The $H^k$ Complexity Ladder: Sheaf Cohomology as a Pre-Diagnostic for Algorithmic Hardness, and a Graded Alternative to the P=NP Question

This paper proposes the H^k Complexity Ladder: a graded stratification of algorithmic hardness by the Betti numbers of the constraint sheaf of a problem instance. The framework provides a polynomial-time pre-diagnostic — computable before any search is attempted — that routes each instance to the cheapest available algorithm.

The four-rung ladder:

• $H^0 = 0$: no solution exists; detect in $O(n+m)$ via union-find.
• $H^1 \neq 0$, $H^2 = 0$: local obstruction, solvable by the Forge ISA at $\beta^*(\rho)$ in polynomial time.
• $H^2 \neq 0$: meta-obstruction, structural change needed; conjectured NP-hard.
• $H^k \neq 0$, $k \geq 3$: higher obstructions; conjectured #P/PSPACE.

The H^k routing algorithm (key practical contribution): compute $\chi = |V| - |E| + |F|$ in $O(n+m)$, infer $\beta_1$ and $\beta_2$, route the instance to the correct algorithm before spending any effort solving it.

The critical temperature $\beta^*(\rho) = \tfrac{3}{8}\ln!\bigl(\tfrac{1}{1-\rho}\bigr)$, where $\rho = \beta_1/|E|$ is the $H^1$ load factor, is computable in $O(|V|+|E|)$ from the problem topology — no hyperparameter search required.

Calibration on $k$-XOR-SAT: the cohomology is analytically known. Experiment x420 confirms $\dim H^0/n \to \max(0, 1-\alpha)$ and $\dim H^1/n \to \max(0, \alpha-1)$ to four decimal places for $n$ up to 1000. The spectral gap $\lambda_{\min}(A^\top A)$ dips near the phase transition $\alpha^* = 1$, consistent with $\beta^*(\rho) \to \infty$ at the critical point.

On P=NP: the H^k framework reframes the question geometrically. For any specific instance, "what is $\beta_2$?" is more useful than "is P=NP?" regardless of how the binary question is resolved. The graded $H^k$ profile is instance-specific and polynomial-time computable.

Three sheaves were tested for Boolean 3-SAT (experiments x420b–d); all three were incorrect models for different reasons. The correct model requires the cavity sheaf ${0, 1, }$ of Mézard-Montanari (2009), whose $H^2$ is conjectured to jump at the known SAT threshold $\alpha^ = 4.267$. This remains the open frontier.

The paper completes the TRS complexity picture alongside the Forge ISA (Paper 419) and the Meld ISA (Paper 454): $H^0$ (tree) = Origami/classical, $H^1$ (loop) = Forge/BPP, $H^2$ (surface) = Meld/BQP.

Keywords

sheaf cohomology, H^k complexity, Betti numbers, P vs NP, algorithmic hardness, phase transition, XOR-SAT, Euler characteristic, routing algorithm, Forge ISA, critical temperature, complexity ladder, TRS framework

Related identifiers (Cites)

Paper 419 Forge ISA: doi:10.5281/zenodo.20694527

Paper 418 Barren Plateaus are H¹: doi:10.5281/zenodo.20694455

Paper 417 H¹=0 Performance Condition: doi:10.5281/zenodo.20684509

The Meld (Paper 416): doi:10.5281/zenodo.20684838

Krishnagopal & Bianconi 2022: doi:10.1103/PhysRevE.104.064303