RLVR is a Kernel, Not a Function: Statistical Inference for pass@$k$ Crossovers

Chen Yang; Jun Chen

stat.MLSep 18, 2026

RLVR is a Kernel, Not a Function: Statistical Inference for pass@ $k$ Crossovers

Authors: Chen Yang, Jun Chen

Organizations: Department of Statistics, Texas A&M University · Division of Computational Biology, Department of Quantitative Health Sciences, Mayo Clinic

Abstract

Reinforcement learning with verifiable rewards (RLVR) often improves pass@1 while falling behind its base model at larger sampling budgets $k$ , a crossover read as evidence that RLVR only sharpens existing capability. We identify two limits to this reading. First, a visible crossing need not be statistically established: comparing models on the same prompts, we build confidence bands across sampling budgets $k$ that require evidence of both an early gain and a later loss. Across five public RLVR pairs no crossing is statistically established in the initial evaluations, while a 32k-token evaluation on fresh prompts locates a reversal with first loss between 11 and 61 samples; power analysis shows why failure to detect a crossing need not mean no crossing, and why more prompts can help more than more answers per prompt. Second, base success alone does not determine what RLVR does to a prompt: prompts with the same base success rate have different post-RL success rates, and these differences repeat across independent generation halves. The relationship is a conditional distribution---a Markov kernel---rather than a single curve, and fitting it predicts crossings in independent generations for the same prompts and corrects the simple model's power estimates. Theory further shows how losses on a minority of the hardest prompts can overturn an early lead even when training improves other prompts, separating evidence that a crossover exists from claims about what it means for capability.

Figures & tables

Figure 1: Four claims, not one: effects of the token limit. DeepScaleR vs. its base, 1000 mixture prompts, three per-response token limits:

\hat{D}(k)

with 95% pointwise and simultaneous (sup-

t

) bounds

L(k),U(k)

; the strip colours each

k

by the strongest supported claim. A visual sign change appears at 16k, dominance is rejected at 16k and 32k, and a statistically supported sign change (dark-red cell) exists only at 32k; the new-prompt confirmation is in Table 1 .

DeepScaleR: initial evaluation
pair / token limit	visual sign change	dominance rejected ( $p$ )	crossing supported	first-loss $\kappa\in$	follow-up status / design
8k	none	no (.99)	no	$[28,\infty)$	—
16k	$k{=}27$	yes, unadj. (.029) ∗	no	$[14,\infty)$	no (fresh 16k)
DeepScaleR: follow-up checks
16k fresh	$k{=}23$ (slice)	no (.91)	no	$[9,\infty)$	same token limit
32k mixture	$k{=}15$	yes (.009)	yes	$[9,33]$	adaptive

Table 1: Statistical evidence, grouped by the role of each measurement. The adaptive 32k crossing evidence is mixture-specific; the pre-specified 32k run confirms a crossing on fresh prompts. The fresh check at the same 16k limit does not replicate the original rejection. All bands and

\kappa

intervals are computed on the dense grid

k=1,\dots,K

(Appendix G ); the “visual sign change” column reports the smallest integer

k

at which the point estimate

\hat{D}(k)

is negative, “none” if it never is within

K

, and for the 16k fresh row the DeepScaleR-corpus slice (

m{=}250

; the pooled estimate stays positive throughout). ∗ Unadjusted; not family-wise significant under Holm over the five discovery tests (

\alpha{=}.05

, threshold

.01

). The 32k rows are a separate pre-specified family.

Figure 2: The operator is a kernel, not a function (ProRL-16k, $1000$ prompts, $n{=}n^{\prime}{=}256$ ). (a) $(\hat{p}_{i},\hat{q}_{i})$ against the best deterministic mean link; 40% of prompts fall outside its 95% binomial envelope (5% expected). (b) Residuals from two disjoint generation halves (cross-fitted mean removed); binomial noise is independent across halves, so the diagonal ( $\rho{=}.93$ ) is replicable heterogeneity. (c) The kernel fitted on generation half $A$ predicts half $B$ on the same prompts and vice versa: predicted crossing $67$ vs. $62$ held out; the inset magnifies the point crossings, not statistical evidence of a crossing.

