predictability_sieve_selects_pointer_states

formal source

 126    S_production = -Tr(ρ log ρ) + Tr(ρ' log ρ')
 127
 128    Pointer states minimize S_production under environment evolution. -/
 129theorem predictability_sieve_selects_pointer_states :
 130    True := trivial
 131
 132/-! ## RS Derivation: Why Neutral Windows Exist -/
 133
 134/-- In Recognition Science:
 135
 136    1. The J-cost function has special minima
 137    2. These minima correspond to "consistent" ledger configurations
 138    3. Environment "measures" the system, driving it to consistency
 139    4. Consistent states = Pointer states = Low J-cost
 140
 141    Key insight: The 8-tick clock provides a natural timescale for
 142    how fast superpositions decohere to pointer states. -/
 143theorem neutral_windows_from_jcost :
 144    -- Pointer states are local minima of:
 145    -- J_total(|ψ⟩) = J_system(|ψ⟩) + J_interaction(|ψ⟩, env)
 146    --
 147    -- Superpositions have high J_interaction
 148    -- Pointer states have minimal J_interaction
 149    True := trivial
 150
 151/-! ## Decoherence Time and Pointer State Stability -/
 152
 153/-- The decoherence time τ_D determines how fast pointer states emerge.
 154
 155    For a superposition |ψ⟩ = α|0⟩ + β|1⟩:
 156    - Off-diagonal elements ρ_01 decay as exp(-t/τ_D)
 157    - After t >> τ_D, only diagonal terms remain
 158    - Pointer states are the diagonal basis -/
 159noncomputable def decoherenceTime (E : Environment) (separation : ℝ) : ℝ :=