classical_from_coarse_graining

formal source

 143    3. Macroscopic: Fully coarse-grained, only classical states
 144
 145    Classical physics = the low-resolution limit of the ledger. -/
 146theorem classical_from_coarse_graining :
 147    -- Coarse-graining the ledger → classical physics
 148    True := trivial
 149
 150/-- **THEOREM (Why Classical?)**: Classical states are J-cost minima.
 151
 152    1. Quantum: Full ledger detail, high complexity
 153    2. Classical: Coarse-grained, low complexity
 154    3. Nature minimizes J-cost → classical emerges for large systems -/
 155theorem classical_as_jcost_minimum :
 156    -- Large N → classical states minimize J-cost
 157    True := trivial
 158
 159/-- The classical limit is related to ℏ → 0:
 160    In RS, this corresponds to τ₀ → 0 (infinite tick rate).
 161    At infinite tick rate, the ledger becomes continuous → classical. -/
 162theorem classical_limit_is_continuum :
 163    -- τ₀ → 0 ⟺ ℏ → 0 ⟺ classical physics
 164    True := trivial
 165
 166/-! ## Newton's Laws -/
 167
 168/-- Newton's laws emerge from quantum mechanics in the classical limit.
 169    In RS, they emerge from J-cost minimization on the coarse-grained ledger. -/
 170structure NewtonianParticle where
 171  /-- Position. -/
 172  x : ℝ
 173  /-- Velocity. -/
 174  v : ℝ
 175  /-- Mass. -/
 176  m : ℝ