QuantumClassicalCrossover
QuantumClassicalCrossover packages the four parameters that control the continuous quantum-to-classical transition inside the Recognition Science J-cost model. Workers modeling decoherence in many-body systems cite the structure when they need to vary system size N, temperature T, coupling strength, and decoherence time tau_D together. The declaration is a bare record definition whose only content is the four fields plus an inline remark on ledger coarse-graining.
claimA structure consisting of a natural number $N$ (number of particles), a real number $T$ (environment temperature), a real number $g$ (coupling strength), and a real number $tau_D$ (decoherence time).
background
The module QF-011 derives classical emergence from many-body J-cost minimization. Single-particle superpositions carry low J-cost while correlated many-body superpositions carry J-cost quadratic in N; for large N the minimum is therefore attained by product states. The structure collects the external knobs that tune the location of this minimum: system size, temperature, coupling, and the resulting decoherence timescale.
proof idea
The declaration is a direct structure definition with zero proof lines; it simply records the four fields N, T, coupling, and tau_D together with a comment block on RS ledger coarse-graining.
why it matters in Recognition Science
The definition supplies the parameter interface required by the QF-011 derivation that classical behavior arises when product states minimize total J-cost. It therefore sits inside the Recognition Science program that obtains all physics from the single J-functional equation and the phi-ladder. No downstream theorems yet consume the structure.
scope and limits
- Does not derive an explicit formula for tau_D in terms of N, T, and g.
- Does not prove the quadratic scaling of entangled-state J-cost.
- Does not specify numerical ranges for the crossover.
- Does not connect the parameters to the phi-ladder or the eight-tick octave.
formal statement (Lean)
127structure QuantumClassicalCrossover where
128 /-- System size (number of particles). -/
129 N : ℕ
130 /-- Environment temperature. -/
131 T : ℝ
132 /-- Coupling strength. -/
133 coupling : ℝ
134 /-- Decoherence time. -/
135 tau_D : ℝ
136
137/-! ## The RS Interpretation -/
138
139/-- In RS, classical emergence is about **ledger coarse-graining**:
140
141 1. Microscopic: Full quantum ledger, all superpositions tracked
142 2. Mesoscopic: Partial coarse-graining, some quantum effects
143 3. Macroscopic: Fully coarse-grained, only classical states
144
145 Classical physics = the low-resolution limit of the ledger. -/