Abstract

We present a geometric-topological framework for understanding Bitcoin's Segregated Witness (SegWit) upgrade as a fundamental transformation from quadratic complexity O(n²) to linear complexity O(n) in transaction validation.

This paper formalizes the pre-SegWit "Quadratic Sighash Problem" and demonstrates how SegWit introduced a geometric restructuring that preserved topological invariants while eliminating pathological validation growth.

Why It Matters: Beyond Cryptocurrency

The geometric solution to double spending provides a template for any system requiring:

Path Uniqueness: Ensuring singular evolution of state

Temporal Ordering: Establishing causal structure on events

Consensus Topology: Creating simply connected decision spaces

Metric Stability: Preventing manipulation of distance functions