Residual Stopping and Pointwise Inference
for Nonuniform Entropic Self-Transport
ScottSchubr
LuminaryOdditiesLLC
October18,2026
Abstract
Westudyobservableresidualstoppingforthehalf-self-Sinkhorniterationwithiidsamplesfrom
a fixed positive smooth density on the one- or two-dimensional torus. For the normalized periodic
Gaussian, assumethat 𝜀 → 0, 𝑛𝜀3𝑑/2 → ∞, and 𝑛𝜀𝑑/2+3 → 0. Startingfromtheconstantscaling,
westopatthefirstprojectivesquare-root-massresidualbelow𝑡 𝑛 = 𝑛−1/2𝜀−𝛾 ,where0 < 𝛾 < 𝑑/4is
fixed. ThestoppedscorehasthesamepointwiseGaussianlimit,centeredatthetruedensityscore,as
theexactlyscaledempiricalestimator. Itsattainedr√esidualliesbetween𝑐𝑡 𝑛 and𝑡 𝑛 withprobability
tendingtooneforeveryfixed0< 𝑐 <1/2,although 𝑛𝑡 𝑛 →∞. Thestoppingindexisasymptoticto
(𝑑/4−𝛾)log(1/𝜀)/log2inprobability. Forthecoefficient-onethreshold,uniformsamplingalready
givesadifferentscorelimitwhen𝛾 ≥ 𝑑/4. Theproofretainstheiteration-dependentempirical-norm
errorandusesanormalized-rowcancellationtocontrolnonuniforminitializationindimensiontwo.
Keywords: entropic self-transport; Sinkhorn iteration; statistical stopping; score estimation; pointwise
inference.
1 Introduction
Symmetrickernelscalinggivesanempiricalentropicself-transportscorethroughitssmoothoff-sample
extension. Anumericalsolverisnormallyterminatedusinganormalizationresidual,whereasstatistical
inferencedependsontheerrorinaspatialderivative. Thispaperidentifiesafamilyofresidualtolerances
thatpreservesthepointwiselimitinglawalongtheactualsamplediteration.
Thetolerancecanexceed𝑛−1/2
byadivergingfactor. Weprovethisforafirst-hitrulecheckedfrom
theinitialscaling,ratherthanforaprescribediterationnumber. Wealsodeterminetheleadingstopping
count,controltheresidualattainedatthathit,andgiveacounterexampleattheuppertoleranceexponent.
Thesameobservationsdeterminethescale,iterates,residualandqueriedscore.
1.1 Relatedwork
Kokot’sdissertation[2],Chapter3,givesGaussian-RKHSscalingdifferentiability,theself-diagonalplus-
resolventderivative,andafixed-regularizationHilbertcentrallimittheorem. ItsSection4.4.1alsogives
first-orderpopulationmatchingaftertwosymmetricupdates. Weusetheseestablishedmechanismsand
providethe joint bandwidth–sample–iteration estimatesneeded forresidual stopping. Mordant [3]stud-
ies decreasing-regularization self-transport score asymptotics and the half-self iteration in a Euclidean
quadratic-costmodel. OurstatisticallimitisprovedfortheexactperiodicGaussian.
WormellandReich[4]analyzeempiricalbistochasticnormalization,resolventsandspectralconver-
gence,includinganalternativesymmetricscalingiteration. ChengandLanda[5]showthatapproximate
nonuniform normalization at statistical accuracy suffices for graph-Laplacian consistency. Maeda, Yao
andNitanda[7]combinesamplinganditerationerrorsfortwo-sampleSchrödingerbridges. Spectraldis-
crepancystoppingisdevelopedin,amongotherworks,CelisseandWahl[6]. Theseresultsestablishthe
generalstatisticalmotivationforearlytermination.
1

Thepresentconclusionsconcernthespecificnonlinearfirstresidualhit: itsadmissibleexponentrange,
attained residual and leading count, together with preservation of the exact estimator’s true-score limit.
The earlier residual-certificate paper [8] gives bounds for arbitrary positive candidates and empirical
sharpnesswitnesses. Itdoesnotgivetheactual-trajectorystoppingtheoremprovedhere. Thebandwidth
windowsbelowaresufficient;nooptimalsamplingrangeorwall-clockcomplexityclaimismade.
2 Model and main theorem
Let 𝑑 ∈ {1,2} and let 𝑓 ∈ 𝐶6(T𝑑) be a fixed probability density with min 𝑓 > 0. Set 𝜇 = 𝑓 𝑑𝑥 and let
˝
𝜇 𝑛 = 𝑛−1 𝑖 𝑛 =1 𝛿 𝑋 𝑖 foriid 𝑋 𝑖 ∼ 𝜇. Lebesguemeasureontheunittorushastotalmassone. Thekernelis
(cid:213)
𝑘 𝜀 (𝑧) = (2𝜋𝜀)−𝑑/2𝑒−|𝑧+ℓ|2/(2𝜀), 𝜆 𝑘 = 𝑒−2𝜋2𝜀|𝑘|2. (1)
ℓ∈Z𝑑
Wealsowrite 𝑘 𝜀 (𝑥,𝑦) = 𝑘 𝜀 (𝑥 − 𝑦). Here𝜆 𝑘 arethfleFouriermultipliersand 𝐻 𝜀 = exp(𝜀Δ/2) isconvo-
lutionagainstLebesguemeasure. Write𝐾 𝜈 𝑏(𝑥) = 𝑘 𝜀 (𝑥−𝑦)𝑏(𝑦)𝑑𝜈(𝑦) and𝐾 𝑛 = 𝐾 𝜇 . Throughout,
𝑛
𝜀 = 𝜀 𝑛 → 0, 𝑛𝜀3𝑑/2 → ∞, 𝑛𝜀𝑑/2+3 → 0. (2)
Constants may depend on this fixed 𝑓 and on fixed exponents. There is no assertion uniform over an
unboundedclassofdensityderivativenorms.
Letb𝑎bethepositiveempiricalself-scale,b𝑎𝐾 𝑛 b𝑎 = 1. Atthesamplesitesitexistsuniquely: thestrictly
convexcoercivefunction
(cid:213) (cid:213)
1 1
Φ(𝑢) = 2𝑛2 𝑘 𝜀 (𝑋 𝑖 − 𝑋 𝑙 )𝑒𝑢 𝑖 +𝑢 𝑙 − 𝑛 𝑢 𝑖
𝑖,𝑙 𝑖
hasauniqueminimizeranditsgradientequationsarethescalingequations. Coercivityandstrictconvex-
ityfollowalreadyfromthepositivediagonalterms. Extendthescalebyb𝑎(𝑥) = 1/𝐾 𝑛 b𝑎(𝑥) awayfromthe
sites. Startingwith𝑏 = 1,usetheaveragedsymmetriciteration[1,3]
0
(cid:8) (cid:9)
𝑏 𝑗+1 = 𝑏 𝑗 /(𝐾 𝑛 𝑏 𝑗 )
1/2,
𝑠 𝑗 = 2∇log𝐾 𝑛 𝑏 𝑗 , b𝑠 = 2∇log𝐾 𝑛 b𝑎. (3)
Onlysitevaluesareneededforeachconvolution. Define
(cid:13) (cid:13)
𝐹 𝑗 = (𝑏 𝑗 𝐾 𝑛 𝑏 𝑗 )1/2, 𝑆 𝑗 = 𝜇 𝑛 𝐹 𝑗 , 𝑟 𝑗 = (cid:13)𝐹 𝑗 /𝑆 𝑗 −1 (cid:13) 𝑛 , ∥𝑢∥2 𝑛 = 𝜇 𝑛 |𝑢|2. (4)
Both𝑟 𝑗 and 𝑠 𝑗 areinvariantunderpositivescalarrescalingof𝑏 𝑗. Forfixed𝛾 > 0,put
𝑡 𝑛 = 𝑛−1/2𝜀−𝛾, 𝐿 𝑛 = max{3,⌈4log𝑛⌉},
(cid:0) (cid:1)
𝜏 𝛾 = min {𝑗 ∈ {0,...,𝐿 𝑛 } :𝑟 𝑗 ≤ 𝑡 𝑛 }∪{𝐿 𝑛 } . (5)
Thebudgetmakestherulefiniteoneverysample. Set
„
√ (2𝜋𝜉 )2
𝐴 𝑛 = 𝑛𝜀𝑑/4+1/2, 𝐽 𝑑 = 1 𝑑𝜉 > 0. (6)
R𝑑 (1+𝑒2𝜋2|𝜉|2)2
Theorem2.1. Assume(2)andfix0 < 𝛾 < 𝑑/4. Then
𝜏
𝛾
𝑑/4−𝛾
−→ inprobability. (7)
log(1/𝜀) log2
Foreveryfixed0 < 𝑐 < 1/2,
P{𝜏 𝛾 < 𝐿 𝑛 , 𝑐𝑡 𝑛 ≤ 𝑟 𝜏 ≤ 𝑡 𝑛 } → 1. (8)
𝛾
2

Foreachfixed𝑥 ∈ T𝑑 ,
(cid:13) (cid:13)
𝐴 𝑛 (cid:13)𝑠 𝜏 (𝑥)−b𝑠(𝑥)(cid:13) → 0 inprobability, (9)
𝛾 (cid:18) (cid:19)
𝐴 𝑛 {𝑠 𝜏 𝛾 (𝑥)−∇log 𝑓(𝑥)} ⇒ 𝑁 𝑑 0, 𝑓 4 ( 𝐽 𝑥 𝑑 ) 𝐼 𝑑 . (10)
Theexactempiricalscoreb𝑠satisfiesthesamecenteredlimit.
