Pre-migration documentation. This site reflects the pre-migration state of the protocol. It’s mostly current, but a few edges may not match ZERA at launch. We’re finalizing the new, detailed ZERA docs now. Thanks for your patience.
Homomorphic Time Evolution
Structure‑Preserving Homomorphisms and Epoch Transitions
Each epoch maintains a state commitment that evolves through structure‑preserving homomorphisms, ensuring group‑law compatibility while preserving privacy.
3.1 Structure‑Preserving Homomorphism
Each epoch T_i maintains a state commitment C_i. A structure‑preserving homomorphism updates commitments while retaining group‑law compatibility:
φ_H: S_i → S_(i+1), S_(i+1) = φ_H(S_i)
The homomorphism preserves the group law:
φ_H(a · b) = φ_H(a) · φ_H(b), ∀ a, b ∈ S_i
3.2 Causal Ordering
Mapping epochs to slices of Minkowski space provides a causal ordering that excludes superluminal double‑spends. This ensures that:
- Events are ordered by their temporal relationship
- No transaction can be double‑spent across causally separated epochs
- The system maintains consistency under relativistic constraints
Mathematical Properties
Group Law Preservation
The homomorphism φ_H ensures that (a · b) maps to φ_H(a) · φ_H(b), maintaining the algebraic structure across epochs.
Causal Consistency
Minkowski space mapping ensures that events respect relativistic causality, preventing temporal paradoxes in the transaction ordering.
State Evolution
Each epoch transition S_i → S_(i+1) preserves the entropy and security properties of the initial state.
Implementation Considerations
- Efficient computation of φ_H for large group orders
- Verification of homomorphic properties in zero‑knowledge proofs
- Synchronization of epoch transitions across distributed nodes
- Handling of concurrent transactions within the same epoch