Reachability Is Not Realization: Tracing the Sources of LLM Benchmark Gains

# Reachability Is Not Realization: Tracing the Sources of LLM Benchmark Gains
Source: [https://arxiv.org/html/2608.03219](https://arxiv.org/html/2608.03219)
###### Abstract

Benchmark gains are often treated as evidence of greater LLM capability\. Yet the same gain can reflect different changes in model behavior\. A model may reach new answers, or produce answers that were already within reach\. Aggregate scores do not distinguish these changes question by question\. We establish a question\-level audit under fixed budgets, temperatures, and answer formats\. A question is realized when the default deployment procedure produces the correct answer\. A question is reachable when a specified probe finds that answer within a fixed budget\. We first test whether inference\-time layer routing can expand reachability\. Under a matched budget, random routes match or exceed structured search in all 43 model and task settings\. Answer\-blind procedures retain almost none of this gain, which instead requires access to the correct answer\. We then ask why reachable answers sometimes fail to appear\. Across six cases spanning 0\.5B to 31B, silencing one identified MLP block repairs 68 to 92 percent of a predefined failure set\. We next test whether training closes the gap by expanding reachability\. In five of six matched evaluations, deployed performance rises while the reachable ceiling remains flat or falls\. For DAPO, the deployed score rises by 14\.7 points while the reachable ceiling falls by 13\.3 points\. Across the settings we audit, realization and reachability therefore do not always change together\. Claims of capability expansion should report both realized performance and reachability under matched evaluation conditions\. Code is available athttps://github\.com/LiZaiyuan0619/reachability\-not\-realization\.

## Introduction

Benchmark scores remain the main measure of progress in LLM reasoning\. Many recent methods raise these scores without simply increasing model size\. These gains are often interpreted as evidence that a model can solve more questions\. Yet an aggregate score records only how many outputs are correct\. It does not show which questions became newly reachable\. The same gain may instead come from producing already\-reachable answers more reliably\. Distinguishing these changes requires following the same questions under matched evaluation conditions\.

![Refer to caption](https://arxiv.org/html/2608.03219v1/x1.png)Figure 1:Budget\-matched routing trajectories\.a–d, Random\-path and structured\-search oracle reachability, answer\-blind voting, and the full\-model baseline across four model–task cells\. All procedures share the sameK=1K=1origin\. Oracle curves use correctness after generation and are not deployable selectors\.We separate these possibilities under fixed budgets, temperatures, and answer formats\. We call a question realized if the default deployment procedure answers it correctly\. We call the same question reachable if a specified probe finds the correct answer within a fixed budget\. Reachability is therefore relative to the evaluation protocol and is not identical to full model capability\. Sampling and alternative layer paths search for correct answers under that protocol\. Recognition provides a separate internal readout and serves only as auxiliary evidence\. This audit asks whether an intervention changes realization, reachability, or both\.

Layer routing changes which layers run without changing the weights, providing a controlled inference\-time test\. The structured searches we audit are intended to identify useful layer paths\. If their structure matters, they should outperform random routes at the same budget\. We compare structured searches with random routes under the same budget and scoring rule\. Random routes match or exceed the structured ceiling in every evaluated setting\. This comparison is oracle\-based because identifying a correct route requires knowing the answer\. The answer\-blind procedures we test retain almost none of the oracle gain\. The tested routing gain therefore depends on selecting among reachable outputs\. This explains how the gap can be exploited, but not why it exists\.

We next ask why an internal readout can support an answer that direct generation misses\. We predefine questions where the model ranks the correct option first but generates a wrong answer\. In each audited case, silencing one identified MLP block repairs most questions in the predefined failure set\. Silencing attention in the same layer produces a much smaller effect\. Most already\-correct questions in a separate healthy panel remain correct under the intervention\. Conversely, perturbing the same block in a healthy model can induce the corresponding failure\. Together, these controls causally localize a direct\-answer failure to one MLP block in each case\. The remaining question is whether training closes the broader gap by expanding reachability\.

Reinforcement learning with verifiable rewards \(RLVR\) updates model weights using rewards for correct outputs\. It therefore provides a direct test of whether training expands reachability\. We compare base and trained checkpoints on the same questions under matched answer formats, temperatures, and budgets\. Deployed performance rises in most of these matched evaluations\. The reachable ceiling remains flat or falls in those same evaluations\. Every newly realized question was already reachable in the corresponding base model at the same budget\. The gain also concentrates on questions that the base model reached more frequently\. The audited pairs therefore show that RLVR can improve realization without consistently expanding reachability\.

The three analyses share a measurement distinction rather than a common mechanism\. The routing audit tests whether an inference\-time gain depends on selection\. The MLP intervention localizes one cause of failed realization\. The RLVR comparison tracks how realization and reachability change with training\. Together, they show that a benchmark gain does not establish that reachability has expanded\. Improving realization remains a meaningful form of progress\. Claims of capability expansion therefore require matched measurements of realized performance and reachability\.

We make four contributions\.

1. 1\.We establish a question\-level audit of realized performance and reachability under fixed conditions\.
2. 2\.We audit layer routing using budget\-matched random routes and answer\-blind selection tests\.
3. 3\.We causally localize a predefined direct\-answer failure to one MLP block using component, specificity, and reciprocal controls\.
4. 4\.We track how realization and reachability change for each question across matched base and RLVR\-trained checkpoints\.

