Lieb-Robinson Bounds and Krylov Unforgeability: A Rigorous Framework

We develop a rigorous mathematical framework proving that the QKD Krylov Detector, a three-layer eavesdropper detection system for Quantum Key Distribution based on Krylov complexity and sidereal filtering, provides information-theoretically grounded security against physically realizable attacks. The central results are: (1) Krylov Locality Theorem: An eavesdropper coupling at distance d from the monitored operator cannot affect the first d Lanczos coefficients. This is an exact algebraic identity, not an approximation. (2) Perturbation Detectability Theorem: The first affected Lanczos coefficient scales as gamma², with computable upper and lower bound constants derived from second-order perturbation theory applied to the Lanczos recursion. (3) Information-Distortion Bound: Eve's accessible information (Holevo quantity) scales as O (gamma²), while the Krylov distortion scales as O (gamma⁴), yielding an explicit tradeoff chi <= const * sqrt (DeltaK). Verified numerically using an explicit Eve ancilla model. (4) Krylov Unforgeability Theorem: Any Hamiltonian perturbation that extracts nonzero information from the quantum channel necessarily distorts the Lanczos coefficients beyond the detection threshold. No physically realizable attack can simultaneously extract information and preserve the clean Krylov signature. (5) Coherent Attack Invariance: The Lanczos coefficients are operator-space quantities independent of any quantum state, rendering coherent and entangled attack strategies unable to circumvent detection. (6) Time-Dependent Unforgeability: Time-dependent coupling strategies gamma (t) produce strictly more distortion than constant coupling at the same average strength, by Jensen's inequality applied via a rigorous stroboscopic argument. All results include full proofs and numerical verification for chain lengths N = 6, 8, 10, 12. This work provides the formal security foundation for the QKD Krylov Detector framework developed across six prior publications. Companion code: https: //github. com/quantumspiritresearch-crypto/qkd-krylov-detector