Model pair	Fitted kernel $k^{\star}$	Same mean, no dispersion
DeepScaleR-16k	$27$	$267$
ProRL-16k	$58$	$845$
GRPO $G=4$	$140$	none
GRPO $G=16$	$168$	none

Table 2: Removing conditional dispersion removes the observed-window crossings. First positive-to-nonpositive crossings of fitted curves; “none” means none through

k=2\times 10^{5}

. These model comparisons are not statistical confirmation of population crossings. Full details are in Appendix E.2 .

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

pair (token limit, $n$ )	base	$\hat{\beta}_{\mathrm{proj}}$ [95% CI]	$\hat{\gamma}$	$\hat{\pi}_{0}$	crossover	model crossing	M2
DeepScaleR (16k, 256)	R1-D	1.26 [1.20, 1.33]	1.23	.08	$k{=}27$ ( $p{=}\mathbf{.029}$ )	[20, 43]	rej.
ProRL (16k, 256)	R1-D	1.58 [1.40, 1.77]	2.56	.23	$k{=}56$ ( $p{=}.20$ )	[55, 70]	rej.
LUFFY (3k, 128) †	Q-Math	2.93 [2.54, 3.38]	4.37	.03	none $\leq 127$	12	rej. ∗∗
ORZ (8k, 128) ‡	Q2.5	4.51 [3.97, 5.41]	14.5	.03	none $\leq 127$	143	rej.
SimpleRL-Zoo (8k, 128) ‡	Q2.5	1.00 [0.96, 1.05]	1.20	.00	none	none	not rej.

Appendix

Table 3: Five public checkpoint pairs (grid EM; prompt-level bootstrap CIs, 2,000 refits).

\hat{\pi}_{0}

: operational excess-zero index. M2: zero-inflation-aware affine-link LRT. DeepScaleR, ProRL, LUFFY and SimpleRL-Zoo use group-relative objectives; ORZ is PPO-based with a critic, so its

G

is nominal (§ 4 ). † Moderate / ‡ severe base-side truncation (ORZ token-limit-confounded; its extreme estimates are upper bounds). ∗∗ Off-policy distillation. “Model crossing” mixes two fits and is not the crossing implied by the

\hat{\beta}_{\mathrm{proj}},\hat{\gamma},\hat{\pi}_{0}

columns, which come from the grid EM with

F

a nonparametric (NPMLE) mixing law. The bracketed entries are

95\%

profile-likelihood sets computed under the

\mathrm{Beta}(a,b)

-parametrized working model of § B.5 (DeepScaleR-16k

[20,43]

; ProRL-16k

[55,70]

, whose own point estimate under that fit is

\hat{k}^{\star}{=}60

); the single numbers are point estimates of the model crossing under the grid EM (LUFFY

12

, ORZ

143

). The two fits differ in how

F

is modelled, so their crossings are not directly comparable — see the

G

-sweep in § D for the grid-EM crossing of ProRL-16k.

	coverage of true $\kappa_{\mathrm{full}}$ / fraction unbounded
$\beta$	$\pi_{0}=0$	$\pi_{0}=.1$	$\pi_{0}=.2$
$1.00$	$1.00\,/\,1.00^{\dagger}$	$.93\,/\,.00$	$.97\,/\,.00$
$1.05$	$.92\,/\,.21$	$.96\,/\,.00$	$.94\,/\,.00$
$1.20$	$.95\,/\,.00$	$.94\,/\,.00$	$.94\,/\,.00$
$1.40$	$.93\,/\,.00$	$.94\,/\,.00$	$.93\,/\,.00$

Appendix

Table 4: Profile-LR coverage of the full-model crossing

\kappa_{\mathrm{full}}

, including the atom-driven

\pi_{0}>0

\beta\approx 1

cells (300 replicates per cell,

m{=}1000

n{=}n^{\prime}{=}64

G{=}8

\alpha=.05

). Membership of

+\infty

is decided by the likelihood ratio against the no-crossing boundary region (

D_{\mathrm{model}}\geq 0

throughout), not by any finite-

\kappa

proxy. † At

\beta=1

\pi_{0}=0

the true crossing is

+\infty

; coverage is the fraction of sets containing it. At

\beta=1

\pi_{0}>0

the true crossing is finite and the set correctly closes (unbounded fraction

0

); the

21\%

unbounded rate at

\beta=1.05

\pi_{0}=0

is honest weak-identification uncertainty near the boundary, and coverage of the finite truth there is