√
Inparticular 𝑛𝑟 𝜏 → ∞inprobability. For𝜀 = 𝑛−𝜅 thesufficientpowerwindowsare
𝛾
𝑑 = 1 : 2/7 < 𝜅 < 2/3, 𝑑 = 2 : 1/4 < 𝜅 < 1/3.
Forfixed0 < 𝛾 < 𝛾 < 𝑑/4,subtractiononthesamesamplegives
1 2
𝜏 −𝜏 𝛾 −𝛾
𝛾 1 𝛾 2 −→ 2 1 inprobability. (11)
log(1/𝜀) log2
This compares update counts, with no claim about cost per update. The upper exponent obstruction is
provedinSection7.
3 Population geometry and the empirical scaling branch
Throughoutthissection, 𝑑 ∈ {1,2}, 𝑓 isthefixeddensityinthemaintheorem,and 𝜇 = 𝑓 𝑑𝑥. Constants
can depend on 𝑓 and 𝑑. Write 𝐻 𝜀 for convolution against the kernel in (1) with respect to Lebesgue
measureandwrite
(cid:213)𝑛
1
𝐾 𝜇 𝑢 = 𝐻 𝜀 (𝑓𝑢), 𝐾 𝑛 𝑢 = 𝑛 𝑘 𝜀 (·,𝑋 𝑖 )𝑢(𝑋 𝑖 ).
𝑖=1
Let𝑎 denotethepositivepopulationscale,𝑎𝐾 𝜇 𝑎 = 1,andput
p
𝑣 = 𝑎−1 = 𝐾 𝜇 𝑎, 𝑤 = 𝐻 𝜀 𝑓, 𝑞 = 𝑎 𝑓, 𝑑 1 = log𝑣− 1 2 log𝑤.
Allboundsbelowholdas𝜀 ↓0.
Lemma3.1(Populationapproximation). Thepositivepopulationscaleisuniqueandsatisfies
∥𝑞−1∥ 𝐶2 ≤ 𝐶𝜀, ∥𝑑 1 ∥ 𝐶2 ≤ 𝐶𝜀, ∥−2∇log𝑎−∇log 𝑓∥ ∞ ≤ 𝐶𝜀. (12)
Inparticular,𝑎,𝑣,𝑤 areboundedaboveandbelow,andtheir𝐶2 normsareboundeduniformlyforsmall
𝜀.
p
Proof. Set𝑔 𝑓 = 𝑓. Thescalingequationbecomes
F 𝜀 (𝑞) := 𝑞𝐻 𝜀 (𝑞𝑔 𝑓 )−𝑔 𝑓 = 0.
We solve it in the Banach algebra 𝐶2(T𝑑). First, ∥𝐻 𝜀 𝑔 𝑓 −𝑔 𝑓 ∥ 𝐶2 ≤ 𝐶𝜀∥𝑔 𝑓 ∥ 𝐶4 by the heat semig √ roup
identity. Second, (𝐼 +𝐻 𝜀 )−1 isuniformlyboundedon𝐶2. Indeed,itsFouriermultiplieris1−𝐹( 𝜀𝑘),
where
1
𝐹(𝜉) = .
1+exp(2𝜋2|𝜉|2)
The function 𝐹 is Schwartz. Periodization of its rescaled inverse Fourier transform gives a convolution
kernelwhose𝐿1normisboundedindependentlyof𝜀. Thesameboundholdsaftercommutingderivatives
throughconvolution.
At𝑞 = 1,
𝐷F 𝜀 (1)𝑢 = (𝐻 𝜀 𝑔 𝑓 )𝑢+𝐻 𝜀 (𝑔 𝑓 𝑢) = 𝑔 𝑓 (𝐼 +𝐻 𝜀 )𝑢+𝐸 𝜀 𝑢,
3

√
where ∥𝐸 𝜀 ∥ 𝐶2→𝐶2 =𝑂( 𝜀). Tocheckthelastbound,differentiatethecommutator [𝐻 𝜀 ,𝑀 𝑔 𝑓 ]𝑢through
ordertwo. Eachresultingtermisacommutatorwithaderivativeof𝑔 𝑓;its
√
kernelcontainsthedifference
ofthatderivativeattwopoints. TheGaussianfirstabsolutemomentis𝑂( 𝜀),soeachtermisbounded
√
by𝐶 𝜀∥𝑢∥ 𝐶2 . Themultiplicationerror𝐻 𝜀 𝑔 𝑓 −𝑔 𝑓 hasthestronger𝑂(𝜀) bound. Consequently𝐷F 𝜀 (1)
hasauniformlyboundedinverseforsmall𝜀. TheremainingterminF
𝜀
(1+𝑢) is𝑢𝐻
𝜀
(𝑔
𝑓
𝑢),boundedby
𝐶∥𝑢∥2 withthecorrespondinglocalLipschitzestimate. Acontractiononaballofradius𝐶𝜀 therefore
𝐶2
yieldsapositivesolutionwith𝑞 = 1+𝑂 (𝜀).
𝐶2
Foruniqueness,twopositivescales𝑎,𝑏 obey
„
(𝑎−𝑏)2
0 ≤ ⟨𝑎−𝑏,𝐾 𝜇 (𝑎−𝑏)⟩ 𝐿2(𝜇) = − 𝑎𝑏 𝑑𝜇 ≤ 0.
Here ppositivity follows from the nonnegative Fourier multipliers of the kernel. Hence 𝑎 = 𝑏. Finally
𝑎 = 𝑞/ 𝑓,𝑤 = 𝑓 +𝑂 (𝜀),andtheelementaryproduct,reciprocal,andlogarithmestimatesin𝐶2 give
𝐶2
everyconclusionin(12). □
LetH = H 𝜀 betherealreproducing-kernelHilbertspaceof 𝑘 𝜀,withfeature 𝑘 𝑦 = 𝑘 𝜀 (·,𝑦). Define
„
𝑆 : 𝐿2(𝜇) −→ H, 𝑆𝑢 = 𝑎(𝑦)𝑘 𝑦 𝑢(𝑦)𝑑𝜇(𝑦), 𝑆∗ℎ(𝑦) = 𝑎(𝑦)ℎ(𝑦),
𝐶 = 𝑆𝑆∗, 𝑃 = 𝑆∗𝑆, 𝐵 = 1(𝐼 −𝐶), 𝑅 = (𝐼 +𝐶)−1.
2
Forafixedquery𝑥 andcomponent𝜈 ∈ {1,...,𝑑},set
𝑑 1
𝜃 𝑥,𝜈 = 𝜕 𝑥 {𝑎(𝑥)𝑘 𝑥 }, 𝛼 𝑑 = + .
𝜈 4 2
Foravectorsection𝜃 𝑥,innerproductswithaHilbert-spaceelementaretakencomponentwise.
Lemma3.2(Featuregeometry). Theoperator 𝑃 isaself-adjointpositiveMarkovoperator. Moreover,
∥𝑆∥ = ∥𝐶∥ = 1, ∥𝐵∥ ≤ 1, ∥𝑅∥ ≤ 1, ∥𝑣∥ H = 1, 𝐶𝑣 = 𝑣, 𝑆∗𝑣 = 1, (13)
2
sup∥𝑘 𝑦 ∥ H ≤ 𝐶𝜀−𝑑/4, sup∥𝜃 𝑥,𝜈 ∥ H ≤ 𝐶𝜀−𝛼 𝑑, tr𝐶 ≤ 𝐶𝜀−𝑑/2. (14)
𝑦 𝑥
Foreveryfixed𝑥 and𝜈,ℓ ∈ {1,...,𝑑},
„
𝜀𝑑/2+1⟨𝜃 𝑥,𝜈 ,𝑅𝐶𝑅𝜃 𝑥,ℓ ⟩ H −→ 𝑓
𝐽
( 𝑑 𝑥) 1{𝜈=ℓ} , 𝐽 𝑑 = R𝑑 [1+ex
(2
p
𝜋
(2
𝜉
𝜋 1
)
2
2
|𝜉|2)]2 𝑑𝜉. (15)
Every fixed interval of positive length strictly inside (0,1) contains at least 𝑐𝜀−𝑑/2 eigenvalues of 𝐶,
countedwithmultiplicity,forallsuﬀicientlysmall𝜀.
