From finite optimizer control to native learning: support weighting and a failed ordinary confirmation in Qwen3-0.6B

Research question and prior work

Can feedback on different training questions teach a small controller which of a model's verified support updates will improve its native policy? This asks about the consequences of an update rather than the correctness of an individual continuation. It nevertheless belongs to established learning-to-reweight and meta-gradient methods. Ren et al. [1] use example weighting to improve a separate loss; Meta-Weight-Net [2] explicitly learns a small weighting MLP from a separate labeled subset. Xu et al. [3] adapt RL update parameters using subsequent experience. DataRater [4] is especially close: it differentiates normalized batch weights through optimizer updates against a separate training subset. MASS [5] learns from the effect of own synthetic examples on adaptation. Our small-model, verifier-only, persistent native-policy experiment is an adaptation of this family, not an algorithmic novelty claim.

Learning-progress curricula provide an older related perspective [6]. Graves et al. distinguish improvement on the practiced batch, a new batch from the same task, and a different target task. Their task-selection bandit differs from our weighting head, but already treats practice through its consequences for a persistent learner. They also note that immediate target progress can miss prerequisite learning and can be noisy; uniform practice remains a strong comparator. Our conditional derivative therefore tests one specified local feedback rule, rather than a general solution to curriculum learning.

The empirical motivation is a repeated gap between learning on practiced questions and learning on different questions. In our preceding fixed comparison [7], the mixed-prefix method changed practiced accuracy from 33 to 57 of 128 while native accuracy on different questions changed from 246 to 247 of 512; the ordinary continuation reached 67 and 228, respectively. Both the failed primary rule and the larger ordinary work remain part of that result. These observations motivate testing a training rule that receives different-question feedback. They do not identify uniform support weighting as the cause of the gap.

Concurrent work in the same small-model setting [8] finds that greedy-wrong/plurality-right practice underperforms nearest-band greedy-right practice by 1.35 percentage points on a reused validation panel, and that two 144-update continuations add no resolved sampled gain. Mean difficulty-band matching does not match the full selection distribution or generated-token work, and remaining below specified drift monitors does not establish zero drift or a capability ceiling. These scoped results reinforce the distinction between useful practice acquisition and native transfer; their BF16 model with FP32 master parameters, selection procedure and sampled readout differ from our full-FP32 package comparison. Their inherited start also reflects MATH-500-informed checkpoint selection, disclosed in the v2 reproduction note; no new benchmark readout is reported for these arms.

The candidate learns relative weights among four already sampled support questions. It cannot create missing successful traces, choose a new training question, or select a test answer. Queries provide additional allowed training information, so the ordinary controls receive the same question stream and learn from query attempts too. The primary empirical test concerns improvement of this complete procedure under a common native readout. The learned package failed that test. Attributing any improvement specifically to learned assignments needs further controls because the present uniform arm also differs in pooling arithmetic.

The finite-control study asks a distinct question: whether the specified scalar weights can appreciably change the optimizer's action at a fixed state. It is a calibrated mechanism study with deliberately regrouped positive controls. It does not establish useful query credit, representability by the implemented shared head, or mathematical learning. Native outcomes must be reported separately.

The ordinary reference also uses established machinery. Ahmadian et al. [9, Section 2.3] study the same independent-response RLOO estimator in preference training. Their positive experiments use 6.9B/7B policies, learned rewards and simulated larger-model evaluations; none of that supervision or evaluation is imported here. Their ratio-clipping ablation concerns PPO ratios, not our global gradient-norm clipping. Their full-sequence RLHF result does not settle the value of mathematical process feedback in this smaller model.

A previous project-level score-temperature comparison, reported in the October 8 discussion, did not meet its fixed final advancement rule despite a favorable interim matched-score contrast. It used BF16 AdamW at $10^{-5}$, eight questions per update, 512 training tokens and 400 question exposures. The present uniform package changes precision, rate, minibatch size, horizon, ancestry and stream. Its result therefore cannot be attributed to score matching or used to erase that earlier failed comparison. The independent confirmation below tests the current complete package; its original advancement rule also fails.

Parent, data and fixed schedule

Every arm begins from one newly trained, unselected Qwen3-0.6B-Base descendant. The parent uses the supplied 50-update recipe with seed 67041; it generated 130,960 completion tokens and consumed 471.5 allocated GPU seconds. Its checkpoint SHA-256 is 3e37c54d511fe17c4621ebc6ec75efc1af3e0b72c730710eb5ed10788a01d36e. No native evaluation selected this parent. Subsequent training updates all 751,632,384 actor parameters in FP32.

The prospective stream is frozen by a seeded shuffle of unique normalized texts in the supplied MATH training pool. It excludes protected identities, the parent's first 50 questions, the previous ordinary 640-question stream and previous mixed-study 512-question stream. Forty questions are reserved for execution preflight; the next 3,072 distinct identities define the main stream and ordinary extensions. Main prompts must fit 960 tokens. One overlong main prompt was replaced in the same seeded order before freezing, without inspecting an answer or outcome. The reused 512-question development panel is disjoint from all these streams; it is a screening panel with prior use, not fresh confirmation. Reserved confirmation questions, MATH-500 and another benchmark are outside this experiment. A historical, separate compiler-feasibility study inspected aggregate grammar coverage on MATH-500 question text (without answers or model outputs) and used that observation in a method decision. Thus MATH-500 cannot be described as wholly untouched by the research project; any future benchmark report must disclose that limited exposure. No MATH-500 model readout enters this receiver screen.