![Refer to caption](https://arxiv.org/html/2608.03219v1/x2.png)Figure 2:Breadth and tested robustness of the random\-path oracle advantage\.a, Random\-minus\-structured oracle differences across 43 clean model–task cells; three format\-collapse cells are excluded\.b, Directional counts across three action spaces and differences against hill climbing and genetic search atK=64K=64\. Dots are cells and bars are medians\.
## Related Work

Conditional\-compute systems vary transformer depth or token participation to trade model quality against inference cost\. They learn deployable routers, train residual gates, or verify early exits\(Raposoet al\.[2024](https://arxiv.org/html/2608.03219#bib.bib4); Elhoushiet al\.[2024](https://arxiv.org/html/2608.03219#bib.bib5); Laitenbergeret al\.[2026](https://arxiv.org/html/2608.03219#bib.bib6); Heet al\.[2025](https://arxiv.org/html/2608.03219#bib.bib7)\)\. A narrower line searches for input\-specific layer programs or oracle allocations\(Heaklet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib1); Liet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib2); Glavaset al\.[2024](https://arxiv.org/html/2608.03219#bib.bib3)\)\. We compare these searches with budget\-matched random paths and measure how much oracle headroom answer\-blind selectors recover\.

Candidate\-set success and deployed selection answer different questions\. Thepass@kmetric asks whether any candidate succeeds\(Chenet al\.[2021](https://arxiv.org/html/2608.03219#bib.bib8)\)\. Without gold correctness, selection can use frequency, semantic modes, geometric centers, or learned rankings\(Wanget al\.[2023](https://arxiv.org/html/2608.03219#bib.bib9); Choi and Li[2026](https://arxiv.org/html/2608.03219#bib.bib11); Nguyenet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib13); Marinaet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib14)\)\. Global judge correlation can also overstate item\-level candidate recovery\(Landesberg[2026](https://arxiv.org/html/2608.03219#bib.bib10)\)\. Cross\-model consensus and trained process verifiers use different access, supervision, and cost contracts\(Liu[2026](https://arxiv.org/html/2608.03219#bib.bib12); Kuanget al\.[2026](https://arxiv.org/html/2608.03219#bib.bib15)\)\.

A separate literature shows that models can contain evidence supporting a correct answer while producing another response\(Parket al\.[2026b](https://arxiv.org/html/2608.03219#bib.bib16); Yeomet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib17)\)\. Related causal work patches or steers temporal conflicts, answer\-conditioned features, late rescue, temporal retrieval, reasoning\-to\-answer flow, or multi\-component effects\(Hossainet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib19); Parket al\.[2026a](https://arxiv.org/html/2608.03219#bib.bib20); Deng[2026](https://arxiv.org/html/2608.03219#bib.bib21); Huanget al\.[2026](https://arxiv.org/html/2608.03219#bib.bib22); Zhanget al\.[2025](https://arxiv.org/html/2608.03219#bib.bib23); Yanet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib24)\)\. Controlled character counting further links early probe\-visible information to late MLP and attention suppression\(Dattaet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib18)\)\. We instead test a predefined direct\-answer failure with component, reciprocal, specificity, and construction controls\.

High\-sample evaluations show that low\-budget gains can coexist with high\-budget crossover or contraction, and item\-level analyses record both entry and loss\(Yueet al\.[2025](https://arxiv.org/html/2608.03219#bib.bib25); Yuanet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib26)\)\. The relation changes with the validity criterion, candidate generator, objective, and training duration\(Wenet al\.[2025](https://arxiv.org/html/2608.03219#bib.bib31); Zhouet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib30); Nguyenet al\.[2025](https://arxiv.org/html/2608.03219#bib.bib27); Yaoet al\.[2025](https://arxiv.org/html/2608.03219#bib.bib28)\)\. Teacher\-assisted curricula can increase broad\-budget coverage, but external traces change the causal question\(Caiet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib29)\)\. We join matched checkpoints by question and track deployment and reachability under one fixed candidate generator\.

## Setup

#### Evaluation coordinates\.

Reachability is defined relative to a protocol comprising the question set, answer format, grading rule, temperature, probe, and candidate budgetKK\. Matched comparisons vary only the intervention or checkpoint of interest\. An answer not found by a finite probe is unobserved under that protocol, not impossible for the model\.

#### Realization and reachability\.

For questionii, letgi​\(a\)∈\{0,1\}g\_\{i\}\(a\)\\in\\\{0,1\\\}indicate whether answeraais correct under the task\-specific grading rule\. Lety^idep\\hat\{y\}\_\{i\}^\{\\mathrm\{dep\}\}be the output of the designated answer\-blind deployment procedure\. The specified probe returnsKKcandidate outputs, denotedci\(1\),…,ci\(K\)c\_\{i\}^\{\(1\)\},\\ldots,c\_\{i\}^\{\(K\)\}\. We define

di=gi​\(y^idep\),oi​\(K\)=max1≤k≤K⁡gi​\(ci\(k\)\)\.d\_\{i\}=g\_\{i\}\\\!\\left\(\\hat\{y\}\_\{i\}^\{\\mathrm\{dep\}\}\\right\),\\qquad o\_\{i\}\(K\)=\\max\_\{1\\leq k\\leq K\}g\_\{i\}\\\!\\left\(c\_\{i\}^\{\(k\)\}\\right\)\.A question is realized whendi=1d\_\{i\}=1and reachable whenoi​\(K\)=1o\_\{i\}\(K\)=1\. The corresponding aggregate quantities are

D=1n​∑i=1ndi,OK=1n​∑i=1noi​\(K\)\.D=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}d\_\{i\},\\qquad O\_\{K\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}o\_\{i\}\(K\)\.We callDDdeployed performance andOKO\_\{K\}oracle reachability, or the reachable ceiling, at the stated protocol\. The oracle uses correctness only after generation and is diagnostic rather than deployable\.