Proof. Thekernelof 𝑃 relativeto 𝜇 is𝑎(𝑥)𝑘 𝜀 (𝑥,𝑦)𝑎(𝑦). Itissymmetric,positivesemidefinite,andhas
rowsumonebythescalingequation. Jensen’sinequalitygives ∥𝑃∥ ≤ 1,and 𝑃1 = 1givesequality. The
singular-valueidentitiesfor𝑆 yieldtheassertednorms. Also
„ „
∥𝑣∥2 H = 𝑎(𝑦)𝑣(𝑦)𝑑𝜇(𝑦) = 1, 𝐶𝑣 = 𝑎(𝑦)2𝑘 𝑦 𝑣(𝑦)𝑑𝜇(𝑦) = 𝑣,
and𝑆∗𝑣 = 𝑎𝑣 = 1. GaussianFouriersumsgive
(cid:213) (cid:213)
𝑘 𝜀 (𝑦,𝑦) = 𝑒−2𝜋2𝜀|𝑘|2 =𝑂(𝜀−𝑑/2), ∥𝜕 𝑥 𝜈 𝑘 𝑥 ∥2 H = (2𝜋𝑘 𝜈 )2𝑒−2𝜋2𝜀|𝑘|2 =𝑂(𝜀−𝑑/2−1).
𝑘∈Z𝑑 𝑘
TogetherwithLemma3.1,theseprovethesectionbounds. Thetraceformulais
„ „
tr𝐶 = 𝑘 𝜀 (0,0) 𝑎2𝑑𝜇 = 𝑘 𝜀 (0,0) 𝑞2𝑑𝑥 =𝑂(𝜀−𝑑/2).
4

ThereisaconvenientcomparisonwithinthesameHilbertspace. Put
„
𝐶 𝐻 = 𝑘 𝑦 ⊗ 𝑘 𝑦 𝑑𝑦, 𝑅 𝐻 = (𝐼 +𝐶 𝐻 )−1.
Since𝑎2𝑓 = 𝑞2,comparisonofquadraticformsgives
∥𝐶 −𝐶 𝐻 ∥ op ≤ ∥𝑞2−1∥ ∞ ∥𝐶 𝐻 ∥ op =𝑂(𝜀). (16)
Here 𝐶 𝐻 has eige√nvalues 𝜆 𝑘 = exp(−2𝜋2𝜀|𝑘|2): in the complexification of H, its orthonormal Fourier
basisconsistsof 𝜆 𝑘exp(2𝜋𝑖𝑘 ·𝑥). Inparticular ∥𝐶 𝐻 ∥ = 1. Resolventidentitiesgive
∥𝑅𝐶𝑅−𝑅
𝐻
𝐶
𝐻
𝑅
𝐻
∥
op
=𝑂(𝜀).
√
Also𝜃 𝑥,𝜈 = 𝑎(𝑥)𝜕 𝑥 𝑘 𝑥 +(𝜕 𝑥 𝑎(𝑥))𝑘 𝑥,wherethesecondtermhasnorm𝑂( 𝜀𝜀−𝛼 𝑑). Aftermultiplication
𝜈 𝜈
by 𝜀2𝛼 𝑑, the operator replacement therefore has error 𝑂(𝜀) and these section replacements have error
√
𝑂( 𝜀). Since𝑎(𝑥)2 → 1/𝑓(𝑥),theGramlimitreducesto
!
(cid:213) 𝑒−2𝜋2𝜀|𝑘|2 2
𝜀𝑑/2+1 (2𝜋𝑘
𝜈
)(2𝜋𝑘
ℓ
) .
1+𝑒−2𝜋2𝜀|𝑘|2
𝑘∈Z𝑑
This is a Gaussian-dominated Riemann sum. Its limit is 𝐽 𝑑 for equal components and zero for unequal
components,proving(15).
Finally,by(16)andthecompactself-adjointmin–maxprinciple,theorderedeigenvaluesof𝐶and𝐶
𝐻
differbyatmost𝐶𝜀. Insideafixedinterval𝐼 ⊂ (0,1),chooseasmallerintervalatpositivedistancefrom
itsendpoints. Theheateigenvaluesinthatsmallerintervalcorrespondtolatticepointsinafixedannulus
√
after multiplication by 𝜀; their number is bounded below by 𝑐𝜀−𝑑/2. All their perturbed eigenvalues
belongto 𝐼 forsmall𝜀,provingthecount. □
Fortheempiricalstatements,allrandomquantitiesusethesamesample. Define
𝑔 = (𝜇 𝑛 −𝜇)(𝑎𝑘), 𝑔 0 = (𝜇 𝑛 „ −𝜇)(𝑘),
𝜁 = 𝑛−1/2𝜀−𝑑/4, 𝜂 = 𝑛−1/2𝜀−𝑑/2, 𝐶 0,𝑛 = 𝑎(𝑦)2𝑘 𝑦 ⊗ 𝑘 𝑦 𝑑𝜇 𝑛 (𝑦).
Thenotation𝑂 P,H belowmeansaboundinHilbert-spacenorm.
Lemma3.3(Quantitativeempiricalbranch). Under(2),theexactempiricalscaleb𝑎,extendedbyb𝑎𝐾 𝑛 b𝑎 =
1,obeys
b𝑣−𝑣 = 𝑅𝑔+𝑂 P,H (𝜂𝜁), ∥b𝑣−𝑣∥ H =𝑂 P (𝜁), b𝑣 = b𝑎−1, (17)
√
∥b𝑎−𝑎∥ ∞ + 𝜀∥∇(b𝑎−𝑎)∥ ∞ =𝑂 P (𝜂). (18)
Inaddition,
𝐶 −𝑣 ⊗𝑣
∥𝑔∥ H +∥𝑔 0 ∥ H =𝑂 P (𝜁), Cov(𝑔) = 𝑛 , ⟨𝑣,𝑔⟩ H = 0, (19)
∥𝐶 0,𝑛 −𝐶∥ op ≤ ∥𝐶 0,𝑛 −𝐶∥ HS =𝑂 P (𝜂). (20)
Define𝑆 𝑛 ,𝑆∗ 𝑛 ,𝐶b,𝑃 𝑛 byreplacing𝑎,𝜇 in𝑆,𝑆∗,𝐶,𝑃 withb𝑎,𝜇 𝑛. Then
∥𝑆 𝑛 ∥ = ∥𝐶b∥ = 1, ∥𝐶b−𝐶∥ op =𝑂 P (𝜂), (21)
sup∥b𝜃 𝑥,𝜈 −𝜃 𝑥,𝜈 ∥ H =𝑂 P (𝜂𝜀−𝛼 𝑑), b𝜃 𝑥,𝜈 = 𝜕 𝑥 𝜈 {b𝑎(𝑥)𝑘 𝑥 }. (22)
𝑥
On a common event of probability tending to one, evaluation satisfies ∥ℎ∥ 𝐿2(𝜇
𝑛
) ≤ 𝐶∥ℎ∥ H for every
ℎ ∈ H.
5

Proof. The expectation of 𝑎(𝑋)𝑘 𝑋 is 𝑣, and its second moment operator is 𝐶. Hence the covariance
identity follows. Moreover ⟨𝑣,𝑎(𝑦)𝑘 𝑦 ⟩ = 𝑎(𝑦)𝑣(𝑦) = 1, so the claimed orthogonality is exact. The
featurenormboundgivesE∥𝑔∥2 +E∥𝑔 ∥2 ≤ 𝐶𝜁2. Similarly,
H 0 H
1
E∥𝐶
0,𝑛
−𝐶∥2
HS
≤
𝑛
E∥𝑎(𝑋)2𝑘
𝑋
⊗ 𝑘
𝑋
∥2
HS
≤ 𝐶𝑛−1𝜀−𝑑 =𝐶𝜂2.
TheseareHilbert-spacesecond-momentbounds;nofinite-dimensionaloperatorinequalityisneeded.
