The physical realization conjecture for homological complexity

Let $L$ be a computational problem with homological complexity $h(L)$. A physical system is said to solve $L$ efficiently when it computes solutions to $L$ within the relevant efficient resource bounds. Physical realization conjecture. The homological complexity $h(L)$ corresponds to the minimum dimension of a physical system required to solve $L$ efficiently: $h(L)=0$ corresponds to one-dimensional systems, $h(L)=1$ to two-dimensional systems, $h(L)=2$ to three-dimensional systems, and $h(L)\geq 3$ to quantum systems or higher-dimensional physics. This proposes a correspondence between topological obstructions in computational problems and the dimensionality of their physical implementations. The source offers heuristic motivation from topological quantum computation, holography, embodied computation, and complexity theory, but no proof or resolution.