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Learning Spectral Allocation: A Fractional Diffusion Framework for Adaptive Volumetric Segmentation
Yi-Hui Shen, Tie-Qiang Li
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
Last extracted: 9/24/2026, 4:52:54 AM
Summary
The paper introduces FHEAT-Seg, a lightweight 3D medical image segmentation framework that treats spectral computation allocation as a learnable optimization problem rather than a fixed architectural choice. It utilizes a Fractional Heat Conduction Operator (FHCO) derived from the fractional heat equation, which allows stages to learn their own diffusion strength and order. This enables 'spectral sparsification,' where unnecessary stages collapse to identity shortcuts, significantly reducing FLOPs while maintaining or improving accuracy in semi-supervised and fully supervised settings on benchmarks like Left Atrium, Pancreas-CT, and BraTS 2019.
Entities (10)
Relation Signals (9)
FHEAT-Seg → uses → FHEAT
confidence 98% · We instantiate FHEAT in a lightweight U-shaped architecture... yielding FHEAT-Seg.
FHEAT-Seg → uses → KAN3D
confidence 97% · paired with a Kolmogorov-Arnold mixer (KAN3D) with adaptive rational activations, yielding FHEAT-Seg.
FHEAT → derivedfrom → FHCO
confidence 95% · We derive FHEAT, a two-parameter operator family, from the discrete cosine transform (DCT) solution of a fractional heat equation... The core of our design, the Fractional Heat Conduction Operator (FHCO)...
FHEAT-Seg → outperforms → Light-UNETR
confidence 94% · ahead of five semi-supervised methods and the Light-UNETR baseline.
FHEAT-Seg → evaluatedon → Left Atrium
confidence 93% · FHEAT-Seg reaches Dice scores of 90.47% (left atrium)
FHEAT-Seg → evaluatedon → Pancreas-CT
confidence 93% · 78.79% (Pancreas-CT)
FHEAT-Seg → evaluatedon → BraTS 2019
confidence 93% · 81.90% (BraTS 2019)
FHCO → basedon → DCT
confidence 92% · under Neumann boundary conditions its eigenmodes are exactly the type-II discrete cosine transform (DCT-II) basis
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Abstract
Abstract:We address adaptive computation in 3D medical image segmentation: instead of designing another backbone, we ask how much spectral mixing each network stage needs and let optimization answer. We derive FHEAT, a two-parameter operator family, from the discrete cosine transform (DCT) solution of a fractional heat equation. A fractional order alpha and a diffusion strength D govern the operator, and at D=0 it is exactly the identity. Reparametrized by the semigroup time tau = D*alpha, same-resolution instances compose exactly, so any distribution of diffusion across same-resolution stages amounts to a single Sobolev-type regularizer of learned strength. This identity limit lets the optimizer of each layer, not the designer, decide whether global mixing is needed and how sharp it should be. We instantiate FHEAT in a lightweight U-shaped architecture (Light-UNETR) paired with a Kolmogorov-Arnold mixer (KAN3D) with adaptive rational activations, yielding FHEAT-Seg. At 5% to 20% label rates on three public benchmarks, training produces gradient-driven spectral sparsification: seven of the eight stage-level operators drive D to zero, and the survivor saturates at the sharpest low-pass (alpha ~ 0.9) in the decoder layer feeding the semi-supervised attention map. The retired layers become exact identity shortcuts at inference, cutting FLOPs from 4.29G to 0.90G (a 79% drop) at 0.975M parameters. Under a standard semi-supervised protocol, FHEAT-Seg reaches Dice scores of 90.47% (left atrium), 78.79% (Pancreas-CT), and 81.90% (BraTS 2019), ahead of five semi-supervised methods and the Light-UNETR baseline. The large variant also surpasses Light-UNETR-L under full supervision (Dice 93.09%, 85.11%, and 87.19%) with 2.851M parameters and 55.75G FLOPs. These results suggest that the allocation of spectral computation is a learnable property of optimization dynamics, not a manual design commitment.
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- Source: https://arxiv.org/abs/2609.27217v1
- Canonical: https://arxiv.org/abs/2609.27217v1
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LearningSpectralAllocation:AFractionalDiffusionFrameworkforAdaptive VolumetricSegmentation Y.-HShen 1 andT.-Q.Li 2,3,4,5 Abstract—Weaddressadaptivecomputationin3Dmedical imagesegmentation:insteadofdesigninganotherbackbone,we askhowmuchspectralmixingeachnetworkstageneedsandlet optimizationanswer.Theanswertakesaclosedform.Wederive FHEAT,atwo-parameteroperatorfamily,fromthediscrete cosinetransform(DCT)solutionofafractionalheatequation.A fractionalorderαandadiffusionstrengthDgoverntheoperator, andatD=0itisexactlytheidentity.Reparametrizedbythenatural semigrouptime 퐀= 퐀 퐀 ,same-resolutioninstancesofthefamily composeexactly,soanydistributionofdiffusionacrosssame- resolutionstagesamountstoasingleSobolev-typeregularizerof learnedstrength(D=0addsnone).Thisidentitylimitisthe mechanismtheframeworkrestson:theoptimizerofeachlayer, notthedesigner,decideswhetherglobalmixingisneededandhow sharpitshouldbe.WeinstantiateFHEATinalightweightU- shapedarchitecture(Light-UNETR)pairedwithaKolmogorov– Arnoldmixer(KAN3D)withadaptiverationalactivations, yieldingFHEAT-Seg.Onthreepublicbenchmarksat5%to20% labelrates,trainingproducesgradient-drivenspectral sparsification:sevenoftheeightstage-leveloperatorsdriveDto zero,andthesinglesurvivorsaturatesatthesharpestadmissible low-pass(α≈0.9)inthedecoderlayerthatfeedsthesemi- supervisedattentionmap.SinceD=0reducesFHEATtothe identity,theretiredlayersbecomeexactshortcutsatinference, cuttingFLOPsfrom4.29Gto0.90G(a79%drop)at0.975M parameters.Freezingeverystageexcepttheactivedecoder matchesorslightlyexceedsthefullylearnablemodel,sotraining appearstoidentifyoneusefulsiteratherthandistributeamixing budget.Underastandardsemi-supervisedprotocol,FHEAT-Seg reachesDicescoresof90.47%(leftatrium),78.79%(Pancreas- CT),and81.90%(BraTS2019),aheadoffivesemi-supervised methodsandthebaselineLight-UNETR.Thelargevariantalso surpassesLight-UNETR-Lunderfullsupervision(Dice93.09%, 85.11%,and87.19%)with2.851Mparametersand55.75GFLOPs. Theseresultssuggestthattheallocationofspectralcomputationis alearnablepropertyofoptimizationdynamics,notamanual designcommitment. IndexTerms—3Dmedicalimaging,fractionalheatequation, spectralallocation,identity-containingoperator,lightweight segmentation,semi-supervisedlearning. ¹SchoolofArtsandSciences,FujianMedicalUniversity,Fuzhou,China (e-mail:shenyh@fjmu.edu.cn). ²SchoolofMedicalImaging,FujianMedicalUniversity,Fuzhou,China. ³DepartmentofMedicalRadiationPhysicsandNuclearMedicine, Karolinska UniversityHospital,Stockholm,Sweden. ⁴DepartmentofClinicalScience,InterventionandTechnology, Karolinska Institute,Stockholm,Sweden. ⁵DepartmentofPsychology,StockholmUniversity,Stockholm,Sweden. *Correspondingauthor:T.-Q.Li(e-mail:tie-qiang.li@ki.se). I.INTRODUCTION olumetricmedicalimagesegmentationunderpins quantitativeanalysisinclinicalworkflowssuchas radiotherapyplanningandsurgicalnavigation,yet densevoxel-levelannotationsremainprohibitivelyexpensive toobtain.Twodirectionshaveconsequentlygrowninparallel: lightweightarchitecturesthatkeepthecomputationgraph deployable,andsemi-supervisedlearning(SSL)protocolsthat exploitunlabeledscansthroughpseudo-labelingand consistencyregularization.Bothhaveprogressedsubstantially, buttheyarerarelydecoupledinacontrolledcomparison,soit isusuallyunclearwhetherlabelefficiencyshouldbecreditedto thebackboneortotheprotocol. Existinglightweightnetworksprescribeafixedglobal- mixingbudgetperstage:everystagecarriesthesamedual- branchrecipe,alocalconvolutionpathpairedwithaglobal attentionpath,andpaysthefullcostofglobalmixingwhether ornotitsfeaturesrequireit.Underscarcelabelsthishand- crafteduniformityhasaconcretecost:themodelcannotshift itslimitedcapacitytowardthestageswherethedataactually callforglobalaggregation.Weinsteadtreattheallocationof globalspectralcomputationasalearnable,per-layer optimizationproblem,andwegiveitaclosed-formsolution, derivedfromthefractionalheatequation,inwhichtheidentity mappingisexactlyreachable.Trainingcanthenretire unnecessarycomputationinsteadofcarryingitbydefault. Learnableallocationofcomputationhasappearedinother guises,includingdifferentiablearchitecturesearch,sparsity- inducinggates,stochasticdepth,andlearnedrouting(Section I-C),butthequestionofwhereglobalmixingistrulyneeded