ConsiderG 𝑛 (𝑧) = 𝑧−𝐾 𝑛 (1/𝑧) inaneighborhoodof𝑣 inH. Thefunction𝑣 isboundedbelow,and
G 𝑛 (𝑣) = −𝑔, 𝐷G 𝑛 (𝑣) = 𝐼 +𝐶 0,𝑛 , ∥(𝐼 +𝐶 0,𝑛 )−1∥ ≤ 1.
For ℎsmallenoughtokeep𝑣+ℎpositive,thereciprocalremainderis
(cid:18) (cid:19)
ℎ2
𝐾 .
𝑛
𝑣2(𝑣+ℎ)
Its H norm is bounded by𝐶𝜀−𝑑/4∥ℎ∥2 H on an eventwhere ∥𝐶 0,𝑛 ∥ ≤ 2. Indeed, write the remainder as
thefeatureoperatorwithweights𝑎 appliedto (𝑎ℎ)ℎ/(𝑣+ℎ);itsoperatornormisboundedby ∥𝐶 0,𝑛 ∥1/2,
while ∥𝑎ℎ∥ 𝐿2(𝜇
𝑛
) ≤ ∥𝐶 0,𝑛 ∥1/2∥ℎ∥ H and ∥ℎ∥ ∞ ≤ 𝐶𝜀−𝑑/4∥ℎ∥ H. The same argument gives the local
Lipschitz remainder bound. Since 𝜀−𝑑/4𝜁 = 𝜂 → 0, a contraction on a ball of radius 𝑂 P (𝜁) yields a
positivesolutionand
b𝑣−𝑣 = (𝐼 +𝐶 0,𝑛 )−1𝑔+𝑂 P,H (𝜂𝜁).
Forprecision,onecanfirstrestrictto ∥𝑔∥ ≤ 𝑀𝜁 forfixed 𝑀,performthecontractionforallsufficiently
large𝑛,andthenlet𝑀increase. Theresolventidentityand(20)replace(𝐼+𝐶
0,𝑛
)−1by𝑅atcost𝑂
P,H
(𝜂𝜁).
The positive solution is the exact empirical branch: at sample sites its reciprocal satisfies the finite
scalingequations,whosepositivesolutionisuniquebythesamepositive-semidefiniteenergyidentityas
inLemma3.1. Offsample,b𝑣 = 𝐾 𝑛 b𝑎 fixesthestatedextension. Thereisnoextensiontoarbitrarysigned
measuresinthisargument.
Thefeatureandderivative-featureboundsappliedto ℎ =b𝑣−𝑣 show
√
∥ℎ∥
∞
+ 𝜀∥∇ℎ∥
∞
=𝑂
P
(𝜂).
Takingreciprocals,usingthedeterministiclowerboundon 𝑣,proves(18). Theempiricaloperator 𝑃 𝑛 is
positive,self-adjoint,andMarkovon𝐿2(𝜇
𝑛
);hencethenormidentitiesin(21)followasinthepopulation
case. Alsob𝑎/𝑎 = 1+𝑂 P (𝜂) uniformly. Comparingquadraticformsgives
∥𝐶b−𝐶 ∥ ≤ 𝑂 (𝜂)∥𝐶 ∥ ,
0,𝑛 op P 0,𝑛 op
which proves the remaining operator bound. Differentiating (b𝑎 − 𝑎)(𝑥)𝑘 𝑥 and using (18) proves (22).
Finally,since𝑎 isboundedbelow,
„
∥ℎ∥2
𝐿2(𝜇
𝑛
)
≤ 𝐶 𝑎2ℎ2𝑑𝜇
𝑛
=𝐶⟨ℎ,𝐶
0,𝑛
ℎ⟩
H
≤ 𝐶∥ℎ∥2
H
onthesameevent. Thisprovesthesimultaneousevaluationassertion. □
4 Initialization and the sampled trajectory
Alloperatorsbelowusethesameobservations. Write𝑒 𝑗 = log(𝑏 𝑗 /b𝑎)and𝐵 𝑛 = (𝐼−𝑃 𝑛 )/2,withempirical
operatorsasinLemma3.3. Offsample,𝑃 𝑛 𝑢(𝑥) = b𝑎(𝑥)𝐾 𝑛 (b𝑎𝑢)(𝑥)isaMarkovrow. Theexactlogupdate
is
Ψ𝑛 (𝑒) = 1
2
{𝑒−log(𝑃 𝑛 (exp𝑒))}, 𝑒 𝑗+1 = Ψ𝑛 (𝑒 𝑗 ). (23)
Its uniform norm is nonincreasing. For small ∥𝑒∥ , Taylor expansion, Jensen’s inequality and the PSD
∞
Markovpropertygive
∥Ψ𝑛 (𝑒)−𝐵 𝑛 𝑒∥ 𝑛 ≤ 𝐶 ∥𝑒∥ ∞ ∥𝑒∥ 𝑛 , ∥𝐵 𝑛 ∥ op ≤ 1 2 . (24)
6

Lemma 4.1 (Two-update initialization). Let 𝑣 = 1/𝑎, 𝑤 = 𝐻 𝜀 𝑓, and 𝑑 1 = log𝑣 − 1 2 log𝑤. Put 𝜁 =
𝑛−1/2𝜀−𝑑/4,𝜂 = 𝑛−1/2𝜀−𝑑/2 and𝑢 = 𝑒 −𝑑 . Then
1 1
𝑢 = 𝑎𝑅𝑔− 1𝑔 /𝑤+𝑂 (𝜂𝜁), (25)
2 0 𝐿2(𝜇 𝑛 ),P
∥𝑢∥
𝑛
=𝑂
P
(𝜁), ∥𝑢∥
∞
=𝑂
P
(𝜂),
∥𝐵 𝑛 𝑑 1 ∥ 𝑛 =𝑂 P (𝜀2+𝜀3/2𝜂), (26)
𝑒 2 = 𝐵 𝑛 𝑢+𝐸 2 , ∥𝐸 2 ∥ 𝑛 =𝑂 P (𝜀2+𝜀3/2𝜂+𝜀𝜁 +𝜂𝜁), (27)
∥𝑒 2 ∥ 𝑛 =𝑂 P (𝜀2+𝜁), ∥𝑒 2 ∥ ∞ =𝑂 P (𝜀2+𝜂). (28)
Proof. Thefirstiterateis𝑏
1
= (𝐾 𝑛1)−1/2. Expand𝑒
1
= logb𝑣−
2
1 log𝐾 𝑛1,usingLemma3.3,𝐾 𝑛1 = 𝑤+𝑔
0
,
andtheuniformvalueerrors𝑂 P (𝜂). Aquadraticlogarithmicremainderhassample 𝐿2 normatmostits
uniformerrortimesits 𝐿2 error. Thisproves(25).
For(26),Lemma3.1gives ∥𝑑 ∥ ≤ 𝐶𝜀. Thus
1 𝐶2
|𝑑 (𝑦)−𝑑 (𝑥)| ≤ 𝐶𝜀dist(𝑥,𝑦), ∥(𝐼 −𝑃)𝑑 ∥ ≤ 𝐶𝜀2.
1 1 1 ∞
Thesecondestimatefollowsfromtheheat-semigroupidentityappliedto𝑎𝑓𝑑 −𝑑 (𝑥)𝑎𝑓,usingbounded
1 1
𝐶2 normsof 𝑎𝑓. Replacethenormalizedrowweights b𝑎(𝑦) by 𝑎(𝑦). Theirrelativechangeis𝑂 P (𝜂), so
itscostisthisfactortimestheempiricalrowmeanof |𝑑 (𝑦)−𝑑 (𝑥)|. TheGaussian-imagebound
1 1
√
dist(𝑥,𝑦)𝑘
𝜀
(𝑥−𝑦) ≤ 𝐶 𝜀𝑘
2𝜀
(𝑥−𝑦)
andtheRKHSvalueestimateatbandwidth2𝜀 makethismean𝑂 P (𝜀3/2),uniformlyin𝑥. Denominators
areboundedbelowsince𝐾 𝑛 𝑎 = 𝐾 𝜇 𝑎+𝑂 P (𝜂) uniformly. Theweightcostistherefore𝑂 P (𝜀3/2𝜂).
Fortheremainingfixed-𝑎 empiricalrow,itscenterednumeratoris
𝑁 𝑛 (𝑥) = (𝜇 𝑛 −𝜇) 𝑦 {𝑘 𝜀 (𝑥−𝑦)𝑎(𝑦)[𝑑 1 (𝑦)−𝑑 1 (𝑥)]}.
