From Pair-Generated Fock Structure to Physical Particle Semantics

From Pair-Generated Fock Structure to Physical Particle Semantics studies the representation-theoretic status of the faithful two-particle seed generated by the current-constrained Gaussian family developed in the preceding papers. For each positive Gaussian parameter, the first nontrivial sector is P2Ψa=Va A‾a†Ω, and the normalized seed Ξa=Va−1P2Ψa is faithful to the Gaussian modulus. Canonical coefficient readouts recover the pair kernel Ka, while the sector norm is controlled by its Hilbert–Schmidt size. The paper first establishes the exact mathematical representation of this generated pair. The completed Bose-symmetric two-particle space is shown to be unitarily equivalent to an independently defined exact degree-two Fock subspace. This unitary intertwines the completed Lorentz, translation, and Poincaré actions, together with the associated shadow and crossing operations. The generated pair therefore has exact Bose symmetry, Fock grading, and spacetime-covariant representation structure. A second result concerns the underlying finite Gaussian coordinate bank. Every coordinate admits a nonzero analytic one-particle Hilbert/Fock realization. These states transport injectively to a retained constrained carrier, remain nonzero, satisfy the established Hodge-type constraint, and admit a total relational-response evaluation. Thus analytic realization extends across the full finite coordinate bank. The paper then separates these positive representation-theoretic results from the stronger question of asymptotic particle identity. A previously supplied particle dictionary reaches only a proper subset of the coordinate bank, and the current full-Gaussian asymptotic junction considered here is obstructed by an exact mismatch between the native kinetic and null asymptotic translation generators. The resulting hierarchy is Fock representation≠analytic carrier realization≠full asymptotic particle identity. The negative asymptotic result is specific to the present carrier and dynamical route; it does not exclude later physical realizations constructed with different data. The principal theorem chain is machine-checked in Lean 4. The paper explicitly distinguishes mathematical representation, analytic realization, particle-semantic qualification, and asymptotic dynamics as separate formal layers. This is the third paper in the series, following Closed Matter-Current Constraints and Gaussian Physical Moduli and Pair-Generated Completion of Gaussian Physical States.