The Renewal Trap and the Irreversibility of Decoherence: A Self-Contained Information-Theoretic Proof Without Thermodynamic Assumptions

We prove that decoherence is practically irreversible through a purely information-theoretic mechanism that invokes neither the second law of thermodynamics nor Hilbert space dimensionality arguments. In branching decoherence dynamics, the frame variable (the relative phase between pointer sectors) is stored as a ramp secret in an access structure induced by redundant environment records. A bounded-memory reversal agent faces a scouting–recovery complementarity: measuring fragments to identify records induces back-action that degrades subsequent frame recovery. This tradeoff yields a meta-Nyquist gap governing the agent's net coherent-information gain rate. When pointer redundancy exceeds a critical threshold, the gap turns negative and the agent enters a renewal trap — an absorbing regime in which coherent qubits are lost faster than useful fragments can be incorporated. The main result (the Informational Irreversibility Theorem) shows that frame recovery probability is exponentially close to chance for any adaptive LOCC protocol. Five extensions establish universality: channel independence via Lindbladian spectral gaps, robustness under partial quantum error correction, a passive–active duality between the meta-Nyquist threshold and the fault-tolerance threshold, an optimal redundancy analysis revealing that quantum Darwinism maximizes the informational cost of reversing classicality, and a thermodynamic separation via an explicit pure-state model with zero entropy. Additional results include a categorical duality functor, tight spectral bounds for physical noise channels, multi-party verifier networks, and an adversarial extension connecting to Byzantine fault tolerance.