Finite Stieltjes Operator Realization
Abstract
Finite positive atomic Stieltjes moments have positive Hankel truncations and an explicit positive diagonal operator realization.
Theorem 1.1 (Positive atomic moments generate Hankel and operator positivity).
Proof. Machine-checked in Lean as D5/S3/Constants/FiniteStieltjesOperatorRealization.finite_stieltjes_operator_realization (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let I be finite, and let x and w be nonnegative real node and weight families. Define mu at n as the finite sum of w(i) times x(i) to the n-th power, and define the order-k Hankel matrix by the moment at p+q. Every such truncation is positive semidefinite; its zero coefficient vector explicitly attains equality.
On the real Euclidean space indexed by I, multiplication by x is an explicit diagonal nonnegative operator U. The vector v has coordinates sqrt(w(i)). Every moment is the inner product of U to the n-th power applied to v with v, and the zero state attains equality in operator nonnegativity.
The proof identifies each Hankel truncation with the Gram matrix of the vectors sqrt(w(i)) x(i)^p and applies Mathlib’s Gram positivity theorem. It formalizes the unconditional finite positive-atomic core only; no Riemann-hypothesis or square-folded-xi representation is assumed or claimed.
The public declarations diagonal_operator_nonnegative and moment_eq_operator_pairing retain the specified diagonal operator and rootWeightVector. For the scalar input map J(u)=u v, the adjoint identity gives J* U^k J=mu(k) id for every k. Coordinatewise multiplication by real nodes is symmetric, so nonnegative quadratic pairings give a positive operator. The finite-prefix error is the maximum of the absolute moment differences over k from zero through the chosen prefix length.
These finite atomic realizations also exhibit discontinuity of exact prefix criteria. For epsilon>0, the prefix (1,1,1,1+epsilon) has positive Hankel blocks and ordinary ranks both equal to one. A symmetric realization would force C J(1)=J(1) from its first three moments, contradicting its fourth moment. It approaches the unit atom at node one. For weights (1-delta,delta) at nodes (1,2), with 0<delta<1, the moments are 1+delta(2^k-1). Every fixed prefix approaches the unit atom, while the order-one Hankel determinant is delta(1-delta)>0. The explicit two-dimensional realization and the Gram rank lower bound give minimum dimension two, compared with one at the limit; both spectra lie in [1,2].
Without a common spectral upper bound, fix any natural r and set t=n+2. Use nodes (1,t^2) and weights (1-w,w), where w=1/t^(4r+3). Both weights are positive. Uniformly for k<=2r+1, the absolute difference of the k-th moment from one is at most 1/t. The next moment equals 1-w+t and tends to positive infinity. For r>=2, the two distinct nodes and positive weights make the ordinary Hankel ranks at orders r and r-1 both equal to two. These are consequences of the finite atomic suppliers, adjoint and Gram rank identities, and scalar estimates; they introduce no further representation assumption.