At 𝑥 = 𝑋 𝑖 its own diagonal vanishes. Conditional on 𝑋 𝑖 = 𝑥, its mean is𝑂(𝜀2/𝑛) and its variance is at
most „
𝐶𝜀2 𝐶
𝑛
𝑘
𝜀
(𝑥−𝑦)2dist(𝑥,𝑦)2𝑑𝜇(𝑦) ≤
𝑛
𝜀3−𝑑/2.
Averagingconditionalsecondmomentsgives∥𝑁 𝑛 ∥ 𝑛 =𝑂 P (𝜀3/2𝜁+𝜀2/𝑛). Theuniformdenominatorerror
is𝑂 P (𝜂) and multiplies a population numerator𝑂(𝜀2). This proves (26) without an independent-query
assumption.
Taylor-expand (23) at deterministic 𝑑 1 . The tilted row𝑄 𝑛,𝑑 1 ℎ = 𝑃 𝑛 (𝑒𝑑 1 ℎ)/(𝑃 𝑛 𝑒𝑑 1 ) differs from 𝑃 𝑛
by𝑂(𝜀) insample 𝐿2 operatornorm,byboundedmultiplicationandthecontractionof𝑃 𝑛. Itsquadratic
remainder in 𝑢 has norm at most 𝐶 ∥𝑢∥ ∞ ∥𝑢∥ 𝑛 . Also ∥Ψ𝑛 (𝑑 1 )−𝐵 𝑛 𝑑 1 ∥ 𝑛 ≤ 𝐶𝜀2. These give (27) and
its 𝐿2 consequence. In the uniform norm Ψ𝑛 is Lipschitz with constant one. The coarser row estimate
∥Ψ𝑛 (𝑑 1 )∥ ∞ =𝑂 P (𝜀2+𝜀𝜂) uses ∥𝑑 1 ∥ ∞ 𝑂 P (𝜂) forweightandempirical-rowerrors. With ∥𝑢∥ ∞ =𝑂 P (𝜂)
thisproves(28). □
Lemma4.2(Commontrajectoryandscoreproxies). Therearenonnegative𝐸 𝑛 = 𝑜 P (1)and𝑀 𝑛 =𝑂 P (1),
commontoallintegers 𝑗 ≥ 2,suchthat
√ (cid:13) (cid:13)
(cid:13) 𝑒 (cid:13) 𝑗 = 𝑆∗ 𝑛 𝐵𝑗𝑅𝑔+𝐸 𝑗,𝑛 , 𝑛(cid:13)𝐸 𝑗,𝑛 (cid:13) 𝑛 ≤ 𝐸 (cid:10) 𝑛 +𝑀 𝑛 𝜀1/2− (cid:11) 𝑑/42 −𝑗, (29)
𝐴 𝑛 (cid:13)𝑠 𝑗 (𝑥)−b𝑠(𝑥)−𝑉 𝑗 (𝑥)(cid:13) ≤ 𝐸 𝑛 +𝑀 𝑛 𝜀1/2−𝑑/42 −𝑗, 𝑉 𝑗 (𝑥) = 2 𝜃 𝑥 ,𝐶𝐵𝑗𝑅𝑔 . (30)
Thescoreremainderis uniformin 𝑥; vector-sectioninnerproductsarecomponentwise. Ateachfixed 𝑥,
foreverydeterministic 𝐽 𝑛 → ∞,
(cid:13) (cid:13)
sup 𝐴 𝑛 (cid:13)𝑠 𝑗 (𝑥)−b𝑠(𝑥)(cid:13) → 0 inprobability. (31)
𝑗≥𝐽
𝑛
7

Proof.(cid:13)Le(cid:13)t𝛿 𝑛 = ∥𝑒 2 ∥ ∞ . Ononeeventofprobabilitytendingtoone,(24)anduniform-normnonexpansion
imply (cid:13)𝑒 𝑗 (cid:13) 𝑛 ≤ (3/4)𝑗−2∥𝑒 2 ∥ 𝑛 forall 𝑗 ≥ 2. Iteratingtheremainderyields
(cid:13) (cid:13)
(cid:13) (cid:13)
sup(cid:13)𝑒 𝑗 −𝐵 𝑛 𝑗−2𝑒 2 (cid:13) =𝑂 P {(𝜀2+𝜂)(𝜀2+𝜁)}. (32)
𝑗≥2 𝑛
√
Everyerrorin(27)and(32)vanishesaftermultiplicationby 𝑛under(2):
√ √ √
𝑛𝜀2 → 0, 𝑛𝜀3/2𝜂 = 𝜀3/2−𝑑/2 → 0, 𝑛𝜂𝜁 = (𝑛𝜀3𝑑/2)−1/2 → 0.
√
Theremainingtermsareboundedby𝜀1−𝑑/4,𝜀2−𝑑/4,𝜀2−𝑑/2 and 𝑛𝜀4,alltendingtozero.
Thecommutator
𝑞 𝑛 (𝑥) = 𝑔 0 (𝑥)−𝑣(𝑥)𝑔(𝑥) = (𝜇 𝑛 −𝜇) 𝑦 𝑘 𝜀 (𝑥−𝑦)[1−𝑣(𝑥)𝑎(𝑦)]
haszerodiagonalat𝑥 = 𝑋 𝑖. Itscoefficientisboundedby𝐶dist(𝑥,𝑦),
√
itsconditionalbiasis𝑂(𝜀/𝑛),and
itsconditionalvarianceis𝑂(𝜀1−𝑑/2/𝑛). Consequently ∥𝑞 𝑛 ∥ 𝑛 = 𝑂 P ( 𝜀𝜁). Using 𝑣/𝑤 −𝑎 = 𝑂(𝜀) and
𝑅− 1𝐼 = 𝐵𝑅 in(25)gives
2
√
𝑢 = 𝑆∗𝐵𝑅𝑔+𝑂 (𝜀𝜁 +𝜂𝜁 + 𝜀𝜁).
𝑛 𝐿2(𝜇 𝑛 ),P
The intertwining 𝐵 𝑛 𝑆∗ 𝑛 = 𝑆∗ 𝑛 𝐵b , with 𝐵b = (𝐼 −𝐶b)/2, is exact. Both half-filters have norm at most 1/2.
Telescoping their powers contributes at most 𝐶𝑗2 −𝑗𝜂∥𝑔∥ H = 𝑂 P (𝜂𝜁) uniformly in 𝑗. The commuta-
tor retains its factor 2 −𝑗 . This proves (29) with common random bounds. In 𝑑 = 2 its fixed-𝑗 root-𝑛
commutatortermisnotnegligible.
The exact score difference is 2∇log(𝑃 𝑛 (exp𝑒 𝑗 )). The derivative row is 𝑆∗ 𝑛 b𝜃 𝑥, of norm at most
𝑂 P (𝜀−𝑑/4−1/2). E(cid:13)xp(cid:13)anding the exponential and the logarithm bounds its nonlinear score remainder
by 𝐶𝜀−𝑑/4−1/2𝛿 𝑛 (cid:13)𝑒 𝑗 (cid:13) D𝑛 . After m E ultiplication by 𝐴 𝑛, this is bounded by the root-𝑛 version of (32).
