vs. vote-based selection, and run-ensemble vs. single-policy sampling ([20], Section 3.7). Our bounded contributions are: (1) the controlled optimizer-clock and active-step loss comparison (`Arms 1–4`), (2) the $2 \times 2$ (`Horizon: 512 vs. 1,024` $\times$ `Bonus: \beta = 0.0 vs. 0.25`) stopped-tree factorial (`Arms 5, 6, 12, 13`) and cap census (`Table 2A`), and (3) the 100-question accumulation replication (`Arm 11`), `MDL-Maj@3` vs. `MaxLen-Maj@3` voting (`Table 4`), donor-prefix splicing and tie-gated low-margin fork audit (`Tables 5A & 5B`), and six-arm 3-seed Online Counterfactual Fork-Pair GRPO study (`Arms 14–19`, `Table 5C`).

---

## 3. Diagnostic Analysis of Baseline Mechanics

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

Evaluating pretrained `Qwen3-0.6B-Base` greedily on `MATH-500` (`base_math500`, $14.80\% = 74/500$) and `GSM8K-200 + AMC23-40` (`base_ext240`, $12.50\% = 30/240$: $13.50\%$ on `GSM8K-200`, $7.50\%$ on `AMC23-40`) reveals a bimodal length distribution (`PASS 1`):
- **Short Mode ($|y| < 30$ tokens):** $426/500$ `MATH-500` responses ($85.2\%$) emit $<30$ tokens (`6–29` tokens, typically immediate `\boxed{...}<|endoftext|>`) at **$11.03\%$ accuracy ($47/426$)**.
- **Multi-Step Mode ($|y| \ge 30$ tokens):** $74/500$ `MATH-500` responses ($14.8\%$) generate $\ge 30$ tokens at **$36.49\%$ accuracy ($27/74$)**. After 50 RLVR updates (`baseline_s42` and `baseline_s101`), $100.0\%$ ($219/219$ and $221/221$) of correct boxed responses contain $\ge 30$ characters before `\boxed{}`.

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

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

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

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

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

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

### 3.3 Surface-Token Divergence in Mixed Groups

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

---

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

### 4.1 Pre-Boxed Derivation Gate (`rfind`)

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

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

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

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

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

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

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

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

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

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

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

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

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

1. **Proposition 1 (Fixed-Tree Shared-Prefix Summation Identity at $\beta = 0$):**
   For any pair $k$ with $\tau_k > 0$, because $u^{(k)} = y^{(k,b)}_{1:\tau_k}$ is shared across $b \in \{1, \dots, B\}$, summing the $\beta = 0$ prefix gradient across its $B$ leaves yields:
   $$
   \sum_{b=1}^B A_{\text{prefix}}^{\beta=0}(k) \nabla_\theta \log \pi_\theta(u^{(k)} \mid x) = \frac{B(\hat{V}_k - \bar{R})}{\sigma_R + 10^{-4}} \nabla_\theta \log \pi_\theta(u^{(k)} \mid x) = \left(\sum_{b=1}^B \frac{R_{k,b} - \bar{R}}{\sigma_R + 10^{-4}}\right) \nabla_\theta \log \pi_\theta(u^{(k)} \mid x),
   $$
   matching the summed prefix gradient of leaf-level GRPO on that realized tree.

