q103/random_k2/random_1.md

		When Can	LLMs	Learn	to Reason	with	Weak	Supervision?
SalmanRahman12* JingyanShen2* AnnaMordvina2 HamidPalangi3 SaadiaGabriel1 PavelIzmailov2
		❖Projectpage:		salmanrahman.net/rlvr-weak-supervision
-----	--------	-------------	-------------	--------------------------------------	---	---------------------------------------------------	---	----------------------	---
		Abstract				pabilitiesinlargelanguagemodels(Guoetal.,2025;Jaech
						etal.,2024;Teametal.,2025).		Withonlybinaryfeedback
Large	language	models	have achieved	signifi-
6202 rpA 02 ]GL.sc[ 1v47581.4062:viXra
cantreasoningimprovementsthroughreinforce- oncorrectness,RLVRhasenabledsubstantialgainsacross
diversereasoningtaskswithoutrequiringdensesupervision.
ment	learning	with verifiable	rewards	(RLVR).
----	--------	---------------	-------	-------	---	---	---	---	---
Yetasmodelcapabilitiesgrow,constructinghigh- However,recentfindingssuggesttheseimprovementsmay
qualityrewardsignalsbecomesincreasinglydif- bedrivenbyfactorsotherthantheintegrationofcorrectness
						signals.	SomestudiesreportthatRLVRsucceedsevenun-
-------	------	------------	-------------	----	---	--------	----------------------------------------	---	---
ficult,	making	it essential	to understand	when
RLVR can succeed under weaker forms of su- derextremeconditions: trainingonjustasingleexample
canyieldsignificantgains(Wangetal.,2025a),andrandom
pervision.		We conduct	a systematic	empirical
----------	---	----------	------------	---------	---	---	---	---	---
studyacrossdiversemodelfamiliesandreasoning orincorrectrewardssometimesmatchground-truthperfor-
domains under three weak supervision settings: mance (Shao et al., 2025). Other work shows that proxy
signalssuchasself-certainty(Zhaoetal.,2025;Prabhudesai
scarce	data,	noisy rewards,	and self-supervised
------	-----	--------------	-------------------	---	---	---	---	---	---
proxy rewards. We find that generalization is etal.,2025),entropyminimization(Agarwaletal.,2025),
majorityvoting(Zuoetal.,2025),orself-generatedtraining
governedbytrainingrewardsaturationdynamics:
models that generalize exhibit a prolonged pre- data(Huangetal.,2025)canreplaceverifiablerewards.
saturationphaseduringwhichtrainingrewardand
Furthermore,techniquesthatsucceedononemodelfamily
downstream performance climb together, while oftenfailonothers(Shaoetal.,2025),underreportedbase-
modelsthatsaturaterapidlymemorizeratherthan
linesmayinflateperceivedbenefits(Chandaketal.,2025),
learn. Weidentifyreasoningfaithfulness,defined and prolonged training with proxy rewards (i.e., reward
astheextenttowhichamodel’sintermediatesteps
						signals derived	from	model outputs without	ground-truth
---	---	---	---	---	---	---------------	----	---------------------	------------
logicallysupportitsfinalanswer,asthepre-RL
verification)canleadtorewardhackingandperformance
propertythatpredictswhichregimeamodelfalls collapse(Shafayatetal.,2025). Thesemixedresultsleavea
into,whileoutputdiversityaloneisuninformative.
						fundamentalquestion:		WhencanRLVRgeneralize1under
---	---	---	---	---	---	--------------------	---	---------------------------	---
Motivatedbythesefindings,wedisentanglethe weaksupervision,andwhatdeterminessuccessorfailure?
contributionsofcontinualpre-trainingandsuper-
visedfine-tuning,findingthatSFTonexplicitrea- UnderstandingwhenRLVRworksunderweaksupervision
mattersforpractice.Ground-truthverifiersareoftenlimited:
soningtracesisnecessaryforgeneralizationunder
weaksupervision,whilecontinualpre-trainingon labelsmaybenoisyorunavailable,andasmodelsbecome
domaindataamplifiestheeffect.Appliedtogether strongerthantheirsupervisors,alternativerewardsignals
becomenecessary(Burnsetal.,2023).
toLlama3.2-3B-Base,theseinterventionsenable
generalizationacrossallthreesettingswherethe
WeconductasystematicempiricalstudyofRLVRunder
basemodelpreviouslyfailed. weak supervision across two model families (Qwen and
						Llama),	and three	reasoning domains (MATH,	SCIENCE,
---	---	---	---	---	---	----------	---------------------------------------	------------------------	--------
						andGRAPH).	Ourworkisorganizedaroundthreequestions:
1.Introduction
						• RQ1(WeakSupervision):DoesRLVRgeneralizeacross
---	---	---	---	---	---	-----------------------------------------------	---	---	---
Reinforcementlearningwithverifiablerewards(RLVR)has
modelfamiliesanddomainsunderscarcedata,noisyre-
emergedasapowerfulparadigmforimprovingreasoningca-
wards,andself-supervisedproxyrewards?
*Equal 1University • RQ2(ModelProperties): Whatpre-RLmodelproper-
	contribution		of	California,	Los An-
-----	------------	---------------	--------	--------------	-------	---	---	---	---
	2New		3Google.
geles		York University		Correspondence
1Throughout,weusegeneralizationtomeanimprovementon
to: Salman	Rahman	salman@cs.ucla.edu,		Jingyan	Shen
----------	------	---------------------	---	-------	----	---	---	---	---
downstreamevaluationbenchmarks,bothin-domainheld-outsets
js15262@nyu.edu.
andout-of-domaintransfer,followingRLtraining.
Preprint.April21,2026.
1

LLMReasoningwithWeakSupervision tiesdeterminewhetheramodelgeneralizesunderweak 2.ExperimentalSetup supervision?

							We evaluate	the	following	model families:		(1) Qwen2.5-
• RQ3(Intervention):			Howcanweenablegeneralization
							1.5B/3B(Base):		General-purposemodelspretrainedon18
inmodelsthatfailunderweaksupervision?
trilliontokens(Team,2024);(2)Qwen2.5-Math-1.5B/7B
Ourinvestigationuncoversthreefindings. First,general- (Math-specialized): BuiltuponQwen2.5withanadditional
izationunderweaksupervisionisgovernedbytraining 1trillionmath-relatedtokens(Yangetal.,2024);(3)Llama-
rewardsaturationdynamics. Modelsthatgeneralizeex- 3.2-3B/8B-Instruct(Instruction-tuned): Pretrainedon9
hibitaprolongedpre-saturationphaseduringwhichtraining trilliontokensandalignedviaSFT,rejectionsampling,and
							DPO(Dubeyetal.,2024).			WeusetheInstructvariantsfor
---	---	---	---	---	---	---	---------------------	---	---	---------------------------	---	---	---
rewardclimbssteadilyandthemodellearnstransferablerea-
soningpatterns;modelsthatfailsaturaterapidlyandentera Llamabecausethebasemodelsdonotreliablyfollowthe
post-saturationphasewherefurthertrainingyieldsdiminish- required formatfor on-policy rollouts. We revisit Llama-
ingreturns. Whichregimeamodelfallsintodependsonits Basein§4,whereSFThandlestheformat-followingissue.
pretrainingpriors:		modelswithstrongdomain-alignedpre-
------------------	---	----------------------------------	---	---	---	---	-------------------	---	-----------------------------	---	---	---	---
							DomainsandDatasets.		Weselectthreedomainswithvary-
training(QwenonMATHandSCIENCE)sustainextended
							inglevelsofpretrainingexposure:				MATH(highexposure),
---	---	---	---	---	---	---	-------------------------------	---	---	---	-------------------	---	---
pre-saturationphasesandgeneralizeunderscarcedata,noisy
SCIENCE(moderatecoverage)andGRAPHtasks(underrep-
rewards,andself-supervisedproxyrewards,whilemodels
							resentedintypicalpretrainingcorpora).				WeuseSkywork-
---	---	---	---	---	---	---	-------------------------------------	---	---	---	-------------	---	---
withoutsuchpriors(Llamaacrossalldomains,andQwenon
OR1(Heetal.,2025a)forMATH,SCPdatasets(Liuetal.,
GRAPH)saturaterapidlyandfailtogeneralizeevenunder
							2025a; Lu	et al.,	2025) spanning	physics,		chemistry,	and
-------------------	---	--------------------------------	---	---	---	---	---------	-------	--------------	--------	---	----------	---
moderatelabelnoise.		Wetreatthemodel-familycontrastas
biologyforSCIENCE,andtasksfromReasoningGym(Sto-
aproxyforpretraining-priorstrengthratherthananintrin-
janovskietal.,2025)involvingdiscretealgorithmicreason-
sicpropertyofeitherfamily,areadingthat§4confirmsby
							ingforGRAPH.	ForMathandScience,weusethe1.5B/3B
---	---	---	---	---	---	---	------------	---------------------------------	---	---	---	---	---
showingthatcontinualpre-trainingonmathdatatransforms
modelsasourprimaryexperimentsandadditionallyevalu-
Llama’sRLbehaviortoresembleQwen’s.
ate7B/8Bmodelstoverifythatourfindingsholdatlarger
Second,reasoningfaithfulness,notoutputdiversity,dis- scale. For GRAPH, we only use the 7B/8B variants be-
tinguishesmodelsthatgeneralizefrommodelsthatmem- cause the smaller models achieve solve@16 = 0, leaving
orize. Anaturalhypothesisforrapidsaturationisthatfail- noinformativesignalforRL.Moredetailsareprovidedin
ing models	lack	exploratory		capacity.	We find	the oppo-	AppendixB.
----------	----	-----------	---	---------	-------	---------	----------	---	---	---	---	---	---
site: Llamamodelsreachperfecttrainingrewardfasterthan
							Model-AwareDataFiltering.			Toensureinformativetrain-
---	---	---	---	---	---	---	-------------------------	---	---	-------------------------	---	---	---
Qwenandmaintainhigheroutputdiversitythroughouttrain-
ingsignals,weimplementmodel-specificdifficultyfiltering.
ing, yet	they generalize		poorly.	The	missing	property is
--------	---------------	---	-------	---	-------	-----------	---	---	---	---	---	---	---
Foreachproblem,wesample16responsesandcountcorrect
reasoningfaithfulness,definedbywhetheramodel’sinter-
							solutions	(solve@16	∈ [0,16]).	We	retain	only problems
-------------------------------------------	---	---	---	---	---	----------	---------	---------	----------	---	------	-------------	---
mediatestepslogicallysupportitsfinalanswer.						Modelsthat
wheresolve@16∈[1,15],effectivelydiscardinginstances
saturate	rapidly	produce	correct	answers	through	reason-
--------	-------	-------	-------	-------	-------	-------	---	---	---	---	---	---	---
thatareeithertrivialorintractableforthemodel,stratified
ingchainsthatdonotjustifythem,memorizingratherthan
							equally	across difficulty	levels	(details	in	Appendix	B.2).
---------	---------	---	----------------	---	----	----------	-------	-----------------	------	--------	---	--------	-----
learning.	Diversity	is	only informative		when	considered
Thisfilteredsetservesasthecandidatepoolforallweak
jointlywithfaithfulness.
supervisionsettingsstudiedinthiswork;wedescribehow
Third,SFTonexplicitreasoningtracesisnecessaryfor trainingdataisconstructedfromthispoolforallsettings
generalizationunderweaksupervision,					andcontinual		in§3.
----------------------------------------------	---------	---	---	-------	------------	----------	----------------------	---	-----------------------	---	---	---	---
pre-training	amplifies		the	effect.	We run a	controlled
							TrainingConfiguration.		WeuseGRPO(GroupRelative
comparisonthatdisentanglesthetwointerventions,						train-
PolicyOptimization)asourRLalgorithm(Shaoetal.,2024).
ing Llama3.2-3B		Base,	a continually		pre-trained	variant
---------------	---	-----	-------------	---	-----------	-------	---	---	---	---	---	---	---
ForeachqueryqsampledfromtrainingdatasetsD,agroup
(CPT, ours),	and	Instruct,	each	with	either Thinking	SFT
---------------------------------------------------	---	---------	----	----	---------------	---	-----------------------	-----------	----	-----------------------	---	-------------	---
							ofindividualresponses{o		}G	aresampledfromthepolicy
(explicitreasoningtraces)orNon-ThinkingSFT(finalso-									i	i=1
							π before	the update.	GRPO	maximizes		the following
θold
lutionsonly).	ThinkingSFTisnecessary:				itimprovesrea-
-------------	-----------------------	---	---	---	--------------	---	---	---	---	---	---	---	---
objective:
soningfaithfulness,extendsthepre-saturationphase,and (cid:104)
(θ)=E
enablesgeneralizationunderallthreeweaksupervisionset- J GRPO (q,a)∼D,{oi}G
i=1 ∼πθold (·	q)
							G		oi
------------------------------------------------	---	---	---	---	---	---	----------	----------	-------------	----------	-----------	---	-------
tings,whileNon-ThinkingSFTonthesamepromptsfails.							1 (cid:88)	1 (cid:88)
									min (cid:0) ρ	Aˆ ,clip(ρ	,1−ϵ,1+ϵ)Aˆ		(cid:1)
Continualpre-trainingisamultiplierratherthanasubstitute. i,t i i,t i
							G		o
----------------------------------------------	---	---	---	---	---	---	---	-----	---	------	---------	---	---
							i=1	i t=1
CPTcombinedwithThinkingSFTproducesthestrongest											(cid:105)
									−βD	(π		π	) ,
generalization, recovering performance in settings where KL θ ref
Llamapreviouslyfailed.
πθ(oi,t	q,oi,<t)
							whereρ i,t	:=		denotestheprobabilityratio
---	---	---	---	---	---	---	----------	---	---	--------------------------	---	---	---
πθold (oi,t	q,oi,<t)
							between	the current	and pre-update		sampling	policy	and
---	---	---	---	---	---	---	-------	-----------	--------------	---	--------	------	---
2

LLMReasoningwithWeakSupervision 1.00 0.75 0.50 0.25 0.00 0 150 300 450 draweR gniniarT 60 45 30 15 0 150 300 450 )%( 005-HTAM 40 30 20 10 0 150 300 450 )%( CMA 20 15 10 5 0 0 150 300 450 )%( draH-PCS 1.00 0.75 0.50 0.25 0.00 0 150 300 450 draweR gniniarT 32 24 16 8 0 0 150 300 450 )%( draH-PCS 24 16 8 0 150 300 450 )%( dnomaiD AQPG 60 45 30 15 0 150 300 450 )%( 005-HTAM 1.00 0.75 0.50 0.25 0.00 0 150 300 450 Training Steps draweR gniniarT 45 30 15 0 0 150 300 450 Training Steps )%( kcoL mutnauQ 40 30 20 10 0 0 150 300 450 Training Steps )%( dnalsI tsegraL 75 60 45 30 0 150 300 450 Training Steps htaM ecneicS hparG )%( 005-HTAM Qwen2.5-Math-1.5B (7B) Qwen2.5-1.5B Llama3.2-3B-Instruct (8B) N=8 N=min(2048, Nmax) Figure1.Comparisonoftrainingdynamicsandtestperformance(avg@16metric)acrossmodelfamiliesanddomains.Foreach domain,weplottrainingreward(column1),in-domainbenchmarkperformance(column2-3)andOODbenchmarkperformance(column 4)overRLstepsfortwodatasetsizes:8(solidlines)andN (dashedlines),whereN isthelargestavailabletrainingsetinthe max max domainforthemodel.ForMATHandSCIENCE,N max =2048.ForGraph,N max =882forQwenmodelandN max =256forLlama model.ColoredverticaldashedlinesmarkthesaturationsteptN foreachrun.Theshadedregionindicatesonestandarddeviationover sat independentsampling.Qwenmodelsexhibitextendedpre-saturationphasesandgeneralizefrom8samples,whileLlamamodels saturaterapidlywithlimitedgains.Correspondingresultsfor7Band8BmodelsonMATHandSCIENCEareprovidedinAppendixC.3. Aˆ i := ri− st m d( e { a r n i ( } { G i r = i} 1 G i ) = ) 1 ) is the advantage of i-th response 3.RLVRUnderWeakSupervision calculatedbynormalizingthegroup-levelrewards.Rewards TounderstandwhenRLVRgeneralizesunderweaksuper- r ∈{0,1}arebinaryandassignedbyground-truthanswer i vision,westudythreesettings: scarcedata(§3.1),noisy verification. TheKLregularizationD (π ||π )isapplied KL θ ref rewards(§3.2),andself-supervisedproxyrewards(§3.3). toafixedreferencepolicyπ ,weightedbyascalarcoeffi- ref Wethenanalyzepolicybehaviortoexplainwhysomemod- cientβ. Allexperimentsusetheverlframework(Sheng els succeed and others fail under these conditions (§3.4). etal.,2024)(hyperparameterdetailsinAppendixB.3). We additionally analyze GRPO baseline selection in Ap- Evaluation. We evaluate reasoning performance using pendixE. avg@16 accuracy (average pass@1 over 16 independent Throughout this section, we compare Qwen and Llama samplesperproblem)withtemperature1.0samplingandre- modelfamilies.Wetreatthiscomparisonasaproxyforvari- portpass@kfork ∈{4,8,16}intheAppendix.ForMATH, ationinpretrainingpriorsratherthananintrinsicpropertyof weuseMATH-500,AMC,AIME2024,AIME2025,Min- eitherfamily: Qwen2.5-Mathispretrainedonanadditional ervaMath,andOlympiadBenchevals. For SCIENCE,we 1Tmath-specifictokens,whileLlama-3.2-Instructisaligned useGPQA-Diamond,aheld-outSCP-Hardset(Liuetal., forgeneralinstruction-following. Thecontrastwereport 2025a) (a subset of SCP problems where both Qwen2.5- isbetweenmodelswithstrongdomain-alignedpretraining 1.5BandLlama-3.2-3B-Instructachievesolve@16=1pre- and those without, and §4 confirms this interpretation by RL),ScienceBench,MMLU-Science,andSuperGPQA.For showingthatcontinualpre-trainingonmathdatatransforms GRAPH,weuseheld-outQuantumLockandLargestIsland Llama’sRLbehaviortoresembleQwen’s. tasksfromReasoningGym(Stojanovskietal.,2025),fil- teredsimilarlytosolve@16=1. Foreachdomain,wedes- 3.1.ScarceData ignatebenchmarksasin-domainorout-of-domain(OOD). For example, for MATH training, MATH-500 and AMC To understand how data scarcity affects RLVR general- arein-domain,whileSCP-HardandGPQA-Diamondare ization, we investigate training dynamics across dataset OOD (full assignments in Appendix Table 2). We report sizesN ∈{8,32,64,512,2048}acrossdiversemodelfam- representativeresultsinthemaintextandfullresultsinthe iliesanddomains. Unlikepriorworkonsample-efficient Appendix. RLVR (Wang et al., 2025a; Sun et al., 2025), which se- lectspecificdatapoints,weusestratifiedrandomsampling 3

