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

Homomorphic Property

The homomorphism φ_H preserves the group operation, ensuring that commitments can be combined and evolved while maintaining their mathematical structure.

Epoch Transitions

Each epoch T_i represents a discrete time step in the system, with state transitions governed by the homomorphic evolution function.

Mathematical Properties

Group Law Preservation

The homomorphism φ_H ensures that (a · b) → 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

Next: Commitment Layer →

← Algebraic Entropy Foundation

Zero-Knowledge Proofs

Thank You

Special thanks to the community members and selfless volunteers who contributed reviews, feedback, and technical insights to make this documentation possible.

Błażej and Jai Santos

•Cryptographic Reviewers•Protocol Contributors