The proof-of-work requirement for a block (finding a nonce such that SHA256(block_header) < target) is equivalent to requiring the attached 2-cell to have scalar curvature R satisfying ∫_{f_i} R dA ≥ κ, where κ is the difficulty parameter.
The hash function acts as a random oracle mapping the block data to a point on a cryptographic manifold. The target defines a submanifold of acceptable solutions. Finding a nonce is finding a geodesic to this submanifold.
The expected work is exponential in the number of leading zero bits, analogous to the exponential decay of curvature in hyperbolic space.
Corollary 4.1 (Exponential Security). The probability that an attacker with less than 50% hash power can reverse a transaction with z confirmations drops exponentially in z, as stated in Section 11 of the whitepaper.
This follows from the Gambler's Ruin analysis in the whitepaper, interpreted as random walk on the 1-skeleton of the blockchain complex.
The deficit z is the distance between the honest chain tip and the attacker's fork tip on this graph.
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Theorem 3.1 (Proof-of-Work as Curvature).
The proof-of-work requirement for a block (finding a nonce such that
SHA256(block_header) < target) is equivalent to requiring the attached 2-cell to have scalar curvature R satisfying ∫_{f_i} R dA ≥ κ, where κ is the difficulty parameter.
The hash function acts as a random oracle mapping the block data to a point on a cryptographic manifold. The target defines a submanifold of acceptable solutions. Finding a nonce is finding a geodesic to this submanifold.
The expected work is exponential in the number of leading zero bits, analogous to the exponential decay of curvature in hyperbolic space.
Lemma 3.3 (Double-Spend as Boundary Dispute).
A double-spend attempt corresponds to two different 1-chains γ₁, γ₂ sharing a boundary vertex (the UTXO being spent) but with different coboundaries.
Only one can be included in the boundary of a valid 2-cell (block).
Corollary 4.1 (Exponential Security). The probability that an attacker with less than 50% hash power can reverse a transaction with z confirmations drops exponentially in z, as stated in Section 11 of the whitepaper.
This follows from the Gambler's Ruin analysis in the whitepaper, interpreted as random walk on the 1-skeleton of the blockchain complex.
The deficit z is the distance between the honest chain tip and the attacker's fork tip on this graph.