The Harmonic Funnel: A Dissipative Dynamical Framework for Quantum State Resolution

We propose a dissipative dynamical model for the selection of a definite outcome from a quantum superposition undergoing decoherence. An accessibility scalar S, defined via the quantum relative entropy to the pointer basis, contracts under accumulated environmental flux Φ. For Markovian pure dephasing channels with exponential coherence decay, we derive the exact equation of motion dS/dΦ = -2S ln S from the Lindblad master equation. In the regime where the initial relative entropy D₀ is large compared to log(I_max/I_i), this reduces to the approximate exponential form S ≈ S₀ exp(-Φ/Λ₀) with Λ₀ = 1/(2D₀). A cost functional C_i = I_i/S, with I_i = -log|c_i|², follows from an information-capacity constraint. Configurations are postulated to be eliminated when C_i exceeds the initial maximum cost C* = I_max/S₀, yielding elimination fluxes Φ_i* = Λ₀ log(I_max/I_i) and an explicit, amplitude-dependent collapse timescale. The Born rule is recovered through two structurally distinct routes: (i) shift invariance and multiplicativity applied to selection probabilities, and (ii) first-passage time competition in a high-noise limit, with barriers fixed by Gleason's theorem (for Hilbert space dimension ≥ 3; the qubit case follows independently from invariance). A gravitational extension is proposed as a working hypothesis connecting the model to a companion paper via the lapse function and predicting a shortening of collapse timescales by the factor N(x) = √(1-2GM/rc²) if factorisation holds. Key predictions include amplitude-dependent collapse times, sequential elimination in qutrits observable via quantum state tomography, and possible gravitational modifications in strong-field regimes.