2. **Proposition 2 (Conditional Expectation and Subset Qualification on Shared-Prefix Pairs $S = \{k : \tau_k > 0\}$):**
   Under i.i.d. Bernoulli($q$) continuations from $u^{(k)}$, $\mathbb{E}[\tilde{V}_k \mid u^{(k)}] = q + \beta q(1-q)$. Across a batch of $K$ pairs where $S = \{k \in \{1,\dots,K\} : \tau_k > 0\}$ is the set of pairs with a non-empty shared prefix, the sum of prefix coefficients actually applied to shared prefixes is:
   $$
   \sum_{k \in S} \sum_{b=1}^B A_{\text{prefix}}(k) = \frac{B \sum_{k \in S} (\hat{V}_k - \bar{R}) + B \beta \sum_{k \in S} (M_k - \hat{V}_k)}{\sigma_R + 10^{-4}}.
   $$
   When all $K$ pairs have $\tau_k > 0$ ($S = \{1,\dots,K\}$, as holds on every post-warmup step of `Stopped-Tree` where $\tau_k = \min(\tau_{\text{causal}}, T_{\text{EOS}}) \ge 1$), the first term $B\sum_{k=1}^K (\hat{V}_k - \bar{R}) = 0$ vanishes identically. However, when some pairs have $\tau_k = 0$ ($S \subsetneq \{1,\dots,K\}$, as in `Root-Restart` when early EOS occurs), $B\sum_{k \in S}(\hat{V}_k - \bar{R}) = -B\sum_{k \notin S}(\hat{V}_k - \bar{R})$ need not vanish: for example (`PASS 2`), with $K=4$ pairs where one shared pair $S=\{1\}$ has rewards $(0,0)$ ($\hat{V}_1 = 0$) and three root-restarted pairs $\{2,3,4\}$ have $(1,1)$ ($\bar{R} = 0.75$), the numerator at $\beta=0$ on $S$ is $2(0 - 0.75) = -1.5 \ne 0$. What holds for *every* subset $S \subseteq \{1,\dots,K\}$ is that the **bonus increment** from adding $\beta > 0$ on shared prefixes is nonnegative:
   $$
   \Delta_\beta \sum_{k \in S} \sum_{b=1}^B A_{\text{prefix}}(k) = \frac{B \beta \sum_{k \in S} (M_k - \hat{V}_k)}{\sigma_R + 10^{-4}} \ge 0.
   $$

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

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

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

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

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

---

## 5. Experimental Results, Resource Accounting, and Statistical Analysis

### 5.1 Chronology Disclosure, External Benchmark, and Qualified Resource Accounting

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

**Auditable Per-Step & Per-Pair Resource Derivation and Analytical Dense-Work Proxies (`training_records.json` & `PASS 3`):**
`training_records.json` stores each run's `config`, `effective_early_eos_mode`, and `step_ledger` (`[pl0, pl1, l0, l1, r0, r1, cat_id]`). `PASS 3` verifies unpadded/trimmed and padded/untrimmed token counts:
- **Unpadded Counts (`gen`, `pref`, `leaf`, `bwd_out`, `bwd_full`):** `gen` sums Stage 1+2 generated tokens; `pref` sums unpadded prefill; `leaf` sums $\sum_{i=1}^8 L_i(s)$, satisfying $D_{\text{S1,leaf}}(s) + D_{\text{S2,gen}}(s) = \text{leaf}(s)$ and $0 \le L_b - \text{pl}_b \le \text{max\_tokens} - \text{pl}_b$; `bwd_out` and `bwd_full` sum active-step backward tokens.
- **Padded Prefill & Untrimmed Steps (`pad_pref`, `untrim_gen`):** Left-padding Stage-2 prompts to batch width and stepping until `done.all()` gives `pad_pref` and `untrim_gen`.
- **Analytical Dense-Work Proxies:** With $P = 751,632,384$, $\text{OutTFLOPs} = (2 D_{\text{gen}} + 6 D_{\text{bwd,out}}) P / 10^{12}$, $\text{FullTFLOPs} = [2(D_{\text{pref}} + D_{\text{gen}}) + 6 D_{\text{bwd,full}}] P / 10^{12}$, and $\text{PadFullTFLOPs} = [2(D_{\text{pad\_pref}} + D_{\text{untrim\_gen}}) + 6 D_{\text{bwd,full}}] P / 10^{12}$ are **analytical dense-work proxies rather than hardware FLOP counts** ($P$ includes `tok_emb.weight`, prefill applies `out_head` on the last token, and quadratic attention and $O(P)$ AdamW steps are omitted).