The linear proxy is 2
b𝜃
𝑥
,𝐶b𝐵𝑗𝑅𝑔
up to the trajectory error. Replacing sections and operators costs
𝑂 P (𝜀− (cid:13) 𝑑/4−1/ (cid:13) 2𝜂𝜁), uniformly in 𝑗,𝑥. This proves (30). At fixed 𝑥, the covariance identity for 𝑔 gives
E[𝐴2 𝑛 (cid:13)𝑉 𝑗 (𝑥)(cid:13)2] ≤ 𝐶4 −𝑗 . Summation over 𝑗 ≥ 𝐽 𝑛 controls the supremum of the proxies. The common
error tends to zero on this tail, since 𝜀1/2−𝑑/4 ≤ 1 for 𝑑 ≤ 2. This proves (31), which is pointwise in 𝑥,
notaspatialsupremumlaw. □
(cid:10) (cid:11) p
Lemma 4.3 (Residual comparison). Put 𝑄 𝑗 = 1 𝑔,𝐶𝐵2𝑗𝑔 , 𝑞 𝑗 = 𝑄 𝑗 and 𝐷 𝑛 = (𝑛𝜀3𝑑/2)−1/4. With
4
common 𝐸 𝑛 = 𝑜 P (1) and 𝑀 𝑛 =𝑂 P (1),simultaneouslyfor 𝑗 ≥ 2,
√ (cid:12) (cid:12)
𝑛(cid:12)𝑟 𝑗 −𝑞 𝑗 (cid:12) ≤ 𝐸 𝑛 +𝑀 𝑛 {𝜀1/2−𝑑/42 −𝑗 +𝐷 𝑛 𝜀−𝑑/82 −𝑗}. (33)
Proof. Theexactmassfieldis 𝐹 𝑗 = exp(cid:13) {(𝑒 (cid:13)𝑗 +log𝑃 𝑛 (exp𝑒 𝑗 ))/2}. Itslinear partis (𝐼 +𝑃 𝑛 )𝑒 𝑗 /2, with
remainderboundedinsample𝐿2by𝐶𝛿 𝑛 (cid:13)𝑒 𝑗 (cid:13) 𝑛 . Dividingby𝑆 𝑗 centersthisfield,withthesameremainder
order. Intertwining and (𝐼 +𝐶)𝑅 = 𝐼 reduce the linear term to 𝑆∗𝐵𝑗𝑔/2, with the common trajectory
𝑛
errorand𝑂 P (𝜂𝜁). Itsmeanobeystheexactidentity
(cid:10) (cid:11) (cid:10) (cid:11)
1,𝑆∗ 𝑛 𝐵𝑗𝑔 𝑛 = b𝑣−𝑣,𝐵𝑗𝑔 =𝑂 P (𝜁22 −𝑗),
√
because𝑆 𝑛1 =b𝑣 and 𝐵𝑣 = 0. Itisroot-𝑛negligiblesince 𝑛𝜁2 = 𝜂 → 0. Finally
(cid:13) (cid:13)
(cid:12)
(cid:12)
(cid:13)
(cid:13)𝑆∗ℎ
(cid:13)
(cid:13) −∥𝑆∗ℎ∥
(cid:12)
(cid:12) ≤
(cid:13)
(cid:13)𝐶b−𝐶
(cid:13)
(cid:13)
1/2
∥ℎ∥ .
𝑛 𝑛 𝐿2(𝜇) H
op
√ √
Apply this to 𝐵𝑗𝑔 and use 𝑛 𝜂𝜁 = 𝐷 𝑛 𝜀−𝑑/8. This proves (33). No uniformly root-𝑛 negligible norm
comparisonisassertedfortheearlyiterates. □
8

5 Spectral concentration and the first hit
Theproxy𝑄 𝑗 isdefinedusingthedeterministicpopulationoperator,notarandomspectraldecomposition
ofthesamplematrix.
Lemma5.1 (Hilbertquadraticconcentration). Let𝜆 𝑙 betheeigenvaluesof𝐶 and 𝛽 𝑙 = (1−𝜆 𝑙 )/2. For
𝑗 ≥ 1,put𝑇 𝑗 = tr(𝐶2𝐵2𝑗). Then
E𝑄 𝑗 = 4 𝑇 𝑛 𝑗 , ( V E a 𝑄 r𝑄 𝑗 ) 𝑗 2 ≤ 𝐶 max𝑙 𝑇 𝜆 𝑗 2 𝑙 𝛽 𝑙 2𝑗 + 𝐶𝜀− 𝑛 𝑑 𝑇 /2 𝑗 4 −𝑗 . (34)
The same calculation applies to deterministic spectral projections. On a band with 𝛽 ∈ [𝛽 𝐻 ,𝛽 𝐻,2 ] ⊂
(0,1/2) offixedpositivewidth,itsrelativevarianceisboundedby
𝐶(𝜀𝑑/2+𝑛−1)(𝛽 𝐻,2 /𝛽 𝐻 )2𝑗. (35)
Proof. ForcenterediidHilbertvectors𝑍 𝑖 ofcovarianceΣ,expansionoftheirsquaredsample-meannorm
gives
Var
(cid:13)
(cid:13)𝑛−1
˝
𝑖 𝑍 𝑖
(cid:13)
(cid:13)2 =
E∥𝑍∥4
𝑛
−
3
(trΣ)2
+
2(𝑛−1
𝑛
)
3
tr(Σ2)
. (36)
Only equal or reversed off-diagonal index pairs contribute. Terms with a singly occurring inde-
pendent centered variable vanish by conditioning; the diagonal term in (36) is retained. Take
𝑍 𝑗,𝑦 = 1 2 𝐶1/2𝐵𝑗[𝑎(𝑦)𝑘 𝑦 − 𝑣]. The station (cid:13) ary r (cid:13) ank-one term vanishes for 𝑗 ≥ 1, so its covariance
is 𝐶2𝐵2𝑗/4. The feature bound gives sup 𝑦 (cid:13)𝑍 𝑗,𝑦 (cid:13)2 ≤ 𝐶𝜀−𝑑/24 −𝑗 . Bound its fourth moment by this
envelopetimesitssecondmoment,andusetr(Σ2) ≤ ∥Σ∥ trΣ. Thisproves(34).
op
Foraprojectedband,Lemma3.2gives𝑇 𝑗,𝐻 ≥ 𝑐𝜀−𝑑/2 (cid:13) 𝛽2 𝐻 𝑗 ,wit (cid:13) hits𝜆valuesboundedawayfromzero.
Theprojectedindividualvectorhasthesharperenvelope (cid:13)𝑍 𝑗,𝑦,𝐻 (cid:13)2 ≤ 𝐶𝜀−𝑑/2𝛽2 𝐻 𝑗 ,2 . In(36),thisbounds
the diagonal relative term by 𝐶(𝛽 𝐻,2 /𝛽 𝐻 )2𝑗/𝑛, and the covariance term by 𝐶𝜀𝑑/2(𝛽 𝐻,2 /𝛽 𝐻 )2𝑗 . This
proves(35). □
Proposition5.2(Stoppingcount). Forfixed0 < 𝛾 < 𝑑/4,therulein(5)satisfies(7)andhitsitsthreshold
strictlybefore 𝐿 𝑛 withprobabilitytendingtoone. Inparticular𝜏 𝛾 → ∞.
Proof. Put𝑢 ∗ = 𝑑/4−𝛾 > 0. Fixasmallslack0 < 𝛿 < min(𝑢 ∗ ,𝑑/8)/2,write𝑢 − =𝑢 ∗ −𝛿,andlet
(cid:22) (cid:23) (cid:24) (cid:25)
𝑗 − =
𝑢 −log(1/𝜀)
, 𝑗 + =
(𝑢
∗
+𝛿)log(1/𝜀)
.
log2 log2
Choosefixed 𝛽 ∗ < 1/2closeenoughto1/2that,with𝜎 = |log𝛽 ∗ |/log2 > 1,
𝑢 − 𝜎 < 𝑢 ∗ , 𝑢 − (𝜎−1) < 𝑑/8. (37)
Afixedbandwith 𝛽 ≥ 𝛽 ∗ and𝜆boundedawayfromzerohas𝑐𝜀−𝑑/2 eigenvalues,hence𝑇 𝑗 ≥ 𝑐𝜀−𝑑/2𝛽 ∗ 2𝑗 .
Forevery2 ≤ 𝑗 ≤ 𝑗 −,(34)isboundedby
𝐶(𝜀𝑑/2+𝑛−1)(4𝛽2)−𝑗.
∗
The sum of these geometric bounds is 𝑜(1): its largest power is at most 𝐶𝜀−2𝑢 − (𝜎−1) , and (37) and
𝑛𝜀3𝑑/2 → ∞controlbothterms. AChebyshevunionboundthereforegives,withprobabilitytendingto
one,simultaneously √
𝑛𝑞 𝑗 ≥ 𝑐𝜀−𝑑/4𝛽 ∗ 𝑗, 2 ≤ 𝑗 ≤ 𝑗 − . (38)
The smallest lower bound, at 𝑗 −, dominates 𝜀−𝛾 , since 𝑢 − 𝜎 < 𝑢 ∗. The empirical-norm term in (33),
dividedby(38),isboundedinprobabilityby
𝐶𝐷
𝑛
𝜀𝑑/8(2𝛽
∗
)−𝑗 − ≤ 𝐶𝐷
𝑛
𝜀𝑑/8−𝑢 − (𝜎−1) → 0.
9

Theotherroot-𝑛 errorsare 𝑜 P (1) or𝑂 P (1) for 𝑑 ≤ 2, whereasthesmallestproxylowerbounddiverges.
Thusnoactualthresholdhitoccursfor2 ≤ 𝑗 ≤ 𝑗 −,and𝑟 2 /𝑡 𝑛 → ∞.
