The Fibonacci Spectral Reconstruction Theory

The Fibonacci Spectral Reconstruction Theory (FSRT) studies a self-adjoint operator satisfying the quadratic reconstruction relation K2=K+IK2=K+I. The operator's spectrum is restricted to {ϕ,−ϕ−1}{ϕ,−ϕ−1} where ϕ=(1+5)/2ϕ=(1+5)/2 is the golden ratio, making KK automatically bounded with ∥K∥=ϕ∥K∥=ϕ. The spectral decomposition K=ϕP+−ϕ−1P−K=ϕP+−ϕ−1P− establishes a one-to-one correspondence between Fibonacci reconstruction operators and orthogonal projections on the Hilbert space. All powers of KK satisfy Fibonacci recursion Kn=FnK+Fn−1IKn=FnK+Fn−1I, and the associated reconstruction channel R(X)=P+XP++P−XP−R(X)=P+XP++P−XP− is a completely positive, trace-preserving, unital map whose fixed algebra is the commutant of KK. The double-commutator flow suppresses non-invariant components with a spectral gap of (ϕ+ϕ−1)2=5(ϕ+ϕ−1)2=5. The theory generalizes to a full metallic reconstruction family Kn2=nKn+IKn2=nKn+I. All rigorous results follow from the single polynomial constraint; physical, geometric, and information-theoretic interpretations are explicitly identified as conjectural extensions.