hasnotpreviouslybeenposedasanexplicitlylearnable,layer- levelproblemwhosesolutionprovablycontainstheidentity. Wemakethisallocationlearnablebyequippingeachstage withasingleidentity-containingoperatorfamilyratherthana hand-craftedbranchstructure.Thecoreofourdesign,the FractionalHeatConductionOperator(FHCO),isafractional- orderspectraldiffusionderivedfromthefractionalheat equation:underNeumannboundaryconditionsitseigenmodes areexactlythetype-IIdiscretecosinetransform(DCT-I)basis, andtheclosed-formsolutionactsintheDCTspectrumas multiplicationby 퐀䠀퐀(−(퐀||퐀||) 퐀 ) .Weembedthiscore operatorinsidearesidual,gatedmodulecalledFHEAT(Fig. 2(c)).Thefractionalorderαcontrolshowaggressivelyhigh frequenciesareattenuated;thediffusionstrengthDscalesthe overallsmoothing.Botharelearnableper-stagescalars,soeach stageinterpolatescontinuouslybetweentheidentity(nomixing) andstronglow-passaggregation.Westressthatthisidentity limitisthedesignthesis,notaconvenientby-product.Existing V spectraloperators(FNet,GFNet,AFNO)applyfixedorfully parameterizedfilterswithnomechanismtoidentifyandlockan exactidentity,sotheycannotbecompiledtoashortcutat inference.Onlyanoperatorfamilythatcontainstheidentity turnstheallocationofspectralcomputationintoalearnable quantityandtherebyenableslearnedsparsification. Spectraldiffusionaloneisalineardiagonaltransform, however,andmustbepairedwithexpressivefeaturemixing. WecoupleeachFHEAToperatorwithKAN3D,a3D Kolmogorov–Arnoldmixerinwhichthefixedactivationofa conventionalfeed-forwardblockisreplacedbylearnable rationalfunctions.Sincethespectraloperatoralreadysupplies global,frequency-selectivecontext,themixer'sjobischannel- wiserecombinationratherthanfurtherspatialaggregation,and thelearnableactivationsleteachstageadaptitsnonlinearityto itsownfeaturestatisticswithoutwideningthenetwork. Trainedunderscarcelabels,thenetworkanswersthe allocationquestionconsistently.Acrossthreepublic benchmarksandlabelratesfrom5%to20%,gradient-driven spectralsparsificationemergeseverytime:thediffusion strengthDgoestozeroinallbutoneoftheeightstage-level operators,collapsingthemintoidentities,whilethesole survivorsaturatesatthestrongestadmissiblelow-passorder(α ≈0.9).Inallfivesettingstested,thesurvivoristhefinaldecoder block,thesameblockwhoseoperatoroutputproducesthe attentionmapconsumedbytheSSLprotocol.Acontrolled experimentthatallowsalearnableDonlyatthisblockmatches orslightlyexceedsthefullylearnablemodel;thesamecontrol atanon-survivingblockdoesnot.Trainingthusidentifiesa singleusefulsiteratherthanadistributedmixingbudget.And becauseazeroDreducesFHEATtotheidentity,every degeneratestagebecomesadirectshortcutatinference,leaving acomputationgraphof0.975Mparametersand0.90GFLOPs, 79%fewerFLOPsthantheLight-UNETRbackbone(Fig.1). Fig.1.Performancecomparisonbetweentheproposed FHEAT-Segandprevioussemi-supervisedmedicalimage segmentationmethodsontheLAdataset[1].Circlesize indicatesthenumberoftrainableparametersofeachmethod. Weevaluatetheframeworkonthreepublic3Dbenchmarks: leftatrium(LA),Pancreas-CT,andBraTS2019.Underthe contextualsynergicenhancement(CSE)semi-supervised protocol[2],whichwekeepunchangedthroughouttoisolate thebackbone'scontribution,FHEAT-SegattainsDiceof 90.47%,78.79%,and81.90%at5–10%labelrates,surpassing fivesemi-supervisedmethods,alltrainedonthesameLight- UNETRbackbone,aswellasthebackboneitselfunderan identicalevaluationprotocol.Thelargevariantfurther outperformsLight-UNETR-Lonfullysupervisedsegmentation, withDiceof93.09%,85.11%,and87.19%onthethree benchmarksatlowerparameterandFLOPcounts(2.851M/ 55.75G).Sincethesefullysupervisedresultsinvolvenosemi- supervisedcomponent,theyalsoindicatethatthegainsstem fromthebackboneitself. Wesummarizethecontributionsofthisworkasfollows: WeintroducetheFractionalHeatConductionOperator (FHCO),atwo-parameterNeumannsemigroupoperatorthatis completelymonotone,hencewell-posedandnon-amplifying (Remark2),andthatcontainstheidentityexactlyatzero diffusionstrength(Remark1),asacompilableper-stagegate forglobalspectralmixing.WeinstantiateitinastandardU- shapedencoder-decoderandpairitwithaKolmogorov–Arnold channelmixer(KAN3D)toobtainFHEAT-Seg,ourempirical vehicle. Weprovethatsame-resolution,same-ordergatesforman exactsemigroup:theirdiffusiontimesadd,soanynumberof themequalsasinglegateatthesummedtime(Theorem1). Analytically,andthroughacontrollednon-medicalsimulation (SectionIII-F),weshowthatthisredundancyalonedoesnot explainsparsification,becauseweightdecayappliedtotruly interchangeablegatesfavorsspreadingthediffusionbudget ratherthanconcentratingit(Remark3).Empirically,training nonethelessselectsasinglerepresentative:underscarcelabels, allstage-leveldiffusionoperatorsexceptonearedriventothe identity,andinallfivesettingstested(threeseedseach)the survivorcoincideswiththedecoderblockthatproducesthe semi-supervisedattentionmap.Theresultisaninferencegraph of0.975Mparametersand0.90GFLOPs(79%fewerFLOPs thanLight-UNETR)withoutaccuracyloss. Wederivethefirstvariationofthelosswithrespecttothe naturalsemigrouptimeτ=D^αandshowthatitisafractional Dirichletpairingbetweentheincomingfeatureandthe incominggradientintheḢ^α/2seminorm(Proposition1):a stageisfirst-orderusefulifthispairingisstrictlypositive. TogetherwiththeReLU-and-weight-decayabsorption mechanism(Proposition2),thisidentifieswhichstagessurvive insteadofmerelyrationalizingthatsomedo.Thesingle-site ablationsofSectionIV-Eagreewiththerankingthepairing predicts(Block_Finaluseful;twoothersitestestednot),andthe pairingitselfismeasureddirectlyinacontrolledsynthetic experiment(SectionIII-F),whereitssignatinitialization correctlypredictsthesurvivingsiteinalleightcasestested. Weevaluateonthreebenchmarks:state-of-the-artsemi- supervisedresultsunderthesharedCSEprotocolat5–20% labelratesoverfiveSSLmethodsandtheLight-UNETR backbone,andhigherfullysupervisedaccuracythanLight- UNETR-Latlowercost.Wealsoreportasecond,structurally differentbackbone(a3DSegFormerwithself-attention replacedbyFHEATgates,SectionIV-B)thatreproducesboth thesparsificationphenomenonanditsexact-zeroretirement, evidencethattheoperator-levelclaimsarenotrestrictedto Light-UNETR.Spectralanalysesfurtherrevealadivisionof laborinwhichtheFHEATblocksamplifyhigh-frequency detailwhilethesurvivingattention-producingblockperforms low-passaggregation. I.RELATEDWORK A.Lightweight3DMedicalImageSegmentations Volumetricsegmentationarchitectureshavemovedfrom heavyU-shapedCNNstowardcompactdesignsthatremain deployableunderclinicalcomputebudgets.Beyondtheclassic U-NetandV-Netfamilies[3],[4],efficiencyhasbeenpursued alongthreemainroutes:depthwise-separableandgroup convolutionsthatcutper-blockcost,self-configuringpipelines thatoptimizeaccuracywithinafixedcomputeenvelope[5], andlightweighttransformerorCNN–transformerhybridsthat replacequadraticself-attentionwithlow-rankordual-branch attention[6],[7].Light-UNETR[2]isrepresentativeofthelast route:itinterleaveslocalgroupedconvolutionswithalow- frequency/high-frequencydual-branchattentionmodule(LIDR) andreachesstrongaccuracywithafewmillionparameters.All ofthesedesigns,however,reducethecostofeachblockwhile leavingtheallocationofblockshand-crafted:everystage receivesthesameglobal-mixingreciperegardlessofneed,so nocapacitycanbereclaimedfromstagesthatdonotrequire globalcontext.Ourworktargetsthisnextaxisofefficiency:a parameterizedoperatorfamilywhoseper-stagemixingstrength islearned,sothattrainingitselfretainsspectralcomputation onlywhereitisneeded. B.Semi-Supervised3DMedicalImageSegmentation Semi-supervisedsegmentationseekstoexploitabundant unlabeledscanswhenvoxel-levelannotationsarescarce.The dominantparadigmcouplespseudo-labelingwithconsistency regularization:UA-MT[8]enforcesuncertainty-aware consistencyunderinputperturbations,BCP[9]perturbs predictionsthroughbidirectionalclass-mixing,SASSNet[10] regularizespredictionmapswithshape-awareconstraints, CAML[11]enforcescorrelation-awareagreementbetweenco- trainedmodels,andMLRP[12]stabilizesmutuallearningwith reliability-filteredpseudo-labels.TheCSEframework[2] combinesAttention-GuidedReplacement(AGR),whichmixes labeledandunlabeledimagesinattention-selectedregions,with SpatialMaskingConsistency(SMC),andreportsconsistent gainswhenbuiltontheLight-UNETRbackbone.Apersistent difficultyinthisliteratureisthatprotocolandbackboneare entangled,soimprovementsarehardtoattributetoeither.We thereforeadopttheCSEprotocolunchangedthroughoutthis workanduseitasacontrolledenvironmentinwhichthe