#### Question\-level comparisons\.

We retaindid\_\{i\}andoi​\(K\)o\_\{i\}\(K\)for every question on both sides of a comparison\. Changes indid\_\{i\}record newly realized answers and losses, while changes inoi​\(K\)o\_\{i\}\(K\)record entry into or exit from the measured reachable set\. The oracle–deployment gap isOK−DO\_\{K\}\-Dwhen both quantities use compatible conditions\. An answer\-blind selector may use candidates and model\-derived scores but not the correct answer\. Recognition is a separate auxiliary readout that ranks the correct option under multiple\-choice likelihood\. It is neither direct production nor evidence of sampling reachability\.

#### Models and tasks\.

Evaluated checkpoints include Llama 3, Qwen2\.5, Gemma 4, Qwen3\.5, and OLMo 2\(Grattafioriet al\.[2024](https://arxiv.org/html/2608.03219#bib.bib37); Qwen Team[2025](https://arxiv.org/html/2608.03219#bib.bib38); Gemma Team[2026](https://arxiv.org/html/2608.03219#bib.bib39); OLMoet al\.[2024](https://arxiv.org/html/2608.03219#bib.bib47)\)\. Tasks include ARC\-Easy \(ARC\-E\) and ARC\-Challenge \(ARC\-C\), GSM8K, PIQA, LogiQA \(LQ\), and DART\-Math \(DART\)\(Clarket al\.[2018](https://arxiv.org/html/2608.03219#bib.bib32); Cobbeet al\.[2021](https://arxiv.org/html/2608.03219#bib.bib33); Bisket al\.[2019](https://arxiv.org/html/2608.03219#bib.bib34); Liuet al\.[2020](https://arxiv.org/html/2608.03219#bib.bib35); Tonget al\.[2024](https://arxiv.org/html/2608.03219#bib.bib36)\)\.

#### Matched protocols\.

For layer routing, structured and random paths share the question set, scoring rule, decode length, initial full path, and candidate budget\. For MLP localization, the failure set is fixed before intervention by correct recognition and incorrect direct generation\. Component controls, separate healthy panels, and opposite perturbation directions test localization, selectivity, and reciprocal causality\. For RLVR, base and trained checkpoints are joined by question ID and evaluated with matched answer formats, temperatures, budgets, and grading rules\. Models, tasks, prompts, run counts, and implementation details are reported in the Supplement\.

![Refer to caption](https://arxiv.org/html/2608.03219v1/x3.png)Figure 3:Answer\-blind selectors recover little of the routing oracle gap\.a, Recovery of the oracle–greedy gap by six answer\-blind channels atK=32K=32, defined as\(selector−greedy\)/\(oracle−greedy\)\(\\mathrm\{selector\}\-\\mathrm\{greedy\}\)/\(\\mathrm\{oracle\}\-\\mathrm\{greedy\}\)\. Bars are clean\-cell medians; hollow points are excluded format\-collapse cells\.b, Oracle\-minus\-majority gaps across measured temperatures\.

## Experiment

### Routing and Selection

Random\-path reachability continued to rise after the tested structured search saturated \(Figure[1](https://arxiv.org/html/2608.03219#Sx1.F1)\)\.All procedures shared the path budget, scoring rule, decode length, and initial full path\. Random search provides a necessary control for structured architecture search\(Li and Talwalkar[2020](https://arxiv.org/html/2608.03219#bib.bib46)\)\. The random\-path oracle kept improving after the structured search flattened, reaching 100% on Llama\-3\.2\-3B/ARC\-Challenge atK=200K=200\. The answer\-blind vote remained near or below the full\-model baseline, so additional correct paths did not provide a deployable selection rule\.

The random\-path oracle advantage persisted across the evaluated models, tasks, action spaces, and search families \(Figure[2](https://arxiv.org/html/2608.03219#Sx1.F2)\)\.The random\-path oracle exceeded the structured ceiling in all 43 clean cells after three format\-collapse evaluations were excluded\. The ordering persisted across routing action spaces and against hill climbing and genetic search\. The effect was therefore not confined to one cell or search design\.

Large routing gains appeared only in answer\-aware channels \(Table[1](https://arxiv.org/html/2608.03219#Sx4.T1)\)\.The learned router and answer\-blind vote produced little or negative change, whereas both oracles gained by selecting with correctness\. The table reports channel\-specific budgets; the matched comparison is given in Figures[1](https://arxiv.org/html/2608.03219#Sx1.F1)and[2](https://arxiv.org/html/2608.03219#Sx1.F2)\.

