Dice Question Streamline Icon: https://streamlinehq.com

Ultimate homological characterization of complexity classes

Show that every natural complexity class C admits a homological characterization of the form C = { L : h(L) ∈ S_C } for some set S_C ⊆ ℕ ∪ {∞} of allowed homological complexities.

Information Square Streamline Icon: https://streamlinehq.com

Background

Having introduced homological complexity h(L) and demonstrated its use to distinguish P from NP, the authors conjecture a unifying principle: that all natural classes can be specified by permissible ranges of h(L).

They present supporting evidence from their earlier characterizations (e.g., P via h(L)=0, NP by h(L)≥1 on hard instances, PSPACE by polynomially bounded h on length‑restricted inputs, EXP by infinite h).

References

Conjecture [Ultimate Homological Characterization] Every natural complexity class \mathcal{C} can be characterized as: \mathcal{C} = {L : h(L) \in S_\mathcal{C}} for some set S_\mathcal{C} \subseteq \mathbb{N} \cup {\infty} of permitted homological complexities. This grand unification conjecture is supported by:

— A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework (2510.17829 - Tang, 2 Oct 2025) in Conjecture (Ultimate Homological Characterization), Subsection "Future Research Directions" of Section 10 (Conclusions and Future Work)