Recursive Ultrametric Structures for Quantum-Inspired Cryptographic Systems

Quantum cryptographic systems demonstrate that strong security properties can arise from the structure of state space rather than from computational hardness alone. However, most existing quantum cryptographic protocols rely on physical quantum states and specialized hardware, limiting practical deployment. This paper explores a purely mathematical alternative based on recursive ultrametric structure derived from the Recursive Division Tree (RDT) algorithm and the associated recursive-adic valuation. In this framework, information is represented by discrete recursive depth levels, inducing a non-Archimedean metric space with discrete (quantized) states and ultrametric separation. Through systematic empirical analysis of integers in the range $[2, 10^5]$, we demonstrate that recursive depth exhibits discrete saturation behaviour, partitioning integers into stable depth shells with quantized population distributions. We show that recursive depth is statistically independent of Collatz stopping time (partial correlation $r = 0.002$ after controlling for $\log n$), establishing it as a structurally distinct complexity measure. We show that recursive depth induces an ultrametric geometry satisfying the strong triangle inequality, and formalize this structure through embedding into a Hahn series field. The resulting framework supports depth-based entropy generation and cryptographic mixing primitives, supported by exploratory validation using the RDT-PRNG pseudorandom number generator, which demonstrates strong statistical randomness properties in standard test suites including no persistent FAILED classifications across repeated Dieharder runs and Quality 4.0 rating on SmokeRand. These evaluations assess structural entropy and mixing behavior rather than cryptographic security. This work positions recursive ultrametric cryptography as a quantum-inspired but software-native approach that complements post-quantum cryptographic methods.

This work presents an empirical structural analysis of recursive ultrametric state spaces and does not propose a complete cryptographic protocol or claim formal security guarantees.