.92

Figure 3: The inference machinery, checked against its own claims. Left: size of the dominance test (2,000 replicates,

m{=}400

K{=}6

B{=}999

; bars are

\pm 1.96

Monte Carlo standard errors). The top four cells vary the null mean : the test is nominal at the all-binding boundary and conservative everywhere inside

H_{0}

, confirming that

D\equiv 0

is the worst case among mean drifts. The bottom three fix

D\equiv 0

and vary only the nuisance correlation ; all three sit at nominal, which no procedure calibrated to a single “

D\equiv 0

law” could achieve — the prompt-level multiplier bootstrap re-estimates the covariance from the data at hand. Middle: level and power against the number of prompts; the configuration fitted to ProRL-16k needs

m\approx 1000

for power near one, while a genuinely shallow crossing is never detected at these sample sizes. Right: coverage of three interval constructions for

\log k^{\star}

m{=}2000

. At the weak-identification boundary

\beta{=}1

the delta method covers

0\%

and the Wald-form Fieller set

53\%

, while profile-likelihood inversion holds

.95

; the

m{=}500

cells behave the same way.

Figure 4: Source-specific analysis of the 32k DeepScaleR result. Left: source-specific

D(k)

curves. Right: their contrast. Shading uses one 95% critical value jointly over both curves, the contrast, and all

k

. Both point estimates cross, but neither source has a statistically supported crossing separately; only the pre-specified 50/50 mixture does.

Figure 5: GRPO and SFT produce different response curves; group size has little effect. Monotone Bernstein-EM fits over the central 20–80% difficulty range: (a) success-rate change

\hat{T}(p)-p

, with average logit slopes in the legend; (b) local logit slope; (c) GRPO

-

SFT contrasts with 95% simultaneous bands. The supported ordering is integrated, not pointwise over the whole range.

pair	$m$	$S$	$\bar{S}_{\mathrm{null}}$	$z_{\mathrm{adj}}$	placebo $z$	$p_{\mathrm{det}}$	$p_{\mathrm{zi}}$	$\widehat{\mathrm{sd}}$
DeepScaleR-16k	1000	.0119	.0001	8.4	1.0	.001	.001	.108
ProRL-16k	1000	.0251	.0039	8.3	1.0	.001	.001	.146
DeepScaleR-8k	1000	.0143	.0005	9.4	0.8	.001	.001	.117
SimpleRL-Zoo	1000	.0020	.0003	7.2	0.8	.001	.001	.041
SimpleRL-Zoo-7B	1000	.0482	.0366	4.8	$-0.3$	.001	.001	.108
GRPO $G{=}4$	1000	.0615	.0390	6.8	$-0.6$	.001	.001	.150

Appendix

Table 5: Split-generation residual covariance test.

\bar{S}_{\mathrm{null}}

: mean of

S

under the fitted deterministic-spline null — the sign-definite bias term of the calibration paragraph; it ranges from

1\%

76\%

S

, largest where the fitted deterministic links are most contorted.

z_{\mathrm{adj}}=(S-\bar{S}_{\mathrm{null}})/\hat{\rm se}(S)

; inference is by the calibrated

p_{\mathrm{det}}

p_{\mathrm{zi}}

(

B{=}800

parametric-bootstrap replicates;

.001

is the resolution floor

1/(B{+}1)

), not by

z

\widehat{\mathrm{sd}}=\sqrt{S-\bar{S}_{\mathrm{null}}}

estimates the root-mean-square dispersion

\smash{\sqrt{\mathbb{E}\,\mathrm{Var}(Q\mid P)}}

— a global summary, not a pointwise function. The 170-prompt hard-tail slices (

n^{\prime}{=}2048

) are omitted: their unadjusted

z\approx 1.9

equals the placebo’s and we treat them as uninformative rather than confirmatory.