The main seed is 42641202, head initialization seed 45641202, stream seed 43641202 and paired bootstrap seed 46641202. Each episode uses four support and four different query questions, with eight own-policy attempts per question and a 1,024-token completion cap. No episode reuses a question within this stream. There are 128 main episodes: 1,024 question presentations, 8,192 sampled attempts and 256 scheduled actor updates. All supplied training answers are used only by the programmatic verifier. No larger model supplies targets, reasoning, judgments or selection.

The learned receiver and uniform reference

M and U sample from the full softmax law at temperature 0.8. Their loss scores the same temperature-0.8 token probabilities and includes both EOS and cap-ending completions. For a question with eight rewards, the leave-one-out advantage of attempt j is

$$ A_j=r_j-\frac{1}{7}\sum_{k\ne j}r_k. $$

Let g_i be the gradient of the negative mean advantage-weighted completion log probability for support question i. M forms the support gradient $g=\sum_{i=1}^4 a_i g_i$. U uses uniform weights. An exactly zero advantage avoids its redundant backward pass; it does not alter M/U's scheduled optimizer clock.

The 65,921-parameter head receives a detached, L2-normalized 1,024-dimensional final prompt representation and four support statistics: mean verifier reward, population standard deviation of RLOO advantages, mean completion length divided by 1,024, and EOS fraction. A 64-unit tanh hidden layer produces $z_i=\log(2)\tanh(s_i)$, then $a=\operatorname{softmax}(z)$. The final linear layer starts at zero, giving uniform initial weights. The head uses CPU float64 Adam, rate 0.003, epsilon $10^{-8}$, betas (0.9, 0.999), no weight decay and gradient-norm clip 1. Shared features and tanh restrict its representable weights; a closed logit-box analysis is only an outer envelope.

All actor updates use AdamW with rate $10^{-6}$, epsilon $10^{-8}$, betas (0.9, 0.999), weight decay 0.01 and global gradient-norm clip 1. Moments persist across support and query phases. M pools detached support gradients in float64 blocks and casts the pooled gradient to FP32. U accumulates its uniform contribution in FP32. This arithmetic-order difference is part of the implemented package contrast and prevents a uniquely controller-assignment interpretation without a later control.

After committing the support update, each actor samples its four query groups from the updated policy and computes their mean RLOO loss gradient q. M contracts q through the conditional support-update derivative to update the head for future episodes. Both M and U then apply the ordinary query actor update. Query content and outcomes are training information, and the head's influence is delayed until a later support phase. This differs from Meta-Weight-Net [2, Algorithm 1 and Equations 3–5], which uses a provisional update to train its weight network and then recomputes the current classifier update with the new weights. Here the committed support step remains in place, and queries also train the actor directly.

Execution order for one receiver episode: support attempts generate weighted gradients, a committed support step produces query attempts, and query credit updates both the actor and a head used only in later episodes.

One episode of M. Solid blue arrows show actor training, green arrows show detached support features and weights, and dashed brown arrows show conditional credit to the head. Queries are training data. The pullback fixes the old state and support gradients, excludes the later query actor step, and uses the declared stored-zero-second-moment surrogate. This diagram records execution order and does not report a capability result.

What the derivative estimates

Condition on the current actor, old optimizer moments, support trajectories, support gradients and detached features. Write the support optimizer map as $\theta'=U(\theta,g,m,v)$. For a differentiable real-arithmetic reference optimizer with consistent moments, the local reward-loss adjoint estimator is

$$ \widehat{\partial_{z_i}L_Q} =a_i(g_i-g)^\top (\partial_g U)^\top q. $$

Under ideal independent sampling from the matched full-support law, RLOO is an unbiased estimator of the gradient of negative expected verifier reward on the four query questions. A leave-one-out baseline is independent of the corresponding attempt, and the expected score of that attempt is zero. Applying the fixed support-map Jacobian therefore estimates the conditional one-support-step reward derivative. This reasoning does not make the sampled scalar score loss itself an unbiased reward estimate, and it does not turn the derivative into a guarantee about greedy accuracy.

More precisely, for fixed support draws $S$, head parameters $\phi$, four query questions $x_q$ and $\theta'(\phi,S)=U(\theta,\sum_i a_i(\phi,S)g_i,m,v)$, the local objective is

$$ F_S(\phi)=-\frac14\sum_{q=1}^4\mathbb E_{y\sim P_{\theta'(\phi,S),0.8}(\cdot\mid x_q)}R(y,x_q). $$

Here $P$ is the capped stopped-trajectory law: all sampled tokens through first EOS, or all 1,024 tokens when no EOS occurs. Cap-ending attempts remain in this probability space and in the score; forced padding after an already observed EOS does not. Under the ideal assumptions, $\mathbb E_Q[(\partial_\phi\theta')^\top\widehat q\mid S]=\nabla_\phi F_S$. This is the gradient of expected verifier reward, not the derivative of the expectation of the numerical score-loss value.

No transformer Hessian is required after conditioning on the support gradients. This is a deliberately truncated object. It excludes how the head changed the current actor, feature representation, support distribution or moments through earlier episodes, and it excludes the subsequent query actor step. Under this conditioning the current support sampling law is independent of the head: the head changes weights after support draws are fixed. Averaging the ideal conditional estimator over those draws therefore targets a one-support-step expected objective, subject to the score and differentiability assumptions above.

For a longer learning trajectory, the objective must distinguish executed sampled updates from ideal expected-gradient updates. Vuorio et al. [10, Sections 2–3] separate derivatives of the update map conditioned on sampled data from score terms for the earlier data distributions. An expected-return Hessian is not automatically the derivative of a finite sampled update. DiCE [11] addresses derivatives of expected stochastic objectives; invoking it alone does not supply the omitted learning-history terms or select the intended objective. Neither literature result removes the numerical qualifications of our implementation.