![Figure 1: Three-seed MATH-500 accuracy and 2,048-token cap hits across arms (seeds 42, 101, and 7).](supplement/figure-1.png)

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

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

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

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

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

![Figure 2: External benchmark accuracy (GSM8K-200 + AMC23-40) and output vs. full-sequence training TFLOPs.](supplement/figure-2.png)

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

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

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

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

2. **Complete Three-Seed $2 \times 2$ Factorial (`Arms 5, 6, 12, 13`), Evaluation Batch-Size Sensitivity Audit, and Retained Token-512 Prefix Audit (`gt512_prefix_audit`, Rows 18–26 of Table 2B):**
   In `Arms 12` and `13`, we hold `--split-tokens 160`, `zero_cont_token_mass=False`, and `terminal_dup` identical to `Arms 5` and `6`, expanding `--tokens 512 -> 1024` under $\beta = 0.0$ (`Arm 12`) and $\beta = 0.25$ (`Arm 13`).
   - **Evaluation Batch-Size Disclosure and 950-Item Sensitivity Audit (`batch=32` vs. `batch=64`, `Table 5B`):** During training (`code/run_experiment.py`), `--batch` affects only post-training `evaluate_dataset` (`Arms 5, 6, 12` used `--batch 32`; `Arm 13` used `--batch 64`). Because left-padded `bfloat16` reductions are non-associative across batch shapes, our 950-item paired audit on `Arm 12` `seed=7` (`Table 5B`, `PASS 4`) records $905/950$ ($95.3\%$) grade agreement ($45/950 = 4.7\%$ flips: $10/210$ `Val210`, $10/240$ `Ext240`, $25/500$ `MATH-500`) and a net difference of $-0.32\text{ pp}$ ($479/950$ vs. $476/950$: $0.00, -0.83, -0.20\text{ pp}$). Because $45/950$ grades flip and this audit does not bound unretained checkpoints, we retain the batch-size mismatch between `Arm 13` and `Arms 5, 6, 12` as an explicit limitation (Section 6).
   - **Retained Evidence for Token-512 Prefix Extractions (`gt512_prefix_audit` in `training_records.json` & `PASS 3`):** Across 3 seeds ($1,200$ rollouts/arm), $266/1,200$ ($22.2\%$) of `Arm 12` and $292/1,200$ ($24.3\%$) of `Arm 13` rollouts exceed 512 tokens ($512 < L \le 1024$). Decoding `t512 = tok.decode(ids[:512])`, we stored `[step, branch_idx, L, r_fin, rfind_512, ext_512, ans_gold, r_512, first_box_tok, retained_txt]` in `training_records.json` (retaining full `t512` for all $35$ rollouts with `rfind_512 >= 0` and the 60-char tail plus `first_box_tok` for the $523$ rollouts with `rfind_512 == -1`). `PASS 3` runs `rfind("\\boxed")` and `extract_final_candidate` on all $35$ boxed prefixes, checks the $523$ unboxed entries (`first_box_tok == -1` or `>= 510`, `ext_512 == ""`), and regrades all $558$ `ext_512` strings: in `Arm 12`, $90/266$ ($33.8\%$) finish with $R=1$ and **$87/90$ ($96.7\%$) fail at token 512 while the final answer passes** ($84/90$ `rfind_512 == -1`, $3/90$ unclosed/wrong `\boxed`, $3/90$ pass at 512); in `Arm 13`, $118/292$ ($40.4\%$) finish with $R=1$ and **$107/118$ ($90.7\%$) fail at token 512 while passing at completion** ($103/118$ `rfind_512 == -1`, $4/118$ wrong `\boxed`, $11/118$ pass at 512; median first `\boxed` at token $560$).
   - **Single-Factor 1,024-Token Horizon (`Arm 12` vs. `Arm 5`) & $2 \times 2$ Interaction (`Rows 18–26` of Table 2B):** On `MATH-500`, `Arm 12 - Arm 5` is `[+2.40, +2.80, +1.60] pp` (mean $+2.27\text{ pp}$, two-level $95\%$ CI $[-0.73, +5.13]\text{ pp}$; Q-clustered $[+0.07, +4.53]\text{ pp}$; Flat3N $[+0.13, +4.47]\text{ pp}$), with $+0.48\text{ pp}$ ($[-5.40, +6.03]$) on `Val210` and $+0.83\text{ pp}$ ($[-4.17, +5.83]$) on `Ext240`. At 1,024 tokens, `Arm 13 - Arm 12` ($\beta = 0.25$ vs. $0.0$) is **$-1.07\text{ pp}$** (`[-1.40, -2.40, +0.60]`, **$[-4.20, +2.07]\text{ pp}$**) on `MATH-500`, **$-0.16\text{ pp}$** (**$[-4.76, +4.76]\text{ pp}$**) on `Val210`, and **$-2.08\text{ pp}$** (**$[-6.67, +2.78]\text{ pp}$**) on `Ext240`, and the $2 \times 2$ interaction $[(\text{Arm 13} - \text{Arm 12}) - (\text{Arm 6} - \text{Arm 5})]$ spans zero across all three benchmarks (**$-1.80, +3.17, -1.11\text{ pp}$**).

