Crystal Money: A Block Complex for Geometric Ledgers

Abstract

A blockchain is a chain for a cryptographic reason: each block must hash to its predecessor to anchor proof-of-work. Remove that requirement and the linear structure has no geometric necessity. This paper proposes crystal-currency: a distributed ledger whose validity condition is a geometric consistency constraint rather than a computational puzzle, capital stake, or authority signature. The constraint derives from the Fano plane $\mathrm{PG}(2,2)$ — the unique projective plane on seven points — whose automorphism group $\mathrm{PSL}(2,7)$ of order 168 acts rigidly on the seven orbit types of the three-qubit Pauli group under the Clifford group. The natural data structure is not a chain but a block complex: a growing simplicial complex in which each confirmed block adds a tetrahedron (3-simplex) subject to a Fano consistency condition. We define Proof of Volume (PoV), a consensus primitive requiring all seven orbit types to be simultaneously attested, and prove three results: (i) PoV cannot be satisfied by fewer than seven geometrically distinct parties, since $\mathrm{PG}(2,2)$ admits no proper sub-plane; (ii) combined with Proof of Stake, orbit-aware slashing graded by Fano distance $d_F(\mathcal{O}_i, \mathcal{O}_j)$ makes even two-party cross-orbit collusion detectable on-chain; (iii) for permissioned central-bank digital currency (CBDC), the circle orbit $\mathcal{O}_0$ intersects every Fano line, giving the central bank geometric sovereignty that cannot be outvoted or out-staked. The open cryptographic problem is reduction of orbit-type unforgeability to the Hidden Subgroup Problem over $\mathrm{PSL}(2,7)$. Keywords crystal currency, blockchain, block complex, simplicial complex, Fano plane, $\mathrm{PG}(2,2)$, $\mathrm{PSL}(2,7)$, proof of volume, proof of stake, orbit-aware slashing, Fano distance, CBDC, central bank digital currency, geometric sovereignty, hidden subgroup problem, distributed ledger, Clifford group, Pauli group