The one-step objective also differs from terminal learning performance, even with an ideal estimator. Wu et al. [12] show that accurately optimizing short-horizon objectives can select poor long-run learning schedules in stochastic, ill-conditioned problems. Their learning-rate result is prior evidence for the distinction, not a diagnosis of our bounded support-weight controller or a theorem about its native endpoint.

The executed FP32 operator needs a further qualification. Where the stored second moment is positive, we evaluate the real-arithmetic AdamW differential with stored quantities and the recorded global-clipping norm. The reconstructed float64 pooled gradient, FP32 installed gradient, stored moments and recorded norm need not be exactly consistent with one real-arithmetic recurrence; this branch is also a numerical evaluation of the reference differential, not an exact derivative of hardware execution. Where it has rounded to exactly zero, we hold that second-moment branch fixed and differentiate the first-moment branch. This explicitly declared surrogate is neither the derivative of exact FP32 rounding nor an unbiased estimate of the full executed training algorithm's gradient. Original strict failures and the revised preflight are retained and charged. The contraction checks that actor parameters, saved query gradients and post-support moments remain unchanged before the query update.

Competitive ordinary control and work matching

H uses a separate competitive ordinary recipe: temperature-0.8, top-p-0.9 generation, raw temperature-1 summed completion scores, and sample-standardized binary rewards followed by whole-trajectory filtering. All eight rewards enter the mean and sample standard deviation; a cap-ending response then receives zero loss coefficient. Retained responses include their terminal EOS token in the scored sequence. H uses the same actor optimizer settings and question order, but skips an actor phase when every executed loss advantage is zero. M/U execute every scheduled phase with explicit zeros where needed. H is therefore a recipe comparison, not a one-variable objective ablation.

M stops at episode 128. U and H each expose episode 128 and then continue in blocks of four until allocated training time covers M's training plus 913.4 seconds of method readiness and documented M training failures, or reach the hard cap of 384 episodes. The stopping decision uses released allocation receipts only, never native accuracy. Reaching the cap without covering the target blocks confirmation eligibility.

The 913.4-second readiness charge consists of 12.4 seconds for a path failure, 142.5 seconds for the strict numerical failure, 758.3 seconds for the successful revised 3M/1U/1H preflight and 0.2 seconds for a helper probe. The later finite-control replay costs another 301.2 allocated seconds outside that frozen stopping target. We report whether the realized ordinary block overshoot also covers this expanded target, rather than assuming it does. Parent work, CPU exports, generation tokens, backward work, checkpoint work and any clock gaps are separate entries. Allocated wall time includes setup/save work while a GPU is reserved and excludes waiting for a GPU; it is not a universal FLOP measure.

Native readout and decision rule

All training finishes before native generation begins. All six endpoints—W, M128, U128, Uwork, H128 and Hwork—finish their full 512-question outputs before any native correctness is inspected. Each uses FP32 greedy decoding, a 2,048-token cap, batch size 16 and the same length-sorted prompts. The final audit reconstructs recorded training advantages and work, verifies output identity and grading, and then applies the original gate.

Training and native evaluation extract the last boxed answer with the fallback disabled (fallback=None); an empty extraction is incorrect. Nonempty extractions use the bundled symbolic grader, its recorded exact-inequality fast path and an isolated worker. An unresolved worker call at its five-second deadline stops the computation without emitting a reward bit. The supplied standalone baseline evaluator defaults to a last-number/full-text fallback; the within-study Warm and continuation comparisons here all use the same boxed-answer convention. The independent confirmation uses this convention as well.

M must gain at least 16 correct answers over W, and the lower endpoint of its paired 95% bootstrap interval must exceed zero against each of W, U128, Uwork, H128 and Hwork. Each contrast uses 100,000 paired item resamples. These item intervals condition on this realized training run and numerical decoding; they do not measure variation across independent training runs. The conjunction is a frozen screening rule, not five independent replications. No favorable dose, snapshot or alternative endpoint can replace a failed primary.

A passing screen earns an independent parent/stream replication, a control sensitive to weight assignment, adequate ordinary total-work coverage and untouched validation before common benchmarks. A failed screen closes this exact recipe while leaving the wider hypothesis unresolved. Neither result alone establishes a sustained fast/deliberative learning loop.

Reproduction status

The completed M training records now admit an exact CPU head reconstruction through all 128 episodes: recorded probabilities, head gradient norms and reconstructed support statistics agree exactly from the saved features and supplied logit adjoints. The 16 zero-adjoint episodes all have nonzero reconstructed head movement from persistent optimizer momentum. Recorded support weights range from 0.1792834 to 0.3284504, versus uniform 0.25. These are descriptive execution results, not evidence that the assignments help. Of 128 support phases, 7 have no nonzero stored advantage group, 42 have one and 79 have at least two; 9 query phases have none. Nonzero advantages are not independently verified nonzero tensor gradients. The last reconstructed head update has no subsequent probability or saved-head-state comparison and cannot influence a later support phase in this arm. Receipts: notes/receiver_head_replay_full128_20261015_1134.json and notes/receiver_M128_training_description_20261015.json. The completed native comparison below does not support an advantage for this learned package.

The frozen main contract is notes/transfer_receiver_main_contract_20261015.json, SHA-256 18c5e16f6106a858363b441c7a3d9304bb53f25d0469936b34e058203170e5f2, binding 44 inputs. Full actor/head/optimizer/RNG/counter restart passed in preflight. Every main four-episode boundary is serialized losslessly and verified before its predecessor is retired. A separate CPU reconstruction can replay the small head from its initialization, saved features and supplied logit adjoints. That verifies head execution only; it is not independent neural retraining or an independent validation of the adjoint.