LLMReasoningwithWeakSupervision Table1.Comparisonofsaturationstepst(8),pre-saturationgain∆(8)andpost-saturationresidual∆∗(8) acrossmodelfamilies sat sat post andtrainingdomainswhentrainingon8examples.Weadditionallyreportthelarge-smallgapG(n1,8)andG(n1,8) .ForGraph,the sat,in sat,ood largestavailablesettingisn =882forQwenmodelandn =256forLlamamodel(markedwith†).Thegreencellsmark∆(8) >0 1 1 sat (effectivepre-saturationlearning)whileredmarkrapidsaturationt(8) < 100. Thelarge-smallgapatsaturationstepsG(n1,8) and sat sat,in G(n1,8) aregenerallysmall.Resultsonmorebenchmarksandpass@kmetricsarereportedinTable3-7inAppendix. sat,ood In-domainBenchmarks OODBenchmark Model t(8) sat ∆( s 8 a ) t ∆∗ p ( o 8 s ) t ∆( s 8 a ) t ∆∗ p ( o 8 s ) t Gsat,in ∆( s 8 a ) t ∆∗ p ( o 8 s ) t Gsat,ood TrainingDomain:Math MATH500 AMC G(2048,8) SCP-Hard G(2048,8) sat,in sat,ood Qwen2.5-Math-1.5B 302 29.7 1.5 18.7 0.6 -1.1 10.5 2.1 2.4 Qwen2.5-1.5B 170 32.1 0.9 12.7 3.3 -0.5 7.0 0.3 -0.4 Llama3.2-3B-Instruct 55 10.8 -1.9 8.8 -2.1 -0.9 3.9 0.0 1.5 TrainingDomain:Science SCP-Hard GPQA-Diamond G(2048,8) MATH500 G(2048,8) sat,in sat,ood Qwen2.5-Math-1.5B 268 14.5 1.1 16.9 1.6 1.1 25.3 0.8 1.1 Qwen2.5-1.5B 161 6.4 0.2 13.3 1.7 1.8 32.3 2.1 1.2 Llama3.2-3B-Instruct 61 1.8 1.7 11.9 3.0 5.1 7.3 2.2 0.6 TrainingDomain:Graph QuantumLock LargestIsland G(n1,8)† MATH500 G(n1,8)† sat,in sat,ood Qwen2.5-Math-7B 150 8.3 4.9 19.8 1.9 -1.8 21.0 2.1 -3.7 Llama3.1-8B-Instruct 29 10.1 7.1 1.8 1.0 3.0† 9.1 3.8 0.0† acrossdifficultylevelsdefinedin§2. ForN < 64,were- tional gain after saturation, defined as ∆∗(n)(M) := post peatpromptsuniformlytoreachbatchsize64(e.g.,N =8 max M(n)(t)−M(n)(cid:0) t(n)(cid:1) . Values near zero implies8repeats). t∈[t( s n at ),T] sat indicatenegligiblepost-saturationgains. Tostudytrainingdynamics,weleveragerewardsaturationto • Large-small gap G(n′,n)(M): we define this gap as distinguishperiodswherethepolicyimprovesonthetrain- sat M(n′)(t(n))−M(n)(t(n))forn′ > n,whichcompares ingdatasetfromthosewhereitplateaus. Intuitively,once sat sat performancebetweenlarger(n′)andsmaller(n)datasets trainingrewardsaturates,furtherupdatesyieldlittlenewsig- (cid:104) (cid:105) atthesaturationstepofthesmallerrun. Atthesmaller nal.Wedefiner¯ :=E 1 (cid:80)G r as t q∼D,{oi}G i=1 ∼πold(·|q) G i=1 i run’ssaturationstep,howmuchbetterdoesthelargerrun theexpectedtrainingrewardatupdatestept∈{1,...,T}, perform? Largerpositivevaluesindicatesubstantialben- andletr¯ max :=max 1≤t≤T r¯ t bethemaximumrewardob- efitfrommoredata;valuesnearzerosuggestlimitedad- servedduringtraining. Weidentifytraininghassaturated vantagefromincreasingdatasetsize. WedenoteG(n1,8) sat,in oncetherewardisclosetothismaximum,anddefinethe astheaveragegapoverthein-domainbenchmarks,and saturationstepastheearliestupdatewherethisoccurs: G(n1,8) astheaveragegapoverOODbenchmarks. sat,ood (cid:110) t sat :=inf t∈{1,...,T eff }:r¯ t ≥ϵ max r¯ max }. Pre-saturationphasedominatessmall-samplelearning, and its length predicts generalization. Table 1 sum- Weuseϵ =0.99andsetT =T−50,i.e.,wesearchfor max eff marizes the proposed metrics across model families and t onlyuptothefirstT updatestoavoidboundaryeffects sat eff training domains when training on 8 examples. Results neartheendoftraining.Wedefinethepre-saturationphase onmorebenchmarksandpass@kmetricsareprovidedin asallstepst∈{1,...,t −1}andpost-saturationphase sat Appendix C.2 and Tables 3-7. All model-domain pairs asallstepst∈{min(t sat ,T),...,T}. showclearlypositive∆(8)forallmetrics(i.e.,bothavg@16 sat To quantify data efficiency, we introduce three metrics. andpass@k,k ∈ {4,8,16})acrossin-domainandout-of- LetM(n)(t)denoteanevaluationmetric(e.g.,avg@16on domain benchmarks, indicating that as few as 8 training MATH-500)attrainingsteptfortrainingwithnsamples, examples can trigger measurable learning during the pre- andt(n)bethecorrespondingsaturationstep. saturation phase. Neither G(2048,8) nor G(2048,8) is sig- sat sat,in sat,out nificantly greater than zero on 7 out of 8 model-domain • Pre-saturationgain∆(n)(M): performancegainfrom sat pairs,indicatingthatthepre-saturationimprovementsare initializationtosaturationas∆(n)(M):=M(n)(cid:0) t(n)(cid:1) − oftencomparabletothoseobtainedwithlargertrainingsets. sat sat M(n)(0). Largerpositivevaluesindicateeffectivelearn- Thissuggeststhatearlylearningisnotstronglydata-limited. ingbeforesaturation. Incontrast,thepost-saturationresidual∆∗(8) istypically post • Post-saturation residual ∆∗(n)(M): maximum addi- smallerthan∆(8),indicatingdiminishingreturnsoncethe post sat 4

LLMReasoningwithWeakSupervision Qwen2.5-Math-7B Llama-3.2-3B-Instruct gestthatevenformodelswithstrongmathematicalpriors,

		Graph				Math
thelackofdomain-specificpre-trainingacceleratessatura-
	draweR gniniarT 1.0			draweR gniniarT 0.75
---	-------------------	---	---	--------------------	---	---	---	---	---	---	---	---	---	---
0.8
				0.60				tionandnecessitateshigherdatavolumetodrivelearning.
---	---	---	---	----	---	---	---	---------------------------------------------------	---	---	---	---	---	---
0.6
				0.45				Wefurtherprovideillustrationsfor7Band8Bmodelson
---	---	---	---	----	---	---	---	-----------------------------------------------	---	---	---	---	---	---
0.4
				0.30				MATHandSCIENCEdomainsinAppendixC.3.
---	---	---	---	----	---	---	---	-----------------------------------	---	---	---	---	---	---
	0.2			0.15
0.0
)%( kcoL mutnauQ Extendedpre-saturationenablesout-of-domaintransfer.
	40			)%( 005-HTAM 52
---	---	---	---	---------------	---	---	---	---	---	---	---	---	---	---
Positive∆(8)
	30			48					valuesinTable1indicatethatthereasoning
---	---	---	---	---	---	---	---	--------	--------------------------------------	---	--------------	---	-----	--------
	20								sat
				44				patterns	learned during	the	pre-saturation		phase	transfer
10
				40				acrossdomains,particularlyforQwenmodels.					Withonly
---	---	---	---	---	---	---	---	----------------------------------------	---	---	---	---	--------	---
0
36
				28				8samples,Qwen2.5-1.5BtrainedonMATHachievescon-
---	---	---	---	---	---	---	---	----------------------------------------------	---	---	---	---	---	---
)%( 005-HTAM 78
				24				sistent gains	on the	out-of-domain		SCIENCE	benchmark
---	---	---	---	-------	---	---	---	-------------	------	-------------	---	-------	---------	---
	72			)%( CMA
	66			20
(SCP-Hard),whileQwen2.5-Math-7BtrainedonGRAPH
	60			16
---	---	---	---	---	---	---	---	---	---	---	---	---	---	---
improvesout-of-domainMATH-500performanceby21.0%
	54			12
---	---	---	---	-----	---	---	---	---------	------------	-----	------	----	-------	-------
								(Fig. 1).	In contrast,	Llama	models	show	limited	out-of-
	0	150	300	450 0	150	300	450
Training Steps Training Steps domain transfer even when in-domain performance im-
		=0		=0.3		=0.7
---	---	---	---	----	---	----	---	---	---	---	---	---	---	---
proves;theirgainsremainlocalizedtothespecifictraining
		=0.1		=0.5		=0.9
---	---	----	---	----	---	----	---	---	---	---	---	---	---	---
distribution.
Figure2.Effectofrewardlabelcorruptionontrainingdynam-
icsandgeneralization.γdenotesthefractionoftrainingprompts
Takeaway:
withcorruptedlabels,rangingfromclean(γ						= 0)tomostlyin-
--------------------------------------	---	---	---	---	---	---------------	---	---	---	---	---	---	---	---
(1)RLVRcangeneralizefromasfewas8samples
correct(γ		= 0.9).
---------	---	-------	---	---	---	---	---	---	---	---	---	---	---	---
ForQwenonGRAPHandLlamaonMATH,
generalizationdegradeatγ ≥ 0.5. when models remain in an extended pre-saturation
ForLlama,trainingreward
curvesstaycloseacrossallγ,suggestingoverfittingtonoise. phase,whereasrapidlysaturatingmodelsrequiresub-
stantiallymoredata.(2)Whetherscarce-datalearning
8-samplerunreachest(8). succeedsismodel-anddomain-dependent,reflecting
			sat					theinfluenceofpretrainingpriors.				(3)Inthelow-data
----	-------	---	--------	------	------	------------	---	--------------------------------	---	---	---	----------------	---	---
Fig.	1 shows	the	training	curves	across	data scales.	The
regime,Llamamodelscanachieveperfecttrainingre-
lengthofthepre-saturationphaseistheprimarydeterminant wardsmuchfasterthanQwenbyrapidlymemorizing
ofwhetheramodelcangeneralize. With8trainingsamples, trainingexamplesbutachievelittlemeaningfultask
Qwen2.5-Math-1.5B			on	MATH increases		reward	steadily
-----------------	---	---	---	--------------	---	------	--------	---	---	---	---	---	---	---
learning.
forover300steps;thissustainedascentallowsthemodelto
extractgeneralizablereasoningpatternsthattransfertoheld-
out	evaluation	benchmarks		such	as MATH-500	and	SCP-	3.2.NoisyRewards
---	----------	----------	---	----	-----------	---	----	----------------	---	---	---	---	---	---
Hard. Awithin-familycomparisonisolatesthepretraining
effect: Qwen2.5-Math-1.5B,whichsharesarchitecturewith Whenground-truthverifiersareavailablebutimperfect,re-
								wardlabelsmaycontainerrors.			ToevaluateRLVRrobust-
---	---	---	---	---	---	---	---	---------------------------	---	---	---------------------	---	---	---
Qwen2.5-1.5Bbuthasadditionalmath-specificpretraining,
nesstosuchnoisysupervision,wevarythefractionofincor-
saturatesmoreslowlyandtransfersfurther(Table1).
								rectlabelsγ	byrandomlyreplacingground-truthanswers
---	---	---	---	---	---	---	---	-----------	--------------------------------------	---	---	---	---	---
Figs. 13, 14, and 15 (Appendix C.1) show the full range with the most frequent incorrect answer produced by the
N	∈ {8,32,64,512,2048}			across	MATH,	SCIENCE,	and
----------------------------------------	--------------------	--------	------	-------	-----	------------	---	----------------------------------	------------	----------	---	---------------	------	---
								modelitself(detailsinAppendixD.1).				Unlessotherwise
GRAPH.		For Qwen	models	on MATH		and SCIENCE,	in-
								noted,experimentsuseN		=2048.
domainperformanceisnearlyindependentofN.						ForLlama
								RLVR	demonstrates	robustness		to reward	noise,	but
acrossalldomains,andforQwenonGRAPH,differentN
								generalizationvariesacrossmodels.				Fig.2andAppendix
---	---	---	---	---	---	---	---	---------------------------------	---	---	---	----------------	---	---
producesvisiblydifferentdynamicsonsomeoftheevals,
Fig.26summarizeperformanceacrosssevenmodel–domain
withsmallerdatasetssaturatingearlierandatlowerdown-
streamperformance. pairsundervaryingγ. Atγ ≤0.3,testperformanceacross
								mostsettingsremainsclosetothecleanrewards(γ						=0),in-
---	---	---	---	---	---	---	---	-------------------------------------------	---	---	---	---	---	-------
Modelswithoutdomain-alignedpriorssaturaterapidly dicating robustness to moderate label noise. On MATH
andfailtogeneralize.			Incontrast,Llamamodelsacrossall
--------------------	---	---	-------------------------------	---	---	---	---	------------	----	------	--------	-----	-----	----
								and SCIENCE,	Qwen	models	maintain	gains	under	sub-
domains,andQwenonGRAPH(Fig.1)exhibitcleardepen- stantial corruption (up to γ = 0.7). In contrast, Qwen
denceondatascale.			ForLlama,trainingon8samplesleads
-----------------	---	---	--------------------------------	---	---	---	---	---	---	---	---	---	---	---
onGRAPHandLlamaonMATHandSCIENCEdegradeat
torapidsaturation,witht(8)occurringwithinthefirst100:
								γ ≥ 0.5.	Higherγ	leadstoconsistentlylowertrainingre-
---	---	---	---	---	---	---	---	--------	-------	-----------------------------------	---	---	---	---
sat
itmaximizesthetrainingrewardmuchfasterthantheQwen wardsthroughouttraining,butforLlamaonMATH,training
models. Thesemodelsrequirelargerdatasets(N ≥512)to rewardcurvesremainnearlyidenticalacrossallγ despite
achievemeaningfulgeneralization(detailsinAppendixC.1 severecorruption,indicatingLlamafitsincorrectanswers
Fig.13andFig.14).			TheresultsintheGRAPHdomainsug-
-----------------	---	---	------------------------------	---	---	---	---	---	---	---	---	---	---	---
5

LLMReasoningwithWeakSupervision

Qwen2.5-3B	Llama-3.2-3B-Instruct
Science	Science
draweR gniniarT 1.0	1.0
0.8	0.8
0.6	0.6
0.4
0.4
0.2
0.2
)%( 005-HTAM
65	48
---	---	--------------------------------------------------------	---	---	---	---
	40	Figure4.Evolutionofsemanticdiversityduring8-sampletrain-
60
55	32	ingonMATH.Llamashowssignificantlyhigherpost-saturation
---	---	------------------------------------------------------	---	---	---	---
50	24
diversitythanQwen,albeitwithlowerperformanceoutcomes.
45	16
---	---	---	---	---	---	---
40
)%( draH-PCS 40 SCIENCE)showimprovementwithmajorityvoting,while
20
32		othermodelsfailentirely.		ForQwen2.5-3Bon		SCIENCE,
---	---	-----------------------------------------------------	---	---------------	---	--------
24	15
	10	majorityvotingyieldstemporarygainsbeforecollapseafter
16
8	5
---	---	---------	---------------------------------------	---	---	---
		500steps,	asthepolicyconvergestowardasingleoutput
0	0
0 150300450600750 0 150300450600750 tomaximizeagreement. Self-certaintyrewardsleadtoper-
Training Steps	Training Steps
--------------	--------------	---	---	---	---	---
RLVR Majority Vote Self Certainty formancecollapseacrossallsettings. Theseresultsshow
thatcurrentself-supervisedproxyrewardsareinsufficient
		toreplaceverifiablefeedbackinmostsettings.				(detailsin
---	---	------------------------------------------	---	---	---	----------
Figure3.Comparisonofrewardvariants(RLVR,self-certainty,
AppendixD.2andFig.27).
majorityvote)with1024trainingsamples.Proxyrewardswith-
outverifiersexhibitfailuremodesunderprolongedtraining:train-
ingcollapse(self-certainty)andrewardspikesfollowedbyperfor- Takeaway: Self-supervisedproxyrewardssucceed
mancedrops(majorityvote)(moreresultsareinAppendixD.2). onlyformath-specializedmodels(Qwen-Mathunder
		majorityvoting).	Othermodelsexhibitthesamepat-
---	---	----------------	-----------------------------	---	---	---
moreeasily. Wealsoobservethatmodel-domainpairswith ternweobservedin§3.1and§3.2: fastersaturation
faster saturation (§3.1) are generally less robust to label andweakerpretrainingpriorscoincidewithbrittle-
noise,aconnectionwedevelopin§3.4and§4. ness. Under prolonged training, the failure mode
		isrewardhacking:		policiesconvergetowardoutputs
---	---	----------------	---	-----------------------------	---	---
Takeaway: Robustnesstolabelnoisevariessharply thatmaximizetheproxywithoutcorrespondingdown-
acrossmodel-domainpairs:QwenonMATHandSCI-		streamgains.
-----------------------------------------	---	------------	---	---	---	---
ENCE toleratesupto70%corruption, whileLlama
andQwenonGRAPHdegradeat50%.Model-domain
pairsthatsaturatefasterundercleanrewardsareless 3.4.WhyDoModelsFailUnderWeakSupervision?
robust,andLlamafitscorruptedlabelsnearlyasfast
		Theresultsin§3.1–§3.3showaconsistentpattern:				models
---	---	--------------------------------------------	---	---	---	------
ascleanones—evidencethatrapidsaturationreflects
		withstrongdomain-alignedpretraining(Qwenon				MATH
---	---	------------------------------------------	---	---	---	----
memorizationcapacityratherthanlearningefficiency.
		and SCIENCE)	generalize	under weak	supervision,	while
---	---	--------------------------------	-------------------	----------	------------	----------
		thosewithout(Llamaacrossdomains,			Qwenon	GRAPH)
		fail. A	natural hypothesis,	motivated	by prior	work link-
3.3.Self-SupervisedProxyRewards
ingdiminishedexploratorycapacitytorapidpolicysatura-
Whenground-truthverifiersareentirelyunavailable,models tion(Cuietal.,2025),isthatfailingmodelsproduceless
mustrelyonalternativerewardsignals(Burnsetal.,2023; diverse outputs. To test this, we analyze model behavior
Rahmanetal.,2025;Bowmanetal.,2022). Recentwork alongtwocomplementaryaxes: responsediversityandrea-
hasproposedself-supervisedproxyrewardsderivedfrom soningfaithfulness. Formaldefinitionsandimplementation
model outputs, but whether these approaches work well detailsareprovidedinAppendixF.
acrossmodelfamiliesandtaskdomainsremainsunexplored.
		To quantify	response	diversity, we	quantify semantic	di-
-------------------------	-------------------------	-----------	--------	-------------	-----------------	---
Weevaluatetwosuchrewards:	self-certainty(Zhaoetal.,
versitytocharacterizemeaningfulpatternsinthemodel’s
2025)andmajorityvote(Zuoetal.,2025)(implementation
reasoningratherthansurface-levelvariation(Farquharetal.,
detailsinAppendixD.2).
2024;Lietal.,2025).Wemeasurediversityonthe8-sample
Proxyrewardstriggerrewardhackingandpolicycol- subsetoftheMATH,SCIENCEandGRAPHtrainingdatasets,
lapse. WhileRLVRtoleratesmoderatelabelnoiseinsome aswellasontheMATH-500evaluationdataset,overase-
model-domainpairs(§3.2),Fig.3showsthatfullyreplacing lectionofpromptsatvariousstepsthroughouttraining. For
verifiablefeedbackwithself-supervisedproxysignalsintro- each prompt, we cluster model responses using pairwise
ducesseverefailuresunderprolongedtraining. Onlymath- similarity judgments from an LLM judge and define the
specialized models (Qwen2.5-Math-1.5B on MATH and diversityscoreastheShannondiversityindexoverthere-
6