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

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

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

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

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

**Complete Specification of the $k=3$ Voting and Tie-Breaking Rules (`_eval_k3_triplets` in `verify_supplement.py`):**
For each test question with $k=3$ candidates $(c_i, L_i, \text{ext}_i)_{i \in \{0,1,2\}}$ (`seed=42, 101, 7` in Cross-Seed Greedy; draws $0, 1, 2$ in Single-Seed Sampled):
1. **String Normalization & Empty-Answer Handling:** Each extracted string is normalized via $e_i = \text{normalize\_text}(\text{ext}_i)$. An empty string ($e_i = \texttt{""}$, when a rollout hits the 2,048-token cap without `\boxed{...}`) **never matches another candidate to form a majority**: if $e_0 \ne \texttt{""}$ and $(e_0 = e_1 \lor e_0 = e_2)$, all rules select candidate $0$; else if $e_1 \ne \texttt{""}$ and $e_1 = e_2$, all rules select candidate $1$.
2. **Tie-Breaking on `1-1-1` Ties:**
   - **`Rand-Maj@3`:** Exact uniform expectation $\frac{1}{3}(c_0 + c_1 + c_2)$.
   - **`MaxLen-Maj@3`:** Selects the longest candidate with $e_i \ne \texttt{""}$ (or longest overall if all $e_i = \texttt{""}$), breaking equal-length ties by lowest index $i \in \{0,1,2\}$ via `max(range(3), key=lambda i: (1 if e_i else 0, L_i))`.
   - **`MDL-Maj@3`:** Defines candidate $i$ as **eligible** if $e_i \ne \texttt{""}$ and $15 \le L_i < 2048$ (a post-hoc window avoiding $<30$-token short guesses and 2,048-token cap hits). Among eligible candidates, `MDL-Maj@3` selects the shortest $L_i$; **if none is eligible (`fb` in Table 4), it falls back to $\arg\min_i L_i$**, breaking ties by lowest $i \in \{0,1,2\}$ via `min(range(3), key=lambda i: (0 if (e_i and 15 <= L_i < 2048) else 1, L_i))` (`fb = 0` in all 4 sampled runs, `fb = 1` on `Val210`/`Ext240` in `Arm 11`, `fb = 2` on `MATH-500` in `Arms 5, 7, 9, 11`, `fb = 4` in `Arm 12`, `fb = 5` in `Arm 13`, `fb = 8` in `Arm 1`).

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

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

Table 4 and `verify_supplement.py` support two bounded observations:
1. **`MDL-Maj@3` vs. `MaxLen-Maj@3`:** On retained 3-seed greedy ensembles, `MDL-Maj@3` exceeds `MaxLen-Maj@3` with positive nominal paired $95\%$ CIs on `MATH-500` in `Arm 5` (**$+4.60\text{ pp}$, $[+2.00, +7.40]$**), `Arm 11` (**$+3.40\text{ pp}$, $[+1.00, +5.80]$**), and `Arm 13` (**$+4.00\text{ pp}$, $[+1.60, +6.60]$**) and on `Val210` in `Arm 12` (**$+4.76\text{ pp}$, $[+1.43, +8.10]$**), while differences on `Arms 1, 7, 9`, `Ext240`, and sampled $k=3$ runs span zero.
2. **Cross-Seed Greedy ($3\times$ Train Cost) vs. Single-Seed Sampled ($1\times$ Train Cost) `MDL-Maj@3` [20]:** Both evaluate $k=3$ rollouts/item (`TestTok` $= 285.4\text{k}$ vs. $273.9\text{k}$ on `Ext240`; $369.4\text{k}$ vs. $350.4\text{k}$ on `Val210`), but Cross-Seed Greedy trains three checkpoints ($2,514.0\text{ OutTFLOPs}$, $3,081.8\text{s}$) vs. one ($838.0\text{ OutTFLOPs}$, $1,027.3\text{s}$). Comparing 3-seed greedy `MDL-Maj@3` against the mean of the two sampled `MDL-Maj@3` runs (`seed=42, 7`) gives $+0.95\text{ pp}$ on `Val210` (nominal $95\%$ CI **$[-3.81, +5.48]$**) and $+2.92\text{ pp}$ on `Ext240` ($64.17\%$ vs. $61.25\%$, **$[-1.25, +7.08]$**), remaining an unresolved descriptive contrast.

