When Can Depth Replace Precision? A Resource Theory of Quantized Neural Computation

When can additional low-bit residual computation replace missing numerical precision for a fixed input-output map?

We model a quantized residual system over a fixed horizon as a pure schedule selecting fields from a declared low-bit operation library, and use relaxed controls to characterize its infinite-depth limit.

The distance from the target to the closed relaxed reachable set is the exact structural floor: no increase in depth can remove it for that library.

Pure schedules approach the relaxed class at rate $O(D^{-1})$ under bounded-variation time dependence and $O(D^{-\vartheta}+D^{-1})$ under Holder dependence of exponent $\vartheta$.

Execution arithmetic can reverse this conclusion: full-state write-back introduces a $D\rho_z$ penalty and can freeze residual updates, whereas increment error feedback replaces this growth by a bounded carry term and obeys an exact common-lattice conservation law.

A fixed-teacher converse makes this rate sharp: for coherent depth-$L$ first-order high-precision comparators, accuracy matching requires $D=\Theta(L)$.

Learned codebooks add a metadata resource, while state-dependent routing introduces hybrid event conditions.

Verified primal and dual bounds yield feasible, impossible, or unresolved decisions before training.

Companion software implements the workflow, and Lean 4 machine-checks the exact discrete core.

Depth replaces precision only relative to a declared library, horizon, execution semantics, and routing model.