pair	$\hat{\beta}$	$\hat{\pi}_{0}$	$k^{\star}$	$\hat{\phi}$	$\hat{\beta}$	$\hat{\pi}_{0}$	$k^{\star}$	$\Delta\ell$
	$\phi=\infty$ (working model)			$\phi$ profiled
DeepScaleR-16k	1.26	.085	36	6.5	0.88	.028	54	$+2969$
ProRL-16k	1.58	.232	32	1.9	0.72	.120	80	$+6257$
GRPO $G{=}4$ ( $n{=}256$ )	2.08	.000	13	1.4	0.58	.000	85	$+13624$

Appendix

Table 6: Beta-binomial refit: the working model (

\phi{=}\infty

) against the same model with the conditional precision

\phi

profiled inside the grid EM.

k^{\star}

is the model-implied crossing under each fit;

\Delta\ell

is the log-likelihood gain from profiling

\phi

, computed in-sample on the full counts — not the held-out quantity of Table 8 , which carries the same symbol but scores untouched generations.

pair	kernel $k^{\star}$	mean-collapsed $k^{\star}$	degraded mass	lock-in bin	$\Pr(Q<10^{-3})$
DeepScaleR-16k	$27$	$267$	$.045$	$[10^{-4},10^{-3})$	$1.00$
ProRL-16k	$58$	$845$	$.049$	$[10^{-4},10^{-3})$	$.97$
GRPO $G{=}4$	$140$	none	$.044$	$[0,10^{-4})$	$.948$
GRPO $G{=}16$	$168$	none	$.056$	$[0,10^{-4})$	$.952$

Appendix

Table 7: Mean-collapse check.

k^{\star}

values are first positive-to-nonpositive crossings on the integer grid

k\leq 2\times 10^{5}

; “none” means no crossing there. “Degraded mass” is the total fitted

p

-marginal mass on nodes with

\mathbb{E}[Q\mid P{=}p]<p

— the

4

–

6\%

quoted in § 5 . The last two columns give the lowest base-difficulty bin whose conditional mass below

q=10^{-3}

exceeds

.9

, and that mass; the bin differs between the public pairs and the controlled arms because their

p

-marginals do.

pair	det-spline	zi-affine	zi-BB $\pm$ se	kernel $\pm$ se	$\hat{\phi}$	BB $\hat{\pi}_{0}$
DeepScaleR-16k	$-88$	$-80$	$998\pm 100$	$967\pm 100$	6.3	.025
ProRL-16k	$-6$	$-1612$	$562\pm 56$	$515\pm 56$	1.9	.127
DeepScaleR-8k	$29$	$-28$	$299\pm 36$	$275\pm 36$	5.4	.063
SimpleRL-Zoo	$6$	$0$	$29\pm 8$	$7\pm 8$	68.0	.003
SimpleRL-Zoo-7B	$0$	$-1238$	$361\pm 34$	$340\pm 35$	1.9	.000
GRPO $G{=}4$	$0$	$-3051$	$1054\pm 69$	$1058\pm 72$	1.4	.000

Appendix

Table 8: Held-out generation prediction: total

\Delta\log p(d_{B}\mid c,d_{A})

relative to the zero-inflated spline model (positive favours the row model); prompt-level standard errors for the two heterogeneity models.

\hat{\phi}

: profiled beta-binomial precision (large

=

near-deterministic); BB

\hat{\pi}_{0}

: the atom remaining once dispersion is allowed. h 170-prompt hard-tail slices (

n^{\prime}{=}2048

): the split test is uninformative there, but held-out prediction still favours dispersion.

pair	$A\to B$ pred./obs.	$B\to A$ pred./obs.	cross-fitted pred./obs.	mean fold curve MAE
DeepScaleR-16k	$26/32$	$33/25$	$29/28$	$.0062$
ProRL-16k	$74/56$	$61/68$	$67/62$	$.0013$
GRPO $G{=}4$	$--/--$	$--/115$	$--/--$	$.0015$
GRPO $G{=}16$	$--/--$	$--/--$	$--/--$	$.0025$
SFT	$--/--$	$--/--$	$--/--$	$.0015$
SimpleRL-Zoo	$--/--$	$--/--$	$--/--$	$.0063$

