Pair-Generated Completion of Gaussian Physical States

Pair-Generated Completion of Gaussian Physical States develops the operator-theoretic reconstruction of the Gaussian current-constrained family introduced in the preceding paper. Starting from the all-orders Fock recurrence of the normalized Gaussian states, the paper isolates a symmetric pair kernel Ka and uses it to construct a quadratic pair-creation operator on the finite-occupation core. Its graph closure is shown to coincide with both the maximal square-summable coefficient realization and the Hilbert adjoint of the corresponding quadratic annihilation operator, fixing the closed generator including its domain. The closed pair creator reconstructs the Gaussian state sector by sector: P2nΨa=Van!(A‾a†)nΩ,P2n+1Ψa=0. Thus a single quadratic generator produces the entire even-particle hierarchy. The degree-two component is already sufficient to identify the constitutive Gaussian modulus. After removing the nonzero vacuum amplitude, Ξa=Va−1P2Ψa=A‾a†Ω, the map a↦Ξa is injective, and canonical two-particle coefficient readouts recover the entries of the pair kernel. Finite pair data therefore determine the underlying Gaussian parameter and completed-state label without replacing the completed state by a finite truncation. The paper also derives the Cayley relation between the pair kernel and the positive one-particle precision, an exact determinant formula for the norm of the pair-generated exponential family, and the vacuum normalization identity. A coefficientwise exponential gives a closed operator realization of the completed state. The role of Hilbert-space completion is explicit: finite pair truncations converge in norm to the exact current-constrained Gaussian state, while every nonzero finite truncation remains outside the joint current kernel. The resulting structure is finitely generated but nontruncatably realized. The principal theorem chain is machine-checked in Lean 4. All results are stated at the established finite regulator. The paper does not claim regulator removal, a pair generator on the full continuum carrier, or asymptotic particle semantics. This is the second paper in the series, following Closed Matter-Current Constraints and Gaussian Physical Moduli.