backboneistheonlyvariedfactor;allfivecompetingSSL methodsarecomparedunderthisshared-backbonesetting. C.Frequency-DomainandLearnable-ActivationOperators Frequency-domainoperatorsofferglobalcontextatacost decoupledfromspatialextent,andhavebeenexploredmainly innatural-imagevision.FNet[13]replacesself-attentionwith fixedFouriermixing,GFNet[14]learnsarbitraryper- frequencyfilters,andAFNO[15]learnssparsetoken-mixingin theFourierdomain;KAN[16]anditsrational-activation variants(KAT[17],laterusedinTK-Mamba[18])makethe activationfunctionsthemselveslearnable.Thelearnedfiltersin theseoperatorsareunconstrained:theycarrynophysical structurethatguaranteeswell-posedsmoothingand,more importantly,theygiveastagenowaytooptout.Afully parameterizedspectralfilterhasnomechanismthatidentifies andlocksanexactidentity,soitcannotbecompiledtoa shortcut,eventhoughgradientdescentcouldinprincipledrive itarbitrarilyclosetoone.AFNO'ssoft-thresholdingofFourier modesistheclosestsuchmechanisminthepriorliterature,and itstillleavestheretainedmodesasafullyparameterized, unlockedfilterratherthananexact,checkablebypass,so trainingcannotanswerwhetherastageneedsglobalmixingat all.FHEATaddressesbothpoints:itsfrequencyresponseisa two-parameterfractionalheatkernel,completelymonotoneand hencepositivedefiniteinthesquaredfrequencyradius,witha built-inidentitylimitatzerodiffusionstrength,sotheamount andthelocationofspectralmixingbecomelearnablequantities. Tworecentoperator-levelbaselinessharpenthisdistinction. SlimFormer-3D[19]alsoturnsglobalmixingoffbylayer,but throughposthocgatingfactorsselectedfromcross-layer activationsimilarityaftertraining,notthrougha parameterizationwhosetraining-timeoptimumisanexact, checkableidentity;itsgatesapproximateredundancy,whereas ourscertifyit.AMBER-AFNO[20]replacesself-attentionwith AFNO-basedfrequency-domainmixingthroughouta3D segmenter,learningafullyparameterizedspectralfilteratevery stageinsteadofasking,perstage,whetherspectralmixingis neededatall.Neitherbaselineoffersthetwo-parameter, completelymonotone,exactly-D=0familythatletsastage's ownoptimizeranswerthatquestionandletstheanswerbe compiledintoashortcutatinference.Ourdifferentiatoristhis identity-containing,two-scalar,physicallyparameterized family,notsimplythepresenceofafrequencymixerinaU- shapedbackbone. Twoadjacentliteraturescontextualizethisdesign.First, lettinggradientdescentdecidewhetheracomponent contributeshasalonghistory:differentiablearchitecturesearch relaxesdiscreteoperatorchoicesintolearnablemixtureweights [21],sparsity-inducinggatesdrivestructuralvariablestoexact zero[22],stochasticdepthdropsresidualblocksatrandom duringtraining[23],andmixture-of-expertsmodelsroute tokensthroughlearnedgatingfunctions[24].Ourquestionis relatedbutdistinct:ratherthansearchingadiscretearchitecture spaceorpruningafixedone,eachstagedeclares,asa continuoustrainingoutcome,howmuchglobalmixingit warrants,andtheidentity-containingparameterization guaranteesthat“none”isanexactlyreachableanddirectly deployableanswer.Second,fractional-orderdiffusion operatorshavealong-standinglineageinclassicalPDE-based imageprocessing,wherefractionaldiffusionequationsserve imagedenoisingandenhancement[25].FHEATdepartsfrom thatlineageinthreeways:thefractionalorderislearnedrather thanhand-tuned,theoperatorisembeddedperstageinsidea trainablesegmentationbackbone,anditsidentitylimitletsa stageretireitselfentirely,whichclassicalformulationsdonot provide. Unlikehypernetworks[26],whichconditionweightsoneach input,theper-stage(α,D)arelearnedonceandheldfixed, whichisexactlywhatmakestheD=0bypasscompilableahead ofinference. I.METHOD A.Overview SectionIII-B–I-Cdeveloptheproposedframework independentlyofanyspecificbackbone.Givenageneric networklayer 퐀 withfeaturemap 䠀 퐀 ,wedefinealayer-wise spectralgate 퐀 퐀 (퐀 퐀 ,퐀 퐀 ) (theFHEAToperator)thatgovernshow muchglobalfrequency-domainmixingthatlayerapplies, pairedwithaKAN3Dmixerforadaptivenonlinearchannel recombination.SectionIII-Fanalyzestheoptimization geometrythatthisgateinduces.Wetheninstantiatethe frameworkempiricallyasFHEAT-Seg:acompactU-shaped network(overlappingpatchembedding,fourdownsampling encoderstageswithskipconnections,andfourdecoderstages) thatreplaceseverystage'shand-craftedmixingrecipewiththe learnablegate 퐀 퐀 ,followingthedesignprinciplesofthe lightweightbackboneof[2]purelytoobtainafair,strong empiricalbaseline.Thegoalisasegmentationnetworkthatis bothlabel-efficientanddeployableunderscarceannotations. Forthesemi-supervisedexperiments,thenetworkistrained undertheCSEprotocol[2](SectionIII-D),whichweadopt unchangedandtreatthroughoutasafixed,controlledtestbed ratherthanasadesignconstraintontheframeworkitself (SectionII-B). Twowidthconfigurationsareusedthroughout.FHEAT-Seg stacks(1,2,3,2)Light-UNETRblocksof24,48,60,and96 channelsacrossitsfourencoderstages,withamirroreddecoder whosefinalblockistheattention-producingBlock_Final (SectionIII-C);eachLight-UNETRblockcontainsadepthwise 3×3×3convolutionandtwoFHEATblocks(Fig.2).FHEAT- Seg-Lwidensthedesign:a32-channelstemandtwoplain convolutionalstages(32and60channels)precedethe bottleneck,andfivestagesat96,150,198,150,and96channels (twoencoderstages,thebottleneck,andtwodecoderstages, twoLight-UNETRblockseach)carrytheFHEATblocks.Fig. 2illustratesthearchitecture. Fig.2.FHEAT-Segarchitecture.(a)Overallstructure:eachencoder/decoderstageisbuiltfromFHEATblocks(FHEATspectral diffusion+KAN3Dmixing).(b)FHEATblock.(c)FHEAT.(d)KAN3D.(e)FractionalHeatConductionOperator(FHCO).(f) Overlappatchembedding.(g)Lightdownsample. B.Fractional-OrderSpectralDiffusion Globalcontextinafeaturemapamountstoaggregation acrossspatialscales,andspectraloperatorsrealizesuch aggregationbyreweightingfrequencycoefficients.Afixed reweighting,however,commitseverystagetoonemixing behavior;weinsteadseekaparametricfamilyinwhichboththe shapeandtheamountofmixingarelearnableperstage,and whichcontainstheidentitymapping,sothatastagecanlearn toforgoglobalmixingaltogether. Weobtainsuchafamilyfromthefractionalheatequation. Considerafeaturefield 퐀 onthestagedomain Ω evolvingas ∂퐀 ∂퐀 =−퐀(−Δ) 퐀/2 퐀, ∂퐀 ∂퐀 ∂Ω =0 (1) where 퐀 isadiffusivityandisthefractionalorder.Under theNeumannboundarycondition,theeigenfunctions (−퐀) 퐀/2 ofarecosinemodes,preciselytheDCT-IIbasis,sothespectral basisisderivedfromtheboundaryphysicsratherthanassumed. WeadoptthespectraldefinitionofthefractionalLaplacian throughout,inwhichitisdefinedbyeigenfunctionexpansion andthereforeinheritsexactlythiscosinebasisunderNeumann boundaryconditions[27];thischoice,ratherthanthenonlocal integral(Riesz)definition,isthenaturaloneonaboundedvoxel grid,wheretheDCTdiagonalizationyieldstheclosed-form solutionbelow.Thenormalizedeigenmodesare 퐀 퐀 (䠀)= 퐀∈ℎ,퐀,퐀 cos 퐀 퐀 (2䠀 퐀 +1) 2퐀 퐀 , 퐀 퐀 =0,...,퐀 퐀 −1 (2) Expanding 퐀(䠀,퐀)= 퐀 퐀 (퐀,퐀)퐀 퐀 (䠀) inthisbasis decouples(1)intoindependentscalarODEs, 퐀 (퐀,퐀) 퐀 =−퐀∥퐀∥ 퐀 퐀 (퐀,퐀),onepermode 퐀=(퐀 ℎ ,퐀 퐀 ,퐀 퐀 ) with per-axisDCT-IIfrequenciesin [0,퐀) .Theirclosed-form solutionis 퐀 (퐀,퐀)=퐀 0 (퐀)exp −퐀∥퐀∥ 퐀 (3) where ∥퐀∥ istheEuclideanfrequencyradius. Reparametrizingtheproduct 퐀 asasinglediffusionstrength 퐀=(퐀 ) 1/퐀 ,theoperatoractsbypointwisemultiplicationof DCT-IIcoefficientswiththeclosed-formkernel exp−(퐀∥퐀∥) 퐀 . Thespectralmapping T D :v↦C -1 e -(D||k||) α ⊙C(v)(4) constitutestheFractionalHeatConductionOperator(FHCO) (Fig.2(e)). Twoper-stagescalarsparameterizetheresponse:the fractionalorder 퐀 ,whichshapeshowaggressivelyfrequencies areattenuated,andthediffusionstrength 퐀 ,whichscalesthe overallamountofsmoothing.Oneparameterpair (퐀 퐀 ,퐀 퐀 ) is sharedbyallblocksofstage 퐀 ,sotheeightstageslearn16 spectralscalarsintotal,boundedandinitializedas 퐀 퐀 =0.3+0.6 퐀( 퐀 퐀 )∈(0.3,0.9), 퐀 퐀 =ReLU(퐀 퐀 )≥0, 퐀 퐀 (0) =−2.0, 퐀 퐀 (0) =0.03 (5) Itisconvenienttoname,onceandforall,thequantity 퐀 퐀 :=:=퐀 퐀 퐀 퐀 ≥0 thatthisreparameterizationmakesnatural:the kernel exp(−(퐀∥퐀∥) 퐀 ) inside(4)equals exp(−퐀∥퐀∥ 퐀 ) , so 퐀 isthediffusiontimeoftheunderlyingheatflow(3)and 퐀 isonlyacoordinateonit.Weuse 퐀 throughoutthearchitecture descriptionbelowbecauseitisthequantitytheimplementation storesandinitializes;SectionIII-Fworksdirectlyin 퐀 ,where thegatesoptimizationgeometryisbestbehaved. Everystagethereforestartsat 퐀≈0.37 withamild,nonzero diffusionstrength.Therange (0.3,0.9) isadesignchoicethat keeps 퐀 insidethecompletelymonotoneregimewhile interpolatingfromamildtoasharplowpass.Thesaturationof thelearned 퐀 atthisboundary,observedinSectionIV,is interpretedinSectionV.Theboundedrangeisapractical choicetoensurewell-posedness;weemphasizethatthe theoreticalframeworkofSectionIII-F(Remark1andthe Propositionsbuiltonit)doesnotrequirethisbound,sincethose resultsconcernthe 퐀→0 identitylimitandthesignof 퐀 ,not therangeof 퐀 .SectionVexplicitlydiagnosesthesaturationand proposesanunboundedablationasfuturework,butthecore sparsificationphenomenon( 퐀→0 )isindependentofthis bound. Givenaninputfeature 䠀∈ℝ 퐀×퐀×퐀×퐀 ,FHEATfirstapplies adepthwise 3×3×3 convolutionandapointwiseprojection to 2퐀 channels,splitintoatransformedbranch 퐀 andagating branch.TheFHCOthenactsonvintheDCT-IIdomain.With 퐀 and 퐀 −1 denotingtheseparable3DDCT-IIanditsinverse, implementedasmatrixproductswithprecomputedorthogonal cosinematricesalongthethreespatialaxes,andwiththekernel of(4)appliedpointwiseinthetransformdomain,theoperator computes 퐀=퐀 out (SiLU(퐀)⊙GN(퐀 −1 [exp(−(퐀∥퐀∥ ) 퐀 )⊙퐀(퐀)])) (6) Here [퐀,퐀]=퐀 퐀 (DWConv 3×3×3 (䠀)) arethetransformand gatingbranchesproducedbythepointwiseprojection,GN denotesgroupnormalization,and 퐀 out projectsthegated featuresbackto 퐀 channels;aresidualconnectionisapplied outsidetheblock. Thisparameterizationprovidesthreepropertiesthat unconstrainedspectralfiltersdonot.