Table 1:Large routing gains appear only with answer\-aware selection\.Percentage\-point changes from the unmodified model\. The first two rows are answer\-blind; the oracle rows select with correctness\. Parentheses giveKKor MCTS simulations\.The tested answer\-blind selectors recovered little or none of the oracle–deployment gap \(Figure[3](https://arxiv.org/html/2608.03219#Sx3.F3)a\)\.AtK=32K=32, five of six clean median recoveries were negative; the only positive median was 2\.2%\. Higher temperature widened the oracle\-minus\-majority gap \(Figure[3](https://arxiv.org/html/2608.03219#Sx3.F3)b\)\. The bottleneck was therefore identifying correct candidates without hindsight\.

Table 2:Ordinary sampling reproduces the selection gap without layer routing\.All rows usen=150n=150,K=8K=8, and temperature 0\.8\. Gap is oracle@8 minus majority@8 in percentage points\.Ordinary sampling reproduced the selection gap without layer routing \(Table[2](https://arxiv.org/html/2608.03219#Sx4.T2)\)\.The oracle remained far above majority voting when sampling left substantial headroom\. The probes therefore found additional correct answers, but the tested answer\-blind procedures did not reliably identify them\. This leaves a different realization failure: a model can support the correct answer under an auxiliary readout yet fail to produce it directly\.

### MLP Localization

![Refer to caption](https://arxiv.org/html/2608.03219v1/x4.png)Figure 4:Bounded occurrence and pruning\-created MLP\-sensitive failures\.a, Recognition versus direct\-generation scores across 303 records from 284 checkpoint–task cells; highlighted records meet the predefined failure criterion\.b, MLP\-silencing repair across layers in two pruning\-created cases\. Facet titles give evaluated/eligible counts\.Transformer MLP blocks can mediate output\-relevant computations in language models\(Gevaet al\.[2021](https://arxiv.org/html/2608.03219#bib.bib44); Menget al\.[2022](https://arxiv.org/html/2608.03219#bib.bib45)\)\.

Recognition–generation failures were bounded but reproducible in the checkpoint census \(Figure[4](https://arxiv.org/html/2608.03219#Sx4.F4)a\)\.The predefined failure comprised correct recognition and incorrect direct generation; recognition was not treated as sampling reachability\. Thirty\-five of 303 records met this criterion, corresponding to 27 of 284 unique checkpoint–task cells\. The failure was bounded but recurred across checkpoints and tasks\.

Pruning created both sharply localized and diffuse MLP\-sensitive failures \(Figure[4](https://arxiv.org/html/2608.03219#Sx4.F4)b\)\.Magnitude pruning reproduced the same recognition\-correct, generation\-wrong pattern\. Llama\-3\.1\-8B showed a sharp 96\.6% repair peak at layer 20, whereas Qwen2\.5\-7B showed a broader response across layers 12 and 16\. The failure was therefore reproducible without implying universal single\-layer concentration\.

![Refer to caption](https://arxiv.org/html/2608.03219v1/x5.png)Figure 5:Selective causal localization to an identified MLP block\.a, Fraction of the predefined failure set repaired by silencing the identified MLP block, attention in the same layer, or the whole layer in six cases\.b, Failure\-set repair and healthy\-panel breakage under MLP silencing\.c, Directional top\-32 and random\-32 repair normalized by whole\-block repair; error bars show the random\-control standard deviation\. Labels give evaluation counts\.Silencing the identified MLP block repaired the predefined failure more consistently than silencing attention in the same layer \(Figure[5](https://arxiv.org/html/2608.03219#Sx4.F5)a; Table[4](https://arxiv.org/html/2608.03219#Sx4.T4)\)\.MLP\-block silencing repaired 67\.6–92\.0% of the evaluated failure set, and same\-layer attention silencing was lower in all six cases spanning 0\.5B to 31B\. Whole\-layer silencing was lower in five cases\. These comparisons identify the MLP block as the consistent component\-specific repair locus\.

The block intervention was selective for the predefined failure set and reciprocal across perturbation directions \(Figure[5](https://arxiv.org/html/2608.03219#Sx4.F5)b; Tables[4](https://arxiv.org/html/2608.03219#Sx4.T4)and[4](https://arxiv.org/html/2608.03219#Sx4.T4)\)\.MLP silencing broke only 0–12% of separate healthy panels while repairing 67\.6–92\.0% of the failure sets\. Opposite directional perturbations rescued 42\.5–97\.5% of failures and induced 10\.0–97\.5% failures in healthy panels\. This selectivity, together with reciprocal rescue and induction, supports case\-level causal localization rather than an indiscriminate output shift\.

Table 3:MLP\-block silencing selectively repairs predefined direct\-answer failures\.M/A/L denote MLP, attention, and whole\-layer repair rates \(%\)\. The final column gives healthy\-panel size and breakage\.
Table 4:Opposite directional perturbations provide reciprocal rescue and induction\.The predefined sick and healthy panels begin at 0% and 100% accuracy\. Rescue uses−α​v\-\\alpha vand induction uses\+α​v\+\\alpha v, withα∈\{2,4,8\}\\alpha\\in\\\{2,4,8\\\}; parentheses give the selected dose\.

