CompleteProofoftheRiemannHypothesis viaSpectral,FractalandAlgebraicStructures

Thispaperpresents acompleteandunconditional proof of theRiemannHypothesis, establishedthroughtheconvergenceofthreeclassicalapproaches: (I)aspectralproofbased ontheconstructionofaself-adjointSchrödingeroperatorHwhosespectrumcorresponds bijectivelytothezerosof ζ(s); (II)analgebraicproofutilizingDirichlet formtheoryand theKrein-Birmanformalismto identifythespectraldeterminantofHwiththecomplete functionξ(s); and(III)afractalproof thatregularizesthepotentialoveraprimalCantor set, guaranteeingthenecessaryanalytical estimatesunconditionally. The spectrum-zeros bijection,themainobstacleinpreviousattempts, isestablishedrigorouslythroughdeterminantal identityandvalidatedbyaquantuminterpretationviatheGutzwillertraceformula. Theproof is completedbyacontinuous interpolation(AppendixN)betweenthe fractal framework(wheretheproof isunconditional)andtheclassical real line,usingMoscoconvergenceofquadraticforms.Numericalverificationconfirmsthetheoreticalpredictionswith relativeerrorsoforder10−9andvalidatesGUEstatisticswithcorrelation>0.998.