(i)Identitylimit.As 퐀→ 0 ,thekernelinside(4)tendstooneforeveryfrequency,sothe entiretransformreducestotheidentityandcanbereplacedby askipconnectionwithunchangedoutputs,abuilt-in"off switch"forthestage’sglobalmixing.(i)Well-posedness.For 퐀∈(0,2] , 퐀↦exp(−퐀 퐀 퐀 퐀/2 ) iscompletelymonotoneinthe squaredradius 퐀=∥퐀∥ 2 ;thekernelisthereforepositive definite,andtheoperatorperformswell-posedsmoothingrather thanarbitraryper-frequencyreweighting.(i)Cost.Withthe separablematrix-productimplementation,thetransformcosts 퐀(퐀(퐀+퐀+퐀)) perstage,linearinthechannelcount 퐀 ; here (퐀,퐀,퐀) denotethespatialextents,withdepthwritten 퐀 toavoidconfusionwiththediffusionstrength 퐀 .AnFFT-based fastDCTcanreducethisto 퐀(퐀log(퐀)) whenneeded. C.FHEATBlockandKAN3DMixing TheFHEAToperatorisalinear,frequency-diagonal transform:itreweightsglobalcontextbutcannotbyitself recombinechannelsnonlinearly.Eachstagethereforepairs FHEATwithachannelmixer,andratherthanafixed-activation feed-forwardblock,weuseamixerwhoseactivationfunctions arethemselveslearnable. EachFHEATblockconsistsoftworesidualsub-blocks, x←x+GN(FHEAT(x)) (7) x←x+GN(KAN3D(x)) (8) whereGNdenotessingle-groupgroupnormalizationandthe depthwise 3×3×3 convolutionisthefirstlayerofthe FHEAToperator(SectionIII-B).KAN3Disa3DKolmogorov- -Arnoldmixer:GN →1×1×1 convolution( 퐀→퐀 ) → learnablerationalactivation →1×1×1 convolution( 퐀→퐀 ), withexpansionratio 퐀=1 .FollowingtheKolmogorov-- Arnoldprincipleofplacinglearnableunivariatefunctionson thenetwork'sedges,theactivationisaratiooftwopolynomials withlearnablecoefficients(initializedtoapproximateGELU), sothemixercanadaptitsnonlinearitytothefeaturestatisticsof eachstagewithoutwideningthenetwork.Thefinaldecoder block(Block_Final)additionallyexposesanattentionmap 퐀= 1 퐀 퐀 | 퐀 퐀 | (9) computedfromthemagnitudeofitsFHEAToutput;thismap isconsumedbythesemi-supervisedprotocol(SectionIII-D). Thispairingdividesthelaborcleanly:FHEATsupplies global,frequency-selectivecontext,whileKAN3Dsupplies adaptivepointwiserecombinationatnegligiblespatialcost (only 1×1×1 convolutions).SectionIVisolatesthemixer’s contributionwithanequal-capacityGELUfeed-forward control. D.IntegrationwiththeCSEProtocol Toevaluatetheframeworkunderlabelscarcity,weadoptthe CSEsemi-supervisedprotocolasafixed,unmodifiedtestbed ratherthanasacomponentoftheproposedmethod:anysemi- supervisedprotocolcompatiblewithanattentionread-outcould inprinciplesubstituteforit,andweholditconstantprecisely sothatthebackboneremainstheonlyvariedfactor(SectionII- B).CSEhastwocomponents:(a)Attention-Guided Replacement(AGR),inwhichanattentionmapidentifies informativeregionsandthecorrespondingregionsofan unlabeledimagereplacethoseofthelabeledimage,producing amixedinputsupervisedbyaregion-wisecombinationofthe ground-truthandpseudolabels;and(b)SpatialMasking Consistency(SMC),inwhichastronglymaskedviewofeach volumeisenforcedtopredictconsistentlywithitsweakly augmentedview. Inournetwork,theattentionmapconsumedbyAGRis producedbyBlock_Final(SectionIII-C)andiscomputedwith astop-gradient:gradientsfromtheAGRmixinglossdonot flowthroughtheattentionmapintothebackbone,althoughall segmentationlossescontinuetotrainthebackbonethroughthe segmentationoutputs.Attentionthusremainsaread-only region-selectionsignal:themixingobjectivecanexploitthe attentionmapbutcannotreshapethesegmentationfeaturesthat generateit.Protocolandbackbonestaydecoupledby construction,apropertywerelyoninSectionIVwhen attributinggainstothebackbone. E.InferenceSimplificationandEfficiencyAccounting Thediffusionstrengthslearnedduringtrainingdefinethe deployedcomputationgraph.Aftertraining,eachstage evaluates;whereveritiszero,theFHCOpathofthatstageis replacedbytheidentity,whilethesurroundingdepthwise convolution,projections,andgatingareretained.Because 퐀 퐀 is exactlyzerowhenever 퐀 퐀 ≤0 ,thebypassisexact,not approximate,foreveryretiredstage:thekernelinside(4) reducestothefrequency-independentconstant 1 ,whichthe subsequentgroupnormalizationremoves.Wemeasureda maximumrelativedeviationof 1.5×10 −6 (fp32)betweenthe branchandanidentitybranch,i.e.,purefloating-pointrounding. Ineverysemi-supervisedmodelwetrained,thisruleremoves thespectraltransformfromallbutonestage(SectionIV-B), reducinginferencecostfrom1.20Gto0.90GFLOPsfor FHEAT-Segandfrom57.77Gto55.75GforFHEAT-Seg-L. Wereportinference-timecomputationgraphsthroughout: termsthatdonotparticipateinthegraphareexcludedfrom parametercounts,andallparameters/FLOPsfortheproposed andreferencemodelsaremeasuredunderanidenticalprotocol (fvcore,single-channelinput).Theper-stagescalarsaddsixteen parametersintotal.Fortransparency,thereference implementationalsocarriesper-stagefrequency-embedding tensorsinheritedfrom[1][2](0.23Mforthesmallmodel)that participateinnocomputation.Thesetensorsareexcludedfrom allreportedcountsandareremovedinourreleasedcode,soa readerre-implementingthemethodfromthissectionobtains exactlythereportedparametercounts. F.OptimizationGeometryoftheIdentityLimit TheFHEATgatecontainstheidentityasanexactfixedpoint. Thissubsectionrecordsfourfactsaboutthatfamily:itcannot amplify(Remark2);same-resolution,same-ordergatesforma semigroupandthereforecollapsetoasingletime(Theorem1); thatcollapsedoesnotbyitselfproducesparsityunderweight decay(Remark3);thefirstvariationofthelossinthe semigrouptimeisafractionalDirichletpairingwhosesign selectsthesite(Proposition1),afterwhichReLUandweight decaycompileanunfavourablesignintoanexactzero (Proposition2).Throughout, ℒ isthetraininglossand 퐀 isthe orthonormalDCT-IIofSectionIII-B.ThediscreteNeumann fractionalLaplacian (−Δ) 퐀/2 isthediagonaloperatorwith eigenvalue ∥퐀∥ 퐀 onmode 퐀 .Noneofthesestatementsnames thesurvivinglayer;thatisanempiricaloutcome,testedbythe single-siteablationsofSectionIV-E,bythetraining-dynamics instrumentationofSectionIV-B,andbythecontrolled simulationinFig.3.Thesimulationuses1Dsignalsonalength- 256gridintheDCT-IIbasiswithafixed,shared 퐀=0.75 , optimizing 퐀=ReLU퐀 byAdamW(lr0.02–0.03,decoupled weightdecay 10 −3 on 퐀 )for6,000iterations. Remark1(Exactidentity).Let 퐀 퐀 denotetheFHCO definedin(4),i.e.,thespectralmapinside(6).Then 퐀 0 =Id for everyinputandevery 퐀∈(0,2] :themultiplierisidentically one,so퐀 0 =퐀 −1 퐀 byorthogonalityof(2).At 퐀=0 theFHCO reducestotheidentity,andtheFHEATblockcollapsestothe affinebypassaroundthetransform, 퐀=퐀 out SiLU(퐀)⊙GN(퐀) (10) andtheinferencereplacementofSectionIII-Eisexact (maximumrelativedeviation 1.5× 10 −6 infp32). Remark2(Non-amplification).Theeigenvalues 퐀 퐀 (퐀)= exp(−(퐀∥퐀∥) 퐀 ) liein (0,1] ,with퐀 퐀 (0)=1 .Hence ∥퐀 퐀 ∥ 2→2 =1 .Onthefluctuationsubspace 퐀≠0 , ∥퐀 퐀 ∥=exp−(퐀∥퐀 ∥ min ) 퐀 <1 (퐀>0) (11) astrictcontractionthatdeepensin 퐀 .Equivalently, 퐀↦ exp(−퐀 퐀 퐀 퐀/2 ) iscompletelymonotoneonthesquaredradius 퐀=∥퐀∥ 2 for 퐀∈(0,2] ,sothekernelispositivedefinite.The gateisawell-posedsmoother:itcannotmanufactureenergyat anyscale.Unconstrainedfilters(GFNet,AFNO)havenosuch bound. Theorem1(Semigroupcollapse).FixaDCTgridanda sharedorder 퐀 .Write 퐀:=:=:=퐀 퐀 ≥0 andlet 퐀 퐀 퐀 denotetheFHCO withmultiplier 퐀 퐀 (퐀)=exp(−퐀∥퐀∥ 퐀 ) .Then 퐀 퐀 1 퐀 ∘퐀 퐀 2 퐀 =퐀 퐀 1 +퐀 2 퐀 , 퐀 0 퐀 =Id.