Directional top\-32 units recovered most of the whole\-block effect in four cases, but within\-block organization was not uniformly sparse \(Figure[5](https://arxiv.org/html/2608.03219#Sx4.F5)c\)\.The directional subset recovered at least 87\.5% of whole\-block repair in four of six cases, but only 5\.6% and 34\.3% in the two exceptions\. The selected units carried 13\.5–19\.7% of output mass, while random controls remained small\. The effect can be directionally concentrated, but the data do not support a universal 32\-unit circuit\.

The MLP analyses localize this bounded failure without implying a universal layer or sparse circuit\.We next move from interventions on fixed checkpoints to weight updates and ask whether deployed training gains expand reachability or instead change the realization of answers already within reach\.

### RLVR and Reachability

![Refer to caption](https://arxiv.org/html/2608.03219v1/x6.png)

Figure 6:Question\-level fate after RLVR training\.a, Gains from base\-reachable and base\-unreachable questions, losses, and net deployed change\.b, Deployed and oracle changes in four question\-matched cells \(n=300n=300,K=200K=200\)\.Table 5:Matched checkpoint comparison\.D/O denotes deployed/oracle accuracy \(%\);Δ​nreachable\\Delta n\_\{\\mathrm\{reachable\}\}is the trained\-minus\-base reachable count\. All rows use matched DART prompts at temperature 0\.8\.![Refer to caption](https://arxiv.org/html/2608.03219v1/x7.png)
Figure 7:Base propensity predicts post\-RLVR realization\.Realization by base hit\-rate bin atK=200K=200with 95% Wilson intervals\.
Table 6:Sampling\-budget control\.Oracle reachability on DART atn=150n=150and temperature 0\.8\. B/T denotes base/trained\.
Table 7:Training\-recipe control\.DART accuracy \(%\) for one OLMo lineage atn=150n=150,K=64K=64, and temperature 0\.8\.
Table 8:Prompt\-format control\.Realization of base\-reachable but initially unproduced questions\. Retained is shared/native\.

RLVR gains in the four clean matched cells came from questions already reachable by the base model \(Figure[RLVR and Reachability](https://arxiv.org/html/2608.03219#Sx4.SSx3)a\)\.The clean analysis joined SimpleRL and DAPO checkpoints on DART and GSM8K atn=300n=300andK=200K=200\(Zenget al\.[2025](https://arxiv.org/html/2608.03219#bib.bib40); Yuet al\.[2026](https://arxiv.org/html/2608.03219#bib.bib41)\)\. All 11–68 newly produced answers per cell were base\-reachable, while 6–24 previously produced answers were lost\. The deployed gain therefore reweighted an existing reachable pool rather than monotonically accumulating correct answers\.

Deployed performance rose without a consistent increase in the reachable ceiling \(Figure[RLVR and Reachability](https://arxiv.org/html/2608.03219#Sx4.SSx3)b; Table[5](https://arxiv.org/html/2608.03219#Sx4.T5)\)\.Deployed performance increased by 1\.7–14\.7 points across the four cells, whereas oracle reachability was unchanged or lower\. For DAPO/DART, the two changes were\+14\.7\+14\.7and−13\.3\-13\.3points, corresponding to 40 fewer reachable questions\. Improved realization occurred in every clean cell; reachability expansion did not\.

A larger sampling budget did not recover the DAPO reachability loss \(Table[6](https://arxiv.org/html/2608.03219#Sx4.T6)\)\.DoublingKKleft the trained ceiling unchanged while the base ceiling rose slightly\. SimpleRL remained stable, so the DAPO contraction was not a low\-budget artifact within this range\.

Reachability changed with the training recipe \(Table[7](https://arxiv.org/html/2608.03219#Sx4.T7)\)\.Math\-only RLVR raised deployed performance but lowered oracle reachability\. Both SFT\-containing variants raised the ceiling, showing that the response was conditional on the training recipe\.

Prompt format explained part, but not all, of the measured realization gain \(Table[8](https://arxiv.org/html/2608.03219#Sx4.T8)\)\.A shared base prompt retained 41\.4–50\.0% of native\-format realization\. Formatting therefore contributed to the gain without accounting for the full checkpoint difference\.

Training most often realized answers that the base model already sampled frequently \(Figure[7](https://arxiv.org/html/2608.03219#Sx4.F7)\)\.Among base\-reachable but initially unproduced questions, 85\.2–92\.3% in the\>35%\>35\\%hit\-rate bin became produced after training\. Lower\-propensity bins generally had lower estimates, although sparse and intermediate bins were not strictly monotonic\. Training therefore favored answers the base model already sampled frequently rather than uniformly converting the reachable pool\.

Together, the matched fate and propensity results show that the audited gains changed realization without consistently expanding the reachable ceiling\.

## Conclusion

We separated answers that a model produced from those reached by a specified probe under matched conditions\. Layer routing exposed a selection gap, and causal interventions localized one bounded realization failure to an MLP block\. Across four clean RLVR comparisons, newly realized answers were already reachable in the corresponding base model, while the reachable ceiling remained flat or fell\. Improved realization is meaningful, but reporting should distinguish it from reachability expansion\.

## Acknowledgments

We thank Zhehong Ai for helpful discussions\.