Appendix

Table 9: Two-fold generation-split validation of model-implied crossings. Entries are the first positive-to-nonpositive crossing; “–” means no crossing through the available held-out window (

k=127

, except SimpleRL-Zoo at

k=63

). The two public crossings survive without count reuse: the cross-fitted ensemble misses their held-out locations by

1

and

5

draws. Across the unaveraged directions,

11/12

predictions match held-out crossing presence; the exception is one shallow GRPO

G{=}4

fold. The lower block runs the spline-link beta-binomial through the same protocol, and is the criterion on which we prefer the kernel for

D(k)

and

k^{\star}

: its cross-fitted crossings sit

5

and

13

draws from the held-out values against the kernel’s

1

and

5

, and on the cross-fitted curves its MAE is

.0049

and

.0042

against the kernel’s

.0004

and

.0005

. On mean fold MAE the two tie on DeepScaleR-16k (

.0062

each, to four decimals) and the kernel leads on ProRL-16k. This ordering is the reverse of the per-prompt held-out likelihood of Table 8 , where the beta-binomial wins; the two criteria target different estimands and we report both. MAE compares each prediction with its untouched fold over every integer

k

in that pair’s available window, then averages the two direction-specific MAEs; it is not the smaller error obtained after fold errors cancel in the pooled curves.

pair	$m$	pass@1	trunc.	$z$	placebo $z$	cal. $p$	$\Delta\ell$	held-out $\Delta$
		base $\to$ RL	base/RL				(kernel)	kernel / beta-bin.
SimpleRL Llama-3.1-8B	$1000$	$.079\to.159$	$.17/.16$	$12.2$	$0.3$	$\leq.0025$	$+637$	$+66\pm 18$ / $+89\pm 18$
SimpleRL DeepSeek-Math-7B	$1000$	$.033\to.075$	$.47/.39$	$7.1$	$1.4$	$\leq.0025$	$+177$	$+32\pm 14$ / $+50\pm 13$

Appendix

Table 10: Non-Qwen replication.

z

: studentised split-generation residual covariance; placebo

z

is the same statistic on independent base quarters, which carries no operator signal. Calibrated

p

is the smaller of the deterministic-spline and zero-inflated-spline parametric-bootstrap

p

-values, and

.0025

is the floor at

400

null draws.

\Delta\ell

: kernel minus zero-inflated-spline log-likelihood. Held-out

\Delta

: gain over the zero-inflated spline on untouched generation halves,

\pm 1

SE.

checkpoint	public position	test $p$	95% set for $k^{\star}$	verdict
DeepScaleR	inversion-free at 8k c	$.029$ (16k)	$[20,43]$	dom. rejected at 16k b
ProRL	boundary expansion, no inversion ( Liu et al., 2025 )	$.20$ (16k)	$[55,70]$	neither claim established
SimpleRL-Zoo	zero-RL family; base superiority at large $k$ ( Yue et al., 2025 )	$1.0$	excl. $k^{\star}{\leq}2{\times}10^{4}$	no crossover
ORZ	zero-RL family ( Yue et al., 2025 )	$1.0$	n/a a	no crossing; confounded
LUFFY	off-policy; gains over on-policy RLVR	$1.0$	n/a a	no crossing $k{\leq}127$ ; out of scope

Appendix

Table 11: Re-analysis of published positions (dominance test of Theorem 1 , 1000 regenerated paired prompts; Fieller sets under the working model). a Set not reported where the model is rejected (LUFFY) or the pair is token-limit-confounded (ORZ). b Not family-wise significant (Holm threshold

.01

); the crossing criterion gives

\kappa\in[14,\infty)

; the fresh-prompt confirmation failed (dominance

p=.91

). c The public position is the absence of a reported inversion at an 8k token limit; the

p{=}.99

quoted elsewhere for this cell is our own 8k dominance test (Table 1 ), not an external claim.