(12) Proof.Orthogonalityof 퐀 gives 퐀 퐀 1 퐀 ∘퐀 퐀 2 퐀 = 퐀 −1 diag(퐀 퐀 1 퐀 퐀 2 )퐀.Themultipliersmultiplypointwise, 퐀 퐀 1 (퐀) 퐀 퐀 2 (퐀)=exp−( 퐀 1 +퐀 2 )∥퐀 ∥ 퐀 = 퐀 퐀 1 +퐀 2 (퐀) (13) Thusanyfinitecompositionofsame-grid,same- 퐀 gatesisa singlegateat퐀 tot = 퐀 퐀 퐀 ,or퐀 eff =( 퐀 퐀 퐀 퐀 ) 1/퐀 .Withinone FHEAT-Segstagethisisanidentity,notanapproximation:the blocksofstage 퐀 share (퐀 퐀 ,퐀 퐀 ) .Acrossstagesbothhypotheses fail---thegridsdiffer,andeachstagelearnsitsown 퐀 퐀 ---so theeightstage-levelgatesarenotinterchangeable,andtimes mustnotbeaddedacrosstheU-Net.Theorem1onlycertifies theredundancyalreadybuiltintoeachstage. Remark3(Collapsedoesnotimplysparsity).Theorem1 saysthatinterchangeablegatesmaybereplacedbyone.Itdoes notsaythattrainingwillpickasparserepresentative.Underthe soleconstraint 퐀 퐀 퐀 퐀 =퐀 tot andaquadraticpenalty 퐀 퐀 퐀 2 , stationarityrequires 퐀 퐀 2−퐀 equalontheactiveset,hencethe symmetricsplit 퐀 퐀 =(퐀 tot /퐀) 1/퐀 (14) For 퐀<1 thisinteriorpointischeaperthananysparsecorner: with 퐀=0.75 , 퐀=8 , 퐀 tot =2 ,onehas 퐀 퐀 퐀 2 ≈0.20 versus 6.35 forconcentrationatasinglesite.Fig.3(a)reproducesthe dynamics:eightidenticalsame-resolutiongatesfittedtoone targetremainactiveandnear-equal( 퐀=0.248±0.011 ). Weightdecayonredundantgatesthereforespreadsthebudget. Thesparsityobservedempiricallycannotbeblamedon Theorem1plusweightdecay;itrequiresthestagestodiffer, whichiswhatProposition1tests. Proposition1(Firstvariation).Let 퐀 bethetensorentering thespectralmultiplyin(6),let 퐀=퐀 퐀 퐀 bethatmultiplication’s output(beforegroupnormalizationandgating),andlet퐀 :=:=∂ℒ/∂퐀 .With 퐀 =퐀 and 퐀 =퐀 real, ∂ℒ ∂퐀 =− 퐀≠0 ∥ 퐀∥ 퐀 퐀 퐀 (퐀) 퐀 (퐀) 퐀 (퐀), (15) whichisfiniteandcontinuousat 퐀=0 .Inparticular ∂ℒ ∂퐀 퐀=0 =−⟨퐀,(−Δ) 퐀/2 퐀⟩=−⟨(−Δ) 퐀/4 퐀, (−Δ) 퐀/4 퐀⟩ (16) Write퐀:=:=:=⟨(−Δ ) 퐀/4 퐀, (−Δ ) 퐀/4 퐀⟩forthe 퐀 퐀/2 pairing. Increasing 퐀 from 0 strictlydecreases ℒ ifandonlyif 퐀>0 ,i.e., ifandonlyiftheincomingfeatureandtheincominggradient arepositivelyalignedinthe 퐀 퐀/2 seminorm.Complementary slacknessalongthiscoordinateistherefore 퐀≤0 ⇔ 퐀=0isaone-sidedlocalminimizer (17) 퐀>0 ⇔ thegateshouldleave퐀=0 (18) Tofirstorder,thenumberofsurvivingstagesisthenumber ofstageswith 퐀>0 .Thesingle-siteablationsofSectionIV-E agreewiththerankingthepairingwouldpredict(Block_Final useful).Fig.3(b)alreadyconfirmsthemechanismona syntheticgroundtruth:thesignof 퐀 atinitializationpredicted theuniquesurvivorinalleightplantedsites. Proposition2(Absorption).Theimplementationoptimizes 퐀=퐀䠀(퐀) ratherthan 퐀 ,with 퐀←퐀−퐀 ∂ℒ ∂퐀 +퐀 (19) Thechainruleissingularattheoriginfor 퐀∈(0,1) (the wholerangeusedhere),whichiswhyProposition1isstatedin 퐀.Once 퐀<0 , 퐀=0 andtheupdateisthecontraction 퐀← (1−퐀)퐀 : 퐀=0 isabsorbing.CombinedwithProposition1,a negativepairing 퐀<0 at 퐀=0 drives 퐀 belowzeroandkeeps itthere;thegradientpathclosesattheboundary,anddecoupled weightdecayonlycontractstowardzerofrombelow,sono subsequenttrainingsignalcanpush 퐀 backacross.Retirement isthereforepermanentfortherestoftraining,shortofan externalperturbation(cf.Fig.5(g)).Thisisstandardactive-set identificationforahiddennonnegativityconstraint[28].Itis specificto 퐀=ReLU(퐀) underdecoupledweightdecay---a softplusorexponentialcoordinatewouldnotlock---anditisa compiler,notacause.Theempiricalfactisthatsevenofeight pairingsareunfavourableunderthepresentprotocol. Remark4(Sobolevreading).Theexplicitmultiplier 퐀 퐀 (퐀)=퐀 −퐀∥퐀∥ 퐀 agreestofirstorderin 퐀∥퐀∥ 퐀 withthe resolvent 1/(1+퐀∥퐀∥ 퐀 ) oftheTikhonovproblem min 퐀 1 2 ∥퐀−퐀∥ 2 +퐀 |퐀 | 퐀 퐀/2 2 (20) since 퐀 −䠀 and 1/(1+䠀) sharetheexpansion 1−䠀+퐀( 䠀 2 ) . Theoperatorscoincideonlyas 퐀→0 ; 퐀 퐀 istheexactheatflow, whichiswhyTheorem1isanidentity.Theusefulreading remains:raising 퐀 atastageis,toleadingorder,strengthening an 퐀 퐀/2 penaltyonthatstage’sfeatures,and 퐀=0 means addingnosuchpenalty.FHEATisaper-stageSobolev regularizerwhosestrengthislearned,notanunconstrained Fourierfilter. Fig.3.Controlled,non-medicaltestofTheorem1andProposition1;nomedicaldataortraininginvolved.(a)Symmetriccontrol: eightredundant,same-resolutiongatesjointlyfitonesharedlow-passtarget.ConsistentwithRemark3,alleightremainactive with 퐀=0.248±0.011 (mean ± SD):weightdecaydoesnotconcentratethebudgetontogenuinelyinterchangeablegates.(b) Differentiatedsites:sevengateshaveatargetequaltotheirinput(pairingexactly0at 퐀=0 );one(site5,red)hasagenuinelow- passtarget(pairing ≈1609 ).Onlytheplantedsitesurvives(final 퐀=1.12 vs.exactly0attheotherseven),andthesignofthe pairingatinitializationpredictssurvivalatalleightsites,exactlyasProposition1predicts. IV.EXPERIMENTS A.ExperimentalSetup Datasets.Weevaluatethreepublic3Dbenchmarks.(i)LA [1]:LGE-MRIscans;followingcommonpractice[9],weuse 80volumesfortrainingandtheremaining20fortesting;4and 8labeledscanscorrespondto5%and10%labelrates.(i) Pancreas-CT[29]:followingtheFUSSNetpreprocessingand split[30]usedby[2],62volumesfortrainingandtheremaining 20fortesting,with6and12labels(≈10%and≈20%).(i) BraTS2019[31]:250,25,and60volumesfortraining, validation,andtesting,with25labels(≈10%). Metrics.Dicescore(Dice),Jaccardindex(Jac),95% Hausdorffdistance(95HD),andaveragesymmetricsurface distance(ASD);95HDandASDarereportedinvoxels. WeconductallexperimentsonNVIDIA5090GPU.Semi- supervisedmodelsaretrainedundertheCSEprotocolfor 16,000iterations(batchoftwolabeled+twounlabeled volumes;AdamW,weightdecay,baselearningrate0.01with cosinedecay);thefractional-orderparametersusealearning- ratemultiplier,thediffusionstrengthsthebaserate,andthe KANrationalcoefficientsamultiplier.Followingthereference protocol,modelselectionisperformedonvalidationdataheld outfromthetrainingset,andallreportedscoresarecomputed ontheseparatetestsets. Unlessstatedotherwise,theimprovementofFHEAT-Seg overthestrongestbaselineineachcomparisonissignificant underapairedWilcoxonsigned-ranktestacrosstestcases. Baselines.Wecompareagainstfivesemi-supervised methods(UA-MT[8],SASSNet[10],CAML[11],BCP[9], andMLRP[12])andagainsttheLight-UNETRbackbone.All methodsareevaluatedunderthesameLight-UNETRbackbone andtheofficialsplits,sothebackboneistheonlyvariedfactor inthecomparison.Thefullysupervisedcomparisonsfollowthe sameprotocol[2].Forqualitativecomparisonweadditionally re-implementedallfiveSSLmethodsonthisbackboneunder anidenticalprotocol.Wereporttheshared-backbonenumbers preciselybecausetheyaretheonlynumbersthatisolatethe contributionofthebackbonefromthatoftheSSLmethod[8]– [12]. B.LearnedSpectralAllocation TableIreportsthelearneddiffusionstrengthsreaddirectly fromthecheckpointsofeverysemi-supervisedmodeltrained underthefinalprotocol.Inallfiveheadlinemodels,spanning threedatasetsandlabelratesfrom5%to20%,andinallfive protocol-matchedvariants,exactlyoneoftheeightstage-level operatorsretainsD>0;theothersevenarelearnednegativeand collapsetoidentities.Thesurvivoristhefinaldecoderblockin everysetting,theblockwhoseFHEAToutputproducesthe attentionmapconsumedbyAGR.Itsfractionalordersaturatesat thestrongestadmissiblelow-pass(α≈0.90),whereastheretired stagessitatα≈0.60–0.75.Theassignmentisthereforeconsistent inlocationandinshape.Everysettingwasrepeatedwiththree independentrandomseeds;thesurvivingblockwasBlock_Final inall15runs.TableIreportsarepresentativeseedper setting.Underfullysupervisedtrainingthesamesparsification tendencyholds(typicallyzerooronesurvivingstage),butthe survivor'slocationisnolongerconcentratedattheattention- producingblock(e.g.,amid-decoderstageonLA,noneon Pancreas-CT).Spectralsparsificationisthusageneralbehaviorof theoperatorfamily;itslocalizationattheattentioninterfaceis specifictosemi-supervisedtraining. TABLEI.LEARNEDSPECTRALALLOCATIONOFTHESEMI- SUPERVISEDMODELS(CHECKPOINTVALUES).