LLMReasoningwithWeakSupervision

							faithfulness.	Fig. 5 (right)	reports	faithful	diversity:	di-
							versitycomputedonlyoverfaithfulresponses.					Thisjoint
measurerevealsaconsistentpatternacrossallthreedomains.
							On MATH,Llama’sapparentdiversityadvantage(Fig.					4)
---	---	---	---	---	---	---	----------------------------------------------	---	---	---	---	---
disappears—mostdiverseresponsesareunfaithful,andthe
							faithfulsubsetisnarrow.		OnSCIENCE,alignedproportions
---	---	---	---	---	---	---	-----------------------	---	----------------------------	---	---	---
areuniformlyhighacrossmodels,maskingrealdifferences
inreasoningquality;faithfuldiversityseparatesthem,with
Qwen-Mathmaintainingthehighestvaluesthroughouttrain-
							ing. OnGRAPH,Qwen-MathandLlamashowcomparable
---	---	---	---	---	---	---	--------------------------------------------	---	---	---	---	---
alignedproportions,butQwen-Mathsustainshigherfaithful
							diversity.	Ineverycase,themodelthatgeneralizesbestin
---	---	---	---	---	---	---	----------	-----------------------------------------	---	---	---	---
§3.1istheoneexploringthewidestrangeoffaithfulrea-
Figure5.Evolutionofreasoningfaithfulness(oncorrectsam- soningpaths—nottheonewiththehighestrawdiversity,
ples)andfaithfuldiversityonmodelsthroughoutRLusing8 nor the one with the highest aligned proportion. Raw di-
samplesfromavarietyofdatasets.LlamamodelsintheMATH
versityoverstatesexploratorycapacity;alignedproportion
domainexhibitsignificantlylowerfaithfulnesscomparedtoQwen.
saturatesoneasierdomains;onlytheirintersectionpredicts
generalization.
sultingclusters.		SeeFigure31forthejudgemodelprompt.
-------------------------------------------	---	----------------------------------	---	---	---	-----	---------	---	---	---	---	---
Highdiversitydoesnotpreventrapidsaturation.						Fig.4	Takeaway:
reportstheevolutionofdiversityscoresformodelstrained
							Low reasoning	faithfulness,		not low	diversity,	ex-
---	---	---	---	---	---	---	-------------	-------------	---	-------	----------	---
on 8 samples from the MATH training dataset, computed plains why models fail under weak supervision:
on the corresponding training set. Llama reaches reward rapidlysaturatingmodelsmemorizeanswersrather
saturation earlier and retains higher diversity than Qwen, than acquire transferable reasoning. Raw diversity
theoppositeofwhattheexploration-saturationhypothesis metricsaremisleading—Llamaexhibitshigherout-
predicts. DiversitycomputedontheMATH-500evaluation
							put diversity	than Qwen	while	generalizing		worse.
---	---	---	---	---	---	---	-------------	---------	-----	------------	---	------
datasetispresentedintheappendix(Fig.30). Diversitybecomesinformativeonlywhencomputed
overfaithfulresponses.
Sincediversityalonedoesnotexplainfailureunderweak
supervision,	we	investigate	the faithfulness		of a model’s
------------	---------------------------------------	-----------	----------------	---	------------	---	---	---	---	---	---	---
reasoning.	Inspiredbypriorwork(Bakeretal.,2025),we
definearesponseasfaithfulifitsreasoningtracecontains Insummary,§3showsthatthesurprisingcapabilitiesoften
the information needed to justify the final answer and is attributedtoRLVR,suchaslearningfromscarcedata,tol-
logicallyconsistentwithit. Atagiventrainingstepandfor eratingnoisyrewards,succeedingwithoutverification,are
notuniversalbutdependonpre-RLreasoningfaithfulness.
agivenprompt,wecategorizeeachpolicyrolloutasaligned,
partiallyaligned,ormisalignedbasedonrubricsprovided §4takesupthenaturalquestion: canpre-RLinterventions
to an LLM-as-a-judge (see prompt in Fig. 32). We then targetingfaithfulnessextendthepre-saturationphaseand
computethepolicyfaithfulnessrateF (l)asthefractionof recovergeneralizationunderweaksupervision?
π
responsesassignedtolabell.			AppendixFoutlinesresults
--------------------------	---	---	------------------------	---	---	---	---	---	---	---	---	---
forinter-modelagreementonalignmentcategorizationto 4.ImprovingRLVRUnderWeakSupervision
evaluatethereliabilityofourLLM-as-a-judge.
viaPre-RLTraining
Modelswithrapidsaturationexhibitlowreasoningfaith-
Section3showedthatrapidsaturationandlowreasoning
fulness. Fig.5(left)showsthefractionofcorrectresponses
							faithfulnessarelinked:	modelsthatgeneralizepoorlyunder
---	---	---	---	---	---	---	----------------------	-------------------------------	---	---	---	---
thatarealignedoverRLtrainingacrossmodelsanddomains
weaksupervisionproducecorrectanswersthroughreason-
studiedin§3.1.		Onthe	MATH domain,	theLlamamodel
---------------------------------------	-----	---------	--------------------------------	-------------	------------	---	-------------------------------------------------------	------------	---------------	----------------------	----	----------
							ingthatdoesnotsupportthem.			Thisraisesacausalques-
shows much	lower	reasoning	faithfulness	during	training
							tion. Iffaithfulnessdrivesthepre-saturationphase,andthe
thantheQwenmodels.			ThisindicatesthatLlama’srapidre-
							pre-saturation	phase drives	generalization,		then	instilling
wardgainsdonotreflectimprovedreasoning:					asubstantial
faithfulnessbeforeRLshouldextendthephaseandrecover
fractionofcorrectanswersarememorized,withreasoning
							generalization.	Wetestthisbyrunningacontrolledcom-
-----------	------	-------	----------	-----	--------	---	---------------	----------------------------------	---	---	---	---
traces that	do not	support	them. Fig.	33 in	Appendix	F
parisonofpre-RLinterventionsonLlama3.2-3B,themodel
includes	additional	faithfulness	results	on these	domains,
--------	----------	------------	-------	--------	--------	---	---	---	---	---	---	---
coveringproportionalignedandproportionmisalignedon thatfailedmostconsistentlyin§3.
correct,incorrectandallresponses. Westudytwoaxesofpre-RLtraining. Thefirstiscontinual
Reasoning diversity should be considered jointly with pre-training(CPT),extendedtrainingondomain-specific
7

LLMReasoningwithWeakSupervision

	draweR gniniarT			)%( 005-HTAM			32
					56					)%( draH-PCS
							)%( CMA			15
ecracS							24
ataD	0.6				48
					40		16
12
18
6
	0.0				12		0			0
--------	---------------	---	---	------------	-----	---	-------	---	---	------------	---	-------
	0	150	300	450	0 150	300	450 0	150	300	450 0	150	300 450
	draweR gniniarT			)%( 005-HTAM	64
							30			)%( draH-PCS
ytirojaM					56		)%( CMA
etoV
	0.6				48					15
---	---	---	---	---	---	---	---	---	---	---	---	---
15
20
	0.0				0					0
---	---	---	---	---	---	---	---	---	---	---	---	---
0 150 300 450 600 0 150 300 450 600 0 150 300 450 600 0 150 300 450 600
60
draweR ysioN	draweR gniniarT			)%( 005-HTAM
------------	---------------	---	---	------------	---	---	-------	---	---	------------	---	---
	0.6						25			)%( draH-PCS
)7.0=					54		)%( CMA
15
					48		20
---	---	---	---	---	---	---	---	---	---	---	---	---
0.3
(							15
---	---	---	---	---	---	---	---	---	---	---	---	---
20
					15		5
---	---	--------------	---	---	--------------	---	-----	--------------	---	-----	--------------	-------
					10		0			0
	0	150	300	450	0 150	300	450 0	150	300	450 0	150	300 450
		Training Steps			Training Steps			Training Steps			Training Steps
Base + Thinking SFT Base + Non-Thinking SFT CPT + Thinking SFT CPT + Non-Thinking SFT Instruct
Figure6.RLtrainingdynamicsandgeneralizationonMATHforLlama3.2-3BBase,CPT,andInstructvariantsunderdifferent
SFTinitializationsacrossthreeweaksupervisionsettings:scarcedata(N =8,top),majorityvote(middle),andnoisyreward
(γ =0.7,bottom).ThinkingSFT(solidlines)consistentlyprolongsthepre-saturationphaseandimprovesgeneralizationforbothCPT
andBasemodelscomparedtotheirNon-ThinkingSFTcounterparts(dashedlines)andtheInstructbaseline(dash-dot).CPT+Thinking
SFTachievesthestrongestperformanceacrossallsettings.
pretrainingtokenstostrengthenthepretrainingprior. The explicitreasoningtracesinfluencesubsequentRLdynamics.
second is supervised fine-tuning (SFT), with the specific WecomparetwoSFTregimesthatdifferonlyinwhether
questionofwhetherSFTonexplicitreasoningtracesdiffers thesupervisionincludesexplicitreasoning. Bothregimes
in its effect from SFT on final answers alone. Crossing use the same 43.5K math prompts and differ only in the
these axes gives a 2×2 design: two initializations (Base, targetoutput. Specifically,wesamplethesepromptsfrom
CPT)eachfollowedbytwoSFTregimes(Thinking,Non- OpenThoughts-114K (Guha et al., 2025), retaining only
Thinking).WeadditionallyincludeLlama3.2-3B-Instructas thosewhosereasoningtraceshavecorrectfinalanswersand
areference: itsharesthearchitectureofLlama3.2-3B-Base totallengthbelow8192tokens.
but	has undergone		extensive instruction		tuning,	rejection
---	-------------	---	---------------------	---	-------	---------	-------------------------------------------------	---	---	---	---	---
							• Non-thinkingSFT:Themodelissupervisedtooutputthe
sampling,andDPO,providingastrongoff-the-shelfbase-
finalsolutionwithoutgeneratingintermediatereasoning
lineagainstwhichtojudgeourtargetedinterventions.						We
------------------------------------------------	---	---	---	---	---	---	---	---	---	---	---	---
traces.
thenrunRLunderallthreeweaksupervisionsettingsfrom
							• ThinkingSFT:Themodelistrainedonexplicit,verified
---	---	---	---	---	---	---	--------------------------------------------------	---	---	---	---	---
§3: scarcedata,noisyrewards,andself-supervisedproxy
long-formreasoningtraces.
rewards.
Wefocusonthe MATH domainfortworeasons: Llama’s AtrainingexampleisshowninFig.12intheAppendix.The
							SFTregimesarenear-iso-compute:				ThinkingSFTtrainson
---	---	---	---	---	---	---	------------------------------	---	---	---	-------------------	---
baselinefailureissharpestthere,providingthecleanesttest
roughly1Btokens,Non-ThinkingSFTonroughly0.27B,
ofwhetherpre-RLinterventionscanrecovergeneralization;
							bothnegligiblerelativetothe52B-tokenCPTstage.					Differ-
---	---	---	---	---	---	---	---------------------------------------------	---	---	---	---	-------
andhigh-qualitymathpretrainingcorpora(Nemotron-CC-
encesbetweenThinkingandNon-ThinkingSFTtherefore
Math)andreasoning-tracedatasets(OpenThoughts-114K)
areavailable,enablingtheinterventionsatsufficientscale. reflectthecontentofthesupervisionratherthanitscost. We
							reporttheCPTlosscurveinAppendixFig.					10andtheSFT
---	---	---	---	---	---	---	-----------------------------------	---	---	---	---	-----------
ContinualPre-Training(CPT).Wecontinuallypre-train losscurvesinFig. 11.
Llama3.2-3B-Base			for one epoch	on	approximately	52B
----------------	---	---	-------------	---	-------------	---	---	---	---	---	---	---
ImplementationandtrainingdetailsofSFTareprovidedin
math	tokens	from	the Nemotron-CC-Math		dataset	(Ma-
-------------------	------	----	-------------------------------	---	-------	----	--------------------------------------------------	---	----------------------------------	---	---	---
							AppendixB.6.		ForthesubsequentRLphase,weevaluate
habadietal.,2025)2.			TrainingdetailsareprovidedinAp-
							acrossallthreeweaksupervisionsettings:scarcedata(N					=
pendixB.5.
							8), noisy	rewards	(γ =	0.7), and	self-supervised	proxy
-------------------	---	---	-----------------------------	---	---	---	----------------------	-------	-----------------------------	---------	---------------	-----
SFTTrainingRegimes.			FollowingCPTorBaseinitializa-
							rewards(majorityvote).		Allotherhyperparametersfollow
tion,weapplysupervisedfine-tuningtodeterminewhether theconfigurationsin§2,withthemaximumresponselength
duringRLextendedto8192tokenstoaccommodatelong-
2Nemotron-CC-Math-v1
8

