Bitcoin's Original Topology: The Proof-of-Work Manifold
Theorem 1 (Nakamoto Connectivity): The probability of attacker success drops exponentially because the honest chain forms a simply connected manifold - each block cryptographically welded to its predecessor. The Poisson distribution λ = q/p measures connection strength.
Theorem 2 (Temporal Dimension): Bitcoin's dimension grows through proof-of-work difficulty adjustment. The moving average target maintains compactness - the manifold adapts to preserve boundedness despite changing hardware landscapes.
Theorem 3 (SPV Projection): Simplified Payment Verification works because Merkle branches are local coordinate charts - the full manifold can be verified through careful sampling of its submanifolds.
We extend topological calculus to model the Anticipation Pool as a ship navigating Bitcoin's mempool seas. The vessel's integrity depends on three topological invariants: connectivity (hull integrity), dimension (cargo capacity), and compactness (watertight boundaries).
Bitcoin's Original Topology: The Proof-of-Work Manifold
Theorem 1 (Nakamoto Connectivity): The probability of attacker success drops exponentially because the honest chain forms a simply connected manifold - each block cryptographically welded to its predecessor. The Poisson distribution λ = q/p measures connection strength.
Theorem 2 (Temporal Dimension): Bitcoin's dimension grows through proof-of-work difficulty adjustment. The moving average target maintains compactness - the manifold adapts to preserve boundedness despite changing hardware landscapes.
Theorem 3 (SPV Projection): Simplified Payment Verification works because Merkle branches are local coordinate charts - the full manifold can be verified through careful sampling of its submanifolds.
Abstract:
We extend topological calculus to model the Anticipation Pool as a ship navigating Bitcoin's mempool seas. The vessel's integrity depends on three topological invariants: connectivity (hull integrity), dimension (cargo capacity), and compactness (watertight boundaries).