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Homological time–complexity lower bound relation

Establish that homological complexity yields a universal exponential lower bound on worst‑case time complexity: for every computational problem L with homological complexity h(L) = max{ n ≥ 0 : H_n(L) ≠ 0 }, prove that its time complexity satisfies T_L(n) = Ω(2^{h(L) · log n}).

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Background

The paper defines computational homology groups H_n(L) for each problem L and the derived homological complexity h(L). Building on this framework, the authors posit that higher nontrivial homology should translate into lower bounds on algorithmic time, proposing an explicit exponential relation between T_L(n) and h(L).

They motivate this claim by arguing that nontrivial homology classes encode essential computational obstructions that algorithms must explore, suggesting exponential resource requirements proportional to homological depth.

References

Conjecture [Homological Time Complexity Relation] There exists a polynomial p such that for any computational problem L, the time complexity T_L(n) satisfies: T_L(n) = Ω(2{h(L) * log n}). That is, the homological complexity provides an exponential lower bound on the time complexity. This conjecture is motivated by several deep connections between homological structure and computational requirements:

— A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework (2510.17829 - Tang, 2 Oct 2025) in Conjecture (Homological Time Complexity Relation), Subsection "Homological Complexity Theory" of Section 8 (Theoretical Extensions and Applications)