Closed Matter-Current Constraints and Gaussian Physical Moduli

We study the completed joint matter-current constraint of a finite-regulator gauge-field Fock model and find a rigid multiparticle structure together with a continuous family of exact physical states. Every joint closed-current zero with uniformly bounded total-particle support vanishes. Any nonzero physical zero mode therefore carries nonvanishing coefficients at arbitrarily large particle number, so the completed constraint naturally directs the state space beyond every finite Fock truncation.

The same current problem contains an explicit Gaussian family generated by the internal gauge geometry. Independent strictly positive rescalings of three gauge-factor sectors preserve a symmetric, positive-definite, adjoint-invariant quadratic structure and determine a three-parameter family of Gaussian precision forms over the full positive orthant.

For every positive scale triple, the normalized Gaussian belongs to the domain of every minimal closed matter-current row and is annihilated by that operator. The scale-to-state map is injective. The particle-sector anatomy is exact: the vacuum component is nonzero, odd sectors vanish, the two-particle sector is nonzero, and nonzero even sectors occur above every finite particle cutoff.

Gaussian–Hermite integration by parts yields a precision-weighted moment balance and an explicit raising/lowering recurrence for arbitrary occupation. The same completed current constraint therefore excludes every bounded-particle nonzero solution while preserving a continuous positive-orthant family of exact states.