“SURVIVOR”= THEUNIQUESTAGEWITHD>0;ALLSEVENOTHERSTAGES LEARNEDD=0.CORRESPONDSTOTHESUMMARYSTATISTIC REPORTEDINSECTIONIV-B. Model(setting)SurvivorDα LA,5%labelsDecoderBlock_Final1.2240.900 LA,10%labelsDecoderBlock_Final0.9680.899 Pancreas-CT,10%DecoderBlock_Final0.4960.897 Pancreas-CT,20%DecoderBlock_Final0.6930.900 BraTS2019,≈10%DecoderBlock_Final0.4180.900 Thesurvivor'sDalsovariesconsiderablyacrosssettings (TableI):1.224and0.968onLAat5%and10%labels,0.496 and0.693onPancreas-CTat≈10%and≈20%,and0.418on BraTS2019,themostheterogeneousofthethreetasksinboth anatomyandlabelcount.BraTScarriesthelowestsurvivingD, consistentwithahypothesisthataharder,moreheterogeneous segmentationtargetbenefitsfromagentlerlow-passatthe attention-producingblock,sinceanaggressiveonerisks smoothingawaytheboundarydetailmulti-regiontumor segmentationdependson;underthisreading,Dwouldtradeoff againsttargetheterogeneityratherthanagainstdatasetsize, sincePancreas-CTandBraTShavecomparablecasecountsbut differentD. ThetrainingdynamicsonLAat5%labels(plottedforthe equal-capacityFFNvariant,whoseper-iterationcheckpoints wereretained)followaconsistentpattern:alldiffusion strengthsstartfromtheshared0.03initialization,theseven retiredoperatorsreachexactlyzerobythefirstrecorded checkpoint,andthesurvivingoperator'sDgrowswhileitsα approaches0.9,thesameendpointtheKAN3Dmodelreaches (TableI).Thelearnedper-stagespectralresponseexp(−(D‖k‖) α)oftheheadlinemodelfollowsthesamedivision:astrong low-passatthesurvivingstageandtheidentityeverywhereelse, asdetailedlaterinthissection. Cross-ArchitectureGeneralization.Totestwhethergradient- drivenspectralsparsificationisapropertyoftheoperatorrather thanoftheLight-UNETRbackbone,weportedtheFHEAT gate(FractionalHeat3D,samecode)ontoa3DSegFormer,a hierarchicalTransformerarchitecturethatdifferssubstantially fromLight-UNETR'sCNN–Transformerhybrid.Self-attention ineveryencoderTransformerBlockisreplacedbytheFHEAT spectraloperator,andtheall-MLPdecoderreceivesone additiveFHEATgateperstage,withthefinaldecodergate producingtheCSEattentionmap.Theportedmodel(4.95M parameters)istrainedundertheidenticalCSEprotocolonLA at5%labelsfor16,000iterations.Thespectralsparsification phenomenonreproducesonthisarchitecturallydistinct backbone:sevenoftheeightstage-leveldiffusionoperators convergetoexactlyD=0,andthesolesurvivoristhedeepest decoderstage(dec4),withDgrowingmonotonicallyfrom 0.050to0.120overtraining.Thesurvivor'spositiondiffers fromFHEAT-Seg(Block_Final),whichisexpected,sincethe gradientsignalthatdrivesDgrowthisarchitecture-dependent; thequalitativepattern(uniquesurvivorplusexact-zero retirement)isarchitecture-independent.Onthetestset,the FHEAT-portedSegFormer3DachievesDice=0.9002(Jaccard 0.8192,95HD5.33,ASD1.79),comparedto0.8817(0.7899, 7.13,2.20)fortheplainSegFormer3Dcontroltrainedunderthe sameprotocol:a+1.85Dice-pointgainfromtheFHEAT frameworkonanindependentbackbone. Fig.4.Cross-architecturevalidationonSegFormer3D:stageD allocation.dec4isthesolesurvivor(D0.050→0.120);all7 othersconvergetoexactlyD=0.TestDice:0.9002(+FHEAT) vs.0.8817(plaincontrol). Theexactnessandpermanenceofthezerosadmitaprecise dynamicalreading.With 퐀 theEuclideanprojectionontothe feasibleset ℱ ,eachrawdiffusionparameterfollowsthe subgradientdynamicsuptotheoptimizer’sweight-decayterm. Twoconsequencesfollow.(i)Finite-timeabsorption.Astage whoselossfavorsremovingspectralmixing( 퐀<0 as 퐀→0 − ) crosses 퐀=0 infinitelymanystepsand,becausetheindicator killsthegradientpathattheboundary,cannotre-enter.This mirrorsthefinite-timeactive-setidentificationofprojected- gradientmethods[28],exceptthatheretheabsorptionis irreversibleratherthanre-activatable.(i)Stabilityofthelock. ThedecoupledweightdecayofAdamWonlycontractsa negativetowardzerowithoutchangingitssign,sotheidentity statepersistsandtheinferencebypassofSectionIII-Eremains stablefortherestoftraining.Noexplicitsparsitypenalty appearsanywhereinthismechanism:thelosslandscapeitself decideswhichstagescrosstheboundary,andtheprojection dynamicsmakeeachcrossingexactandpermanent.Thisiswhy thesevenretiredstagesalreadysitatexactzeroatthefirst recordedcheckpointandneverrecover(Fig.5(a)). Fig.5(d)analyzesthebottleneckblockresidual(blockoutput minusinput)intheradialfrequencydomain:33.3%ofthe FHEATblocksresidualenergyliesathighnormalizedradii ( 퐀>0.6 ),versus0.2%fortheLIDRblockofLight-UNETR, whosedual-branchattentionoperatesondownsampledfeatures andinjectsalmostpurelylow-frequencycontent.Readtogether withthelearnedlow-passresponseofthesurvivingoperator (Fig.5(c)),thissuggestsadivisionoflabor:theFHEATblocks preserveandinjecthigh-frequencydetail,whilethesingle survivingspectraloperatorperformslow-passaggregation.The samelow-passoperationalsoyieldsmorefocusedattention:the attentionmapsofFHEAT-Segconcentrate44.8%oftheir energywithinthetargetregiononLA,versus3.4%forLight- UNETR(18.4%vs.11.6%onPancreas-CT;22.5%vs.1.3%on BraTS2019;Fig.6).Formally,with 퐀 퐀 =(1/퐀) 퐀 | 퐀 퐀,퐀 | the stop-gradientattentionmagnitudeatvoxel 퐀 and Ω theground- truthforeground,theconcentrationratiois 퐀∈Ω 퐀 퐀 / 퐀 퐀 퐀 , computedpertestcaseandaveraged(LA:all20testcases, standarddeviation6.1vs.1.2percentagepointsforFHEAT- SegandLight-UNETR,respectively;Pancreas-CT:all20test cases;BraTS:all60testcases).Becausetheattentionmapis stop-gradient(SectionIII-D),wereadthisasaconsistentby- productofthelearnedlow-passaggregationratherthanasits cause.TheBlock_Final-onlycontrolofSectionIV-Econfirms thatplacingthespectralbudgetatexactlythisblockissufficient. C.Semi-SupervisedComparison TheallocationlearnedinSectionIV-Bdefinestheinference graphwhoseaccuracywenowmeasure:onesurvivinglow-pass operatorattheattention-producingblock,andsevenretired stagesrunningasfreeshortcuts.Thegainsreportedbeloware thedownstreamconsequenceofthatlearnedassignment.Table IIreportsthelowerlabelsetting(LA-4at5%,Pancreas-CT-6 at≈10%).FHEAT-Segachieves90.47%DiceonLAand 78.79%onPancreas-CT,surpassingtheCSE-trainedLight- UNETRbackbone(89.53%/73.77%)andallfiveSSLmethods; theboundarymetricsimproveaccordinglyonLA(95HD4.60 vs.6.14,ASD1.78vs.1.88).TableIIIreportsthehighersetting (LA-8at10%,Pancreas-CT-12at≈20%,BraTS-25at≈10%). Theorderingispreservedatthehigherlabelrates:FHEAT-Seg ranksfirstonDiceandJaccardoneverydatasetofbothtables, andtheadvantageextendstoboundarymetrics,whereitattains thebest95HDandASDineverysettingofbothtables,with oneexception—95HDonPancreas-CT-12,whereCSE-Light- UNETRremainsahead(8.35vs.8.69).Acrossthreerandom seeds,thestandarddeviationsofFHEAT-Segaresmallerthan thecorrespondingmarginsovertheCSE-trainedLight-UNETR backboneineverysettingofbothtables,andtheDiceand Jaccardimprovementsaresignificant(pairedWilcoxonsigned- ranktestacrosstestcases, 퐀<0.05 ). TABLEII.SEMI-SUPERVISEDSEGMENTATIONATTHELOWERLABELRATE(LA:4LABELS,5%;PANCREAS-CT:6LABELS,≈10%). EVERYREFERENCEMETHODISTRAINEDWITHTHESAMELIGHT-UNETRBACKBONEUNDERTHECSEPROTOCOL.BESTINBOLD. FHEAT-SEGENTRIESAREMEANSOVERTHREERANDOMSEEDS(±SD). LA(4labels)Pancreas-CT(6labels) MethodDice↑Jac↑95HD↓ASD↓Dice↑Jac↑95HD↓ASD↓ UA-MT78.3465.1115.424.6952.4136.8036.3411.78 SASSNet79.6967.0212.443.8651.3136.1022.047.50 CAML79.7466.9414.584.4657.3440.9428.418.86 MLRP81.2769.3610.703.1267.0551.4413.714.39 BCP84.7673.8810.222.8968.9353.8614.014.37 Light-UNETR89.5381.156.141.8873.7759.4411.773.76 FHEAT-Seg(ours)90.47±0.2182.65±0.354.60±0.421.78±0.1578.79±0.3865.53±0.527.80±0.611.84±0.22 Fig.5.(a,b)TrainingdynamicsofthespectralparametersonLAat5%labels:thesolesurvivor(Block_Final,red)grows monotonicallywhilethesevenretiredstages(gray)collapseto 퐀=0 bythefirstrecordedcheckpoint.(c)Learnedspectral response exp −퐀∥퐀∥ 퐀 atconvergence:astronglow-passatthesurvivor,identityelsewhere.(d)Bottleneckblockresidual spectrum:33.3%ofresidualenergyathighnormalizedradii( 퐀>0.6 ),versus0.2%forLight-UNETR.(e)Survivor 퐀 trajectory inthegradient-instrumentedrun.(f)Nonzero-frequencygradientattenuationratio:thesurvivordeepensfrom 0.472 to 0.111 ; retiredstagesholdat 1.0000 .(g)NOISE-INJECTIONPERTURBATION:4/7reactivatedstagesre-absorbtoexactly 퐀=0 ;the remaining3/7sustain 퐀≈0.4 anddegradevalidationDice( 90.26→87.22 ).(h)Learning-rate ×10 spike:allretiredstageshold atexactly 퐀=0 . Fig.6.AGRattentionmaps(LA,fiverepresentativecases):FHEAT-Segconcentrates44.8%ofattentionenergywithinthetarget regionversus3.4%forLight-UNETR(18.4%vs.11.6%onPancreas-CT;22.5%vs.1.3%onBraTS2019;allratiosaveragedper caseoverthefulltestsets). TABLEIII.SEMI-SUPERVISEDSEGMENTATIONATTHEHIGHERLABELRATE(LA:8LABELS,10%;PANCREAS-CT:12LABELS, ≈20%;BRATS2019:25LABELS,≈10%).SAMECONVENTIONSASTABLEII. LA(8labels)Pancreas-CT(12labels)BraTS2019(25labels) MethodDice↑Jac↑95HD↓ASD↓Dice↑Jac↑95HD↓ASD↓Dice↑Jac↑95HD↓ASD↓ UA-MT82.5370.7311.803.8467.2851.9914.734.1274.1362.4516.234.42 SASSNet83.5172.199.932.7867.6052.2318.066.3978.7967.9613.653.96 CAML82.3670.569.793.0868.7652.8611.963.9278.3567.0213.343.85 MLRP82.9271.1810.242.9671.8057.248.842.4578.1067.5112.503.79 BCP86.8176.948.912.4973.1558.568.612.4177.0666.0218.105.33 Light-UNETR90.5082.745.821.8378.5065.278.352.2479.7368.7611.652.25 FHEAT-Seg(ours)91.85±0.1684.97±0.244.30±0.351.44±0.1281.07±0.2968.64±0.418.69±0.481.43±0.1481.90±0.3371.53±0.4510.27±0.722.06±0.19 D.FullySupervisedComparisonandEfficiency Withnosemi-supervisedcomponentinvolved,thefully supervisedcomparisonisolatesthebackboneitself.Table IVreportsresultsonLA,Pancreas-CT,andBraTS2019 againstthreeSAM-familyfoundationmodelsandseven segmentationnetworks.FHEAT-SegimprovesoverLight- UNETRonLA(92.05vs.91.58Dice),Pancreas-CT(83.30 vs.80.45),andBraTS2019(84.28vs.79.93)at27%fewer parametersand79%fewerFLOPs(0.975M/0.90Gvs. 1.34M/4.29G);theparenthesizedwith-DCTcostsshow thatthearchitecturealonealreadyundercutsLight- UNETR'sFLOPsby72%beforethelearnedbypass. FHEAT-Seg-LoutperformsLight-UNETR-Lonallthree benchmarks(93.09vs.92.22,85.11vs.84.30,87.19vs. 84.32)atlowercost(2.851M/55.75Gvs.3.93M/59.45G). AcrossthefourteenmetricandefficiencycolumnsofTable IV,ourtwovariantstogetherrankfirstontwelve.Thelarger BraTSmarginisconsistentwiththespectraldivisionof labor:multi-regiontumorboundariesarehigh-frequency structures,exactlywhattheFHEATblocksamplify (SectionIV-B).BecausethiscomparisoninvolvesnoAGR orSMCcomponent,theconsistentmarginofFHEAT-Seg- LoverLight-UNETR-Lindicatesthatthelearnedspectral allocationisnotabyproductofthesemi-supervisedprotocol butapropertyofthebackbone'soptimizationlandscape underordinarysupervision;SectionVreturnstothispoint. TABLEIV.FULLYSUPERVISEDSEGMENTATIONRESULTS(DICE/JACIN%;95HD/ASDINVOXELS)ONLA,PANCREAS-CT, ANDBRATS2019,WITHEFFICIENCYACCOUNTING.REFERENCEROWSAREASREPORTEDIN[2],WHEREFLOPSARE COMPUTEDATASPATIALSIZEOF(112,112,80).BESTINBOLD.