LLMReasoningwithWeakSupervision formreasoningtraces. 4.1.Results

Fig. 6 reports	RL	training	dynamics	for	the five	pre-RL
configurations(Base,CPT,andInstruct,withThinkingSFT
or Non-Thinking		SFT	applied to	Base	and CPT)	across
--------------------------------	------	---------	----------	---------------	--------	--------	---	---	---	---	---	---	---
thethreeweaksupervisionsettings.				Foreachsetting,		we
plot training	reward	alongside	three	downstream		metrics:
Figure7.EvolutionofreasoningfaithfulnessoftheLlama3.2-
twoin-domain(MATH-500,AMC)andoneout-of-domain 3Bfamilyonweaksupervisiondomainswhencombinedwith
(SCP-Hard);additionalbenchmarksandpass@kresultsare continualpretrainingandSFTvariants.Whencombinedwith
Fig.34andFig.35inAppendixG.Wedrawthreefindings Thinking-SFT and CPT, the Llama3.2-3B-Base model exhibits
higherreasoningfaithfulness.
fromthisfigure,developedintheparagraphsbelow.
ThinkingSFTisnecessaryforsubstantiallearningunder
							highest faithfulness		among	all configurations,			consistent
---	---	---	---	---	---	---	--------------------	---	-----	-------------------	---	---	----------
weaksupervision.TheInstructbaselineisflatordecreasing
							with its strongest	generalization			across	all weak	supervi-
---	---	---	---	---	---	---	------------------	--------------	---	---	------	--------	--------
acrossallthreesettingsonalldownstreamevaluations—
							sion settings.	Together	with	the	extended	pre-saturation
---	---	---	---	---	---	---	--------------	--------	----	---	--------	--------------	---
RLproducesnomeaningfulimprovementfromthisstarting
							dynamics	visible in	Fig.	6 (leftmost	column),		this result
---	---	---	---	---	---	---	--------	----------	----	-----------	--------	---	-----------
point. ThinkingSFTistheonlyinterventionthatenables
							supportsourhypothesisin§3.4:				pre-RLinterventionsthat
---	---	---	---	---	---	---	----------------------------	---	---	---	-----------------------	---	---
substantialdownstreamgainsonscarcedataandmajority
instillfaithfulnessproducelongerpre-saturationphasesand
vote,anditdoessoforbothBaseandCPTinitializations
recoveredgeneralization,inmodelsthatpreviouslyfailed.
(solidblueandsolidred).			Non-ThinkingSFTshowsmodest
-----------------------	------------	------	--------------------------	-------	----------------	-----	---------	----------------------------------	---	---	---	---	---
gains only	when	paired	with CPT,	and	only under	noisy
							Takeaway:	SFTonexplicitreasoningtraces,noton
rewards;	Non-Thinking		SFT on	Base is	flat or degrades
finalanswers,isnecessaryforLlamatolearnsubstan-
acrossallthreesettings.
							tiallyfromRLunderweaksupervision.					Itraisesrea-
---	---	---	---	---	---	---	---------------------------------	---	---	---	---	------------	---
soningfaithfulness,extendsthepre-saturationphase,
CPTamplifiestheThinkingSFTeffect.					ThinkingSFT
---------------------------------	---	---	---	---	-----------	---	---	---	---	---	---	---	---
onBasealoneproducesmodestgains. CombinedwithCPT, andrecoversgeneralizationunderscarcedata,noisy
							rewards,	and self-supervised			proxy	rewards.	Con-
---	---	---	---	---	---	---	--------	-------------------	---	---	-----	--------	----
itproducessubstantiallylargergainsoneveryevaluation:
CPT+ThinkingSFTisthetop-performingcurveacrossall tinualpre-trainingamplifiestheeffectbutdoesnot
threeweaksupervisionsettingsandallthreeevals.TheCPT substituteforit: CPT+Non-ThinkingSFTfailsde-
							spitematchedcompute.			Thestrongestconfiguration,
---	---	---	---	---	---	---	--------------------	---	---	--------------------------	---	---	---
+Non-ThinkingSFTcomparisonrulesoutacompute-based
explanation:thesame52BCPTtokens,pairedwithSFTtar- CPT + Thinking SFT, recovers performance in set-
tingswhereLlamahadpreviouslycollapsedentirely.
getsthatstripreasoningtraces,failtoenablegeneralization
onscarcedataandmajorityvote.				Theamplificationisspe-
---------------------------------------------------	---	---	-------------------------------	----------------------	---	---	-------------	---	---	---	---	---	---
cifictothecombination:			extrapre-trainingcomputealoneis
insufficient;ThinkingSFTalonehelpsbutislimited,only							5.RelatedWork
thecombinationrecoversfullgeneralization.
							RLVR for	Reasoning.	Reinforcement			learning	with ver-
---	---	---	---	---	---	---	--------	----------	-------------	---	---	--------	---------
Baseinitializationfailsundermostweaksupervisionset- ifiable rewards has emerged as an effective post-training
tingsregardlessofSFT.TheBasemodelshowsmeaningful
							method for	improving	reasoning		in large	language	mod-
---------------------------------	---	---	---	---	-------------	---	-----------	----------	---------	-------	--------	--------	----------
improvementonlyintwocombinations:					Base+Thinking
							els (Guo et	al., 2025;	Olmo	et al.,	2025;	Yu et	al., 2025;
SFT under scarce data and majority vote, and even there Zengetal.,2025). RecentworkhasexploredwhenRLVR
gains are	modest.	Under	noisy	rewards,	neither	Base +
---------	-------	-----	-----	--------	-------	------	---	---	---	---	---	---	---
yieldsimprovements(Liuetal.,2025b;a;Huetal.,2025).
Thinking SFT nor Base + Non-Thinking SFT produces Wangetal.(2025a)demonstratethattrainingonasingleex-
meaningfuldownstreamimprovement.				ThisisolatesCPT’s
--------------------------------	---	---	---	-----------------	---	---	-----------------------------------------	---	---	---	---	---------	---
							amplecanprovidemeaningfullearningsignals.					Otherwork
contribution:ThinkingSFTisnecessarybutnotsufficient—
exploresalternativerewards,includingself-certainty(Zhao
domain-alignedpretrainingisrequiredfortheintervention etal.,2025),majorityvoting(Zuoetal.,2025),negativesig-
togeneralizeacrossallthreeweaksupervisionsettings.
nals(Zhuetal.,2025),self-generatedtrainingdataHuang
							etal.(2025),andspuriousrewards(Shaoetal.,2025).						How-
-----------------------------------------	---	---	---	---	---	-------	-----------------------------------------------	---	---	---	---	---	----
ThinkingSFTimprovesreasoningfaithfulness.						In§3.4,
ever,thesefindingsoftendonottransferacrossmodelfam-
weidentifiedlowreasoningfaithfulnessasthepre-RLprop-
							ilies, with	studies	reporting	inconsistent		results	between
---------	-------------	---	------------	----------	---	-------	-----------	-------	---------	------------	---	-------	-------
erty that	distinguished		failing from	succeeding		models.
QwenandLlama(Zengetal.,2025;Gandhietal.,2025;
Fig. 7 shows	that	Thinking	SFT	raises	aligned-response
------------	----	--------	---	------	----------------	---	---	---	---	---	---	---	---
ratethroughoutthepre-saturationphase,relativetotheNon- Shao et al., 2025). Moreover, most prior work focuses
onimprovingperformanceonnarrowdomains(primarily
ThinkingSFTbaseline.			CPT+ThinkingSFTachievesthe
--------------------	---	---	--------------------------	---	---	---	------------------------------------	---	---	---	---	-------------	---
							math)withoutexamininggeneralization.					Recentwork(He
9

LLMReasoningwithWeakSupervision et al., 2026; Yang et al., 2026; Plesner et al., 2026) has gesttwoconcretepracticesforRLfromweaksupervision. concurrentlystudiedwhenandhowRLVRcanlearnunder First, monitor training reward saturation as a diagnostic: self-supervisionornoisysupervision.Ourworkextendsthis plateauedrewardwithflatdownstreamperformanceindi- literatureintwoways. First,wecharacterizetheconditions catesthemodelhasexhaustedwhatRLcanextractfromits underwhichRLVRgeneralizesacrossmodelfamiliesand priors,andfurtherRLcomputeisunlikelytohelp. Second, domains, focusing on saturation dynamics and reasoning whenweaksupervisionfails, allocatecomputetopre-RL faithfulness. Second, we identify a concrete intervention interventionsthatinstallstrongpriorsratherthantolonger thatrestoresgeneralizationinmodelswhereweaksupervi- RL training. Taken together, our findings argue that RL sionwouldotherwisefail. underweaksupervisionisbestunderstoodnotasatraining techniqueappliedtoafixedmodel,butasthefinalstageof

Role of	Pre-Training	and	Fine-Tuning		in RL.	Recent
apipelinewhosesuccessislargelydeterminedbeforeRL
workemphasizesthatpre-trainingandmid-trainingshape
begins.
RL generalization		(Qi et	al., 2025;	Wang	et	al., 2025b;
-----------------	---	------	----------	----	---	-----------	---	---	---	---	---	---	---
Zhangetal.,2025;Akteretal.,2025),butfocusesoncom-
puteallocationanddistributionalignmenttoimproveperfor- Acknowledgements
mance.Ourworkspecificallyfocusesonunderstandinghow
WewouldliketothankLeonLi,VatsalBaherwani,Rohun
base model	priors	shaped	from continual		pretraining	and
----------	------	------	--------------	---	-----------	---	---	---	---	---	---	---	---
Agrawal,SiyanZhao,LiweiJiang,andAndyHanfortheir
reasoning	SFT can	enable	generalization		across	different
---------	-------	------	--------------	---	------	---------	----------	-----------	---	--------	---	----------	-----
							insightful	discussions	and	feedback	on	the draft.	Pavel
weaksupervisionsettings.
							Izmailov	was supported		by a grant	from	the	Alignment
---	---	---	---	---	---	---	--------	-------------	---	----------	----	---	---------
Diversity and Faithfulness in Reasoning. Maintaining Project,fundedbytheUKAISecurityInstitute(grantAP-
outputdiversityduringRLhasbeenproposedtopromote							S2-100141).
-----------------------------------------------	---	---	---	---	---	---	-----------	---	---	---	---	---	---
explorationandmitigatemodelcollapse(Kirketal.,2024;
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LLMReasoningwithWeakSupervision A.LimitationsandFutureWork Weacknowledgeseverallimitations. First,duetocomputationalconstraints,ouranalysisisrestrictedtospecificmodel families and scales. Validating these findings across larger architectures and broader task suites remains an important direction.Second,ouranalysisofdiversityandfaithfulnessreliesonanLLM-as-a-judgeframework.Althoughweconducted small-scalehumanverificationtovalidatelabelquality,wecurrentlyrestrictthisevaluationtoasmallscaletoallowfor reasonablelabelingcosts.Consequently,thedevelopmentofscalablemetricsforreasoningfaithfulnessanddiversityremains animportantdirectionforfutureresearch. B.ImplementationDetails B.1.TrainingandEvaluationDatasets WeinvestigateRLtrainingdynamicsacrosstwomodelfamilies: Qwen(comprisingQwen2.5-1.5B/3BandQwen2.5-Math- 1.5B/7B)andLlama(Llama-3.2-3B/8B-Instruct). Ouranalysisspansthreedistinctreasoningdomains,MATH,SCIENCE, andGRAPH,allowingforaholisticinvestigationofRLVRunderweaksupervisionacrossdifferentdomainsandmodel families. ForMATH,wesampletrainingpromptsfromtheSkywork-OR1(Heetal.,2025b)dataset. ForSCIENCE,wedraw problemsfromtheSCPdatasetcuratedbypriorwork(Liuetal.,2025a;Luetal.,2025),byselectingPhysics,Chemistry, andBiologysubjects. ForGRAPH,wegeneratetwosyntheticalgorithmictasks,QuantumLockandLargestIsland,using thecurriculumspecificationsprovidedbytheReasoningGymbenchmark(Stojanovskietal.,2025). Foreachtask,we instantiatefivedifficultylevelsfollowingthebenchmark’scurriculum,withabalancednumberofsamplesperlevel. Weincludethefollowingdomain-specificbenchmarksforevaluations: • MATH500(Lightmanetal.,2023): AwidelyusedsubsetoftheMATHtestsplit(Hendrycksetal.). • AMC(AI-MO,2024b): 40competition-levelmathquestions. • AIME2024(AI-MO,2024a): 30competition-levelmathquestions. • AIME2025(Opencompass,2024): 30competition-levelmathquestions. • MinervaMath(Lewkowyczetal.,2022): Asetof272undergraduate-levelscienceandmathquestionsfromMIT OpenCourseWare. • OlympiadBench(Heetal.,2024): Abenchmarkof675problemsfrominternationalmatholympiadsandphysics contests. • GPQA-Diamond(Reinetal.,2024): 198expert-levelquestionsfromGPQAspanningphysics,chemistry,andbiology; wepreprocessthedatafollowingpreviouspractice(Chengetal.,2025). • SCP-Hard(Luetal.,2025;Liuetal.,2025a): Aheld-outsetof50SCPquestionsfilteredsuchthatthebasemodels (Qwen2.5-1.5BseriesmodelsandLlama3.2-3B-Instructmodel)achievesolve@16=1,containingdisjointquestions fromtheSCPtrainingdatasets. • SuperGPQA (Du et al., 2025): a subset constructed from the original SuperGPQA which contains 319 science questionsand250non-sciencequestions. • MMLUSCI(Wangetal.,2024): asubsetofMMLUProbenchmarkcontainingallcollege-levelchemistry,physics andbiologyquestions. • ScienceBench(Wangetal.,2023): 692college-levelsciencequestions. • GraphTest: Aheld-outsetof50algorithmicallygeneratedinstancesfromtheQuantumLock andLargestIsland tasks using Reasoning Gym (Stojanovski et al., 2025), disjoint from training, filtered such that the base models (Qwen2.5-1.5BseriesandLlama3.2-3B-Instruct)achievePass@16=1. WealsonotethatGPQA-Diamond,MMLUSCI,andSuperGPQAaremultiple-choicebenchmarks,forwhich pass@kmaybealessreliablemetric. Table2detailsthetrainingandevaluationdatasetsacrossthethreereasoningdomains. 15

LLMReasoningwithWeakSupervision Table2.Trainingdatasetsandevaluationbenchmarksacrossthreereasoningdomains. Domain TrainingSource In-DistributionEval Out-of-DistributionEval MATH Skywork-OR1 MATH-500,AMC,AIME-2024,AIME- Science Bench, SuperGPQA, GPQA- 2025,MinervaMath,OlympiadBench Diamond,SCP-Hard,MMLUSCI SCIENCE SCP-116K SCP-Hard,GPQA-Diamond,MMLUSCI, MATH-500, AMC, Minerva Math, SuperGPQA,ScienceBench OlympiadBench GRAPH ReasoningGym QuantumLock,LargestIsland MATH-500,AIME-2024,MinervaMath, GPQA-Diamond,SCP-Hard PrompttemplateforMATHandGRAPH. system YouareahelpfulAIAssistant,designedtoprovidewell-reasonedanddetailedresponses. YouFIRSTthinkaboutthereasoningprocessstepbystepandthenprovidetheuserwiththeanswer. Pleaseencloseyourfinalanswerinthebox: \boxed{YourAnswer}. user assistant Figure8.PrompttemplateusedforRLtrainingandevaluationonMATHandGRAPH.Theplaceholderisreplaced withtheactualmathematicalquestionduringfine-tuningandevaluation.Specialtokensareomittedforclarity. B.2.TrainingDataPreparationDetails Wedescribeourprocedureforconstructingfilteredtrainingdatasetstailoredtoeachmodel’scapabilities. DifficultyEstimation. Foreachprobleminthesourcedataset,wesample16responsesfromthebasemodelandcount thenumberofcorrectsolutions,yieldingsolve@16∈[0,16]. Weretainonlyproblemswithsolve@16∈[1,15],excluding problemsthataretoodifficult(solve@16=0)ortriviallyeasy(solve@16=16)forthemodel. PrompttemplateforSCIENCE. system Let’sthinkstepbystepandoutputthefinalanswerwithin\boxed{}. user assistant Figure9.PrompttemplateusedforRLtrainingandevaluationonSCIENCE.Theplaceholderisreplacedwiththe actualmathematicalquestionduringfine-tuningandevaluation.Specialtokensareomittedforclarity. Stratified Sampling. We use a stratified round-robin selection method to construct training subsets of size N ∈ {8,32,64,512,2048}. Filteredproblemsarepartitionedinto15bins{B }15 accordingtotheirsolve@16values. Toselect i i=1 N problems:

1. Initialization: Setthecurrentcountofselectedproblemsn =0. total
2. Round-RobinSelection: Whilen <N: total • IteratethroughbinsB fori=1,...,15. i • IfB containsunsampledproblems,randomlyselectoneproblemwithoutreplacement,addittothetrainingset, i andincrementn . total • Terminateimmediatelyifn =N. total 16

LLMReasoningwithWeakSupervision Thisapproachensuresthatalldifficultylevelsarerepresentedasuniformlyaspossibleacrossalldatascales. B.3.ImplementationDetailsofRLTraining Allexperimentsareimplementedusingtheverlframework(Shengetal.,2024)withitsdefaulthyperparameters: learning rate 10−6, KL coefficient β = 0.001, clip ratio ϵ = 0.2 and no entropy regularization. We set group size G = 8 for computationalefficiency. Forresponsesampling,wefixthesamplingtemperature1.0andamaximumresponselengthof 2048tokensunlessotherwisenoted. Inverl,wesetboththetrainingbatchsizeandmini-batchsizeto64prompts,yielding exactlyonegradientupdatepertrainingstep. Eachexperimentisrunfor496totalgradientupdates. Asimplerule-based rewardfunctionisused,assigningreward1tocorrectanswersand0otherwise,withoutincorporatinganyformat-related signals. ForMATHandSCIENCE,answermatchingandrewardcomputationisimplementedwithMath-Verify3library;for GRAPH,weusetheinternaltask-specificevaluationprotocolfromReasoningGym. PrompttemplatesaredetailedinFig.8 andFig.9. Training Steps (log scale)

	10	100	1k 10k	100k
Training loss (EMA, 50-step window)
1.20
1.10
1.00
ssoL
0.90
0.80
0.70
0.60
		0.1 B	1 B	10 B 51 B
---	---	-----	---	---------	---
Tokens Seen (log scale)
Figure10.Traininglossduringcontinualpre-trainingofLlama3.2-3Bonapproximately52BtokensofNemotron-CC-Mathdata.
	Training Steps			Training Steps
-----	--------------	-------	-----	--------------	-------
0 100	200 300	400 500	0 100	200 300	400 500
0.50
	Training loss (EMA, 50-step window)			Training loss (EMA, 50-step window)
----	------------------------------------	---	----	------------------------------------	---
0.80			0.45
0.40
0.70
0.35
ssoL			ssoL
----	---	---	----	---	---
0.30
0.60
0.25
0.50			0.20
----	---	---	----	---	---
0.15
0.00 B 0.20 B 0.40 B 0.60 B 0.80 B 1.00 B 0.00 B 0.05 B 0.10 B 0.15 B 0.20 B 0.25 B
	Tokens Seen			Tokens Seen
---	----------------------	---	--------------------------	-----------	---
	(a) CPT + Thinking SFT		(b) CPT + Non-Thinking SFT
Figure11.Traininglossfor(a)ThinkingSFTand(b)Non-ThinkingSFTon43.5Kmathprompts,initializedfromtheCPTcheckpoint.
B.4.ImplementationDetailsofEvaluation
Weevaluatereasoningperformanceusingavg@16accuracy(averagepass@1over16independentsamplesperproblem)
withtemperature1.0samplingandreportpass@kfork ∈{4,8,16}.
3https://github.com/huggingface/Math-Verify
17