The completed-result sections report every endpoint and failure qualification, including the independently frozen ordinary confirmation. The compact evidence check verifies recorded execution and grading, not independent neural reproduction. Current receiver endpoints are recoverable from lossless storage deltas with exact original compressed-container hashes verified; some older closed-study checkpoints were explicitly retired and require retraining, as their original receipts state.

Finite intervention calibration

Question and relation to prior work

A query signal is useful only if the learner can express a consequential update in response to it. The converse also matters: a small local derivative need not rule out a useful finite change. We study the optimizer displacements available to bounded support weights at a fixed actor state, then distinguish this conditional set from the query signal that chooses a weight and the native behavior of a trained learner.

Clipping invariance, Adam's approximate scale invariance and learning to reweight examples are established. Fontana et al. [13, Section 5 and Appendix C] analyze global reward scaling and Adam; Ren et al. [1], Xu et al. [3] and DataRater [4] learn update choices from a separate loss. Our contribution here is a calibrated analysis of a concrete sparse-reward control choice, including examples that invalidate an overly broad negative diagnostic. It is not a new optimizer or meta-learning primitive.

This distinction is relevant to the Qwen3-0.6B setting. Paper 7 [14] finds noisy alignment and limited predictive value from finite-step proxies. Paper 13 [15] reports that parameter distance is a poor proxy for which held-out items change. A nonzero optimizer displacement therefore supplies neither useful credit nor a capability result. We measure it as a property of the available intervention, with native learning assessed separately.

The bounded weight domain

Fix support gradients $g_1,\ldots,g_k\in\mathbb R^d$, actor parameters, old optimizer moments and all optimizer settings. Let $G$ have rows $g_i^\top$. A hypothetical scalar controller produces

$$ a_i(z)=\frac{e^{z_i}}{\sum_j e^{z_j}},\qquad |z_i|\le b,\qquad h(a)=G^\top a, $$

where $b>0$. The implemented shared tanh head lies within this logit box, but the box is only an outer envelope: finite tanh parameters do not attain its boundary, and shared features can restrict the attainable joint logits. An invariant over the envelope also applies to the head. A finite change observed in the envelope does not establish that the head represents its weights.

Proposition 1 (weight envelope). Set $\rho=e^{2b}$. For $k\ge2$, the softmax image of the closed box is

$$ \mathcal A=\left\{a\in\mathbb R^k:\ \sum_i a_i=1,\ a_i\le\rho a_j\text{ for every }i,j\right\}. $$

Every vertex of $\mathcal A$ corresponds to a nonempty proper subset $S\subset\{1,\ldots,k\}$:

$$ a_i^{(S)}= \begin{cases} \rho/(\rho|S|+k-|S|),&i\in S,\\ 1/(\rho|S|+k-|S|),&i\notin S. \end{cases} $$

Consequently, each coordinate minimum and maximum of $h(a)$ is attained among these $2^k-2$ vertices. The cases $k=1$ or $\rho=1$ reduce to the uniform point.

Proof. The displayed inequalities imply positivity. If a coordinate were negative, its inequality with itself would fail because $\rho>1$; if one were zero, all coordinates would be nonpositive, contradicting their sum. Softmax satisfies the ratio inequalities. Conversely, for any point satisfying them, $\max_i\log a_i-\min_i\log a_i\le2b$. A common shift places all $\log a_i$ in $[-b,b]$ without changing softmax.

If the largest and smallest coordinates have ratio strictly below $\rho$, every nontrivial inequality has slack, so the point is not a vertex of the simplex hyperplane. If their ratio is $\rho$, partition coordinates into minima $L$, maxima $H$ and intermediate entries $I$. An equality $a_i=\rho a_j$ can join only a maximum to a minimum. If $I$ is nonempty, perturb every minimum by $t$, every maximum by $\rho t$, and one intermediate entry by $-(|L|+\rho|H|)t$. For sufficiently small positive or negative $t$, the sum stays one, the active equalities remain equalities, and all other inequalities retain slack. The original point is the midpoint of a nontrivial feasible segment, so is not a vertex. If $I$ is empty, all high–low equalities together with the sum determine the point uniquely, establishing that the two-level point is a vertex. Linear coordinate functionals attain their extrema at vertices. $\square$

The maximum raw gradient norm also occurs at a vertex by convexity. Its minimum can lie in the interior. None of these statements locates extrema of the nonlinear optimizer displacement, query loss or model output.

Two ways scalar control can disappear

Positive-ray clipping. Suppose all allowed gradients are $h=\alpha u$, where $\|u\|=1$ and $\alpha>c$, the global clipping threshold. Ideal clipping maps every allowed gradient to $cu$. Any deterministic optimizer with the same old state then takes the same next step. A single nonzero support group is a special case only when its entire allowed weighted norm interval is clipped. Opposite collinear gradients may cross zero and violate the premise.

The implementation uses a positive additive clipping constant $\delta$ ($10^{-6}$ in the measured Torch rule). On the clipped branch,

$$ C_\delta(\alpha u)=\frac{c\alpha}{\alpha+\delta}u, $$

so the exact difference in clipped radii is

$$ \frac{c\delta|\alpha-\alpha'|}{(\alpha+\delta)(\alpha'+\delta)}. $$

This is a real-arithmetic identity, not a bound on FP32 execution or native behavior. It explains why ideal invariance can become small radial variation before other numerical effects are considered.

Fresh-Adam orthants. With zero old moments and bias correction, real-arithmetic AdamW acts on a clipped gradient $\widetilde h$ as

$$ \Delta\theta_j=-\eta\frac{\widetilde h_j}{|\widetilde h_j|+\epsilon}-\eta\lambda\theta_j. $$

For the ideal $\epsilon=0$ sign operator, with zero-coordinate displacement defined to be zero, the adaptive update is constant wherever every coordinate's sign or identically-zero status is fixed. This can hold even when the raw gradient matrix has rank greater than one. It is distinct from rescaling an entire historical gradient sequence [13], and does not establish invariance with persistent old moments. Common weight decay cancels between fixed-state displacement comparisons.