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

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

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

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

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

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

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

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

    Replacing $15 \le L_j < 2048$ with $15 \le L_j < \text{eff\_cap}_{i,j}$ (`PASS 4`) yields identical `MDL-Maj@3` counts (`117/210`, `141/240`, `226/500`). Under `MinMargin-Fork@3`, symmetric `MDL-Maj@3` is unresolved (`+2.86 pp`, `+1.25 pp`, `-0.80 pp`) [18] because on the `114` `MATH-500` 3-way ties where all 3 branches complete, picking the shortest branch abandons the greedy root (`9/114` vs. `17/114`), whereas keeping `Branch 0` on ties whenever completed (`CompletedRoot-Maj@3` / `CR-Maj@3`) scores $55.71\%$, $59.58\%$, and $46.80\%$ ($+1.60\text{ pp}$ $[+0.20, +3.20]$ over `MDL-Maj@3` on `MATH-500`). Triggering at the 1st and 2nd $\Delta z_t \le 0.5$ positions with per-slot minimum-margin fallback (`FirstGap05-Fork@3`: $1,571/1,900 = 82.7\%$ strict $\Delta z_t \le 0.5$ [$865$ `B1`, $706$ `B2`] and $329/1,900 = 17.3\%$ fallback with zero below-threshold fallbacks and exact $k/128$ dyadic margins) rescues **$65/471$ ($13.8\%$) single-branch and $103/471$ ($21.9\%$) 2-branch wrong anchors** (vs. $50/471$ and $83/471$ for `MinMargin`), lifting `FirstGap05` `MDL-Maj@3` to **$502/950$ ($52.84\%$, $+2.42\text{ pp}$ $[+0.63, +4.21]$)** and `CR-Maj@3` to **$503/950$ ($52.95\%$, $+2.53\text{ pp}$ $[+1.16, +3.89]$)** across five strata (`Unanimous` $435$, `Root+1` $219$, `Sibling Override` $70$, `Completed Tie` $187$, `Truncated Tie` $39$). For $M \in \{3, 4, 5\}$ branches (`_plurality_cr_mdl` in `verify_supplement.py`), when the maximum non-empty frequency $f_{\max} \ge 2$ is unique, both `CR-Maj` and `MDL-Maj` select that plurality answer; when two answers tie at $f_{\max} = 2$ ($2$-$2$-$1$), they pick the tied answer with the shortest completed ($15 \le L_j < \text{eff\_cap}_{i,j}$) branch; and when $f_{\max} = 1$, `MDL-Maj` picks the shortest completed branch overall while `CR-Maj` retains `Branch 0` if completed ($15 \le L_0 < 2048$) and otherwise falls back to `MDL-Maj`. Stopping at $K=3$ on the $724$ majority items and querying two additional low-margin forks (`FG3, FG4` in $[4, 64)$) strictly on the $226$ 3-way ties (**`TieGated-FG05@3.48`**, $3,302/950 = 3.48$ branches/item) resolves **$137/226$ ($60.6\%$)** of ties ($f_{\max} \ge 2$), reaching **`514/950 = 54.11%` `CR-Maj` (`+3.68 pp`, $[+2.00, +5.37]$; `119/210`, `148/240`, `247/500`)** and **`513/950 = 54.00%` `MDL-Maj` (`+3.58 pp`, $[+1.89, +5.37]$)**, compared to $502/950$ `CR-Maj` and $496/950$ `MDL-Maj` ($497/950$ restricting plurality to completed branches; $587/950$ `Pass`) for unconditioned `Hybrid-Fork@3.52` ($3,341$ branches; $+1.26\text{ pp}$ $[+0.21, +2.32]$ `CR-Maj` difference).