Result	Pre-committed treatment
$\pi_{\varepsilon}$ (the realized excess-zero mass $\rho(p)=\pi_{0}(1-p)^{G}$ ) falls with $G$ as $(1-p)^{G}$ predicts	group-size hazard promoted to mechanism evidence
$G$ -sweep flat or irregular	“tied to $G$ ” language removed; zero-inflation kept as descriptive sensitivity component
weak-KL raises $\hat{\gamma}$ with $\hat{\beta}$ stable (primary contrast $\Delta\gamma>0$ ; realized KL reported)	supports reward-pressure/tempering knob separation
weak-KL moves both $\hat{\beta},\hat{\gamma}$	coordinates reported as empirical, no one-to-one mechanism map claimed
ProRL/DeepScaleR deep-resample candidates mostly revive	“support loss” removed from abstract; $\pi_{0}$ reported as finite-resolution near-zero mass
candidates still zero at $n{=}2048$	only exact binomial upper bounds reported ( $q<1.5\times 10^{-3}$ (95% one-sided exact binomial) at $0/2048$ ); no exact-zero claim

Appendix

Table 12: Pre-specified decision rules and the treatment committed to for each possible outcome, fixed before the corresponding results were examined.

population / repair	$m$	$\inf\hat{D}(1)$	$L(1)$	$\sup\hat{D}(128)$	$U(128)$	$\kappa\in$
as reported	$1060$	$.07596$	$.06547$	$-.01981$	$-.00370$	$[11,61]$
$\leq 5\%$ truncation, both sides	$647$	$.04043$	$.03198$	$-.00927$	$.00028$	$[3,25]$
zero truncation, both sides	$342$	$.01834$	$.01034$	$-.00292$	$.00463$	$[2,\infty)$
$f=0$ , joint repair band	$1060$	$.07596$	$.06421$	$-.01981$	$-.00177$	$[11,73]$
$f=.25$ , failed base truncations	$1060$	$.05709$	$.04562$	$-.10914$	$-.08239$	$[3,4]$
$f=1$ , failed base truncations	$1060$	$.00048$	$-.01361$	$-.11887$	$-.09039$	$[1,2]$

Appendix

Table 13: Truncation sensitivity under failed-output repair. The first three rows use separate bands. The repair rows use a common band jointly over

f

k

, and both admissible overlap endpoints. Repair applies only to failed base truncations. The displayed point values are the minimum

\hat{D}(1)

and maximum

\hat{D}(128)

over missing-overlap completions;

L

and

U

enclose both endpoints. The final row fails the early-positive requirement, although it bounds the first loss from above. “

\infty

” means no supported upper endpoint.

U(k)

need not be monotone: the low-truncation row has negative

U(25)

but positive

U(128)

$k$	$\hat{D}(k)$	$L(k)$	$U(k)$	sign
1	$.0760$	$.0655$	$.0864$	$+$
2	$.0524$	$.0422$	$.0625$	$+$
4	$.0308$	$.0214$	$.0402$	$+$
8	$.0145$	$.0052$	$.0237$	$+$
12	$.0077$	$-.0018$	$.0172$
16	$.0038$	$-.0060$	$.0136$

Appendix

Table 14: 32k confirmation on the pre-specified fresh population (

m{=}1060

n{=}n^{\prime}{=}128

\hat{D}(k)=\text{pass@}k^{\mathrm{RL}}-\text{pass@}k^{\mathrm{base}}

with a

95\%

sup-

t

simultaneous band (

B{=}4000

). “

+

” marks

L>0

and “

-

” marks

U<0

; the crossing criterion requires both, in that order. Band computed on the dense grid

k=1,\dots,128

; the pre-registered run used the sparse grid

\{1,2,3,4,6,8,12,16,24,32,48,64,96,128\}

and gave

\kappa\in[9,64]

(Appendix G ); densifying raises the critical value by

<0.001

and tightens the interval. Rows show a subset of the grid.