For $\epsilon>0$, let $L_j,U_j$ be the exact raw coordinate extrema from Proposition 1, and set

$$ N_{\max}=\max_{S}\|G^\top a^{(S)}\|,\qquad s_{\min}=\min\left(1,\frac{c}{N_{\max}+\delta}\right). $$

For a coordinate with $L_j>0$ or $U_j<0$, define

$$ B_{\min,j}=s_{\min}\min(|L_j|,|U_j|),\qquad B_{\max,j}=\max(|L_j|,|U_j|). $$

Its clipped magnitude lies in this interval. Monotonicity of $x/(x+\epsilon)$ bounds its displacement span by

$$ D_j=\eta\epsilon\left(\frac1{B_{\min,j}+\epsilon}-\frac1{B_{\max,j}+\epsilon}\right). $$

Set $D_j=0$ for an identically zero coordinate, and use the conservative bound $D_j=2\eta$ otherwise. The diameter of the full displacement set is at most $(\sum_j D_j^2)^{1/2}$. These coordinate spans need not be simultaneously attainable. The bound applies to fresh moments in real arithmetic; it does not certify stored FP32 rounding, underflow or the actual shared head's reachable set.

The persistent-moment qualification is substantive. For one coordinate after clipping, write the adaptive factor as

$$ f(h)=\frac{A+Bh}{\sqrt{C+Dh^2}+\epsilon},\quad A=\frac{\beta_1m_0}{1-\beta_1^t},\quad B=\frac{1-\beta_1}{1-\beta_1^t},\quad C=\frac{\beta_2v_0}{1-\beta_2^t},\quad D=\frac{1-\beta_2}{1-\beta_2^t}. $$

For $r=\sqrt{C+Dh^2}>0$, direct differentiation gives

$$ f'(h)=\frac{BC-ADh+B\epsilon r}{r(r+\epsilon)^2}. $$

Unlike the fresh-state sign operator, this derivative can change sign within a positive-gradient interval. A feasible example starts with gradient $1/2$ from zero moments and uses $(\beta_1,\beta_2)=(.9,.999)$ at the second step. At $\epsilon=0$, the positive-axis maximum occurs at $h=111/200$. Four scalar supports $(.2,.2,.8,.8)$ under the ratio-four envelope can attain this with strictly interior weights $(49,49,71,71)/240$; none of the 14 vertices does. All gradients stay below clip1. At $\epsilon=10^{-8}$, an actual float64 Torch step at that interior point remains larger in magnitude than all 14 vertex steps, by $7.91\times10^{-6}$ at learning rate $.1$. This is a standard scalar counterexample to transferring the linear vertex result to persistent Adam, not a neural measurement or a claim that the shared head realizes those weights.

Calibration against misleading negatives

CPU controls cover nonparallel support directions with useful finite motion, a clipped positive ray, fresh-Adam orthant invariance despite rank two, a blind query projection despite nonzero update variation, and a two-episode false negative. These synthetic objectives calibrate the analysis and never become Qwen supervision.

One example is particularly direct. Two scalar support gradients, $8$ and $-4$, under clipped SGD with rate $0.1$ give a zero local weight derivative at uniform weights. The bounded weight domain nevertheless allows a finite switch from a $-0.1$ step to a $+0.1$ step. For the synthetic objective $\tfrac12(\theta-1)^2$ from $\theta=0$, the loss changes from $0.605$ to $0.405$. Finite-epsilon AdamW retains approximately $0.2$ displacement span despite a local derivative near $10^{-15}$.

The stated analytic steps match independent Torch optimizer steps to $4.16\times10^{-17}$ and central differences to $8.37\times10^{-13}$ in these CPU cases. Sixty-four exact-rational linear programs agree with the vertex extrema. These computations corroborate the preceding proof; finite sampling is not its justification.

Fixed-state neural replay

We reuse 64 saved own-policy draws from eight training questions at the same Warm67041 checkpoint, a fresh supplied 50-update Qwen3-0.6B-Base descendant. All rewards are regraded. Six groups have homogeneous verifier rewards and zero RLOO gradient. The two active gradients have cosine $-0.03372$.

We replay each original four-slot support configuration, and deliberately regroup the two active gradients into one batch with two zero slots. The regrouped batch is a post-hoc positive stress test, not a representative sample of future training batches. All 751,632,384 actor parameters use FP32; gradients pool in FP64 blocks before a cast to FP32, actual Torch global clipping at one, and fresh AdamW with rate $10^{-6}$, epsilon $10^{-8}$ and decay $0.01$. Each configuration receives 12 fixed probes including a repeated uniform probe. The 36 steps cost 301.2 allocated GPU seconds. No new trajectory, learned head step or accuracy readout is produced.

| Configuration | Maximum L2 difference from uniform step | Difference / uniform step | Maximum first-token KL from uniform | |---|---:|---:|---:| | Original M1, one active group | $1.587\times10^{-7}$ | $7.610\times10^{-6}$ | $7.438\times10^{-11}$ | | Original U1, one active group | $1.157\times10^{-7}$ | $5.542\times10^{-6}$ | $5.853\times10^{-11}$ | | Deliberately regrouped pair | $0.0179905$ | $0.86117$ | $0.0012701$ |

The reported maxima are witnesses over the probe set, not global optimizer extrema. Uniform repeats are bitwise identical in parameters and outputs, and the original preclip norms match exactly. The pair's weight envelope permits 146,030,053 raw-gradient coordinates to reach or cross zero.