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

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

Table 5C and the exact coefficient replay in `verify_supplement.py` (`PASS 4`) document four descriptive findings (where single-policy two-level $95\%$ CIs span zero):
- **Unconditional `CF-GRPO` (`Arm 14`) vs. `RG-CF-GRPO` Package (`Arm 15`):** With Bessel-corrected sample SD $s_R = \sqrt{\frac{1}{7}\sum_{i=1}^8 (R_i - \bar{R})^2}$ (`s_R < 1e-6` guard, denominator $s_R + 10^{-4}$), all standardized coefficients are dimensionless. In `Arm 14` (`--cf-balance-div-len --cf-fork-hinge-weight 0.5`), $36/74$ ($48.6\%$) of divergent pairs have $(R_{\text{raw}}(k,0), R_{\text{raw}}(k,1))=(1,0)$, activating $13$ steps where all 4 on-policy branches passed ($A_{\text{pref}} = -1.0604$). `Arm 15` changes three package settings (`--cf-rescue-only`, replacing `--cf-balance-div-len` with `--cf-dampen-lose-len`, and `--cf-fork-hinge-weight 0.15`), duplicating the winning branch when $R_{\text{raw}}(k,0)=1 \wedge R_{\text{raw}}(k,1)=0$ (which removes observed $(1,0)$ activations by construction without isolating their causal effect on endpoint accuracy) and scoring $47.67\% \pm 0.83\%$ on `MATH-500` ($+1.20\text{ pp}$ $[-1.67, +4.00]$ vs. `Arm 12`) and $52.56\% \pm 0.75\%$ on `Pool950`.
- **Active-Count Control (`Arm 15` vs. `Arm 16`):** Continuing `Arm 15` to match `Arm 12`'s $18.00\text{ updates/seed}$ ($53/57/60$ problems; `Pool950` diffs `[-20, -12, -6]`) changes `Pool950` by $-1.33\text{ pp}$ ($51.23\% \pm 0.90\%$, $[-2.74, +0.07]\text{ pp}$). Descriptively, $12/14$ ($85.7\%$) of the additional active updates occur at $b_{0,\text{pass}}=3$ (`(3,7)`: $7 \to 16$, `(3,6)`: $4 \to 7$), where duplicating 3 solved roots sets $\bar{R} = 0.875, s_R = 0.3536$ ($A_{\text{pref}}(k^*) = -1.0604$, mean sequence-summed signed token mass [10] $M_{\text{tot}} = \frac{1}{8}\sum_{i=0}^7 [A_{\text{pref}}(i)\text{pl}_i + A_{\text{cont}}(i)(L_i-\text{pl}_i)] = +29.65$ and $+16.24$ tokens in `Arms 15, 16`); on the out-of-sample 27-checkpoint $R_2$ ($7..13/27$, $n=59$) stratum, `Arm 15` vs. `Arm 16` scores $28.00$ vs. $21.33/59$.
- **Stratified 7-Subject `RG-CF-GRPO` (`Arm 18`):** At 1,024 tokens on `math_train_stratified.json[:50]`, `Arm 18` rescues **`13` `b0=0/4` steps** ($10$ post-warmup `(0,1)` groups), scores **$47.13\% \pm 0.58\%$ greedy** ($+2.93\text{ pp}$ over 512-token multi-subject) and **`315/500 = 63.00%` `Pass@3`** on `MATH-500` ($16.33/108$ greedy, $43/108$ `Pass@3` on $R_1$), and unlocks $42$ `MATH-500` items missed by `Arm 15` (**6-model oracle union `Arm 15` $\cup$ `Arm 18` = `350/500 = 70.00%`**; 3-model `Arm 15(101)+17(42)+18(7)` `Pass@3 = 643/950 = 67.68%`).