For the upper bracket use 𝑇 𝑗 ≤ 4 −𝑗 tr(𝐶2) ≤ 𝐶𝜀−𝑑/24 −𝑗 . At 𝑗 +, E[𝑛𝑄 𝑗 ]/𝜀−2𝛾 ≤ 𝐶𝜀2𝛿 → 0, so
+
Markov’sinequalitygives𝑞 𝑗 /𝑡 𝑛 → 0. Theempirical-normerrordividedby𝑡 𝑛 is𝑂 P (𝐷 𝑛 𝜀𝑢 ∗ +𝛿+𝛾−𝑑/8) =
+
𝑂 P (𝐷 𝑛 𝜀𝑑/8+𝛿) = 𝑜 P (1). Theothertermsalsovanishrelativeto𝑡 𝑛,provingahitby 𝑗 +. Moreover 𝑗 + < 𝐿 𝑛
eventually: thesamplingconditionin(2)boundslog(1/𝜀) aboveby2log𝑛/(3𝑑) forlarge𝑛.
Itremainstoexclude 𝑗 = 0,1withoutimposingadelayontherule. Forsmalllogoscillation,replace
𝑒 by𝑒−𝜇 𝑛 𝑒,whichleavestheresidualunchanged. Itslinearcenteredmassfieldis (𝐼 +𝑃 𝑛 )𝑒/2. Onthe
mean-zerosubspaceitsnormliesbetween∥𝑒∥ /2and∥𝑒∥ . Thenonlinearerrorisatmost𝐶 ∥𝑒∥ ∥𝑒∥ .
𝑛 𝑛 ∞ 𝑛
Consequentlytherearefixedlocalconstantssuchthat𝑟(𝑒) iscomparableto ∥𝑒∥ aftercentering,and
𝑛
𝑟(Ψ𝑛 (𝑒)) ≤ 𝐶𝑟(𝑒). (39)
IndeedΨ𝑛killsconstants,centeringisan𝐿2contraction,and(24)controlsitscenteredoutput. Nononlin-
earmean-preservationorglobalresidualmonotonicityisassumed. Since∥𝑒 1 ∥ ∞ = 𝑜 P (1)and𝑟 2 /𝑡 𝑛 → ∞,
(39)implies𝑟 1 /𝑡 𝑛 → ∞andexcludes 𝑗 = 1.
Ifthefixed 𝑓 isnonconstant,uniformconsistency𝐾 𝑛1 → 𝑓 andthelawoflargenumbersgive
p
{Var𝜇 ( 𝑓)}1/2
𝑟 −→ p > 0.
0
𝜇 𝑓
As𝑡 𝑛 → 0,thisexcludes 𝑗 = 0. If 𝑓 = 1,Lemma3.3gives ∥𝑒 0 ∥ ∞ = 𝑜 P (1),so(39)gives𝑟 0 /𝑡 𝑛 → ∞and
excludes 𝑗 = 0too. Thustheunrestrictedfirsthitliesbetween 𝑗 − and 𝑗 + withprobabilitytendingtoone.
Lettingthearbitraryslack𝛿 tendtozeroproves(7). □
Proposition5.3(Attainedresidual). Foreachfixed0 < 𝑐 < 1/2,(8)holds.
Proof. Foreachrealized 𝑔,𝑄 𝑗 isapositivespectralsum. Itslog-convexityimpliesthat 𝑞 𝑗+1 /𝑞 𝑗 isnon-
decreasingandatmost1/2whendefined. Let𝐽 = 𝑗 − fortheslackabove. Fix0 < 𝛽 𝐿 < 1/2,andchoose
𝛽
𝐿
< 𝛽
𝐻
< 𝛽
𝐻,2
< 1/2withsufficientlysmallpositivewidththat
2𝑢 −log(𝛽
𝐻,2
/𝛽
𝐻
)/log2 < 𝑑/2.
The low-𝛽 trace at 𝐽 is at most 𝐶𝜀−𝑑/2𝛽2𝐽 , whereas the high band has trace at least 𝑐𝜀−𝑑/2𝛽2𝐽 . The
𝐿 𝐻
latter concentrates by (35), including its diagonal term. The ratio of the two expectations tends to zero.
Markov’s inequality for the low part and concentration for the high part show that modes 𝛽 ≤ 𝛽 𝐿 con-
tribute𝑜 P (𝑄 𝐽 ). Thus𝑞 𝐽+1 /𝑞 𝐽 ≥ 𝛽 𝐿 −𝑜 P (1). Letting 𝛽 𝐿 approach1/2andusingmonotonicitygives
(cid:12) (cid:12)
sup (cid:12)𝑞 𝑗+1 /𝑞 𝑗 −1/2 (cid:12) → 0 inprobability. (40)
𝑗≥𝐽
Theseratiosaredefinedonaneventofprobabilitytendingtoone,sincetheretainedhighbandhaspositive
energythere.
Uniformly for 𝑗 ≥ 𝐽, (33) divided by 𝑡 𝑛 is 𝑜 P (1). Its empirical-norm term is bounded by
𝑂 P (𝐷 𝑛 𝜀𝑢 − +𝛾−𝑑/8) =𝑂 P (𝐷 𝑛 𝜀𝑑/8−𝛿),andthecommutatortermis𝑂 P (𝜀1/2−𝛿). Atthefirsthit,𝑟 𝜏
𝛾
−1 > 𝑡 𝑛,
while(40)gives
𝑟
𝜏
≥ (1/2−𝑜
P
(1))𝑡
𝑛
−𝑜
P
(𝑡
𝑛
).
𝛾
Thedefinitionsupplies𝑟 𝜏 ≤ 𝑡 𝑛,provingtheclaim. Thisdoesnotassertauniquelimitfor𝑟 𝜏 /𝑡 𝑛;discrete
𝛾 𝛾
crossingmayretainaroundingphase. □
6 Score limits and pointwise inference
Proposition6.1(Exactandstoppedscores). Under(2),ateachfixed𝑥 theexactscoresatisfies
𝐴
𝑛
{b𝑠(𝑥)−∇log 𝑓(𝑥)} ⇒ 𝑁
𝑑
(0,[4𝐽
𝑑
/𝑓(𝑥)]𝐼
𝑑
).
ThesamelimitholdsatthestoppediterateinTheorem2.1.
10

Proof. Thereciprocal-branchexpansionofLemma3.3gives
b𝑠(𝑥)−𝑠 pop (𝑥) = 2⟨𝜃 𝑥 ,𝑅𝑔⟩+𝑜 P (𝐴 𝑛 −1), 𝑠 pop = −2∇log𝑎. (41)
Indeedb𝑠 = 2∇logb𝑣 and the linear derivative is 2∇[𝑎(b𝑣 −𝑣)]. The logarithmic nonlinear remainder is
boundedby𝑂 P (𝜂𝜀−𝑑/4−1/ √ 2𝜁),andreplacingb𝑣 −𝑣 by 𝑅𝑔 hasthesamederivativebound. Aftermultipli-
cationby 𝐴 𝑛 bothare𝑂 P ( 𝑛𝜂𝜁) = 𝑜 P (1).
Theleadingtermin(41)isacenterediidtriangulararray. Itsnormalizedcovarianceis
(cid:10) (cid:11)
4𝜀𝑑/2+1 𝜃
𝑥,𝜈
,𝑅(𝐶 −𝑣 ⊗𝑣)𝑅𝜃
𝑥,ℓ
.
The rank-one subtraction is zero, because 𝑅𝑣 = 𝑣/2 and ⟨𝜃 𝑥 ,𝑣⟩ = ∇[𝑎(𝑥)𝑣(𝑥)] = 0. Lemma 3.2 gives
thelimitingcovariance (4𝐽 𝑑 /𝑓(𝑥))𝐼 𝑑. Anormalizedsummandisboundedineveryfixeddirectionby
𝐶𝜀𝑑/4+1/2 (cid:13) (cid:13) 𝐶
√ 𝑛 ∥𝜃 𝑥 ∥ H su
𝑦
p (cid:13)𝑎(𝑦)𝑘 𝑦 −𝑣(cid:13) H ≤ √ 𝑛𝜀𝑑/4 → 0.
T√he multivariate Lindeberg theorem therefore applies. Finally the population bias is 𝑂(𝜀) and 𝐴 𝑛 𝜀 =
𝑛𝜀𝑑/2+3 → 0,provingtheexact-scorelimit.
