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Physical realization of homological complexity

Determine whether the homological complexity h(L) of a computational problem L corresponds to the minimum physical dimension required to solve L efficiently, specifically: h(L) = 0 corresponds to 1D systems, h(L) = 1 to 2D systems, h(L) = 2 to 3D systems, and h(L) ≥ 3 requires quantum systems or higher‑dimensional physics.

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Background

The authors seek to connect computational homology with physical computation, hypothesizing that topological complexity measured by h(L) dictates the spatial dimensionality needed for efficient physical realization. They cite parallels with topological quantum computing, holography, and embodied computation as motivating evidence.

If true, this would link computational complexity to fundamental physical constraints, providing a physics‑informed interpretation of homological invariants.

References

Conjecture [Physical Realization of Homological Complexity] The homological complexity h(L) of a problem corresponds to the minimum dimension of a physical system required to solve L efficiently. This conjecture is motivated by several independent lines of evidence:

— A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework (2510.17829 - Tang, 2 Oct 2025) in Conjecture (Physical Realization of Homological Complexity), Subsection "Connections to Physics and Natural Computation" of Section 8 (Theoretical Extensions and Applications)