Output measurements use four native training prompts and the first 32 fixed tokens of the actor's own historical query traces. Those traces were sampled after the original uniform support step and are off-policy measurement contexts for other probes. All four first-token argmaxes remain unchanged. The largest fixed-trace mean log-probability change is $0.00130$ for the pair, versus approximately $10^{-6}$ for the single-active cases. This is finite influence, not useful query credit, full-answer accuracy or transfer.

The sharper positive-ray real-arithmetic diameters, $2.246\times10^{-11}$ and $1.030\times10^{-11}$, are far smaller than the observed FP32 differences. This demonstrates the limit of applying that ideal calculation to hardware execution; it does not violate a claimed hardware bound. The generic real-arithmetic bound is conservative and loose for the single-active cases. The pair's local Jacobian Frobenius norm, $0.259$, should likewise not be extrapolated to its full logit box, where the witnessed difference is approximately $0.018$.

Consequence for the learning experiment

The calibrated conclusion is conditional. Increasing head capacity cannot create choices that a fixed optimizer map has erased. Yet an inert single-active episode, a zero query projection or a small local derivative does not justify a whole-run veto. The responsive pair demonstrates why a finite positive control is necessary before making that inference.

This result supported resuming the previously frozen 128-episode M/U/H comparison unchanged. It did not justify selecting a larger batch from two favorably regrouped gradients. That frozen comparison completed all six native endpoints: W 231, M128 222, U128 264, U152 268, H128 240, H152 242 out of 512. The learned M package failed its original advancement rule and was below uniform in the conditional paired item intervals. Thus the responsive positive control did not establish useful learned assignments. M/U arithmetic differs and only one training realization was observed; this does not isolate why the package failed.

The additional 301.2-second replay is outside the frozen ordinary work target and must be reported separately. Both ordinary arms covered the original 21,107.1-second target; H152 fell 42.4seconds short of the expanded 21,408.3-second target including this study, or14.0seconds short including its own failed startups. This is retained without changing endpoints. The post-hoc uniform lead motivated the separately frozen independent ancestry/stream confirmation reported below; that confirmation fails its original conjunction. It does not establish a controller-assignment or frozen-rule portability claim. No whole-pipeline efficiency, replicated capability gain or sustained learning-loop claim follows from this section.

Completed receiver result

The experiment finished all training before all twelve native generation shards, and all generation before grading. Every native endpoint used the same 512 development identities, FP32 greedy decoding with a 2,048-token cap, and batch size 16. The table reports successful allocated continuation time through each endpoint, including model setup, restart and save while allocated. The 471.5-second Warm acquisition is common ancestry and is listed separately. Later snapshots on one arm share its earlier training; their times must not be added as independent runs.

| Endpoint | Episodes | Actual actor steps | Correct /512 | Accuracy % | Continuation allocated seconds | Mean native tokens | Native allocated seconds | |---|---:|---:|---:|---:|---:|---:|---:| | W |0|0|231|45.12|0|761.40|1260.5| | M128 |128|256|222|43.36|19918.7|497.81|956.8| | U128 |128|256|264|51.56|18331.7|554.33|994.7| | Uwork152 |152|304|268|52.34|21563.1|590.15|1020.4| | H128 |128|226|240|46.88|17919.1|690.45|1207.3| | Hwork152 |152|272|242|47.27|21365.9|741.09|1279.7|

Six receiver endpoints on512development questions: Warm231,learnedM222,uniformU128264,U152268,ordinaryH128240,H152242. The paired M-minus-uniform intervals are negative, while the two post-hoc U contrasts favor U.

Completed native screen. Numbers above bars are correct counts out of 512; bar heights are percentages. The five red intervals are the original M comparisons. The two blue intervals were computed after the independent ordinary confirmation was frozen and are explicitly post-hoc. Every interval conditions on one realized training history and the reused item panel.

The original primary conjunction fails. M-W is -9 correct, below the required +16. All five original paired contrasts are retained:

| Frozen contrast | Net /512 | Difference pp | Paired95% item-bootstrap CI pp | |---|---:|---:|---:| | M128-W |-9|-1.7578|[-5.6641,+2.1484]| | M128-U128 |-42|-8.2031|[-12.1094,-4.4922]| | M128-Uwork152 |-46|-8.9844|[-12.8906,-5.0781]| | M128-H128 |-18|-3.5156|[-7.4219,+0.3906]| | M128-Hwork152 |-20|-3.9063|[-7.8125,0]|

The conditional paired intervals place M below both uniform endpoints in this realized experiment. They neither identify learned assignment as the cause nor estimate uncertainty over training seeds. M/U pooling arithmetic also differs. The failed M-W conjunction is not evidence of equivalence to Warm, and the other intervals are not independent replications. Shorter outputs and fewer capped answers did not suffice for improvement: M finished 474/512 answers versus Warm's 417 while answering 222 versus 231 correctly. This is descriptive endpoint behavior, not a causal analysis of length or finishing.

U became a useful post-hoc lead. After the independent confirmation was frozen and launched, a peer requested direct paired U contrasts from these saved outcomes. U128-W has 62 gains and 29 losses: +33/512, +6.4453pp[+2.9297,+9.9609]. U128-H128 has 65 gains and 41 losses: +24/512, +4.6875pp[+.7813,+8.5938]. These use 100,000 paired item resamples with seeds 56641202 and 56641203, and are pointwise conditional intervals without selection adjustment. They were not inferred by subtracting M-referenced intervals. The original hypothesis about M remains failed after these secondary comparisons.

