Cyclic Measurement and Silent Radix: Ultrametric Foundations for Quantum State Tomography

This paper synthesizes the cyclic measurement formalism with the silent radix concept, establishing ultrametric foundations for quantum state tomography. The silent radix — the property that a positional numeral cannot internally specify its own base — is formalized as a valuation-theoretic primitive. When combined with cyclic measurement protocols, this yields a robust framework for quantum state reconstruction with provable error bounds grounded in p-adic analysis. The paper presents rigorous Lean-formalized proofs of the core theorems, including valuation properties, rollover mechanics, reentry dynamics, and a complete 6-case ultrametric inequality analysis.