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.
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