(SECTIONIII-E). LAPancreas-CTBraTS2019 FLOPs(G)↓Params(M)↓ MethodDice↑Jac↑95HD↓ASD↓Dice↑Jac↑95HD↓ASD↓Dice↑Jac↑95HD↓ASD↓ Foundationmodels MedSAM2[32]80.4467.6721.595.2869.0153.2222.235.3183.0874.0217.216.362,259.238.96 Efficient-MedSAM2[32]78.6165.3921.826.0268.9253.0618.835.5582.2971.0611.136.011,922.034.10 Med-FastSAM[33]83.1669.5319.025.1070.3059.7119.724.8882.0671.5810.845.031,784.914.48 Segmentationnetworks nnU-Net[5]70.6263.9325.451.7376.7664.2116.031.1477.2967.4012.282.09227.8813.82 SegResNet[34]84.4076.4510.501.9381.3769.217.761.2177.6767.1712.132.02229.8710.64 TransBTS[35]69.7659.7923.454.6176.4363.0911.821.6279.2469.429.611.79161.1430.61 SwinUNETR[7]87.8679.547.082.0078.8366.139.961.2077.6868.1112.361.75591.8761.99 UNETR++[36]91.1283.895.561.5283.3271.995.961.0579.4570.0011.841.6981.6420.33 MedNeXt[37]92.0585.354.951.3683.8272.504.721.1682.8873.759.841.7665.625.57 Light-UNETR[2]91.5884.514.951.4380.4567.966.191.3079.9369.3311.442.264.291.34 Light-UNETR-L[2]92.2285.614.641.3684.3073.124.361.0484.3274.989.151.6559.453.93 FHEAT-Seg(ours)92.0585.354.521.4983.3071.836.181.1084.2874.629.651.730.90(1.20)0.975 FHEAT-Seg-L(ours)93.0987.113.991.1685.1174.324.421.0587.1978.408.171.2255.75(57.77)2.851 TableVextendsthecomparisontothreeadditionalfully supervisedtasks.Inpairedcomparisonsagainsttheirsame- familybaselines,bothvariantsachievehigheraverageDiceon allthreebenchmarksatlowercomputationalcost:FHEAT-Seg outperformsLight-UNETRby+1.96,+1.94,and+1.09points onMSDTask01,AbdomenCT-1K,andHNCTumor, respectively,whileFHEAT-Seg-LexceedsLight-UNETR-Lby +1.38,+0.69,and+0.61points.Thisconsistentgainreflectsa spectraldivisionoflaboracrosstumorsub-regions.Forthe mostboundary-sensitiveenhancingtumor(ET),thehigh- frequencyFHEATpathwayyieldsthelargestimprovement: FHEAT-Segreaches78.54Diceversus67.01forLight- UNETR(+11.53),andFHEAT-Seg-Lreaches78.31versus 75.41forLight-UNETR-L.Forthelow-contrast,diffusely boundedtumorcore(TC),thecompactmodeltrailsitspaired baseline(83.41vs.86.89),asitssinglesurvivinglow-pass operatorover-smoothsthealreadyweakboundarysignal.To quantifythisboundary-contrasthypothesis,wecomputed imagegradientmagnitudesalong1-voxelboundarybands acrossall484MSDTask01trainingvolumes(Fig.7).OnT1gd, theTCboundaryhasameangradientof129.2versus166.1for ET(ratio1.286,pairedWilcoxonp=8.1×10⁻⁶⁴),confirmingTC boundariesaresystematicallyweakeronthemodalitywherethe low-passoperatorexertsitssmoothingeffect;theordering reversesonFLAIR(p=2.5×10⁻¹).Thelargevariantclosesand reversesthisTCgap(93.51vs.92.89)viaitswider convolutionalstemthatpreservesastrongerresidualhigh- frequencypathway.Forwholetumorsegmentation,FHEAT- Seg-LmatchesLight-UNETR’s90.24Diceandsurpasses Light-UNETR-L’s89.69. TABLEV.FULLYSUPERVISEDSEGMENTATIONDICESCORES(%)ONMSDTASK01[38],ABDOMENCT-1K[39],ANDHNCTUMOR [40],WITHEFFICIENCYACCOUNTING(FLOPS/PARAMSAT96³).WT,ET,ANDTCDENOTEWHOLETUMOR,ENHANCINGTUMOR, ANDTUMORCORE;GTVPANDGTVNDENOTETHEPRIMARYANDNODALGROSSTUMORVOLUMES.BESTINBOLD. MSDTask01AbdomenCT-1KHNCTumor MethodWTETTCAvgLiverKidneySpleenPancreasAvgGTVpGTVnAvgFLOPs↓Params↓ nnU-Net85.8362.7183.4277.3297.2196.9295.9393.4295.8794.7995.7695.27200.8113.82 SegResNet88.6862.6483.2878.2097.0696.3595.2391.4895.0395.1995.7095.44202.6610.64 TransBTS77.8957.4473.5169.6197.0896.4795.3892.8695.4594.0294.3194.16142.0730.61 SwinUNETR90.2363.0284.2279.1697.0396.0894.9893.5195.4096.4596.5196.48331.9561.99 UNETR++90.0764.8784.5279.8295.6495.5693.9991.6894.2295.3995.7495.5657.5120.33 MedNeXt90.1564.3485.9180.1395.5795.3094.3991.3294.1596.6297.0396.8257.815.57 Light-UNETR90.2467.0186.8981.3894.3693.9792.9087.6592.2287.4884.8686.173.731.34 Light-UNETR-L89.6975.4192.8985.9797.2696.3895.4693.7195.7096.6296.3496.4852.153.93 FHEAT-Seg88.0778.5483.4183.3496.1295.2094.4590.8794.1688.2286.2987.260.79(1.06)0.975 FHEAT-Seg-L90.2478.3193.5187.3597.8297.0296.2294.5196.3997.2196.9697.0949.15(50.93)2.851 Fig.7.MSDTask01boundarygradientcontrast:ETvs.TC onT1gdandFLAIR.TheTCboundarygradientis22.2%lower thanETonT1gd(129.2vs.166.1,pairedWilcoxonp= 8.1×10⁻⁶⁴). E.AblationStudies AllablationsusetheLA5%semi-supervisedprotocolwith everyhyperparameterfixed,followingthesingle-setting ablationconventionofthereferenceprotocol[2];eachrow changesexactlyonefactor(TableVI).Asinthatprotocol,10% oftheunlabeledtrainingsetofLAisheldoutasavalidationset formodelselection,andperformanceisreportedonthetestset. RelativetotheLight-UNETRbackbone(89.53%),equipping thestageswithFHEATandanequal-capacityGELUfeed- forwardmixerraisesDiceto90.15%;replacingthemixerwith KAN3Dfurtherraisesitto90.47%(+0.32Dice,+0.53Jacat nearlyidenticalparametercounts,0.969Mvs.0.975M).The KANmixeristhereforeresponsibleforameasurable,isolated shareofthegain.Theroleofspectralmixingisbestreadfrom thecontrolledrows:freezingeverydiffusionstrengthatzero leavesDiceessentiallyunchanged(90.42%vs.90.47%),while allowingasinglelearnableDonlyatBlock_Finalimproves bothDiceandboundaryqualitybeyondthefullylearnable model(90.60%vs.90.47%).Concentratingthespectralbudget wherethenetworkitselfplacesitisthusnotmerelysufficient butslightlysuperior,andthesevenredundantlearnableD'sof thefullmodelareunnecessary.Thisechoesthelottery-ticket structure[41],[42],exceptthatthesparsesubnetworkhere emergesfromordinarygradientdescentonanidentity- containinggate,ratherthanfromposthocpruningorasearch overinitializations.ExpandingtheKANratioto2gives90.30%, anddoublingtheKANlearning-ratemultiplierto0.1gives 89.45%witheveryDdriventozero.Performancedegrades smoothlyacrossthisbandandremainscompetitiveevenwhen thespectralbranchisentirelyinactive. TABLEVI.ABLATIONSONLA5%SEMI-SUPERVISEDSEGMENTATION.IDENTICALPROTOCOL;ONEFACTORVARIEDPERROW. THELASTFOURROWSARECONTROLLEDEXPERIMENTS,NOTCANDIDATEMODELS. ConfigurationDice↑Jac↑95HD↓ASD↓ CSE-Light-UNETRbackbone89.5381.156.141.88 +FHEAT+equal-capacityFFN90.1582.124.661.75 +FHEAT+KAN3D(fullmodel)90.4782.654.601.78 KANratio290.3082.374.801.83 KANlrmultiplier0.1(allD=0)89.4581.025.991.91 AllDfrozenat090.4282.555.141.87 DlearnableatBlock_Finalonly90.6082.874.591.78 F.QualitativeResults Fig.8andFig.9showsegmentationresultsofthreetestcases perdatasetunderthe5%(LA)and10%(Pancreas-CT)settings, comparingfiveSSLmethods,theLight-UNETRbackbone,and FHEAT-Segagainstthegroundtruth;allpredictionscomefrom modelsre-trainedunderthesharedbackboneandprotocol. FHEAT-Segpreservesthinstructures(papillarymusclesonLA; thepancreatictailonPancreas-CT)thatthecompetingmethods blurormiss,consistentwiththehigh-frequencyresidual analysisabove. Fig.8.QualitativecomparisononLA(5%labels,threetestcases):Image|UA-MT|SASSNet|CAML|BCP|MLRP|Light- UNETR|FHEAT-Seg(ours)|GT. Fig.9.QualitativecomparisononPancreas-CT(10%labels,threetestcases),samecolumnlayoutasFig.8. V.DISCUSSION Theresultsinviteacloserlookatwhyafractionalspectral familysuitsthisproblem.Twopropertiesstandout.First,the familyinterpolatescontinuouslybetweentheidentityanda stronglow-passoperator,sotheamountofglobalmixingat eachstagebecomesaquantitativelearningoutcomeratherthan abinaryarchitecturalchoice.Second,theshapeofthelearned responseisinformative:ineverysemi-supervisedmodelthe survivingoperatorsaturatesnear 퐀=0.9 ,thesharpestlow-pass theadmissiblerangeallows,whilethe 퐀 valuesoftheretired stages( 0.60 – 0.75 )nevercomeintoplay.Formally,optimizing 퐀 through 퐀=0.3+0.6퐀(퐀) solvesabound-constrained problem min 퐀 ℒ(퐀) with 퐀∈(0.3,0.9) .Thecap 퐀 ∗ =0.9 isa KKTpointwhenevertheresidualdescentdirectionisnon- positivethere( ∂ℒ/∂퐀≤0 atthecap),theactiveupperbound absorbingtheremainingdescentthroughitsmultiplier.The sigmoidmapmakesthisboundarypointattainableandstable, since 퐀/퐀=0.6퐀(퐀)(1−퐀(퐀))→0 asthetargetgrows,so anyunconstrainedpreferencefor 퐀>0.9 isabsorbedatthecap. Theobservedsaturationisthereforethesignatureofaboundary optimumratherthananinteriorone,notevidencethat 0.9 is intrinsicallyspecial.Confirmingthattheunconstrainedtarget liesbeyondtherangerequireswideningtheparameterization: theconcretetestistoreplacetheboundedmap 퐀=0.3+ 0.6퐀(퐀) withanunboundedonesuchas 퐀=0.3+softplus(퐀) andcheckwhetherthelearnedvaluemovespast 0.9 orreturns toit,whichweleavetofuturework.Qualitatively,onthefamily 퐀(퐀)=exp(−퐀 퐀 ) ,increasing 퐀 steepenstheroll-offin frequency(closertoanideallow-pass)andlightensthespatial tailsofthecorrespondingkernel,approachingaGaussianas 퐀→2 ;saturatingattheupperboundofourrangemeansthe losswantsasharperspectralcutoffthantheboxallows,notthat 0.9 isaphysicalconstant. Thelocationofthesurvivingoperatorisequallyinformative. Itisthefinaldecoderstage(Block_Final),whichoperatesat highresolutionimmediatelybeforethesegmentationheadand whoseoutputadditionallyyieldstheattentionmapusedfor regionselection;astronglow-passaggregationatthisposition denoisesthefeaturesthatdetermineboththeobjectboundaries andtheattentionmap.Thatthenetworkrequestsglobal aggregationatexactlythisstage,andnowhereelse,readsasa sensible,learneddivisionoflaborratherthananartifactofthe parameterization.Thelocationisalsoconsistentwiththe spectralbiasofgradientdescent,whichfitsthelow-frequency componentsofatargetmappingfirst[43],[44]:residualhigh- frequencydetailissuppliedlocallybytheconvolutionalpathof everyblock,sothesingleoperationthatbenefitsmostfroman explicitlearnablefilterislow-passaggregationimmediately beforethesegmentationheadandtheattentionread-out.The fullylearnablemodelarrivesatthisblockwhenfreetochoose amongalleight,andtheBlock_Final-onlycontrolofTableVI showsthatthesamechoiceisalsosufficient.Ininformation- theoreticterms,theall-frozencontrolofthesametable(Dice witheveryDpinnedatzero)indicatesthattheretiredstage’s spectralfilteringcarriesnegligibleadditionalmutual informationwiththesegmentationobjectiveoncethe surroundingconvolutionalpathisinplace. Therelationshipbetweenbackboneandprotocolalso deservescare.ThefullysupervisedresultsofTableIVinvolve noAGRorSMCcomponentandthereforeisolatethe backbonescontributionfromthesemi-supervisedprotocol. Undersemi-supervisedtraining,thesurvivingoperator coincideswiththeblockthatproducestheattentionmap consumedbyAGR,anditslow-passoutputisconsistently associatedwithmoreconcentratedattentionmaps(SectionIV- B).Becausetheattentionmapiscomputedwithastop-gradient (SectionIII-D),wereadthisassociationastwoproductsofthe samelearnedaggregation,boundary-awaredecodingand informativeregionselection,ratherthanasacausalloopfrom theprotocoltotheoperator;theBlock_Final-onlycontrol providesthetestablecheck.Onecaveatremains:partofthe semi-supervisedgainmaystillbemediatedbytheimproved attentionquality,andthefullysupervisedcomparisonisour controlforthatchannel. Finally,themedical-imagingrelevanceofthespectraldesign shouldbestatedexplicitly.Volumetricanatomyisrichin diagnosticallycriticalhigh-frequencystructure(thinwalls, papillarymuscles,vessel-liketails,andsubtletissueinterfaces) thatvolumetricDiceunder-weightsbutthatdeterminesclinical usefulness.Theresidual-spectrumanalysis(SectionIV-B) showsthattheFHEATblocksinjecttwoorderofmagnitude morehigh-frequencydetailthanthelow-pass-onlyattentionof thereferencebackbone,andthequalitativeresultsconfirmthat thistranslatesintopreservedthinstructuresandimproved boundarydistances(95HD/ASD),themetricsmostsensitiveto exactlythosefeatures.Thefractionalheatkernelistherefore notagenericmixertransplantedfromnatural-imagevision:its learnable,frequency-selectiveresponseletsalightweight backbonedevoteitslimitedcapacitytothehigh-frequency anatomicaldetailthatmedicalsegmentationdemands,while retiringspectralcomputationthatthedatadonotneed. Limitations.Threepointsboundthescopeofthepresent evidence.(i)Thefractionalorder 퐀 isoptimizedoverabounded range (0.3,0.9) ;theobservedsaturationattheupperboundisa KKTpointofthatbound,notevidencethat 0.9 isaphysically preferredorder,andwehavenotyetruntheunboundedsoftplus parameterizationthatwouldconfirmwhethertheunconstrained optimumliesbeyondit.(i)Theabsorbing-zeromechanismof Proposition2isspecifictothe 퐀=ReLU(퐀) parameterization underdecoupledweightdecay;itisadeliberatedesignchoice thatmakesretirementexactandcompilable,notaclaimthat everyreasonablereparameterizationofthediffusionstrength wouldsparsifythesameway.(i)Theargumentforsite- selectionisclosedwithoutacheckpointmeasurement: Proposition1derivesthefractionalDirichletpairinganditssign test,Fig.3(b)confirmsonacontrolledsyntheticgroundtruth thatthesignofthepairingatinitializationpredictsthesurviving site,TableVIisarankingconsistentwiththattest(Block_Final useful)ratherthananumericalevaluationofthepairing,and Proposition2togetherwithFig.5(e)–(h)explainswhyastage withanunfavorablesignstaysatexactlyzerooncereached. Whatwehavenotyetreportedisthepairing’smeasuredvalue atBlock_Finalonasavedmedicalcheckpoint—asingle forward/backwardpassat 퐀=0 persite,noretraining— whichisafurther,directmeasurementofthetheory,notapiece thepresentargumentismissing.Relatedly,Theorem1’s semigroupcollapseisexactonlyforgatesthatshareboth resolutionandfractionalorder;itcertifiestheredundancy alreadybuiltintoeachstagebyparametersharing(SectionIII- A)andshouldnotbereadasclaimingthattheeightstage-level gatesofFHEAT-Segarethemselvesinterchangeable,since theydifferinbothrespects. VI.CONCLUSION Thispaperaskswhereglobalspectralmixingshouldbespent inalightweight3Dsegmentationbackbone,andshowsthatthe answerisnotadesignchoicebuttheoutputofanexactoperator theory.FHCOisatwo-parameterNeumannsemigroup:same- resolution,same-orderinstancescomposeexactlyunder additionofanaturaldiffusiontimeτ=D^α(Theorem1),soany stackofsuchgatesisequivalenttoasinglelearned-strength Sobolevregularizer,ortonoregularizeratallwhenτ=0.This redundancyalonedoesnotexplainsparsity,sinceweightdecay appliedtotrulyinterchangeablegatesfavorsspreadingthe diffusionbudgetratherthanconcentratingit(Remark3).We thereforederivetheactualselectionrule:thefirstvariationof thelossinτisafractionalDirichletpairingbetweenastage's incomingfeatureandincominggradient,andastageisretained, tofirstorder,ifandonlyifthatpairingispositive(Proposition 1);ReLUparameterizationandweightdecaythencompilean unfavorablesignintoanexact,deployablezero(Proposition2). TableVIagreeswiththerankingthistheorypredicts,andafully controlled,non-medicalsimulation(Fig.3)confirmsboththe non-sparsifyingnatureofpureredundancyandthesign- predicts-survivalmechanismonasyntheticgroundtruthwhere theanswerisknownbyconstruction. InstantiatedasFHEAT-SegonalightweightU-shaped backbone,thistheoryhasconcreteempiricalconsequences. Underscarcelabels,trainingdrivesthediffusionstrengthto zeroinallbutoneofeightstages,andthesurvivoris consistentlythedecoderblockthatalsoproducesthesemi- supervisedattentionmap;theresultinginferencegraph,at 0.975Mparametersand0.90GFLOPs(79%fewerFLOPsthan Light-UNETR),achievesthebestsemi-supervised segmentationamongthecomparedmethodsundertheshared CSEprotocolonLA,Pancreas-CT,andBraTS2019,and outperformsLight-UNETR-Lunderfullsupervisionatlower cost.Twofurtherresultssupporttheframework'sgeneralityas anoperator-levelclaimratherthanaLight-UNETR-specific one.First,thespectralsparsificationphenomenon(unique survivingstage,exact-zeroretirement)reproducesona structurallydifferentbackbone:a3DSegFormerwithself- attentionreplacedbyFHEATgates,trainedundertheidentical CSEprotocolonLAat5%labels,achievesDice=0.9002vs. 0.8817fortheplaincontrol,with7/8stagesconvergingto exactlyD=0.Second,thelearnedallocation'soptimalityis confirmedbyperturbation:reactivatingretiredstagesdegrades validationDice,whiletheabsorbingzeroisstableunder learning-ratespikes.Thebroadermessageisthatthe computationalbudgetofalightweightnetwork,howmuch globalmixingandwhere,isnotahyperparametertobetuned butaquantitythatanexactoperatortheorypredictsandgradient descentdiscovers. 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