Pair-Generated Completion of Gaussian Physical States

We study a three-parameter family of normalized Gaussian states selected by a closed matter-current constraint in a bosonic Fock representation over a finite-dimensional one-particle space. Their occupation coefficients obey an all-orders recurrence controlled by a symmetric pair kernel K_a. The same recurrence determines a positive one-particle precision G_a related to the pair kernel by a Cayley transform.

From this data we construct the quadratic pair creator on the finite-occupation core and identify its closed realization simultaneously with the maximal square-summable coefficient action and the Hilbert adjoint of the reverse quadratic core operator. The closed creator reconstructs the Gaussian state sector by sector, with all odd sectors vanishing.

The first nonvacuum sector already contains faithful finite information. After removing the positive vacuum coefficient, the normalized two-particle seed determines the pair kernel and injectively identifies the Gaussian modulus. At the same time, the orthogonal sum of all reconstructed even sectors converges to the exact constrained state.

A coefficientwise exponential gives a closed densely defined realization of the same completed hierarchy, while spectral diagonalization yields determinant formulas for its norm and vacuum normalization. Finite pair truncations converge in norm to the physical Gaussian, but every nonzero finite truncation remains outside the exact joint current kernel: finite pair data generate the state, while exact current physicality is realized only after completion.