Harmonic Field Geometry: Axiomatic Foundations of Coprime Convergence - A Mathematical Framework for Bounded Traversal, Resonance, and Deterministic Agreement

This work establishes the mathematical foundations for geometric 
arbitration and deterministic consensus through harmonic field geometry.

We introduce geometric arbitration, a deterministic Byzantine consensus 
primitive based on rotor traversal over finite cyclic groups ℤₙ with 
projection-based geometric selection. Our core results prove that 
constant coprime strides produce complete traversal in exactly n steps, 
and establish bounded convergence guarantees.

The framework is formalized axiomatically with rigorous proofs of 
safety (identical outcomes), liveness (finite bounded termination), 
and completeness (all valid configurations reachable). We prove 
Byzantine fault tolerance under the DLS partially synchronous model, 
tolerating f < n/3 faults with O(n²) message complexity.

This provides the mathematical substrate for the SevenFold Proof of 
Consensus protocol, whose implementation and operational characteristics 
are described in companion publications on Zenodo.