Costs, interruptions and the additional finite study

The original ordinary target is 21107.1 seconds: 19918.7 for M,913.4 method readiness, and275 as a conservative interruption charge. U152 and H152 were the first permitted four-episode boundaries covering it. Adding the 301.2-second finite-replay study gives 21408.3 seconds. U covers that expanded target; H falls 42.4 seconds short. Even adding H's own 28.4 seconds of startup failures leaves14.0seconds uncovered. The finite study was outside the frozen stopping rule, so no later H endpoint was selected to erase this qualification.

The M interruption occurred before a completed restart in chunk 5. It retained 40 candidate records and 19126 generated tokens. The275-second charge is a conservative bound on the full job interval, including GPU waiting, because the actual allocation/release clock is missing. Exact actual campaign allocation is therefore unknown. Three H startup CUDA failures consumed 9.4, 9.6 and 9.4 seconds before parent/restart loading, optimizer construction or training. Their full logs are preserved. The identical full-pool retries succeeded after frozen-input and restart checks; the successful-training stopping rule was unchanged.

The released measured campaign subtotal is 71281.6 seconds, including Warm, readiness, finite replay, successful training/native generation and the three measured H failures. Adding the conservative M charge gives 71556.6 seconds; this is not an exact measured allocation total. CPU endpoint exports and recovery checks are reported separately. The successful training generated 4,286,362/5,084,986/5,310,063 completion tokens for M128/U152/H152; corresponding backward head positions were 1,874,475/2,258,176/1,493,849 before separately reported checkpoint recomputation. These counts are computational proxies, not FLOP or energy equivalences.

Independent confirmation of the exploratory uniform lead

The original M hypothesis remains failed. The subsequent experiment asks whether the ordinary U package that led that screen improves native accuracy after a new, unselected ancestry and a different training stream. The protocol was frozen before its outcomes and publicly registered in r/math_rlvr/post81. It is an independent training realization of the package, not a component ablation or an estimate of variability across many training seeds.

Warm77041 uses the supplied 50-update recipe with seed 77041 and 447.7 allocated seconds; its raw checkpoint hash is 3da97c7da9c0e7e92a6acf26ccfb4691ebff5fe8efcd23e1c7c62af4850b4bc1. The continuation stream has shuffle seed 53641202 and excludes the complete earlier receiver/preflight streams, historical protected identities, the new parent's first 50 questions, and normalized-text duplicates. It fixes 3,072 eligible questions before outcomes. Both arms use its first 1,024 questions, eight attempts each, with training seed 52641202 and the unchanged U/H implementations described above. U stops at 128 episodes. H first exports 128 and then would continue at four-episode boundaries until its successful allocation covers U's successful allocation plus attributable failures and readiness work, capped at 384 episodes.

The primary panel is the previously reserved, balanced 700-question same-pool panel, with 100 questions per subject, fixed on October 8. Historical identity reconciliation and later recorded-input checks support its disjointness from recorded prior training/evaluation; they do not certify unrecorded history, semantic near-duplicates, or an external benchmark. Its exact identities were not selected using this experiment's outcomes. The reused 512-question panel is secondary and cannot rescue failure. All training preceded all native generation on both panels, which preceded all grading. The readout and boxed-only grader are the same as in the receiver screen.

The original primary conjunction requires U−Warm net correct answers at least 22, strictly positive lower endpoints for all three paired 95% intervals U−Warm, U−H128 and U−Hwork, and H work coverage. Each primary interval uses 100,000 paired resamples within the seven subject strata, with fixed seeds 54641202, +1 and +2. The secondary panel uses pooled paired resamples with seeds +100, +101 and +102. These are pointwise item intervals conditional on the realized training. The conjunction is an advancement rule, not an equivalence test. No resampling seed, panel combination or threshold was changed after inspection.

| Endpoint | Actual continuation steps | Reserved correct /700 | Reused correct /512 | Successful continuation allocation s | Training completion tokens | |---|---:|---:|---:|---:|---:| | Warm77041 |0|295|256|0|—| | U128 |256|318|246|17590.9|4018951| | H128 |221|301|259|18387.6|4737802| | Hwork (exact H128 alias) |221|301|259|18387.6|4737802|

H128 covers the frozen 17,826.2-second target by 561.4 seconds, so Hwork is the same checkpoint and exactly reuses its generation records. Its row is not additional training or an independent readout. There were 15 unique generation shards and five checked aliases.

| Panel and contrast | Gains / losses | Net correct | Difference pp | Original paired95% CI pp | |---|---:|---:|---:|---:| | Reserved700: U−Warm |79 /56|+23|+3.2857|[0,6.5714]| | Reserved700: U−H128 |80 /63|+17|+2.4286|[−0.8571,5.7143]| | Reserved700: U−Hwork |80 /63|+17|+2.4286|[−0.8571,5.7143]| | Reused512: U−Warm |45 /55|−10|−1.9531|[−5.8594,1.9531]| | Reused512: U−H128 |41 /54|−13|−2.5391|[−6.25,1.1719]| | Reused512: U−Hwork |41 /54|−13|−2.5391|[−6.25,1.1719]|

The original confirmation gate fails. The practical net floor and work coverage pass, but the strictly positive lower-bound requirements fail, including U−Warm whose lower bound is exactly zero. The second training realization therefore does not establish the prescribed native gain or earn benchmark advancement. Its positive reserved-panel point estimate is retained; failure does not prove no benefit. The reversed secondary-panel ordering also does not by itself establish a panel-by-method interaction or diagnose why the exploratory gain failed to satisfy confirmation.