pair	power at realized design	observed $p$	cheapest design for power $.80$ (total generations)
DeepScaleR-16k	.63	.029	$1.0$ M ( $m{=}4000$ , $n{=}128$ )
ProRL-16k	.21	.20	$4.1$ M ( $m{=}8000$ , $n{=}256$ )
GRPO $G{=}8$	.07	$\geq.20$	$16.4$ M ( $m{=}32000$ , $n{=}256$ )
GRPO $G{=}4$	.01	$\geq.20$	none on the grid

Appendix

Table 15: Kernel-implied power of the dominance test at

\alpha{=}.05

m{=}1000

n{=}n^{\prime}{=}256

(1000 replicates per cell), and the cheapest design on a

(m,n)

grid that reaches power

.80

. The GRPO rows are the controlled training arms of § 6 .

	power at realized design		cheapest design for power $.80$
pair	$\alpha{=}.05$	$\alpha{=}.01$	$\alpha{=}.05$	$\alpha{=}.01$
DeepScaleR-16k	$.630$	$.358$	$1.0$ M ( $m{=}4000$ , $n{=}128$ )	$1.0$ M ( $m{=}4000$ , $n{=}128$ )
ProRL-16k	$.207$	$.074$	$4.1$ M ( $m{=}8000$ , $n{=}256$ )	$4.1$ M ( $m{=}8000$ , $n{=}256$ )
GRPO- $G{=}8$	$.074$	$.016$	$16.4$ M ( $m{=}32000$ , $n{=}256$ )	$16.4$ M ( $m{=}32000$ , $n{=}256$ )
GRPO- $G{=}4$	$.014$	$.003$	—	—

Appendix

Table 16: Kernel-simulated power of the dominance test and cheapest design reaching

.80

. These are an upper bound on the power to meet the two-sided crossing criterion of Proposition 1 , which additionally requires

L(j)>0

at some

j

and

U(\ell)<0

at some

\ell>j

. Cost is total generations

m(n+n^{\prime})

;

1000

replicates per grid cell. “—” means no grid cell reaches

.80

. Realized designs are

m{=}1000

n{=}n^{\prime}{=}256

. The cheapest design coincides on this grid; at DeepScaleR-16k’s cell (

m{=}4000

n{=}128

) power is

.967

\alpha{=}.05

and

.877

\alpha{=}.01

Reinforcement Learning with Verifiable Rewards

Reinforcement learning with verifiable rewards trains models using outcomes checked by code, proofs, simulators, or structured criteria.

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The Invisible Leash: Why RLVR May or May Not Escape Its Origin
Jul 20, 2025Fang Wu, Weihao Xuan, Danny Xing +5RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards
Stanford University, CA, USA · The University of Tokyo, Japan · RIKEN AIP, Tokyo, Japan +2

Oct 29, 2026J/K move · Enter open · S save

There Is No Free Lunch in RLVR: Every pass@1 Gain Has a pass@k Price

Yinuo Liu

RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards

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Reinforcement learning with verifiable rewards (RLVR) makes a language model solve more problems on the first try and fewer when given many tries: it raises pass@1 while the base model overtakes it at large sampling budget k. Is this a failure of RL to add reasoning, or a transient that more training removes? We settle the question with a theorem. The reversal is a price, and we compute its exact rate. Modeling RLVR as an exponential tilt of a base policy over strategies shared across prompts, we prove that under exclusive specialization every update that raises pass@1 strictly lowers the success of some prompt; a strictly positive price on prompts (Stiemke duality) is the exact condition, and being merely the best strategy on a prompt is provably not enough. When the loss reaches the hardest prompts, it forces a pass@k crossover, and the number of crossings is at most the sign variation of the shift in per-prompt success, a totally positive bound that is tight and that recovers the known single-crossover result as its order-preserving case. The crossover is not a bug to patch: a constrained update traces the exact pass@1 versus pass@k frontier, and removing the crossover entirely costs the entire pass@1 gain. The same tilt identity yields entropy collapse and the entropy-performance law with an identifiable exponent, unifying two literatures under one mechanism. Reinforcement cannot widen the reasoning frontier; it can only move along it, at a price we make measurable by a diagnostic that reads the bound, the order-preserving test, and the price from any evaluation log. There is no free lunch, only a price.

Abstract

Reinforcement Learning with Verifiable Rewards

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