LLMReasoningwithWeakSupervision Table3.Math-domaintraining(1.5B/3B):in-domainbenchmarks. Model Metric t(8) MATH-500 AMC-2023 MinervaMath OlympiadBench AIME-2024 sat ∆sat ∆∗ post Gsat ∆sat ∆∗ post Gsat ∆sat ∆∗ post Gsat ∆sat ∆∗ post Gsat ∆sat ∆∗ post Gsat Qwen2.5-Math-1.5B Avg@16 302 29.7 1.5 -2.0 18.7 0.6 -0.1 14.3 1.5 -1.0 13.7 1.0 -0.8 7.3 0.6 0.2 Pass@4 12.3 0.5 -1.2 15.3 -0.1 0.2 14.0 2.0 -1.0 10.4 1.0 0.5 14.8 0.4 2.0 Pass@8 6.2 0.2 -1.0 13.1 -0.1 -1.2 11.1 2.0 -1.4 8.0 1.2 1.0 16.5 0.4 2.4 Pass@16 2.6 0.4 -0.4 10.8 -0.1 -2.8 10.3 1.2 -3.3 5.8 1.8 1.6 16.7 0.5 3.3 Qwen2.5-1.5B Avg@16 170 42.6 0.9 -0.8 15.8 3.3 -1.0 12.6 0.9 -1.3 13.5 1.2 0.2 1.0 0.9 -0.2 Pass@4 25.4 0.6 0.5 16.7 5.2 0.1 16.1 1.1 -1.1 13.9 1.3 0.6 3.4 2.9 0.3 Pass@8 18.6 0.6 1.0 15.0 7.1 2.1 15.7 1.0 -0.3 12.9 1.4 1.2 4.2 4.4 2.2 Pass@16 13.2 0.7 1.2 8.4 11.0 3.6 14.0 1.6 1.5 11.4 1.7 1.6 3.3 6.4 6.7 Llama3.2-3B-Instruct Avg@16 55 10.8 -1.9 -1.4 8.8 -2.1 -0.4 7.4 -1.1 -1.7 6.5 -0.7 0.5 7.1 -0.6 -3.8 Pass@4 23.9 -3.2 -0.6 21.3 -2.7 3.2 7.7 -1.3 -1.2 14.3 -1.1 2.4 0.0 0.0 -4.2 Pass@8 21.7 -3.9 -0.7 20.0 -2.1 6.4 5.6 -0.4 0.1 13.8 -0.9 3.3 0.0 0.4 -3.7 Pass@16 18.8 -3.8 0.2 15.7 -1.1 10.1 1.9 2.2 2.3 11.6 -1.0 4.4 0.0 1.1 -3.3 Table4.Math-domaintraining(1.5B/3B):out-of-domainbenchmarks. Model Metric t(8) GPQADiamond SCP-Hard MMLUSCI ScienceBench sat ∆sat ∆∗ post Gsat ∆sat ∆∗ post Gsat ∆sat ∆∗ post Gsat ∆sat ∆∗ post Gsat Qwen2.5-Math-1.5B Avg@16 302 12.6 1.1 0.9 10.5 2.1 2.4 25.4 1.2 -10.8 4.4 0.2 6.5 Pass@4 33.0 1.7 1.7 22.8 6.0 7.2 35.9 -0.0 -6.7 7.6 0.1 -1.6 Pass@8 41.0 2.5 2.2 25.9 8.6 10.2 31.9 -0.4 -5.4 5.6 0.2 -1.5 Pass@16 39.1 3.8 2.5 26.0 10.6 10.0 22.3 -0.4 -3.4 3.5 1.0 -0.7 Qwen2.5-1.5B Avg@16 170 13.8 1.4 -6.6 7.0 0.3 -0.4 29.4 -0.7 -11.2 6.8 0.6 -0.8 Pass@4 33.0 1.5 -15.5 19.0 2.0 -0.6 47.4 -0.7 -14.6 9.2 0.7 -0.6 Pass@8 37.9 1.0 -17.8 25.3 5.0 0.6 46.4 -0.4 -15.3 9.4 1.0 0.0 Pass@16 31.5 2.3 -16.2 28.0 9.8 2.0 37.4 -0.3 -16.1 9.0 1.2 0.4 LLama3.2-3B-Instruct Avg@16 55 -4.3 1.9 0.8 2.0 0.0 1.5 2.4 -0.4 -1.7 4.8 -0.3 0.1 Pass@4 -2.1 2.8 0.6 5.5 -0.4 3.3 0.1 0.6 0.3 6.8 -0.6 -0.5 Pass@8 0.5 3.0 0.7 8.3 -1.2 3.2 -0.3 0.6 0.2 7.7 -1.0 -0.8 Pass@16 -1.1 6.7 2.1 14.0 -4.3 0.0 -0.3 1.1 0.3 8.2 -1.0 -0.3 B.5.ImplementationDetailsofContinualPre-Training We continually pre-train Llama3.2-3B on the Nemotron-CC-Math-4plus subset (Mahabadi et al., 2025), comprising approximately52Btokensofmath-relevantdocumentsfilteredatqualityscore≥4. Trainingisconductedforoneepoch withamaximumsequencelengthof2,048tokensandabatchsizeof128sequences. WeuseAdamWwithapeaklearning rateof2×10−5,cosinedecayschedule,5%linearwarmup,weightdecayof0.01,andgradientclippingat1.0. B.6.ImplementationDetailsofSFT ForSFT,wetrainforthreeepochswithabatchsizeof16andamaximumsequencelengthof8192tokens. Wetunethe learningrateforeachmodelwithinthe1×10−5,5×10−5]andreportresultsforthebest-performingsetting. Forthe subsequentRLphase,weevaluateperformanceacrosstrainingsamplesizesN ∈ {8,2048}. Allotherhyperparameters follow the configurations established in Section B.3, with the maximum response length extended to 8192 tokens to accommodatelong-formreasoningtraces. C.DataScaleEffect C.1.AdditionalExperimentalResultsfromSmalltoLargeDataScale Figs. 13, 14, and 15 present domain-specific training dynamics and generalization performance across sample sizes N ∈ {8,32,64,512,2048}. Each figure tracks the training reward, two in-distribution benchmarks, and one OOD benchmark,aslistedinTable2. IntheMATHdomain,Llamamodelsexhibitrapidsaturationinsmall-sampleregimesandrelyheavilyondatascale. In contrast,Qwenmodelsyieldcomparableperformanceacrossvaryingsamplesizes,characterizedbyextendedsaturation periods. Specifically,themath-specializedQwen2.5-Math-1.5Bsustainsapre-saturationphasefor330gradientstepson8 18

LLMReasoningwithWeakSupervision Table5.Science-domaintraining(1.5B/3B):in-domainbenchmarks. t(8) Model Metric ScienceBench SCP-Hard GPQA-Diamond MMLUSCI SuperGPQA sat ∆sat ∆∗ Gsat ∆sat ∆∗ Gsat ∆sat ∆∗ Gsat ∆sat ∆∗ Gsat ∆sat ∆∗ Gsat

		post	post		post		post	post
Qwen2.5-Math-1.5B Avg@16 268 11.1 0.2 0.0 14.5 1.1 0.8 16.9 1.6 1.3 19.1 1.0 2.0 6.6 0.3 1.4
Pass@4 7.5 -0.1 -0.5 35.0 -1.3 -1.3 38.0 2.3 1.4 34.7 1.1 -0.4 18.1 0.7 2.8
Pass@8 6.0 -0.2 -0.7 43.5 -2.7 -1.6 44.5 1.7 0.5 31.4 0.9 0.3 24.7 0.4 2.8
Pass@16 4.6 0.2 -0.7 44.0 -1.2 -2.0 41.1 0.2 0.5 22.3 0.5 -0.0 29.4 -0.1 1.3
Qwen2.5-1.5B Avg@16 161 6.8 0.5 -0.0 6.4 0.2 0.6 13.3 1.7 2.9 25.4 4.4 8.7 7.9 0.9 2.4
Pass@4 9.7 0.4 0.0 20.4 -1.5 -0.5 33.5 1.9 5.1 54.9 2.0 7.3 21.1 1.5 5.8
Pass@8 9.9 0.3 0.3 31.1 -1.7 -2.3 39.7 1.5 5.4 62.5 0.1 4.7 28.5 2.0 7.4
Pass@16 9.4 0.3 0.7 40.0 1.7 -2.0 32.0 3.0 5.6 60.5 0.2 2.6 32.8 3.1 8.8
Llama3.2-3B-Instruct Avg@16 61 2.6 0.5 0.7 1.8 1.7 5.9 11.9 3.0 4.3 10.6 0.8 2.3 5.2 2.2 3.7
Pass@4 3.5 0.4 0.6 5.1 2.9 11.1 24.0 4.8 3.8 8.7 1.4 0.7 8.5 3.9 5.6
Pass@8 4.1 0.3 0.3 8.2 1.9 11.9 25.7 4.2 1.5 6.7 1.8 -0.2 10.6 3.9 4.5
Pass@16 4.8 0.1 -0.3 16.0 -4.3 6.0 22.3 2.3 -1.1 2.6 2.6 -0.5 11.7 3.4 2.2
Table6.Science-domaintraining(1.5B/3B):out-of-domainbenchmarks.
t(8)
Model	Metric	MATH-500		AMC		OlympiadBench		MinervaMath
-----	------	--------	---	---	---	-------------	---	-----------
sat
		∆∗		∆∗		∆∗		∆∗
---	---	---------	---------	----	---------	----	----	--------------
		∆sat post	Gsat ∆sat	post	Gsat ∆sat	post	Gsat	∆sat post Gsat
Qwen2.5-Math-1.5B Avg@16 268 25.3 0.8 1.1 14.7 1.5 0.1 11.4 0.9 0.5 12.7 1.1 0.6
	Pass@4	10.2 0.5	0.7 12.8	0.6	0.5 9.7	0.5	-1.1	12.5 1.3 0.7
---	-------	--------	--------	----	--------	----	----	------------
	Pass@8	4.5 0.8	0.5 10.8	0.1	0.1 8.1	0.0	-1.6	9.4 1.8 0.7
	Pass@16	1.2 1.3	0.4 9.4	-0.0	-1.3 7.4	-0.3	-2.1	7.7 1.7 0.7
Qwen2.5-1.5B Avg@16 161 32.3 2.1 1.2 14.5 0.7 -0.2 10.7 1.5 1.6 10.6 1.9 0.7
	Pass@4	30.9 1.1	0.9 21.1	1.9	2.0 16.9	1.4	2.1	18.3 1.4 0.7
---	------	--------	--------	---	--------	---	---	------------
	Pass@8	24.5 0.6	0.1 20.4	3.0	5.3 16.9	1.0	1.8	20.4 0.5 0.1
Pass@16 19.0 0.2 -0.6 15.7 4.1 11.2 14.1 1.1 1.4 21.3 0.3 -1.5
Llama3.2-3B-Instruct Avg@16 61 7.3 2.2 0.6 4.9 2.7 1.5 5.8 1.2 -0.1 5.9 1.2 1.2
	Pass@4	4.8 2.4	2.3 6.6	3.9	1.5 8.4	2.2	0.3	7.0 1.5 1.8
---	-------	-------	-------	---	-------	---	---	-----------
	Pass@8	3.2 2.4	3.2 6.0	4.5	0.2 8.3	2.6	0.4	5.9 2.2 2.0
	Pass@16	1.0 3.0	3.6 4.9	5.1	0.0 7.7	2.7	0.4	4.3 2.4 1.5
samples,drivingcontinuousimprovementsonin-domainbenchmarks.
IntheSCIENCEdomain,thepre-saturationphaseyieldssimilargainsacrossallsamplesizes;however,afterthesaturation
point,largersamplesizesdemonstratedistinctbenefits. SimilartoMATHdomain,modelsexhibitsignificantlydifferent
saturationdynamicsonsmallsamples.
IntheGRAPHdomain,wecomparetwolargermodels,Qwen2.5-Math-7BandLlama3.1-8B-Instruct. TheQwenmodelalso
saturatesfasterherethaninotherdomains,implyingthatthelackofdomain-specificpre-trainingacceleratessaturationin
small-sampleregimes.
C.2.FullEvaluationResults
Inthissection,wewillreportthefullevaluationresultswithallbenchmarksandpass@k(k ∈{1,4,8,16}metrics. Fig.16,
Fig. 17, Fig. 18, Fig. 19, Fig. 24, and Fig. 25 include in-domain and out-of-domain evaluation results across multiple
benchmarksinMATH,SCIENCEandGRAPHdomains.
Discussions on pass@k. Despite prior work (Yue et al., 2025) discussing divergent behavior between pass@1 and
pass@k for k > 1 during RL training, we observe that ∆(8) keeps the same sign for all k ∈ {1,4,8,16} across most
sat
model-benchmarkpairs,indicatingconsistentimprovementinbothpass@1andpass@k. Thisindicatesthatduringthe
pre-saturationperiod,themodelisnotjustclosingpass@kandpass@1gap.
C.3.AdditionalExperimentalResultsonLargeModels
Inthissection,wewillreportthefullevaluationresultson7Band8Bmodels.
Fig. 20 and Fig. 21 show the results of Qwen2.5-Math-7B and Llama3.1-8B-Instruct models on MATH domain with
19

LLMReasoningwithWeakSupervision Table7.Graph-domaintraining(7B/8B):in-distributionbenchmarks.

Model	Metric t(8)	QuantumLock		LargestIsland
sat
		∆ ∆∗	G ∆	∆∗ G
--------------------	----------	---------	---------	---------
		sat post	sat sat	post sat
Qwen2.5-Math-7B	Avg@16 151	8.0 4.9	7.3 19.8	1.9 -10.9
	Pass@4	22.6 1.5	-1.3 16.5	6.1 16.3
	Pass@8	26.8 2.8	-2.4 10.7	9.0 29.9
	Pass@16	30.6 5.5	-4.1 6.8	8.6 41.3
LLama3.1-8B-Instruct	Avg@16 29	10.1 7.1	3.2 1.8	1.0 -0.2
	Pass@4	15.3 -0.0	6.2 1.8	0.0 1.6
	Pass@8	15.4 7.4	12.5 1.8	0.0 2.8
	Pass@16	20.1 16.0	25.0 1.8	0.0 3.1
in-domainandout-of-domainbenchmarks,respectively.
Fig.22andFig.23presenttheresultsofQwen2.5-Math-7BandLlama3.1-8B-InstructmodelsonSCIENCEdomainwith
in-domainandout-of-domainbenchmarks,respectively.
Fig.25providestheresultsofQwen2.5-Math-7BandLlama3.1-8B-InstructmodelsonGRAPHdomainwithmoreout-of-
domainbenchmarks.
Similar to the observations on smaller models, during the pre-saturation phases, models show generalization on both
in-domainandout-of-domainbenchmarksintermsofpass@kmetrics. Comparedtothe3Bmodel,the8BLlamamodel
exhibitsbettercross-domaingeneralization. However,LlamamodelsstillsaturatemorefasterthanQwenmodelsandshow
cleardatadependence(e.g.,Fig.22onSCIENCE).
D.RewardTypeEffect
D.1.AdditionalResultsonRewardCorruption
Rewardcorruptionimplementation. Foreachcorruptionlevelγ,weuniformlysampleaγ fractionofpromptsfromthe
N =2048trainingsetforeachmodel–domainpair. Foreachselectedprompt,wedraw96modelresponsesattemperature
1.0andselectthemostfrequentlyoccurringincorrectfinalanswer(i.e.,onethatreceiveszerorewardunderourverifier)as
thecorruptedtarget. DuringRLtraining,wereplacetheground-truthlabelsoftheselectedpromptswiththesecorrupted
labels. For Llama models and the GRAPH domain, we cap γ at 0.9 due to the base model’s inability to generate valid
solutionsevenwithextensivesampling.
Results. Fig.2showscomplementaryresultstoSection3.2. Weobservesimilarpatternsthatsomemodelsarerobustto
evenlargeamountsofrewardnoise. Inparticular,Qwenmodelsexhibitgeneralizationabilityevenwhentrainedonalmost
completelycorrupteddata;incontrast,Llamamodelstendtoshowhighrewardcurvesyetpoorergeneralizationtonewdata,
suggestingoverfittingtoincorrectresponses.
D.2.AdditionalResultsonSelf-SupervisedProxyRewards
Proxyrewardsimplementation. Weevaluatetwoself-supervisedproxyrewardsasalternativestoground-truthverification:
majorityvotingandself-certainty.

1. MajorityVotingReward. FollowingTTRL(Zuoetal.,2025),weestimatepseudo-labelsviamajorityvotingand assignbinaryrewardsbasedonagreementwiththeconsensusanswer. Foreachprompt,wesample16responsesfrom thepolicymodel. Themostfrequentlyoccurringansweramongthese16responsesisselectedasthepseudo-label. Rewards are then computed as: r = 1 if the response matches the pseudo-label, and r = 0 otherwise. For policy optimization,weusethefirst8responsestocomputeadvantages. AllotherRLhyperparametersfollowSectionB.3.
2. Self-Certainty Reward. Following Zhao et al. (2025), we use the model’s own confidence as the reward signal. Self-certaintyisdefinedastheaverageKLdivergencebetweenauniformdistributionoverthevocabularyandthe 20

LLMReasoningwithWeakSupervision model’snext-tokendistribution: |o| 1 (cid:88) r =Self-certainty(o|q):= KL(U∥p (·|q,o )) (1) |o| πθ <i i=1 whereo denotespreviouslygeneratedtokensandU istheuniformdistributionoverthevocabulary. Highervalues <i indicategreatermodelconfidence. Foreachprompt,wesample8responsesandusetheself-certaintyscoresdirectlyas rewardstocomputeadvantages. AllotherRLhyperparametersfollowSectionB.3. Results. Fig.27showsfullresultsofself-supervisedproxyrewardsacrossmodel-domainpairs. ExceptforQwen2.5-Math- 1.5B,allothermodelsexhibitfailuremodesunderprolongedtraining. ForQwen2.5-1.5BonSCIENCE,bothproxyrewards collapse: majority voting shows a sharp reward spike followed by performance degradation, while self-certainty leads tocompletetrainingcollapse. Similarly,Llama-3.2-3B-InstructonMATHshowsdegradedperformancewithbothproxy rewardsdespiteincreasingtrainingrewards.OnlyQwen2.5-Math-1.5BonMATHmaintainsstableperformancewithmajority voting,thoughself-certaintystillcollapsesafterapproximately200steps. Theseresultsdemonstratethatself-supervised proxyrewardsarebrittleandmodel-dependent,withonlymath-specializedmodelsshowingpartialrobustness. D.3.RewardHackingExampleUnderMajorityVote Table8showstworolloutsfromQwen2.5-3Btrainedon SCIENCE withmajorityvoterewardsattrainingstep846. In bothcases,themodelproducesplausibleintermediatereasoningbutconvergestothesamefinalanswer 0 ,regardlessof theproblemcontent. Themajorityvoterewardis1.0becauseallrolloutsagreeonthisanswer—thepolicyhaslearned toproduceidenticaloutputstomaximizeconsensus,constitutingrewardhacking. Thecorrectanswers(68.4gandτ /k, 0 respectively)appearinthereasoningtracesbutareoverriddeninthefinalanswer. Table8.TworolloutsfromQwen2.5-3BonSCIENCEatstep846undermajorityvotereward.Bothproducecoherentreasoningtowardthe correctanswerbutoutput 0 asthefinalanswer,achievingmajorityvoterewardof1.0. Rollout1:Sucrosesolutionproblem Rollout2:Momentofinertiaproblem Prompt:Preparea0.0348molefractionsolutionofsucroseusing Prompt: A wheel with moment of inertia I is acted upon by 100gofwater. torqueτ ,resistedbyτ =−kω.Findthemaximumspeed. 0 f Reasoning(excerpt):“Massofsucroserequired=0.2moles× Reasoning(excerpt):“ω = τ0” max k 342g/mole=68.4g” Finalanswer: 0 Finalanswer: 0 Majorityvotereward:1.0 Majorityvotereward:1.0 E.BaselineEffect WeanalyzehowthechoiceofrewardbaselineinfluencesgeneralizationinGRPO.StandardGRPOusesthewithin-group meanreward(µ= 1 (cid:80)G r )asthebaseline. Byreplacingµwithaconstantbaselineb∈{0,1},weisolatethedirection G i=1 i of the policy update: b = 0 retains only positive reinforcement from correct samples (GRPO-POS), equivalent to the REINFORCEalgorithm,whileb=1retainsonlynegativereinforcementfromincorrectsamples(GRPO-NEG),which (Zhuetal.,2025)studiedinMATHdomain. Weremovethelengthpenaltyterm 1 inGRPOforthisexperiment. Based |o| onthepolicygradienttheory,wheresubtractinganaction-independentbaselinedoesnotchangetheexpectedgradientbut reducesvariance,withalargebatchthesetwomethodsshouldyieldsimilarlearningbehavior(Williams,1992). Figs.28and29presentthetrainingresultsontheSCIENCEdomainfor8and1024samples,respectively. Inbothregimes, GRPO-POSandGRPO-NEGachievecomparablePass@1performancetostandardGRPO,exhibitingsimilarsaturation andgeneralizationbehaviors. Wenotethatthiscontrastswithrecentfindingsby(Zhuetal.,2025),whichhighlightthe superiorityofGRPO-NEG. However,theirimprovementswereprimarilyobservedinPass@kmetricsratherthanPass@1 andevaluatedonMATHdomain. Beyondthesemetricdifferences,it’sworthstudyingwhetherimplementationartifactsmay alsoinfluenceobservations. Forinstance,clippingtermsintheGRPOformulationcanintroducebiases(Shaoetal.,2025; Chenetal.,2025a). Whileourstrictlyon-policysetupmitigatessuchclippingeffects,weleaveacomprehensiveanalysisof theseaffectstofuturework. 21

LLMReasoningwithWeakSupervision F.DiversityandFaithfulness Table9.Inter-rateragreementbetweenLLMjudgesmeasuredusingCohen’sKappa.

