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Lazy Rendering of Quantum States: A Testable Framework from Spectral-Geometric Decoherence Theory

We present the Lazy Rendering framework, a novel synthesis of decoherence theory, quantum Darwinism, and information-theoretic duality that resolves conceptual paradoxes such as the delayed-choice quantum eraser. By introducing an information-theoretic perspective inspired by computer science metaphors (backend/frontend, lazy evaluation, commits), we provide both conceptual clarity and rigorous mathematical foundations. Key Contributions: Spectral Theory of Commit Threshold (§3.1): We derive the threshold θ for classical emergence as the normalized spectral gap of the Lindblad superoperator: θ_spec = |λ₂|/|λ_max|. This transforms θ from a phenomenological parameter to a calculable quantity. Wave-Particle Duality Bound (§3.2): We rigorously prove that V² + D² ≤ 1 from density matrix positivity (Theorem 1). We explore the connection to information geometry and conjecture that equality corresponds to optimal (geodesic) decoherence in the Bures metric. Quantum Darwinism Formalism (§4): We connect redundancy R_δ and mutual information I(S:E_k) to the commit threshold via accessibility A_S, providing operational definitions for "classical objectivity." Quantum Eraser Resolution (§5): We show how delayed-choice quantum eraser experiments operate in the "lazy rendering regime" (A_S < θ), where which-way information is stored coherently but not redundantly published, allowing reversible erasure. Testable Predictions (§6): We generate specific predictions for NV centers, transmon qubits, and other systems, with full code for numerical verification. The spectral theory (§3.1) and quantum Darwinism formalism (§4) are rigorously proven. The geometric interpretation (§3.2) is presented as a conjecture for future work. All claims are validated numerically (code provided).

By Michael Fellinger