Proposition 5.2 places the stopping index beyond a deterministic sequence tending to infinity with
probability tending to one. Lemma 4.2 then gives (9) at that same-sample random index. Slutsky’s
theoremgivesthestoppedlimit. □
ProofofTheorem2.1. Propositions 5.2, 5.3 and 6.1 prove the count, attained-residual and score asser-
tions. Thejointeventin(8)followsbyintersectionoftheireventsofprobabilitytendingtoone. Itslower
√
boundalsoimplies 𝑛𝑟 𝜏 → ∞. □
𝛾
Corollary6.2(Feasiblepointwiseinference). Let e𝑓(𝑥) = 𝐾 𝑛1(𝑥) andlet𝑥 befixed. Then
q
𝐴 e𝑓(𝑥)
𝑛
√ {𝑠 𝜏 (𝑥)−∇log 𝑓(𝑥)} ⇒ 𝑁 𝑑 (0,𝐼 𝑑 ), (42)
2 𝐽 𝑑 𝛾
𝐴2 e𝑓(𝑥) (cid:13) (cid:13)
𝑛 4𝐽
𝑑
(cid:13)𝑠 𝜏 𝛾 (𝑥)−∇log 𝑓(𝑥)(cid:13)2 ⇒ 𝜒 𝑑 2. (43)
Consequentlytheball q
8 9
> >
> >< > > > 𝑢 ∈ R𝑑 : (cid:13) (cid:13)𝑠 𝜏 𝛾 (𝑥)−𝑢 (cid:13) (cid:13) ≤ 2 𝐴 𝐽 q 𝑑 𝜒 e𝑓 𝑑 2 ( ,1 𝑥 − ) 𝛼 > >= > > >
: 𝑛 ;
haslimitingcoverage1−𝛼 forthefixedtruescore,for0 < 𝛼 < 1. Predetermineddirectionalintervals
followfrom(42).
Proof. The periodic KDE is positive, its bias tends to zero, and its variance at fixed 𝑥 is 𝑂((𝑛𝜀𝑑/2)−1).
Hence e𝑓(𝑥) → 𝑓(𝑥) > 0 in probability. Slutsky’s theorem uses no independence between the plug-in
andthestoppedscore. Thechi-squarelawandcontinuityatitsquantileprovecoverage. □
For𝑑 = 2thecoflnstanthastheclosedform𝐽
2
= (𝜋2/12−log2)/(2𝜋): polarcoordinatesand𝑢 = 2𝜋2𝑟2
reduceitto
(2𝜋)−1 ∞ 𝑢(1+𝑒𝑢)−2𝑑𝑢.
Integrationbyparts,followedbythegeometric-seriesintegralfor
fl
0
∞ 𝑢/(1+𝑒𝑢)𝑑𝑢,gives𝜋2/12−log2.
0
11

7 Upper tolerance exponent
The coefficient of the threshold in (5) is one. The following statement concerns that convention and a
counterexampleinthestateddensityclass.
Theorem7.1. Assume(2)and 𝑓 = 1. If𝛾 ≥ 𝑑/4,thenP(𝜏 𝛾 = 0) → 1. Ateachfixedquery,
𝐴 𝑛 𝑠 𝜏 (𝑥) ⇒ 𝑁 𝑑 (0,𝑊 𝑑 ), 𝑊 𝑑 = 2(4𝜋)−𝑑/2𝐼 𝑑 ,
𝛾
whereas 𝐴 𝑛 b𝑠(𝑥) ⇒ 𝑁 𝑑 (0,4𝐽 𝑑 𝐼 𝑑 ) and𝑊 𝑑 −4𝐽 𝑑 𝐼 𝑑 ispositivedefinite.
Proof. Foruniformsampling,𝑎 = 𝑣 √= 1and𝐶 istheheatcovarianceoperator. Write ℎ = 𝐾 𝑛1−1 = 𝑔
0
.
Theactualinitialmassfieldis 𝐹 0 = 1+ℎ. Itslinearpartis ℎ/2. Wehave ∥ℎ∥ ∞ = 𝑂 P (𝜂) and √ ∥ℎ∥ H =
𝑂 P (𝜁), so its nonlinear 𝐿2 error is 𝑂 P (𝜂𝜁). The empirical-to-population norm change is 𝑂 P ( 𝜂𝜁), by
thesampleheatcovariancecomparison. Theempiricalmeanisexactly 𝜇 𝑛 ℎ = ∥ℎ∥2 H ,because ⟨ℎ,1⟩ = 0.
Itis𝑂 P (𝜁2) = 𝑜 P (𝜁). Thustheactualcenteredandnormalizedresidualsatisfies
𝑟 0 = 2 1 ∥ℎ∥ 𝐿2(𝑑𝑥) +𝑜 P (𝜁).
˝
The squared leading norm has mean (4𝑛)−1 𝑘≠0 𝜆2 𝑘 . Its relative variance is 𝑂(𝜀𝑑/2): in (36) the indi-
vidualsquaredfeaturenormisconstantbytranslationinvariance, soitsdiagonalvariancevanishes,and
˝ ˝
𝜆4 𝑘 /( 𝑘≠0 𝜆2 𝑘 )2 =𝑂(𝜀𝑑/2). Gaussianlatticesumsconsequentlyyield
𝑟
0 −→ 𝑐 𝑑 = 1(4𝜋)−𝑑/4 < 1 inprobability. (44)
𝑛−1/2𝜀−𝑑/4 2
For𝛾 ≥ 𝑑/4thethresholdisatleast𝑛−1/2𝜀−𝑑/4,sotheactualrulestopsatzerowithprobabilitytending
toone.
Here𝑠 0 = 2∇log(1+ℎ). Atafixedquery, 𝐴 𝑛 ∇ℎistightbyitskernelsecondmoment,andℎ(𝑥) → 0.
Itskerneltriangular-arrayCLThascovariance
„
𝑊 𝑑 = 4 (2𝜋𝜉)(2𝜋𝜉)𝑇𝑒−4𝜋2|𝜉|2 𝑑𝜉 = 2(4𝜋)−𝑑/2𝐼 𝑑 .
R𝑑
√
The Lindeberg bound is again 𝐶/( 𝑛𝜀𝑑/4) → 0; the logarithmic denominator is negligible by Slutsky.
Theexact-scorelimitfollowsfromProposition6.1. WritingΛ(𝜉) = 𝑒−2𝜋2|𝜉|2 ,itscovarianceis
„
4𝐽 𝑑 𝐼 𝑑 = 4 (2𝜋𝜉)(2𝜋𝜉)𝑇[Λ(𝜉)/(1+Λ(𝜉))]2𝑑𝜉.
Foreverynonzerodirection,thedifferencefrom𝑊 𝑑 istheintegralofasquarednonzerolinearprojection
againstastrictlypositiveradialweight. Itispositive,provingtheclaim. □
The theorem does not classify arbitrary endpoint multipliers or assert failure for every nonconstant
density. Itestablishestheupperexponentobstructionforauniformguaranteeoverthemodelclasswith
thisrule.
8 Scope of the inference statements
The confidence regions above are pointwise-in-𝑓 asymptotic regions at predetermined spatial queries.
They do not provide uniform finite-sample coverage over the unrestricted positive𝐶6 class. A standard
discontinuous-functionalobstruction[9]makesthatdistinctionnecessary. Forexample,ataquery𝑥 and
0
eachfixedreal𝑞,
𝑞
𝑓 𝑚,𝑞 (𝑥) = 1+ 2𝜋𝑚 sin(2𝜋𝑚(𝑥 1 −𝑥 0,1 ))
12

hastruescore 𝑞𝑒 at𝑥 ,issmoothandliesbetween1/2and3/2forlargeinteger 𝑚,whileits 𝑛-sample
1 0
total variation distance from uniform data is at most 𝑛|𝑞|/(2𝜋2𝑚). For each fixed 𝑛 this tends to zero.
Anyuniformlyhonestconfidenceintervalwouldthereforehavetocovereveryfixed𝑞underuniformdata
withitsnominalprobability;analmostsurelyboundedintervalcannotdoso. Quantitativesmoothnessin-
formationoraspecifiedsmoothedtargetisneededforsuchfinite-sampleclaims. Thisdoesnotcontradict
Theorem2.1,whichfixesthedensitybeforetakingitslimit.
The pointwise result permits confidence regions for a density’s logarithmic gradient and tests of a
zerogradientataprespecifiedpoint. Itdoesnotlocateanunknownmodeorjustifydata-selectedqueries.
Uniformspatiallimits,boundaries,vanishingdensitiesanddiffusion-patherrorcontrolrequireadditional
arguments. At 𝑑 = 3 the two conditions on 𝑛𝜀9/2 in (2) are incompatible; higher dimensions are not
covered. Theleadingupdatelawisanasymptoticiterationcount,andfinite-sampleruntimeimprovements
arenotinferredfromit.
AI assistance
AItoolsassistedpreparationandchecking. Theauthorisresponsibleforthecontent.
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