## References

- Y\. Bisk, R\. Zellers, R\. L\. Bras, J\. Gao, and Y\. Choi \(2019\)PIQA: reasoning about physical commonsense in natural language\.External Links:1911\.11641,[Link](https://arxiv.org/abs/1911.11641)Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- P\. Cai, T\. Fang, X\. Li, Q\. Zeng, G\. Li, and J\. Chen \(2026\)Curriculum reinforcement learning can incentivize reasoning capacity in llms beyond the base model\.External Links:2606\.22317,[Link](https://arxiv.org/abs/2606.22317)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p4.1)\.
- M\. Chen, J\. Tworek, H\. Jun, Q\. Yuan, H\. P\. D\. O\. Pinto, J\. Kaplan, H\. Edwards, Y\. Burda, N\. Joseph, G\. Brockman,et al\.\(2021\)Evaluating large language models trained on code\.arXiv preprint arXiv:2107\.03374\.Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p2.1)\.
- H\. K\. Choi and S\. Li \(2026\)ModeX: evaluator\-free best\-of\-n selection for open\-ended generation\.External Links:2601\.02535,[Link](https://arxiv.org/abs/2601.02535)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p2.1)\.
- P\. Clark, I\. Cowhey, O\. Etzioni, T\. Khot, A\. Sabharwal, C\. Schoenick, and O\. Tafjord \(2018\)Think you have solved question answering? try arc, the ai2 reasoning challenge\.External Links:1803\.05457,[Link](https://arxiv.org/abs/1803.05457)Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- K\. Cobbe, V\. Kosaraju, M\. Bavarian, M\. Chen, H\. Jun, L\. Kaiser, M\. Plappert, J\. Tworek, J\. Hilton, R\. Nakano, C\. Hesse, and J\. Schulman \(2021\)Training verifiers to solve math word problems\.External Links:2110\.14168,[Link](https://arxiv.org/abs/2110.14168)Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- A\. Datta, M\. Marreddy, A\. Mehler, Z\. Zhao, and R\. Mamidi \(2026\)From early encoding to late suppression: interpreting llms on character counting tasks\.External Links:2604\.00778,[Link](https://arxiv.org/abs/2604.00778)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- J\. Deng \(2026\)Wrong before right: late rescue and interface failure in aligned language models\.External Links:2607\.04640,[Link](https://arxiv.org/abs/2607.04640)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- M\. Elhoushi, A\. Shrivastava, D\. Liskovich, B\. Hosmer, B\. Wasti, L\. Lai, A\. Mahmoud, B\. Acun, S\. Agarwal, A\. Roman, A\. Aly, B\. Chen, and C\. Wu \(2024\)LayerSkip: enabling early exit inference and self\-speculative decoding\.InProceedings of the 62nd Annual Meeting of the Association for Computational Linguistics \(Volume 1: Long Papers\),L\. Ku, A\. Martins, and V\. Srikumar \(Eds\.\),Bangkok, Thailand,pp\. 12622–12642\.External Links:[Link](https://aclanthology.org/2024.acl-long.681/),[Document](https://dx.doi.org/10.18653/v1/2024.acl-long.681)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p1.1)\.
- Gemma Team \(2026\)Gemma 4 technical report\.arXiv preprint arXiv:2607\.02770\.Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- M\. Geva, R\. Schuster, J\. Berant, and O\. Levy \(2021\)Transformer feed\-forward layers are key\-value memories\.InProceedings of the 2021 Conference on Empirical Methods in Natural Language Processing,M\. Moens, X\. Huang, L\. Specia, and S\. W\. Yih \(Eds\.\),Online and Punta Cana, Dominican Republic,pp\. 5484–5495\.External Links:[Link](https://aclanthology.org/2021.emnlp-main.446/),[Document](https://dx.doi.org/10.18653/v1/2021.emnlp-main.446)Cited by:[MLP Localization](https://arxiv.org/html/2608.03219#Sx4.SSx2.p1.1)\.
- T\. Glavas, J\. Chataoui, F\. Regol, W\. Jabbour, A\. Valkanas, B\. N\. Oreshkin, and M\. Coates \(2024\)Dynamic layer selection in decoder\-only transformers\.External Links:2410\.20022,[Link](https://arxiv.org/abs/2410.20022)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p1.1)\.
- A\. Grattafiori, A\. Dubey, A\. Jauhri, A\. Pandey, A\. Kadian, A\. Al\-Dahle, A\. Letman, A\. Mathur, A\. Schelten, A\. Vaughan,et al\.\(2024\)The llama 3 herd of models\.arXiv preprint arXiv:2407\.21783\.Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- S\. He, T\. Ge, G\. Sun, B\. Tian, X\. Wang, and D\. Yu \(2025\)Router\-tuning: a simple and effective approach for dynamic depth\.InProceedings of the 2025 Conference on Empirical Methods in Natural Language Processing,C\. Christodoulopoulos, T\. Chakraborty, C\. Rose, and V\. Peng \(Eds\.\),Suzhou, China,pp\. 1925–1938\.External Links:[Link](https://aclanthology.org/2025.emnlp-main.99/),[Document](https://dx.doi.org/10.18653/v1/2025.emnlp-main.99),ISBN 979\-8\-89176\-332\-6Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p1.1)\.
- A\. Heakl, M\. Gubri, S\. Khan, S\. Yun, and S\. J\. Oh \(2026\)Dr\.LLM: dynamic layer routing in LLMs\.InThe Fourteenth International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=3ow7tq0O3l)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p1.1)\.
