Closed Matter-Current Constraints and Gaussian Physical Moduli

Closed Matter-Current Constraints and Gaussian Physical Moduli studies the completed kernel of a matter-current constraint in a finite-regulator gauge-field Fock model. The paper establishes two complementary structural results. First, every joint closed-current state with uniformly bounded total-particle support is zero. Consequently, any nonzero exact current-constrained state must have nonvanishing support at arbitrarily large particle number. Second, despite this strong exclusion result, the completed current kernel contains an injective three-parameter family of normalized Gaussian states parametrized by the positive orthant (0,∞)3. For this Gaussian family, the paper proves exact membership in the domains and kernels of the minimal closed current operators, not merely pointwise cancellation of the corresponding differential expressions. Its Fock-space structure is then determined explicitly: the vacuum component is nonzero, every odd particle sector vanishes, the two-particle sector is nonzero, and nonzero even sectors occur above every finite particle cutoff. Gaussian–Hermite integration by parts further yields an arbitrary-occupation raising/lowering recurrence controlled by a finite pair kernel Kλ, connecting the invariant precision geometry of the Gaussian family to its all-orders Fock coefficients. The resulting picture is that the current constraint simultaneously enforces nontruncatable particle support and preserves a finite-dimensional family of exact completed states. In this sense, finite constitutive data coexist with an intrinsically all-orders realization. The principal theorem chain is machine-checked in Lean 4. The formal development distinguishes algebraic core identities, unbounded-operator domain statements, completed-kernel membership, and fixed-regulator transport as separate mathematical claims. The present paper does not assert regulator removal, asymptotic particle semantics, or a continuum-limit theorem. This is the first paper in a series developing the corresponding pair-generated, representation-theoretic, and celestial structures.