Operational Calculus on Curved Differentials: Optimal N-Complex Bounds and Persistent Homology

TL;DR

This paper develops a canonical operator calculus for curved differentials, providing algebraic insights and stability results for persistent homology in curved differential algebras.

Contribution

It introduces a normal form for curved differentials, establishes criteria for N-complex structures, and connects curvature constraints to persistent homology stability.

Findings

Curvature nilpotency to the n-th power implies a (4n-2)-complex structure.

The framework allows Lipschitz control of barcodes under curvature variation.

A toy example demonstrates the practical application of the theory.

Abstract

We establish a canonical normal form for the iterates of a curved differential in curved differential algebras (CDA). This operator calculus clarifies the underlying algebraic structure of CDAs and bypasses the need for complex combinatorics. Using this framework, we provide sharp criteria for curvature constraints to induce N-complex structures. We demonstrate that, while the nilpotency of the curvature element to the n-th power is insufficient to bound the nilpotency of d to 2n, it fundamentally guarantees a strict (4n-2)-complex structure. On the applied side, we model curvature as a filtration controller on a genuine square zero chain complex. This places us under the standard persistence stability framework and yields a Lipschitz control of barcodes with respect to degreewise curvature variation. A reproducible toy example on a four vertex flag complex illustrates the mechanism