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.