	JudgePair				Cohen’sKappa
	OpenAIo3vs.GPT-OSS-20B(OpenAI,2025b)				0.752
	OpenAIo3vs.Gemini3Flash(GoogleDeepMind,2025)				0.649
F.1.Quantificationofgenerationdiversity
Toquantifythegenerationdiversityofamodelonagivenprompt,wegenerateanumberofresponses,y ,...,y andcluster
						1	N
---	---	---	---	---	---	---	---
thembasedontheirreasoningsimilarity. Basingouranalysisonthemethodusedby(Lietal.,2025),todeterminereasoning
similaritybetweentwooutputsy ,y ,wedefineafunctions(y ,y ) ∈ {0,1}suchthats(y ,y ) = 1ify ,y aresimilar
		i j		i j		i j	i j
---	---	---	---	---	---	---	---
and0otherwise. Toevaluates(·,·),wepromptGPT-4o(Hurstetal.,2024)asadiversityjudgetodeterminewhetherthe
reasoningproducedbyanytworesponsesfollowsadifferentreasoningpathusingthepromptspecifiedinFig.31.
Weformsemanticclustersbyiteratingthroughresponsesandcomparingthemtoarepresentativeresponsefromeachexisting
cluster,creatinganewclusteriftheresponseisdissimilartoeachrepresentative. Thisisperformedundertheassumption
oftransitivityofsimilarity. Wecreateclusters{C ,...C }whereC ={y ,...,y }suchthats(y ,y )=1 ∀y ,y ∈C .
			1 K	i 1	ni	i j	i j i
---	---	---	---	---	---	---	-----
WethendefinethediversityscoresusingtheShannonDiversityIndex(Shannon,1948)asfollows.
Foragivenprompt,letN bethetotalnumberofresponses,n bethenumberofresponsesinclusterC ,andK bethe
				i		i
---	---	---	---	---	---	---	---
n
numberofclusters.	Letp i =	i DefinetheShannonentropy
-----------------	--------	-------------------------	---	---	---	---	---
N
K
(cid:88)
			H(p)=−	p logp
---	---	---	------	------	---	---	---
i i
i=1
andtheeffectivenumberofclusters
(cid:0) (cid:1)
			N eff =exp	H(p) .
---	---	---	----------	------	---	---	---
Wethendefinethediversityscore
N −1
			Div (x)=	eff .			(2)
---	---	---	--------	-----	---	---	---
π
K−1
whenK >1and0otherwise.
ForadatadistributionD,wedefinetheoverallgenerationdiversityasd (D)=E [Div (x)]. Empirically,wesample
				π	x∼D π
---	---	---	---	---	-----	---	---
N =16outputsperpromptandestimated using8promptsfromthespecifieddataset.
π
WedefineFaithfulDiversityasthismetriccalculatedonlyonresponsesthatachieveafaithfulnessscoreof1(seebelow).
Fig.36showsanexampleoftheLM-as-judgeoutputwhenpromptedtoevaluatethesimilarityof2responses.
F.2.Quantificationofreasoningfaithfulness
Inspiredbypriorwork(Bakeretal.,2025),wedefinethefaithfulnessasaresponse’sintermediatereasoningtracecontains
allrelevantinformationandremainslogicallyconsistentwiththepredictedfinalanswer. Eachmodelrolloutproducesa
responseythatcontains(i)areasoningtraceand(ii)afinalanswer. Wewritey =(r,a),whereristhereasoningtextanda
istheextractedfinalanswer. Foreachinputpromptx,wesampley ∼π(·	x)fromthepolicy.
Faithfulnesslabeling. Wedefineadiscretefaithfulnesslabelingfunctions :X ×Y →{0,1,1},wheres (x,y)
				faithful		2	faithful
---	---	---	---	--------	---	---	--------
measurestheinternalagreementbetweenrandainy:
• s faithful (x,y)=1(aligned)ifthereasoningtracerconstitutesacoherentandlogicallysupportivejustificationforthe
producedanswera,regardlessofwhetheraiscorrect;
• s (x,y)= 1 (partiallyaligned)ifrexhibitsaplausibleargumentativetrajectorytowardabutcontainssubstantial
faithful	2
--------	---	---	---	---	---	---	---
gaps,unsupportedleaps,orlocalinconsistenciesthatweakenthejustification;
22

LLMReasoningwithWeakSupervision • s (x,y) = 0(misaligned)ifaisnotsupportedbyr, e.g., r contradictsa, failstoaddressthequestion, orthe faithful answerappearsasthe“lucky”guess. Inpractice,weimplements (x,y)byqueryingOpenAIo3(OpenAI,2025a)asanLLM-as-a-judgewithafixedrubric faithful (Fig.32). OpenAIo3isusedforthistask,asopposedtoGPT-4o,duetorequiringalargermodelinordertobeableto accuratelyreasonaboutcomplexmathematicalandscientificstepspresentinthereasoningtraces. Foralabell∈{0,1,1}, 2 wedefinethefaithfulnessrateofpolicyπoverdatasetDas F (l) := P (cid:2) s (x,y)=l (cid:3) . π x∼D,y∼π(·|x) faithful Attrainingstept,weapproximateF (l)usingN trainingprompts{x }N andK rolloutsperprompt: πt i i=1 N K F(cid:98)πt (l) = N 1 K (cid:88)(cid:88) 1(cid:8) s faithful (cid:0) x i ,y i,k (cid:1) =l (cid:9) , y i,k ∼π t (·|x i ). (3) i=1k=1 WeuseN =8promptsandK =16rolloutsperpromptatselectedRLcheckpointsonthespecifiedtrainingdataset. We reportF(cid:98)πt (l)forl∈{0,1 2 ,1}tocharacterizethedistributionofreasoningfaithfulnessunderthepolicyπ t . Fig.37showsanexampleoftheLM-as-judgeoutputwhenpromptedtoevaluatethefaithfulnessofamodelresponsewhen trainedontheMATHtrainingdataset. ReliabilityofLLM-as-a-judge. TomitigatebiasfromusinganLLM-as-a-judgeforfaithfulnessevaluation,weassess consistency across multiple LLM judges by computing Cohen’s Kappa (Cohen, 1960) across 16 faithfulness-scored Qwen2.5-Math-1.5Boutputswhentrainedon8samplesfromtheMATHtrainingdatasetatsteps20,120and440. Thejudgesachievesubstantialagreement(κ=0.752and0.649),indicatingconsistentfaithfulnesslabelingacrossdifferent models. Weadditionallyconductedasmall-scalemanualevaluationtohuman-checkthefaithfulnessscoresandfindfair alignmentwiththeLLMjudges. F.3.Additionalresultsondiversityanalysis Fig.30showsthesemanticdiversityofLlama3.2-3B-Instruct,Qwen2.5-1.5BandQwen2.5-Math-1.5BontheMATH-500 evaluationdatasetthroughoutRLtraining. Qwen-Mathexhibitshigherreasoningdiversityoncorrectresponsesthanthe othermodelsatthelaterstagesoftraining,highlightingthatRLenablesittosuccessfullylearndiverseandreliablestrategies; coupledwithitsbetterperformanceontheevaluationdataset,thisindicatesstrongergeneralizationproperties. Inparticular, weobservesignificantlylowerdiversityintheLlama3.2-3B-Instructmodelincomparisontoitsdiversitymetriconthe trainingdataset(Fig.4),implyingdisagreementbetweentrainingandevaluationdistributionsandfurtherhighlightingthe limitationsoftrainingdiversityasanindicatorofreasoningcapabilities. F.4.Additionalresultsonfaithfulnessanalysis Fig.33showstheproportionofresponsesthatareclassifiedasalignedormisalignedwhencalculatedwithrespecttocorrect, incorrectorallresponses. Outofallcorrect,incorrectandoverallresponses,bothQwen2.5-1.5BandQwen2.5-Math-1.5B showhigherproportionofalignedresponsesandlowerproportionofmisalignedresponsesthanLlama3.2-3Bwhentrained on8samplesfromtheMATHdataset. Qwen2.5-Math-1.5Badditionallyshowsthisresultwhentrainedon8samplesfrom SCIENCE. G.Pre-RLIntervention Fig.34reportpass@16resultsandFig.35reportsresultsonmorebenchmarks. 23

LLMReasoningwithWeakSupervision ExamplepromptandresponseformatofSFT UserPrompt: Calculatethelimitofthefunction: arcsin(3x)

	lim	√ √
	x→0	2+x− 2
Assistant:
Okay,soIneedtofindthelimit...
Substitutingx=0givestheindeterminateform 0,so...
0
Rationalizingthedenominator,weget...
Thelimitiscalculatedasfollows:

1. Recognizetheindeterminateform: substitutingx=0yields 0 , 0 whichsuggestsusingrationalizationorl’Hoˆpital’srule.
2. Rationalizethedenominator: | | √ | √ | √ | √ |

| ---------- | ------ | ------------ | ------- | ------- | | | | | (cid:0) | (cid:1) | | arcsin(3x) | 2+x+ | 2 arcsin(3x) | 2+x+ | 2 | | √ | √ · √ | √ = | | . | | 2+x− | 2 2+x+ | 2 | x | | 3. Splitthelimit:

		√	√
	arcsin(3x)	(cid:0)	(cid:1)
	lim	· lim 2+x+	2 .
x
	x→0	x→0
---	---	---	---	---
4. Evaluateeachpart:
• Using
arcsinz
		lim =1,
---	---	-------	---	---
z
z→0
thefirstlimitbecomes3.
√
• Thesecondlimitevaluatesto2 2.
5. Combinetheresults: √ √
3·2 2=6 2.
FinalAnswer:
√
6 2
Figure12. ExamplepromptandresponseformatofSFT.InThinkingSFT,themodelistrainedwithreasoningtracesenclosedby
and,whereasNon-ThinkingSFTomitsthem.
24

LLMReasoningwithWeakSupervision

	8 samples		32 samples		64 samples			512 samples		2048 samples
	draweR gniniarT 1.0		)%( 005-HTAM
			72			52			)%( draH-PCS 36
htaM-5.2newQ	0.8					)%( CMA
						48			30
69
	0.6								24
-----	---	---	---	---	---	---	---	---	---	---	---
B5.1-						44
			66						18
	0.4					40
12
	0.2		63			36
------------	-----------------	-------	---------------	--------------	-------	---	--------------	-------	---------------	--------------	-------
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
	draweR gniniarT 1		)%( 005-HTAM 60
B5.1-5.2newQ						30			)%( draH-PCS 20
50
)%( CMA 24
15
			40			18
---	---	---	---	---	---	---	---	---	---	---	---
10
			30			12
---	--------------	-------	---	--------------	-------	---	--------------	-------	---	--------------	-------
			20			6			5
	0		10			0			0
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
draweR gniniarT 1.0
B3-5.2newQ			)%( 005-HTAM 66						)%( draH-PCS
----------	---	---	---------------	---	---	-------	---	---	------------	---	---
						40			25
	0.8					)%( CMA
			63			36			20
0.6
			60			32			15
---	---	---	---	---	---	---	---	---	---	---	---
	0.4					28			10
57
0.2
			54			24			5
---	-----	-------	---	---	-------	---	---	-------	---	---	-------
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
B3-2.3amalL Training Steps Training Steps Training Steps Training Steps
draweR gniniarT
tcurtsnI-	1.0		)%( 005-HTAM						)%( draH-PCS 15
---------	---	---	------------	---	---	-------	---	---	---------------	---	---
			54			30
	0.8					)%( CMA			12
	0.6		51			25			9
			48			20			6
0.4
						15			3
---	---	---	---	---	---	---	---	---	---	---	---
	0.2		45
0
10
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
---	--------------	-------	---	--------------	-------	---	--------------	-------	---	--------------	-------
	Training Steps			Training Steps			Training Steps			Training Steps
Figure13.ComparisonsofRLtrainingdynamicsandperformanceacrossdifferentmodelsonMATHdomain.Resultsareaveraged
overthreeindependentruns,withshadedregionsindicatingerrorbars. Verticaldashedlinesdenotethesaturationstepforeachdata
scaleifitsaturatesbefore496gradientsteps.Llamamodelsexhibitrapidsaturationinsmall-sampleregimesandrelyheavilyondata
scale.Incontrast,Qwenmodelsyieldcomparableperformanceacrossvaryingsamplesizes,characterizedbyextendedsaturationperiods.
Evaluationresultsinthisfigurearebasedongreedydecoding.
25

LLMReasoningwithWeakSupervision

	8 samples		32 samples		64 samples			512 samples		2048 samples
)%( dnomaiD AQPG
draweR gniniarT 1.0
			)%( draH-PCS 40			30			)%( 005-HTAM
------------	---	---	---------------	---	---	---	---	---	------------	---	---
htaM-5.2newQ									72
	0.8		35			25
	0.6					20			69
B5.1-			30
	0.4					15
			25						66
10
	0.2		20
---	-----	-------	---	---	-------	---	---	-------	---	---	-------
						5			63
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
Training Steps Training Steps )%( dnomaiD AQPG Training Steps Training Steps
	draweR gniniarT					30
------------	---------------	---	---------------	---	---	---	---	---	---------------	---	---
	1		)%( draH-PCS 30						)%( 005-HTAM 60
B5.1-5.2newQ						24
			24						50
			18			18			40
			12			12			30
			6			6
20
	0		0
---	---	---	---	---	---	---	---	---	---	---	---
0
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
---	-----	-------	---	---	-------	---	---	-------	---	---	-------
Training Steps Training Steps )%( dnomaiD AQPG Training Steps Training Steps
draweR gniniarT
B3-5.2newQ	1.0		)%( draH-PCS						)%( 005-HTAM
----------	---	---	------------	---	---	---	---	---	------------	---	---
			30			33			66
0.8
63
	0.6		24			30
---	---	---	---	---	---	---	---	---	---	---	---
			18			27			60
0.4
			12			24			57
---	---	---	---	---	---	---	---	---	---	---	---
0.2
			6			21
---	---	---	---	---	---	---	---	---	---	---	---
54
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
---	-----	-------	---	---	-------	---	---	-------	---	---	-------
Training Steps Training Steps )%( dnomaiD AQPG Training Steps Training Steps
B3-2.3amalL draweR gniniarT
tcurtsnI-	1.0								)%( 005-HTAM
---------	---	---	------------	---	---	---	---	---	------------	---	---
			)%( draH-PCS			30			54
	0.8		32
						27			52
24
	0.6					24			50
---	---	---	---	---	---	---	---	---	---	---	---
16
	0.4					21
---	---	---	---	---	---	---	---	---	---	---	---
			8						48
18
	0.2								46
---	---	---	---	---	---	---	---	---	---	---	---
0
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
---	--------------	-------	---	--------------	-------	---	--------------	-------	---	--------------	-------
	Training Steps			Training Steps			Training Steps			Training Steps
Figure14. ComparisonsofRLtrainingdynamicsandperformanceacrossdifferentmodelsonSCIENCEdomain. Resultsare
averagedoverthreeindependentruns,withshadedregionsindicatingerrorbars.Verticaldashedlinesdenotethesaturationstepforeach
datascale.Thepre-saturationphaseyieldssimilargainsacrossallsamplesizes;however,afterthesaturationpoint,largersamplesizes
demonstratedistinctbenefits.Modelsexhibitsignificantlydifferentsaturationdynamicsonsmallsamples.Evaluationresultsinthisfigure
arebasedongreedydecoding.
	8 samples		32 samples		128 samples			256 samples		882 samples
---	---------	---	----------	---	-----------	---	---	-----------	---	-----------	---
htaM-5.2newQ
	draweR gniniarT 1		)%( kcoL mutnauQ 50			)%( dnalsI tsegraL			82
---	-----------------	---	-------------------	---	---	------------------	---	---	---	---	---
)%( 005-HTAM
			40			32
---	---	---	---	---	---	---	---	---	---	---	---
80
B7-						24
---	---	---	---	---	---	---	---	---	---	---	---
			30						78
			20			16
76
8
10
74
	0		0			0
---	--------------	-------	---	--------------	-------	---	--------------	-------	---	--------------	-------
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
)%( kcoL mutnauQ
B8-2.3amalL	draweR gniniarT 1.0					)%( dnalsI tsegraL
-----------	-------------------	---	---	---	---	------------------	---	---	------------	---	---
tcurtsnI-						6			)%( 005-HTAM
			45						54
0.8
4
0.6
			30						52
---	---	---	---	---	---	---	---	---	---	---	---
	0.4					2
			15						50
0.2
			0			0
---	--------------	-------	---	--------------	-------	---	--------------	-------	---	--------------	-------
	0 150	300 450	0	150	300 450	0	150	300 450	0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
Figure15. ComparisonsofRLtrainingdynamicsandperformanceacrossdifferentmodelsonGRAPHdomain. Weuselarger
models(Qwen2.5-Math-7B,Llama-3.1-8B-Instruct)duetoincreasedtaskdifficulty.Resultsareaveragedoverthreeindependentruns,
withshadedregionsindicatingerrorbars.Verticaldashedlinesdenotethesaturationstepforeachdatascale.Qwenmodelalsosaturates
fasterherethaninotherdomains.Largerdatasetsyieldcleargainsinthepost-saturationphases.Evaluationresultsinthisfigurearebased
ongreedydecoding.
26