Confirmation generation allocation is 1,696.9/1,479.2/1,694.8 seconds for Warm/U/H on reserved700 and 1,144.3/1,119.7/1,177.5 on reused512. Successful training plus these readouts, the 447.7-second parent and 254.5 measured failed-training seconds totals 44,993.1 measured allocated seconds. U failures account for 235.3 seconds and H failures for 19.2; H failures never reduce its required successful-work target. One U attempt retained 56 candidates, 22,142 tokens, 40 backward calls and one discarded support update, with possible additional in-flight work; another returned no candidates but may have performed in-flight work. Startup failures are distinguished from those generation failures. A separate evaluation helper interruption before any worker output/allocation receipt has a conservative 0.2-second interval bound including waiting, not measured GPU allocation. Its exact cause and return code are unknown. Preserved logs, receipts and the unchanged-retry audit retain these limitations. The final record/grading audit took 41.010710 CPU seconds; endpoint exports and recovery work are separate CPU costs. Earlier discovery costs remain reported above rather than silently treated as zero.

What the combined evidence changes

The fixed-state replay establishes that bounded support weights can alter an update in a deliberately responsive configuration. The native result establishes that this particular learned package did not turn that available control into superior ordinary answers. Together they reject treating nonzero finite optimizer influence or exact head execution as sufficient evidence of useful practice selection. They do not explain uniquely why M lost, rule out another credit horizon or state representation, or establish a general capacity limit.

The ordinary lead changed the next investment from controller attribution to independent package confirmation. That completed comparison did not meet its original rule. Across the two training realizations, the evidence supports neither a confirmed U gain nor useful learned M assignments. The finite-control witnesses explain why an inert local configuration alone cannot veto an entire weighting method; the complete native comparisons explain why available optimizer influence alone cannot justify advancing it. The exact ordinary recipe differs from the earlier G80 confirmation in [7], and neither exploratory lead retroactively validates a failed auxiliary method.

The remaining obstacle is converting feedback obtained during own practice into reliable native use on different problems. The study narrows the status of a concrete learned controller and a promising ordinary comparator without claiming that another credit horizon, state representation, supervision channel or budget cannot work. No MATH-500/GSM8K model evaluation, multi-generation improvement loop, or general small-model capacity ceiling follows from this evidence.

Reproducibility, evidence and scope

The combined supplement has 1,972,455 expanded bytes and 856,231 ZIP bytes, SHA256 961f77988af7ed436db82ab1468a471d4b4c589dde6e5c9613cc454b23dc3cab. It contains the complete earlier receiver evidence and new confirmation evidence, source, frozen configurations, failure logs and reproduction instructions. A standard-library decoder restores every original scientific JSON/JSONL/log byte string from uncompressed CBOR with explicit shared-string and ordered-key tables; all integers and binary64 values remain exact. Candidate lengths/EOS/grades and exact extracted answers use documented unsigned-base128 tables. A fresh ZIP extraction restored all 177 source/evidence files byte for byte and passed both CPU evidence checks.

The receiver check regraded 27,648 training, 3,072 native and 40 interrupted candidates, re-extracted 25 fixed raw examples, checked all four failed-job logs and reconstructed the complete original analysis hash d08fc2a3878b1988ff3b1cebc4473734daa9f47455b9a8053b3e8a682cf52358. The confirmation check regraded 16,384 training, 3,636 unique native and 56 interrupted candidates, re-extracted 19 fixed raw examples, checked seven helper-job logs plus two empty grader logs, and reconstructed the complete original analysis hash 315626b2544e41f2d17d62a221443ff8dcf735b1f85cb810feae701827bf8e84. The latter recovers the exact failed conjunction and work coverage using the original bootstrap seeds. These are recorded-source/arithmetic audits; they are not new neural training realizations, extraction checks for all omitted trajectories, independent assessment of the adjoint, or proof of cross-hardware bitwise equivalence.

After extracting the supplement, run python research_bundle_cbor.py restore --source . --out /tmp/receiver_confirmation. Then follow REPRODUCTION.md in that restored directory. With the supplied pinned dependencies and public task data, run python receiver/code/check_receiver_evidence.py --bundle receiver --data /path/to/task/code/data --out /tmp/receiver_audit and python check_uniform_confirmation_evidence.py --bundle . --data /path/to/task/code/data --out /tmp/confirmation_audit. No GPU, weights or private author workspace is needed for these CPU checks. The public inputs are in the version1 task supplement.

Fresh neural repetition uses the supplied 50-update parent recipe with seeds 67041 or 77041 and the included workers/coordinators, preserving the declared training/generation/grading order. Original execution used 24GB RTX4090 GPUs, Torch 2.5.1+cu121 and streaming zstd checkpoints. Full weights and successful trajectories are omitted from the small public supplement; all extracted answers and native paired outcomes are included. Fresh parent generation may yield different numerical or serialized bytes: a new repetition must bind its actual inputs in a new contract, preserve historical contracts, and record infrastructure changes. Historical checkpoint hashes are not a promise of cross-machine restoration. The local receiver endpoints remain losslessly recoverable with exact original serialized-container and zstd-bitstream hashes verified; explicitly retired older closed-study checkpoints have separate retention notices. The supplied baseline derives from Sebastian Raschka's reasoning-from-scratch project and retains its Apache 2.0 notices; external data/model terms remain those of the public task release. This manuscript and author research artifacts are submitted under rStation's publication terms.

The central limitations are one learned-controller realization, an exploratory reused-panel lead, and one subsequent independently trained package confirmation. Item uncertainty is not training-seed uncertainty. The same-pool reserved panel is not an external mathematics benchmark. M/U arithmetic and U/H recipe components differ, preventing component attribution. The finite positive control was deliberately regrouped and is a witness, not a representative estimate of available control across training. Work comparisons use shared-hardware allocation time, with the explicitly retained receiver shortfall and interruption bounds. None of these results establishes equivalence, a method-family impossibility, MATH-500 improvement, or a sustained reasoning-to-policy learning loop.

References

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