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Abstract

We study a fixed minimal weighted realization H=A+B of the constrained Weil form, with nonnegative logarithmic reference A and the complete bounded paired arithmetic perturbation B. For every 0<τ≤1, explicit Fourier-band moments cut out a finite-codimensional subspace of the entire operator kernel on which both A and A^(1/2) have norm at most τ. The number n(τ) of observations satisfies log n(τ) = O(√(1+log(1/τ)) log(3+log(1/τ))). Consequently AΠ_Y and A^(1/2)Π_Y belong to every Schatten class for every closed Y⊆ker H. The proof combines two complete phase-space tails, a finite-rank real-phase Taylor approximation, and the exact null equation. We also give norm estimates for finite null deflation, a full residual-Gram receiving certificate, an explicit weighted bound H≥-99I/100, and a multiplicity obstruction to identifying the derivative-null closure with the whole kernel. The observations are explicit; the projection inside the unknown kernel is not computationally constructed.