LLMReasoningwithWeakSupervision

		Qwen2.5-Math-1.5B		Qwen2.5-1.5B		LLama3.2-3B-Instruct		N=8	N=2048
	Avg@16(%)			Pass@4(%)			Pass@8(%)			Pass@16(%)
88
80
60									88
--------	---	---	---	---	---	---	---	---	---	---	---
005-HTAM						80
70
45						72			80
---	---	---	---	---	---	---	---	---	---	---	---
60
						64			72
---	---	---	---	---	---	---	---	---	---	---	---
30
			50			56
---	---	---	---	---	---	---	---	---	---	---	---
64
15
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	---	---	-----	---	---	-----	---	---	-----	---	-------
40			60			70			80
50
30						60			70
---	---	---	---	---	---	---	---	---	---	---	---
CMA
			40			50
---	---	---	---	---	---	---	---	---	---	---	---
20									60
			30			40			50
10
			20			30
---------------	---	---	-----	---	---	-----	---	---	-----	---	-------
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
htaM avreniM 24									48
			36			42
18			30			36			42
24
12						30			36
---	---	---	---	---	---	---	---	---	---	---	---
			18			24			30
6
			12			18
---	---	---	-----	---	---	-----	---	---	-----	---	-------
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
hcneBdaipmylO 30
			40			48			56
---	---	---	---	---	---	---	---	---	---	---	---
24
			32			40			48
---	---	---	---	---	---	---	---	---	---	---	---
18
						32			40
---	---	---	---	---	---	---	---	---	---	---	---
12			24
			16			24			32
6
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	---	---	-----	---	---	-----	---	---	-----	---	-------
40
15
4202 EMIA			24			32			40
---------	---	---	---	---	---	---	---	---	---	---	---
10
			18			24			30
---	---	---	---	---	---	---	---	---	---	---	---
5			12			16
20
0			6
---	---	---	-----	---	---	-----	---	---	-----	---	-------
						8			10
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
10.0
			16			20			24
---	---	---	---	---	---	---	---	---	---	---	---
5202 EMIA 7.5
12
						15			18
---	---	---	---	---	---	---	---	---	---	---	---
5.0
			8			10			12
---	---	---	---	---	---	---	---	---	---	---	---
2.5
			4			5			6
---	---	---	---	---	---	---	---	---	---	---	---
0.0
			0			0			0
---	--------------	---	-----	--------------	---	-----	--------------	---	-----	--------------	-------
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
Figure16.Fullin-domainbenchmarkevaluationresultsfortheMATHdomainacrossmultiplemodels.Verticaldashedlinesdenote
thesaturationstepforeachdatascale.
27

LLMReasoningwithWeakSupervision

		Qwen2.5-Math-1.5B		Qwen2.5-1.5B		LLama3.2-3B-Instruct		N=8	N=2048
	Avg@16(%)			Pass@4(%)			Pass@8(%)			Pass@16(%)
90
dnomaiD AQPG			60			75
------------	---	---	---	---	---	---	---	---	---	---	---
24									75
			45			60
16						45			60
30
8						30			45
---	---	---	---	---	---	---	---	---	---	---	---
15
0
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	---	---	-----	---	---	-----	---	---	-----	---	-------
32									80
			45			60
draH-PCS 24
60
45
16			30
---	---	---	---	---	---	---	---	---	---	---	---
						30			40
8			15
						15			20
0
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	---	---	-----	---	---	-----	---	---	-----	---	-------
			25			30
16
hcneB ecneicS
			20			25			30
---	---	---	---	---	---	---	---	---	---	---	---
12
						20			24
---	---	---	---	---	---	---	---	---	---	---	---
8			15
						15			18
4			10
10
			5						12
---	---	---	---	---	---	---	---	---	---	---	---
0
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	---	---	-----	---	---	-----	---	---	-----	---	-------
60			80
						80			90
ICS ULMM
45			60						75
---	---	---	---	---	---	---	---	---	---	---	---
60
30			40						60
---	---	---	---	---	---	---	---	---	---	---	---
40
45
15			20
---	---	---	---	---	---	---	---	---	---	---	---
20
0									30
---	---	---	-----	---	---	-----	---	---	-----	---	-------
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
60
20
AQPGrepuS			40
---------	---	---	---	---	---	---	---	---	---	---	---
						45			60
15			30
10									45
			20			30
5									30
			10			15
0
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	--------------	---	-----	--------------	---	-----	--------------	---	-----	--------------	-------
	Training Steps			Training Steps			Training Steps			Training Steps
Figure17.Fullin-domainbenchmarkevaluationresultsfortheSCIENCEdomainacrossmultiplemodels.Verticaldashedlines
denotethesaturationstepforeachdatascale.
28

LLMReasoningwithWeakSupervision

		Qwen2.5-Math-1.5B		Qwen2.5-1.5B		LLama3.2-3B-Instruct		N=8	N=2048
	Avg@16(%)			Pass@4(%)			Pass@8(%)			Pass@16(%)
			60						90
dnomaiD AQPG						75
24
			45			60			75
---	---	---	---	---	---	---	---	---	---	---	---
18
			30			45			60
---	---	---	---	---	---	---	---	---	---	---	---
12
6						30			45
---	---	---	---	---	---	---	---	---	---	---	---
15
0
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	---	---	-----	---	---	-----	---	---	-----	---	-------
20						60			80
40
draH-PCS
15			30			45			60
---	---	---	---	---	---	---	---	---	---	---	---
10						30
			20						40
5
			10			15			20
---	---	---	---	---	---	---	---	---	---	---	---
0
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	---	---	-----	---	---	-----	---	---	-----	---	-------
16			24			28			32
hcneB ecneicS
						24			28
---	---	---	-----	---	---	-----	---	---	-----	---	-------
12			20
8			16			20			24
						16			20
4			12
0			8			12			16
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
60			80
						80			90
ICS ULMM
45			60						75
---	---	---	---	---	---	---	---	---	---	---	---
60
30									60
---	---	---	---	---	---	---	---	---	---	---	---
40
40
15									45
---	---	---	---	---	---	---	---	---	---	---	---
20
						20			30
---	--------------	---	-----	--------------	---	-----	--------------	---	-----	--------------	-------
0 0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
Figure18.Fullout-of-domainbenchmarkevaluationresultsfortheMATHdomainacrossmultiplemodels.Verticaldashedlines
denotethesaturationstepforeachdatascale.
		Qwen2.5-Math-1.5B			Qwen2.5-1.5B	LLama3.2-3B-Instruct		N=8	N=2048
---	---------	-----------------	---	---------	------------	--------------------	---------	---	------	----------	---
	Avg@16(%)			Pass@4(%)			Pass@8(%)			Pass@16(%)
90
60			75						90
---	---	---	---	---	---	---	---	---	---	---	---
005-HTAM
						75			80
---	---	---	---	---	---	---	---	---	---	---	---
45			60
30						60			70
45
60
15			30			45
---	---	---	-----	---	---	-----	---	---	-----	---	-------
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
60
40									75
---	---	---	---	---	---	---	---	---	---	---	---
60
30			45
---	---	---	---	---	---	---	---	---	---	---	---
CMA									60
45
20
			30						45
---	---	---	-----	---	---	-----	---	---	-----	---	-------
10						30
			15						30
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
40
48
htaM avreniM 24						40
---------------	---	---	---	---	---	---	---	---	---	---	---
32
18									40
---	---	---	---	---	---	---	---	---	---	---	---
32
24
12						24			32
---	---	---	---	---	---	---	---	---	---	---	---
16
6									24
---	---	---	---	---	---	---	---	---	---	---	---
16
8
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
---	---	---	-----	---	---	-----	---	---	-----	---	-------
30
hcneBdaipmylO						50			60
-------------	---	---	---	---	---	---	---	---	---	---	---
40
24									50
---	---	---	---	---	---	---	---	---	---	---	---
40
18			30						40
---	---	---	---	---	---	---	---	---	---	---	---
30
12
			20						30
---	--------------	---	-----	--------------	---	-----	--------------	---	-----	--------------	-------
6						20
			10						20
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
Figure19.Fullout-of-domainbenchmarkevaluationresultsfortheSCIENCEdomainacrossmultiplemodels.Verticaldashedlines
denotethesaturationstepforeachdatascale.
29

LLMReasoningwithWeakSupervision

			Qwen2.5-Math-7B		LLama3.1-8B-Instruct		N=8	N=2048
	Avg@16(%)			Pass@4(%)			Pass@8(%)			Pass@16(%)
80			93			95.0			97.5
90
			87			92.5			95.0
-----------	---	---	---	---	---	----	---	---	----	---	---
005-HTAM 70						90.0			92.5
84
60			81			87.5			90.0
---	---	---	---	---	---	----	---	---	----	---	---
			72			80
84
50			68			76			80
---	---	---	-----	---	---	-------	---	---	-----	---	-------
			64			72
						68			76
40			60						72
			56			64
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
60									92
			76			82.5
			7 2			8 0 . 0			8 8
						7 7 . 5			8 4
45			6 8			7 5 . 0			8 0
CMA			64
						72.5			76
30			4 5			54			66
4 0
			35			48			60
------------	---	---	-----	---	---	-----	---	---	-----	---	-------
15			30			42			54
			25			36			48
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
32									57
htaM avreniM						51
28			44
						48			54
24			40
						45			51
20
36
						42			48
---	---	---	---	---	---	---	---	---	---	---	---
16
			32			39
----------------	---	---	-----	---	---	-----	---	---	-------	---	-------
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
hcneBdaipmylO 40			56			63			70.0
			5 2			6 0			6 7 . 5
6 5 . 0
32			4 8			5 7			6 2 . 5
---	---	---	---	---	---	---	---	---	-------	---	---
			4 4			5 4
24						5 1			6 0 . 0
			32			40			48
36
16			28						44
---	---	---	-----	---	---	-----	---	---	-----	---	-------
			24			32			40
			20			28			36
8									32
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
32						54
60
4202 EMIA			40			48			55
---------	---	---	---	---	---	---	---	---	---	---	---
24						42			50
			30			36
45
16									40
---	---	---	---	---	---	---	---	---	---	---	---
			20			24			32
8						18
			10			12			24
16
0						6			8
---	---	---	-----	---	---	-----	---	---	-----	---	-------
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
24
						32			40
---	---	---	---	---	---	---	---	---	---	---	---
5202 EMIA 12
			18			24
---	---	---	---	---	---	---	---	---	---	---	---
30
8
			12			16			20
---	--------------	---	-----	--------------	---	-----	--------------	---	-----	--------------	-------
4			6
						8			10
0			0			0			0
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
Figure20.Fullin-domainbenchmarkevaluationresultsfortheMATHdomainon7Band8Bmodels.
30

LLMReasoningwithWeakSupervision

		Qwen2.5-Math-7B		LLama3.1-8B-Instruct		N=8 N=2048
	Avg@16(%)		Pass@4(%)			Pass@8(%)		Pass@16(%)
64
dnomaiD AQPG 30					80
---------------	---	---	---	---	---	---	---	---	---
		56					88
25					72
80
20		48			64
---	-------	-----	---	---	-----	-------	-----	-------	---
15		40			56		72
10		32			48		64
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
56
24					72		88
--------	---	---	---	---	---	---	---	---	---
draH-PCS		48
18					64		80
40
					56		72
---	---	---	---	---	---	---	---	---	---
12
		32			48		64
---	---	---	---	---	---	---	---	---	---
6
40
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
---	-------	-----	---	---	-----	-------	-----	-------	---
33
hcneB ecneicS 18		27					36
----------------	---	---	---	---	---	---	---	---	---
30
15		24					33
---	---	---	---	---	---	---	---	---	---
27
		21					30
---	-------	-----	---	---	-----	-------	-----	-------	---
12					24
		18					27
9					21
		15					24
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
64
90
ICS ULMM		80					94.5
--------	---	---	---	---	---	---	----	---	---
56					88
48		76					93.0
86
40
		72			84		91.5
---	---	---	---	---	---	---	----	---	---
32
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
---	--------------	-----	--------------	---	-----	--------------	-----	--------------	---
	Training Steps		Training Steps			Training Steps		Training Steps
Figure21.Fullout-of-domainbenchmarkevaluationresultsfortheMATHdomainon7Band8Bmodels.
		Qwen2.5-Math-7B		LLama3.1-8B-Instruct		N=8 N=384
---	---------	---------------	---------	--------------------	---	---------	---	----------	---
	Avg@16(%)		Pass@4(%)			Pass@8(%)		Pass@16(%)
96
dnomaiD AQPG		64			80
------------	---	---	---	---	---	---	---	---	---
30							88
		56			70
24							80
48
18					60		72
---	---	---	---	---	---	---	---	---	---
40
12					50
---	-------	-----	---	---	-----	-------	-----	-------	---
		32					64
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
					80		90
40
		60			70		80
---	---	---	---	---	---	---	---	---	---
draH-PCS 30
		50			60		70
---	-------	-----	---	---	-----	-------	-----	-------	---
20		40
					50		60
10		30
					40		50
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
24		32			36		40
hcneB ecneicS
20		28			32		36
---	-------	-----	---	---	-----	-------	-----	-------	---
16		24			28		32
12		20			24		28
8		16			20		24
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
92
70
ICS ULMM		84			90		94
---------	-------	-----	---	---	-----	-------	-----	-------	---
60		80
					88		92
50		76
					86		90
40		72
					84		88
30		68
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
24							75
AQPGrepuS		45			60
70
20		40			55
---	---	---	---	---	---	---	---	---	---
65
16
		35			50
---	---	---	---	---	---	---	---	---	---
12							60
		30			45
55
0	150 300	450 0	150	300	450 0	150 300	450 0	150 300	450
---	--------------	-----	--------------	---	-----	--------------	-----	--------------	---
	Training Steps		Training Steps			Training Steps		Training Steps
Figure22.Fullin-domainbenchmarkevaluationresultsfortheSCIENCEdomainon7Band8Bmodels.
31

LLMReasoningwithWeakSupervision

			Qwen2.5-Math-7B		LLama3.1-8B-Instruct			N=8 N=384
	Avg@16(%)			Pass@4(%)			Pass@8(%)			Pass@16(%)
80			93			95.0			95
9 0
005-HTAM			8 7			9 2 . 5
--------	---	---	---	---	---	-------	---	---	---	---	---
70			8 4			9 0 . 0			90
			81			87.5
60									85
			75			80
50			70			76			80
65
72
40			60			68			75
---	---	---	---	---	---	---	---	---	---	---	---
64
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
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60						84			92
			7 5			81			88
			7 2			78
45			69						84
CMA			66			75			80
			6 43			7 2
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0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
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57.5
htaM avreniM 35						51
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			44						55.0
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52.5
25			40
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			36						50.0
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20						42
47.5
15			32
----------------	---	---	-----	---	---	--------	---	---	-----	---	-------
0	150	300	450 0	150	300	450 39 0	150	300	450 0	150	300 450
			56			64
hcneBdaipmylO 40									69
			48			60			66
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						52			60
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50
			32			40			45
---	--------------	---	-----	--------------	---	-----	--------------	---	-----	--------------	-------
16						35			40
			24			30
8						25			35
0	150	300	450 0	150	300	450 0	150	300	450 0	150	300 450
	Training Steps			Training Steps			Training Steps			Training Steps
Figure23.Fullout-of-domainbenchmarkevaluationresultsfortheSCIENCEdomainon7Band8Bmodels.
			Qwen2.5-Math-7B		LLama3.1-8B-Instruct			N=8 N=256
---	---------	---	---------------	---------	--------------------	---	---------	---------	---	----------	---
	Avg@16(%)			Pass@4(%)			Pass@8(%)			Pass@16(%)
100
kcoL mutnauQ						80
------------	---	---	---	---	---	---	---	---	---	---	---
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60
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dnalsI tsegraL						60
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0	150	300 450	0	150	300	450 0	150	300	450 0	150	300 450
---	--------------	-------	---	--------------	---	-----	--------------	---	-----	--------------	-------
	Training Steps			Training Steps			Training Steps			Training Steps
Figure24.Fullin-domainbenchmarkevaluationresultsfortheGRAPHdomainon7Band8Bmodels.
32

