From Pair-Generated Fock Structure to Physical Particle Semantics

This paper studies the first nontrivial sector of the current-constrained Gaussian family developed in the preceding papers. For each positive scale triple, the normalized two-particle seed is faithful to the Gaussian modulus: canonical degree-two coefficients recover the pair kernel while its norm is determined by the Hilbert–Schmidt norm of that kernel.

The generated pair has an exact completed Bose realization and is unitarily equivalent to an independently defined degree-two Fock subspace. The same unitary intertwines completed Lorentz and translation actions, yielding a Poincaré representation on the full pair space together with associated shadow and crossing operations. These results determine the Fock grading and spacetime-symmetry representation without yet assigning a complete asymptotic particle interpretation to the one-particle coordinates.

A separate analytic construction realizes every coordinate in the sixty-element Gaussian bank as a nonzero one-particle Hilbert/Fock state. These states transport injectively to a retained constrained carrier, remain nonzero, lie in the kernel of its Hodge-type operator, and carry a relational-response functional.

The stronger asymptotic-particle junction remains more restrictive. The previously available particle dictionary covers only a proper subset of coordinates, and the native kinetic and null asymptotic translation generators are inequivalent. Exact Fock degree, analytic carrier realization, and asymptotic particle identity are therefore distinct structures: the first two are realized here, while the last still requires compatible particle data and dynamics.