A Mathematical Framework for Carrier-Based Physical Computation: Constraint Compilation, Engineered Dissipation, and Microwave-Referenced Readout

Abstract

We formulate a mathematical framework for validating candidate carrier-based physical computations. Logical continuation is kept distinct from compilation, physical preparation, open-system evolution, restricted observation, and decoding. For every finite boundary/future relation we construct a diagonal positive-semidefinite penalty Hamiltonian whose zero-versus-positive ground-energy class equals existential continuation. To connect this extensional static representation to candidate physical dynamics, we introduce a typed driven-dissipative architecture motivated by superconducting bosonic hardware. The generator is separated into a hardware-stabilization layer and an instance-dependent semantic problem layer. We define the fixed-point, protected-manifold, and no-spurious-state conditions required for exact dissipative compatibility. Protected violation-removal corrections are required to admit uniform synthesis and to preserve the encoded manifold. A Penalty-Lyapunov condition yields exponential decay of the expected excess penalty above the ground energy and, together with explicit fixed-point and convergence hypotheses, identifies the semantic ground sector as the terminal stationary set. For Hamiltonians differing only by a global scalar shift, normalized state data are identical in the absence of an operational energy reference. We therefore formulate a reference-coupled microwave decoding criterion: if calibrated measurement distributions remain uniformly separated under declared reference perturbations, a classical decoder achieves a quantified worst-case error bound. A four-level hierarchy separates static semantic representation, exact dissipative compatibility, efficient convergence, and robust physical readout, so that mathematical representation remains distinct from scalable physical synthesis.