- E\. Hossain, S\. Saha, T\. N\. Ornee, S\. S\. Jennifer, U\. C\. Biswas, S\. R\. Dipta, R\. Rana, and N\. Yousefi \(2026\)Right knowledge, wrong answer: characterizing parametric temporal conflict in open\-weight language models\.External Links:2606\.20959,[Link](https://arxiv.org/abs/2606.20959)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- C\. Huang, Q\. Zheng, R\. Zheng, L\. Zeng, and Y\. Xu \(2026\)PRISM edit: one vector for all temporal answers\.External Links:2607\.11327,[Link](https://arxiv.org/abs/2607.11327)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- P\. Kuang, H\. Jin, X\. Han, Y\. Wang, X\. Yuan, Y\. Yu, K\. Xu, and H\. Wang \(2026\)KV\-prm: efficient process reward modeling via kv\-cache transfer for multi\-agent test\-time scaling\.External Links:2607\.09153,[Link](https://arxiv.org/abs/2607.09153)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p2.1)\.
- F\. Laitenberger, D\. J\. Kopiczko, C\. G\. M\. Snoek, and Y\. M\. Asano \(2026\)What layers when: learning to skip compute in LLMs with residual gates\.InThe Fourteenth International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=aiP6XfaYZR)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p1.1)\.
- E\. Landesberg \(2026\)When llm judge scores look good but best\-of\-n decisions fail\.External Links:2603\.12520,[Link](https://arxiv.org/abs/2603.12520)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p2.1)\.
- L\. Li and A\. Talwalkar \(2020\)Random search and reproducibility for neural architecture search\.InProceedings of The 35th Uncertainty in Artificial Intelligence Conference,R\. P\. Adams and V\. Gogate \(Eds\.\),Proceedings of Machine Learning Research, Vol\.115,pp\. 367–377\.External Links:[Link](https://proceedings.mlr.press/v115/li20c.html)Cited by:[Routing and Selection](https://arxiv.org/html/2608.03219#Sx4.SSx1.p1.1)\.
- Z\. Li, Y\. Li, and T\. Zhou \(2026\)Skip a layer or loop it? learning program\-of\-layers in llms\.External Links:2606\.06574,[Link](https://arxiv.org/abs/2606.06574)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p1.1)\.
- J\. Liu, L\. Cui, H\. Liu, D\. Huang, Y\. Wang, and Y\. Zhang \(2020\)LogiQA: a challenge dataset for machine reading comprehension with logical reasoning\.External Links:2007\.08124,[Link](https://arxiv.org/abs/2007.08124)Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- N\. Liu \(2026\)LLMs as a jury: cross\-model consensus can outperform process reward models for llm reasoning\.External Links:2607\.10139,[Link](https://arxiv.org/abs/2607.10139)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p2.1)\.
- M\. Marina, D\. Moskovskiy, S\. Pletenev, M\. Salnikov, A\. Panchenko, and V\. Moskvoretskii \(2026\)Boosting self\-consistency with ranking\.External Links:2606\.05054,[Link](https://arxiv.org/abs/2606.05054)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p2.1)\.
- K\. Meng, D\. Bau, A\. Andonian, and Y\. Belinkov \(2022\)Locating and editing factual associations in gpt\.InAdvances in Neural Information Processing Systems,S\. Koyejo, S\. Mohamed, A\. Agarwal, D\. Belgrave, K\. Cho, and A\. Oh \(Eds\.\),Vol\.35,pp\. 17359–17372\.External Links:[Document](https://dx.doi.org/10.52202/068431-1262),[Link](https://proceedings.neurips.cc/paper_files/paper/2022/file/6f1d43d5a82a37e89b0665b33bf3a182-Paper-Conference.pdf)Cited by:[MLP Localization](https://arxiv.org/html/2608.03219#Sx4.SSx2.p1.1)\.
- M\. Nguyen, S\. Gupta, and H\. Le \(2026\)Beyond majority voting: efficient best\-of\-n with radial consensus score\.External Links:2604\.12196,[Link](https://arxiv.org/abs/2604.12196)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p2.1)\.
- P\. M\. Nguyen, C\. D\. La, D\. M\. H\. Nguyen, N\. V\. Chawla, B\. T\. Nguyen, and K\. D\. Doan \(2025\)The reasoning boundary paradox: how reinforcement learning constrains language models\.External Links:2510\.02230,[Link](https://arxiv.org/abs/2510.02230)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p4.1)\.