- **`2x2` Curriculum $\times$ Objective Package Factorial & Post-Adjustment Coefficient Replay (`Arms 15, 17, 18, 19`):** Paired per-seed (`42/101/7`) `PB - RG` contrasts are `[+4,-1,+5]` ($+1.27\text{ pp}$ $[-2.86,+5.56]$) and `[+4,+6,-2]` ($+1.27\text{ pp}$ $[-3.33,+6.03]$) on `Val210` ($56.51\%$, `Pass@3 = 70.48%` in `Arm 19`), `[+2,-4,+3]` ($+0.14\text{ pp}$) and `[+14,+5,+1]` ($+2.78\text{ pp}$ $[-1.67,+7.78]$, `Pass@3 = 76.25%` in `Arm 19`) on `Ext240`, and `[-11,-1,-4]` ($-1.07\text{ pp}$) and `[-7,+3,-24]` ($-1.87\text{ pp}$) on `MATH-500` ($2\times 2$ interactions: `Val210` `[0,+7,-7]` $= 0.00\text{ pp}$, `Ext240` `[+12,+9,-2]` $= +2.64\text{ pp}$ $[-3.33,+8.75]$, `MATH-500` `[+4,+4,-20]` $= -0.80\text{ pp}$ $[-5.60,+3.67]$, `Pool950` `[+16,+20,-29]` $= +0.25\text{ pp}$; all $329/329$ active steps have pre-clipping $\|g\|_2 \ge 179.7 > 1.0$). Replaying `code/run_experiment.py` (`PASS 4`) shows that on `(3,7)` groups, `PB` zeroes the 3 `(1,1)` pairs and $A_{\text{pref}}(k^*)=0$ (reducing mean $M_{\text{tot}}$ from $+29.65/+19.40$ in `Arms 15, 18` to $+5.58/+0.94$ in `Arms 17, 19`), whereas when $b_{0,\text{pass}} < 3$, `PB` applies `torch.clamp(adv_pref, min=0.0)` and sets $A_{\text{cont}}(\text{lose}) = -A_{\text{win}}\min(1, \frac{L_{\text{win}}-\tau}{L_{\text{lose}}-\tau})$. Thus on all post-warmup `(0,1)` groups (`10` in `Arm 18`, `9` in `Arm 19`, e.g., `Arm 19 seed=42 step 33`), **both `RG` and `PB` retain $A_{\text{pref}}(k^*,b) = +1.0604 > 0$**. For continuations on `(0,1)` groups, length damping ($L_{\text{win}}-\tau < L_{\text{lose}}-\tau$) occurs on $3/10$ groups in `Arm 18` ($\sum_{i=0}^7 A_{\text{cont}}(i) \in [0.000, +0.272]$, mean $+0.058$; mean $M_{\text{tot}} = +15.60$) and $5/9$ groups in `Arm 19` ($\sum_{i=0}^7 A_{\text{cont}}(i) \in [-1.0604, +0.4059]$, mean $-0.717$; mean $M_{\text{tot}} = -55.65$; e.g., on `Arm 19 seed=101 step 41`, $\tau=19, L_{\text{win}}=266, L_{\text{lose}}=1024$ gives $A_{\text{cont}}(k^*)=(+1.9440, -0.4778)$ and $\sum A_{\text{cont}} = +0.4059$). Because these online procedures jointly alter step gating, continuation weights, and subsequent trajectories (`hm_pb_cf_grpo_strat_seed42` scored $253/450$), subgroup shifts (`R1`: $16.33$ vs. $10.67/108$; `Val210 R2`: $11.33$ vs. $14.67/30$) and `1/3_tie` length patterns (`Arm 17`: $695.0$ vs. $625.1$ tokens, `MDL` $50/93 = 53.8\%$; `Arm 19`: $461.1$ vs. $561.1$ tokens due to $64/114$ `seed=7` tie picks, `MDL` $31/114 = 27.2\%$ vs. `MaxLen` $46/114 = 40.4\%$) are reported as descriptive package readouts rather than isolated causal effects.

