Classical Matter Occupancy and Joint-Constraint Closure under AASC

WDW 4/10 This paper closes the two classical matter burdens left open by WDW-1 through WDW-3 while preserving them as distinct theorem objects connected by a one-way dependency. The first module occupies the regular Einstein–Klein–Gordon scalar branch. On a closed hyperbolic three-manifold, it constructs an exact nonvacuum datum whose scalar momentum balances the Hamiltonian constraint, eliminates every generalized Killing initial datum, and supplies a timelike-gradient local clock. Retaining the scalar first-order column in the coupled principal symbol, a fixed balanced normal operator yields a compatible loss-two smooth-tame right inverse. This populates every clause of the ScalarReg3 certificate and promotes the previously conditional characteristic-reduction and finite-refoliation results to a nonempty local chart. The scalar clock is internal to the constrained system rather than an external time parameter or preferred foliation. It is future-directed, transverse to the characteristic flow, normalized locally, and descends through the proper-redescription quotient together with an explicit failure boundary. Relational observables are consequently well defined on the certified local clock domain. No global clock, globally preferred time, or unrestricted deparametrization is claimed. The second module uses the occupied scalar chart as an analytic interface to construct a distinct torsion-free tetrad/spin/gauge Standard Model carrier. On the frozen trivial-bundle branch with global gauge group ((SU(3)_c \times SU(2)_L \times U(1)_Y)/Z_6), three generations, one Higgs doublet, and no right-handed neutrino, it derives the graded Dirac–Bergmann carrier and the complete local Lorentz, internal Gauss, spatial-diffeomorphism, and Hamiltonian constraint system. The fermionic second-class sector is reduced explicitly, and the full first-class Dirac-bracket algebra is proved on a common domain. A sufficiently small, nonzero gauge–Higgs configuration with trivial joint internal stabilizer is corrected through the scalar-chart tame splitting to an exact zero of the full coupled constraint map. The resulting linearization admits a compatible upper-triangular tame right inverse, thereby occupying all clauses of the SMReg4B regularity certificate while preserving the local scalar clock by openness. The integrated theorem retains the pure scalar datum and the full Standard-Model-coupled datum as separate constraint zeros related by an exact parametric lift; it does not identify their constraint maps or zero loci. Perturbative Standard Model anomaly cancellation, the original Witten (SU(2)) parity test, and representation descent to the frozen (Z_6) global form are established at their stated levels. Nontrivial bundle sectors, determinant-line and bordism anomalies, large-diffeomorphism and dynamical-gravity anomalies, regulated quantum constraints, operator domains, ordering, physical inner products, and global-clock claims remain open. The resulting source-attached, branch-typed package supplies the complete classical matter handoff to WDW-5 without yet constructing a Wheeler–DeWitt operator or any other quantum object.