BFSS as a Totally Constrained System: Explicit Hamiltonian Structure and Shallow-Kernel World sheet Emergence

This paper provides an explicit canonical formulation of the BFSS matrix model as a totally constrained Hamiltonian system. By promoting time to a dynamical variable and enforcing reparameterization invariance, the Hamiltonian is shown to become a first-class constraint rather than a generator of physical time evolution. Standard BFSS quantum mechanics is recovered only after gauge fixing, revealing it as a deparameterized, shallow-kernel description of a deeper constrained theory. The full constraint algebra is derived explicitly, including the Gauss constraint and the Hamiltonian (reparameterization) constraint, establishing BFSS as a constraint-entangled theory rather than a conventional dynamical system. Within this framework, spacetime, particles, and perturbative strings do not appear as fundamental observables. Instead, they arise only after successive shallow-kernel reductions associated with clock choice, background selection, and effective decoupling. Finally, the paper clarifies the status of matrix string theory by showing that worldsheet dynamics emerges only in a shallow–shallow kernel regime, requiring both deparameterization and an approximately commuting matrix sector. This construction provides a clean canonical foundation for interpreting Matrix Theory as a background-independent, totally constrained system whose familiar dynamical descriptions are emergent and gauge-fixed limits