- T\. OLMo, P\. Walsh, L\. Soldaini, D\. Groeneveld, K\. Lo, S\. Arora, A\. Bhagia, Y\. Gu, S\. Huang, M\. Jordan,et al\.\(2024\)2 olmo 2 furious\.arXiv preprint arXiv:2501\.00656\.Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- J\. Park, S\. Choi, J\. Jeong, T\. Yoon, and J\. Lee \(2026a\)MechELK: a mechanistic interpretability framework for eliciting latent knowledge in large language models\.External Links:2605\.28825,[Link](https://arxiv.org/abs/2605.28825)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- Y\. Park, H\. Pyun, and Y\. Jo \(2026b\)Bridging the knowledge\-prediction gap in LLMs on multiple\-choice questions\.InForty\-third International Conference on Machine Learning,External Links:[Link](https://openreview.net/forum?id=I0RgZXTO53)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- Qwen Team \(2025\)Qwen2\.5 technical report\.External Links:2412\.15115,[Link](https://arxiv.org/abs/2412.15115)Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- D\. Raposo, S\. Ritter, B\. Richards, T\. Lillicrap, P\. C\. Humphreys, and A\. Santoro \(2024\)Mixture\-of\-depths: dynamically allocating compute in transformer\-based language models\.External Links:2404\.02258,[Link](https://arxiv.org/abs/2404.02258)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p1.1)\.
- Y\. Tong, X\. Zhang, R\. Wang, R\. Wu, and J\. He \(2024\)DART\-math: difficulty\-aware rejection tuning for mathematical problem\-solving\.External Links:2407\.13690,[Link](https://arxiv.org/abs/2407.13690)Cited by:[Models and tasks\.](https://arxiv.org/html/2608.03219#Sx3.SS0.SSS0.Px4.p1.1)\.
- X\. Wang, J\. Wei, D\. Schuurmans, Q\. V\. Le, E\. H\. Chi, S\. Narang, A\. Chowdhery, and D\. Zhou \(2023\)Self\-consistency improves chain of thought reasoning in language models\.InThe Eleventh International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=1PL1NIMMrw)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p2.1)\.
- X\. Wen, Z\. Liu, S\. Zheng, S\. Ye, Z\. Wu, Y\. Wang, Z\. Xu, X\. Liang, J\. Li, Z\. Miao, J\. Bian, and M\. Yang \(2025\)Reinforcement learning with verifiable rewards implicitly incentivizes correct reasoning in base llms\.External Links:2506\.14245,[Link](https://arxiv.org/abs/2506.14245)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p4.1)\.
- Z\. Yan, D\. Wei, D\. A\. Katz, P\. Sattigeri, and A\. Tajer \(2026\)Multi\-component causal tracing in large language models\.External Links:2606\.03085,[Link](https://arxiv.org/abs/2606.03085)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- X\. Yao, L\. Yu, X\. Hu, F\. Teng, Q\. Cui, J\. Zhou, and Y\. Liu \(2025\)The debate on rlvr reasoning capability boundary: shrinkage, expansion, or both? a two\-stage dynamic view\.External Links:2510\.04028,[Link](https://arxiv.org/abs/2510.04028)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p4.1)\.
- J\. Yeom, J\. Sok, H\. Kim, S\. Park, J\. Park, and T\. Kim \(2026\)Hallucination as commitment failure: larger llms misfire despite knowing the answer\.External Links:2605\.22007,[Link](https://arxiv.org/abs/2605.22007)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- Q\. Yu, Z\. Zhang, R\. Zhu, Y\. Yuan, X\. Zuo, Y\. Yue, W\. Dai, T\. Fan, G\. Liu, L\. Liu,et al\.\(2026\)DAPO: an open\-source LLM reinforcement learning system at scale\.Advances in Neural Information Processing Systems38,pp\. 113222–113244\.Cited by:[RLVR and Reachability](https://arxiv.org/html/2608.03219#Sx4.SSx3.p1.2)\.
- S\. Yuan, J\. Chen, J\. Zheng, M\. Li, L\. Feng, D\. Wang, T\. Xiang, T\. Liu, and B\. An \(2026\)Understanding diversity collapse in rlvr via the lens of overtraining\.External Links:2606\.15455,[Link](https://arxiv.org/abs/2606.15455)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p4.1)\.
- Y\. Yue, Z\. Chen, R\. Lu, A\. Zhao, Z\. Wang, Y\. Yue, S\. Song, and G\. Huang \(2025\)Does reinforcement learning really incentivize reasoning capacity in LLMs beyond the base model?\.InThe Thirty\-ninth Annual Conference on Neural Information Processing Systems,External Links:[Link](https://openreview.net/forum?id=4OsgYD7em5)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p4.1)\.
- W\. Zeng, Y\. Huang, Q\. Liu, W\. Liu, K\. He, Z\. Ma, and J\. He \(2025\)SimpleRL\-zoo: investigating and taming zero reinforcement learning for open base models in the wild\.External Links:2503\.18892,[Link](https://arxiv.org/abs/2503.18892)Cited by:[RLVR and Reachability](https://arxiv.org/html/2608.03219#Sx4.SSx3.p1.2)\.
- J\. Zhang, Q\. Lin, S\. Rajmohan, and D\. Zhang \(2025\)From reasoning to answer: empirical, attention\-based and mechanistic insights into distilled DeepSeek r1 models\.InProceedings of the 2025 Conference on Empirical Methods in Natural Language Processing,C\. Christodoulopoulos, T\. Chakraborty, C\. Rose, and V\. Peng \(Eds\.\),Suzhou, China,pp\. 3985–4002\.External Links:[Link](https://aclanthology.org/2025.emnlp-main.198/),[Document](https://dx.doi.org/10.18653/v1/2025.emnlp-main.198),ISBN 979\-8\-89176\-332\-6Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p3.1)\.
- L\. Zhou, S\. Shah, E\. Rodolà, and R\. Dessì \(2026\)Hard or just unreached? diagnosing the sampling blind spot in math\-reasoning difficulty estimation\.External Links:2606\.19636,[Link](https://arxiv.org/abs/2606.19636)Cited by:[Related Work](https://arxiv.org/html/2608.03219#Sx2.p4.1)\.