Theoretical Foundations of Krylov-Based Quantum Channel Authentication: Physical Bridge, One-Way Function Property, and Universality

This paper closes three theoretical gaps that remained open in the Krylov-based quantum channel authentication framework developed across Papers 1-8 of this series.

Physical Bridge (Theorem 1): I derive a formal proof that the QBER autocorrelation is proportional to the operator autocorrelation, C_QBER(τ) = α(N) · C_op(τ), establishing the connection between Krylov theory and observable measurement statistics (numerically confirmed with Pearson r = 0.9997).

Physical One-Way Function (Theorem 2): I prove that the Krylov fingerprint map constitutes a physical one-way function with three independent hardness sources: (I) unconditional information-theoretic exponential ill-conditioning of the inverse moment problem (κ_n ~ 32^n), (II) complexity-theoretic reduction to Hamiltonian learning, and (III) physical irreversibility of scrambling dynamics (t_rec ~ 10^765). Hardness source I is unconditional and provides post-quantum security.

Universality (Theorem 3): I establish the universality of the detection framework through systematic numerical verification across 8 Hamiltonian families and 10 perturbation types, achieving 8/8 families and 10/10 perturbation types at the physically relevant distance d = 1.

All three results are detector-agnostic: they characterize the Krylov framework itself, independent of the specific detection algorithm applied to the Lanczos coefficients.

Paper 9 in the Krylov QKD series. See also: qkd-krylov-detector on GitHub.