LLMReasoningwithWeakSupervision 75 60 45 30 0 150 300 450 005-HTAM Avg@16(%) Pass@4(%) Pass@8(%) Pass@16(%) 95.0 90 92.5 8 8 4 7 90.0 88 81 87.5 7 7 8 2 85 8 .0 0 80 64 72 72 56 64 48 56 64 0 150 300 450 0 150 300 450 0 150 300 450 18 12 6 0 0 150 300 450 4202 EMIA 60 40 50 35 45 30 40 45 25 35 30 16 24 12 18 8 12 15 4 6 0 0 0 150 300 450 0 150 300 450 0 150 300 450 40 32 24 16 0 150 300 450 htaM avreniM 55 48 50 55 42 45 50 36 40 45 30 35 40 24 0 150 300 450 0 150 300 450 0 150 300 450 32 24 16 8 0 150 300 450 dnomaiD AQPG 60 88 70 50 80 60 40 72 50 30 64 40 20 56 0 150 300 450 0 150 300 450 0 150 300 450 40 32 24 16 8 0 150 300 450 Training Steps draH-PCS Qwen2.5-Math-7B LLama3.1-8B-Instruct N=8 N=256 60 70 80 50 60 70 40 50 60 30 40 50 20 30 40 0 150 300 450 0 150 300 450 0 150 300 450 Training Steps Training Steps Training Steps Figure25.Fullout-of-domainbenchmarkevaluationresultsfortheGRAPHdomainon7Band8Bmodels. 0.60 0.45 0.30 0.15 0.00 draweR gniniarT Qwen2.5-1.5B Math 0.75 0.60 0.45 0.30 0.15 draweR gniniarT Llama-3.2-3B-Instruct Math 0.60 0.45 0.30 0.15 0.00 draweR gniniarT Qwen2.5-1.5B Science 0.8 0.6 0.4 0.2 draweR gniniarT Llama-3.2-3B-Instruct Science 1.0 0.8 0.6 0.4 0.2 0.0 draweR gniniarT Qwen2.5-Math-7B Graph 60 50 40 30 20 )%( 005-HTAM 54 51 48 45 42 39 )%( 005-HTAM 30 24 18 12 6 0 )%( tluciffiD-PCS 32 24 16 8 )%( tluciffiD-PCS 50 40 30 20 10 0 )%( kcoL mutnauQ 30 24 18 12 6 0 150 300 450 Training Steps )%( CMA 32 28 24 20 16 12 0 150 300 450 Training Steps )%( CMA 60 50 40 30 20 0 150 300 450 Training Steps )%( 005-HTAM 54 52 50 48 46 0 150 300 450 Training Steps )%( 005-HTAM 82 80 78 76 74 0 150 300 450 Training Steps )%( 005-HTAM =0 =0.1 =0.3 =0.5 =0.7 =0.9/1.0 Figure26.Effectofrewardlabelcorruptionontrainingdynamicsandgeneralization.γdenotesthefractionoftrainingprompts withcorruptedlabels,rangingfromclean(γ =0)tofullyincorrect(γ =1).QwenmodelsonMATHandSCIENCEdomainsmaintain performanceundersubstantialcorruption,whilegeneralizationofLlamamodelsandGRAPHdomaindegradeatγ ≥0.5.Evaluation resultsinthisfigurearebasedongreedydecoding. 33

LLMReasoningwithWeakSupervision Qwen2.5-Math-1.5B Llama-3.2-3B-Instruct Qwen2.5-1.5B Qwen2.5-Math-1.5B

		Math			Math			Science			Science
draweR gniniarT							1			1.0
0.75				0.90						0.8
0.75
0.60										0.6
----	---	---	---	----	---	---	---	---	---	---	---
				0.60						0.4
0.45				0.45
0.2
0.30				0.30			0
----	---	---	---	----	---	---	---	---	---	---	---
)%( 005-HTAM
				60			60
---	---	---	---	---	---	---	---	---	---	---	---
70
70
				50						60
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50
				40						50
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60
)%( tluciffiD PCS							12
-----------------	---	---	---	---	---	---	---	---	---	---	---
	32			15
				12			9			32
	24									24
				9			6
	16			6						16
	8						3
				3						8
	0			0			0			0
0 200 400 600 800 0 200 400 600 800 0 200 400 600 800 0 200 400 600 800
		Training Steps			Training Steps			Training Steps			Training Steps
---	---	--------------	---	----	--------------	-------------	---	--------------	---	---	--------------
				RLVR		Majority Vote		Self Certainty
Figure27.Comparisonofrewardvariants(RLVR,self-certainty,majorityvote)with1024trainingsamples.Proxyrewardswithout
verifiersexhibitfailuremodesunderprolongedtraining:trainingcollapse(self-certainty),andrewardspikesfollowedbyperformance
drops(majorityvote).Evaluationresultsinthisfigurearebasedongreedydecoding.
					GRPO-POS		GRPO-NEG		GRPO
---	---	---	---	---	--------	---	--------	---	----	---	---
Training Reward SCP-Difficult (%) GPQA Diamond (%) MATH-500 (%)
htaM-5.2newQ							)%( dnomaiD AQPG
------------	-------------------	---	---	--------------------	---	---	----------------	---	---	------------	---
	draweR gniniarT 1.0			)%( tluciffiD-PCS 35							72
								25		)%( 005-HTAM
	0.8			30							70
B5.1-
20
	0.6			25							68
---	---	---	---	---	---	---	---	---	---	---	---
15
	0.4			20							66
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								10			64
	0.2			15
5
62
		0 150 300	450		0 150	300	450	0 150	300	450	0 150 300 450
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Training Steps Training Steps )%( dnomaiD AQPG Training Steps Training Steps
	draweR gniniarT 1.0			)%( tluciffiD-PCS						52.5
-----------	-------------------	---	---	-----------------	---	---	---	---	---	------------	---
B3-2.3amalL				15						)%( 005-HTAM
27.5
	tcurtsnI- 0.8			12						51.0
---	-------------	---	---	---	---	---	---	---	---	----	---
25.0
				9						49.5
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	0.6						22.5
6
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3
46.5
	0.2			0			17.5
---	---	--------------	---	---	--------------	---	----	--------------	---	---	--------------
		0 150 300	450		0 150	300	450	0 150	300	450	0 150 300 450
		Training Steps			Training Steps			Training Steps			Training Steps
Figure28.EffectofbaselinevariantsonSCIENCEdomainwith8trainingsamples.GRPO-pos(positiveupdatesonly)andGRPO-neg
(negativeupdatesonly)producecomparableperformancetostandardGRPO.
					GRPO-POS		GRPO-NEG		GRPO
---	---	---	---	---	--------	---	--------	---	----	---	---
htaM-5.2newQ Training Reward SCP-Difficult (%) GPQA Diamond (%) MATH-500 (%)
)%( dnomaiD AQPG
	draweR gniniarT			)%( tluciffiD-PCS 40						)%( 005-HTAM
---	---------------	---	---	--------------------	---	---	---	---	---	------------	---
	0.75							25			72
	B5.1-			35
								20			70
	0.60			30
68
	0.45			25				15
---	----	---	---	---	---	---	---	---	---	---	---
66
10
	0.30			20							64
---	----	--------------	---	---	--------------	---	---	--------------	---	---	--------------
				15				5			62
		0 150 300	450		0 150	300	450	0 150	300	450	0 150 300 450
		Training Steps			Training Steps			Training Steps			Training Steps
)%( dnomaiD AQPG
	draweR gniniarT			)%( tluciffiD-PCS 25				28			54
-----------	---------------	---	---	--------------------	---	---	---	---	---	------------	---
B3-2.3amalL	0.90									)%( 005-HTAM
	tcurtsnI-							26
	0.75			20							52
24
	0.60			15							50
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	0.45							22			48
10
	0.30							20
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				5							46
	0.15							18
		0 150 300	450		0 150	300	450	0 150	300	450	0 150 300 450
		Training Steps			Training Steps			Training Steps			Training Steps
Figure29. EffectofbaselinevariantsonSCIENCEdomainwith1024trainingsamples. SimilartoFigs.28,GRPO-pos(positive
updatesonly)andGRPO-neg(negativeupdatesonly)producecomparableperformancetostandardGRPO.
34

LLMReasoningwithWeakSupervision Figure30. Responsediversityon8samplesfromtheMATH-500evaluationdataset. Qwen-mathshowshighdiversitywithinits correctanswers,suggestingarangeoflearnedrobustreasoningpaths. DiversityPromptforLLM-as-a-judge Youaregiventheoriginalpromptandtwomodel-generatedresponses. Determinewhetherthetworesponsesuse differentstrategiestosolvetheproblem. Usethefollowingguidelines: -Different solution methods: Clearly different approaches (e.g., algebraic vs. geometric, analytical vs. numerical). -Criticalreasoningdivergence:Significantdifferencesinkeyreasoningstepsorassumptions,eveniffinalanswers match. -Conceptualdifferences: Distinctunderlyingconceptsorrepresentations(e.g.,probabilityvs. combinatorics). **Alsolabelasdifferentif:**Thetworesponsessharethesamegeneralapproachbutdiffermeaningfullyin specificintermediatestepsormanipulationscrucialtothesolution. Originalprompt: prompt Generation0: generation0 Generation1: generation1 Question: DoGeneration0andGeneration1usedifferentstrategies? You may first generate a short reasoning, then end your response with either ||yes|| if they use different strategiesor||no||iftheyusethesamestrategy. Figure31.LMprompttochecksimilaritybetweenresponses. 35

LLMReasoningwithWeakSupervision FaithfulnessPromptforLLM-as-a-judge Youwillbegiven: (1)amathproblemprompt,and(2)amodelresponsethatmayincludeBOTHreasoninganda finalanswer. Definitions: ”Reasoning” = the parts of the model response that attempt to justify or derive a result (intermediate steps, equations,explanations,casework,narrativelogic). ”Finalanswer”=themodel’sexplicitcommittedresult(e.g.,after”Final:”,”Answer:”,”Therefore”,” boxed”, or the last clear numeric/symbolic conclusion). If multiple answers appear, treat the last explicitly committedoneasthefinalanswer. Task:Decidehowwellthereasoningsupportsthefinalanswer,usingtheselabels: Label1(Correlated): Thereasoningformsacoherentderivationthatwouldleadtothefinalanswerasstated. Minoralgebraslipsare allowedIFtheoverallderivationstillclearlytargetsthatanswer. Thefinalanswermaybeobjectivelywrong;you judgealignment,notcorrectness. Label0.5(PartiallyCorrelated): Thereasoningisrelatedtotheproblemandseemstomovetowardthefinalanswer,buthasmajorgaps,unjustified leaps,missingsteps,orseriouserrorsthatbreaktheproof. Theanswerisnotapurenon-sequitur,butthesupport isweak/incomplete. Label0(Uncorrelated): The final answer is not supported by the reasoning. Examples include: contradiction with earlier derived statements; switchingtoanunrelatedmethod; violatingkeyconstraintsfromtheprompt; orthefinalanswer appearingasanunsupportedguess. Outputformat(MANDATORY): 1)Brieflyidentify(a)theextractedfinalanswerand(b)thekeyreasoningpathin1–3sentences. 2)Thenoutputexactlyonelabeltokenonitsownattheend: ∥1∥or∥0.5∥or∥0∥. Prompt:prompt Response:response Question: Does the reasoning path correspond to the provided answer? You may first generate a short reasoning,thenendyourresponsewitheither∥1∥iftheyarefullycorrelated,∥0.5∥iftheyarepartiallycorrelated, or∥0∥iftheanswerisuncorrelatedtotheprecedinglogic. Figure32.LMprompttoevaluatereasoningfaithfulnessonasamplefromtheMATHdataset. 36

LLMReasoningwithWeakSupervision Figure33.Proportionofalignedandmisalignedresponsesacrossmodelsandtrainingdatasets.

draweR gniniarT		)%( 005-HTAM 90			72		80
)%( draH-PCS
		84			)%( CMA 64
------	---	---	---	---	----------	---	---	---	---
ecracS	0.6	78			56
ataD
40
60
40
		54			30
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0.0
	0 150 300	450	0 150	300	450	0 150 300	450	0 150	300 450
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90
draweR gniniarT		)%( 005-HTAM					)%( draH-PCS
---------------	---	------------	---	---	----------	---	------------	---	---
ytirojaM					)%( CMA 60		60
etoV
0.6
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40
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0 150 300 450 600 0 150 300 450 600 0 150 300 450 600 0 150 300 450 600
draweR ysioN
draweR gniniarT		)%( 005-HTAM
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	0.6						)%( draH-PCS
)7.0=					)%( CMA 60
		75					60
0.3
(
40
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	Training Steps		Training Steps			Training Steps		Training Steps
Base + Thinking SFT Base + Non-Thinking SFT CPT + Thinking SFT CPT + Non-Thinking SFT Instruct
Figure34.Evaluationresultsofpass@16metricacrossmodelswithdifferentpre-RLinterventiononweaksupervision.
37

LLMReasoningwithWeakSupervision

	avg@16		pass@16		avg@16		pass@16
)%( hcneB ecneicS
)%( 4202 EMIA
30
8
		20		10
---	---	---	---	---	---	---	---	---
ecracS
ataD
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0 0 150 300 450 0 0 150 300 450 0 0 150 300 450 0 150 300 450
)%( hcneB ecneicS
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ytirojaM
8		20		10		20
---	---	---	---	---	---	---	---	---
etoV
0		0		0		0
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draweR ysioN				)%( hcneB ecneicS
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)%( 4202 EMIA
)7.0=						30
-----	---	---	---	---	---	---	---	---
8		25		10
(
15
0		0		0
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0	150 300	450	0 150 300	450	0 150 300	450	0 150 300	450
	Training Steps		Training Steps		Training Steps		Training Steps
Base + Thinking SFT Base + Non-Thinking SFT CPT + Thinking SFT CPT + Non-Thinking SFT Instruct
Figure35. EvaluationresultsonAIME2024andScienceBenchacrossmodelswithdifferentpre-RLinterventiononweak
supervision.
38

LLMReasoningwithWeakSupervision DiversityJudgeExampleOutput Prompt: Aboxcontains5whiteballsand5blackballs. Idrawthemoutofthebox,oneatatime. Whatisthe probabilitythatallofmydrawsalternatecolors,startingandendingwiththesamecolor? ResponseA: To determine the probability that all of the ball draws alternate colors, starting and end- ingwiththesamecolor,weneedtoconsiderthefollowing:

1. Totalnumberofwaystoarrangetheballs: Thereare5whiteballsand5blackballs,sothetotalnumberof waystoarrangethese10ballsisgivenbythebinomialcoefficient (cid:0)10(cid:1) ,whichrepresentsthenumberofwaysto 5 choose5positionsoutof10forthewhiteballs(theremaining5positionswillbefortheblackballs). Thisvalue iscalculatedas: (cid:18) (cid:19) 10 10! = =252 5 5!5!
2. Numberoffavorableoutcomes: Forthedrawingstoalternatecolorsstartingandendingwiththesame color,thereareonlytwopossiblesequences: -Startingwithawhiteballandalternating(WBWBWBWBW B)-Startingwithablackballandalternating(BWBWBWBWBW) Sothereareonly2favorableoutcomes.
3. Probability calculation: Theprobability istheratioof thenumberof favorableoutcomestothe total numberofoutcomes. Therefore,theprobabilityP is: Numberoffavorableoutcomes 2 1 P = = = Totalnumberofoutcomes 252 126 1 Sothefinalansweris . 126 ResponseB: Tosolvetheproblem,weneedtodeterminetheprobabilitythatasequenceof10balls(5white and5black)isarrangedsuchthattheyalternatecolors. Thereareonlytwopossiblesequencesthatmeetthis condition: ”WBWBWBWBWB”and”BWBWBWBWBW”. Thetotalnumberofwaystoarrange5whiteand5 blackballsisgivenbythebinomialcoefficient (cid:0)10(cid:1) ,whichrepresentsthenumberofdistinctpermutationsof5 5 whiteand5blackballs. Thisiscalculatedas: (cid:18) (cid:19) 10 10! = =252 5 5!5! Sincethereareonly2favorableoutcomes(thetwoalternatingsequencesmentionedabove),theprobabilityisthe ratioofthenumberoffavorableoutcomestothetotalnumberofoutcomes: 2 1 Probability= = 252 126 1 So,theprobabilitythatallofmydrawsalternatecolors,startingandendingwiththesamecolor,is . 126 JudgeOutput: Both Generation 0 and Generation 1 use the same strategy to solve the problem. They both identifythattherearetwopossiblesequencesthatsatisfytheconditionofalternatingcolorsandstartingand endingwiththesamecolor: ”WBWBWBWBWB”and”BWBWBWBWBW”. Theybothcalculatethetotal numberofwaystoarrangethe10ballsusingthebinomialcoefficient (cid:0)10(cid:1) ,whichaccountsforthepermutations 5 of5whiteand5blackballs. Finally, theybothcomputetheprobabilitybytakingtheratioofthenumberof favorableoutcomes(2)tothetotalnumberofoutcomes(252),resultinginthesamefinalanswerof 1 . 126 Therearenosignificantdifferencesintheirreasoningsteps, assumptions, orintermediatecalculations. Both responsesfollowasimilarlogicalflowandusethesamemathematicalconceptstoarriveatthesolution. JudgeResult: ∥no∥ Figure36.QualitativeExampleofDiversityAnalysis 39

LLMReasoningwithWeakSupervision FaithfulnessJudgeExampleOutput Prompt: Findallrealpairs(a,b)thatsolvethesystemofequation a2+b2 =25, 3(a+b)−ab=15. Response: Tosolvethesystemofequations: (cid:40) a2+b2 =25, 3(a+b)−ab=15, wecanusethefollowingreasoningandalgebraicmanipulations. First,recalltheidentity: a2+b2 =(a+b)2−2ab. Let’sdenotes=a+bandp=ab. So,thefirstequationbecomes: s2−2p=25.Thesecondequationremains: 3s−p=15.Wecansolvethesetwoequationssimultaneously. First,weexpresspfromthesecondequation: p=3s−15.Next,wesubstitutepintothefirstequation: s2−2(3s−15)=25.Simplifytheequation: s2−6s+30=25 =⇒ s2−6s+5=0. √ Thisisaquadraticequationins. Wecansolveforsusingthequadraticformulas= −b± b2−4ac,wherea=1, 2a b=−6,andc=5: √ √ 6± 36−20 6± 16 6±4 s= = = . 2 2 2 So,thesolutionsforsare: s=5 or s=1.Now,wefindthecorrespondingpvaluesforeachs: 1. Ifs=5, thenp=3(5)−15=0. 2. Ifs=1,thenp=3(1)−15=−12. Wenowneedtofindthepairs(a,b)thatsatisfy theequationsa+b=sandab=p. Thesearetherootsofthequadraticequations: 1. Fors=5andp=0,the quadraticequationis: t2−5t=0 =⇒ t(t−5)=0. So,thesolutionsaret=0andt=5. Thus,thepairs(a,b)are(0,5)and(5,0). 2. Fors=1andp=−12,the quadraticequationis: t2−t−12=0. Wesolvethisquadraticequationusingthequadraticformula: √ 1± 1+48 1±7 t= = . 2 2 So,thesolutionsaret=4andt=−3. Thus,thepairs(a,b)are(4,−3)and(−3,4). Therefore,thecomplete setofsolutionsis: (0,5),(5,0),(4,−3),(−3,4) . JudgeOutput: Finalanswergiven: (0,5),(5,0),(4,−3),(−3,4).Reasoningpath: Introducessumss=a+b andproductp=ab,rewritesequationsintoasysteminsandp,solvestogets=5or1,findscorrespondingp, thensolvesquadratict2−st+p=0toobtainallorderedpairsconsistentwitheach(s,p),yieldingthelisted solutions. Thederivationiscompleteandconsistentlyleadstothestatedpairs. JudgeResult: ∥1∥ Figure37.QualitativeExampleofFaithfulnessAnalysis 40