---

## 6. Limitations and Conclusions

1. **Scope and Evaluation Limitations:** Across 50- and 100-question arms (`Arms 1–19`), two-level (`seed x question`) $95\%$ CIs between well-controlled update rules span zero (e.g., `Arm 15 - Arm 12` is $+1.20\text{ pp}$ $[-1.67, +4.00]$ on `MATH-500` and $+0.35\text{ pp}$ $[-2.00, +2.60]$ on `Pool950`). `Arm 13` was evaluated at `--batch 64` vs. `--batch 32` in `Arms 5, 6, 12, 14–19` ($45/950 = 4.7\%$ flips under padding, `Table 5B`), retained as an explicit limitation.
2. **Methodological Conclusions:** Within this `Qwen3-0.6B-Base` regime, (i) $O(P)$ zero-gradient AdamW updates (`Arms 2, 4`) are bitwise equivalent to $0.0$-weighted backward passes on fixed trajectories (`PASS 2`) and cut dense FLOP proxies by $40.1\%$–$43.2\%$; (ii) a stopped tree $\tau_k = \min(\tau_{\text{causal}}, T_{\text{EOS}})$ preserves exact marginals $\Pr(Y_{k,b}=y)=\tilde{\pi}_\theta(y\mid x)$ whereas root resampling biases sibling marginals (`PASS 2`); (iii) at 512 and 1,024 tokens, prefix bonus $\beta = 0.25$ shows no detected benefit over $\beta = 0.0$, while raising `--tokens` to $1,024$ cuts cap-associated active steps from $29.3\%/37.1\%$ to $5.6\%/1.8\%$ (`PASS 3`); (iv) `TieGated-FG05@3.48` lifts test-time `CR-Maj` on the explored checkpoint to $514/950$ ($54.11\%$, $+3.68\text{ pp}$ $[+2.00, +5.37]$); and (v) across six 3-seed 1,024-token Online Counterfactual Fork-Pair GRPO packages (`Arms 14–19`, `PASS 4`), `Arm 15` (`RG-CF-GRPO`) scores $47.67\% \pm 0.83\%$ greedy ($52.20\%$ `MDL`, $61.60\%$ `Pass@3`) on `MATH-500` and $52.56\% \pm 0.75\%$ on `Pool950`, while `Arms 16–19` document descriptive $2\times 2$ (`Curriculum` $\times$ `Objective`) package behavior across post-adjustment coefficients, `R1/R2/R3` strata, and `1/3_tie` voting (`Arm 18` reaching $63.00\%$ `Pass@3` and a $70.00\%$ 6-model oracle union with `Arm 15`).

---

## 7. Reproducibility and Supplement Verification

The supplement (`supplement.zip`) is verified in $\sim 2$ seconds on CPU via `python3 verify_supplement.py`:
- **`PASS 1`:** Verifies 128 evaluations (41,270 grades) + 2,700 sampled-$k=3$ grades in `per_item_grades.json` (43,970 total, 100% SymPy agreement).
- **`PASS 2`:** Executes `generate_bifurcated` ($p=1/2$: `(0.50, 0.50)` vs. `(0.50, 0.25)`), Proposition 2 ($-1.5 \ne 0$), and AdamW zero-gradient equivalence.
- **`PASS 3` (`Tables 1–4`):** Reconstructs 46 training runs (2,502 steps, 10,008 pairs), 558 $>512$-token prefixes, 12 external retraining runs, and Tables 1–4.
- **`PASS 4` (`Tables 5A–5C`):** Regrades all 22,172 extracted answers in `prefix_splice_probe.json` ($1,372$ discordant + $3,700$ branching + $17,100$ `Arms 14–19` responses; **66,142 / 66,142 total across `PASS 1 + PASS 4`**), verifies 15,141/15,636 (96.83%) TF agreement, $k/128$ dyadic margins, `TieGated-FG05@3.48` ($514/950$), and replays all 920 steps (3,680 pairs, 329 active groups with exact `arm_configs`, post-adjustment coefficients, and signed token mass) across `Arms 14–19`.

---

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[21] Shenzhi Wang, Le Yu, Chang Gao, Chujie Zheng, Shixuan Liu, Rui Lu, Kai Dang, Xionghui Chen, Jianxin Yang, Zhenru Zhang, Yuqiong Liu, An Yang, Andrew Zhao, Yang Yue, Shiji Song, Bowen Yu, Gao Huang, and Junyang Lin. Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning. arXiv preprint arXiv:2506.01939, 2025. https://arxiv.org/abs/2506.01939