phi_rung_jcost_energy

formal source

 144  exact Jcost_nonneg hx
 145
 146/-- **THEOREM EN-004.10**: Energy storage at φ-rung x = φ^n is definitionally E_coh × J(φ^n). -/
 147theorem phi_rung_jcost_energy (n : ℤ) :
 148    jcost_energy (phi ^ n) (zpow_pos phi_pos n) =
 149    E_coh_storage * Jcost (phi ^ n) := by
 150  rfl
 151
 152/-- **THEOREM EN-004.11**: Ground state minimizes energy for any positive x. -/
 153theorem phi_coherent_minimizes_jcost_per_energy :
 154    ∀ x : ℝ, ∀ hx : 0 < x,
 155    jcost_energy 1 one_pos ≤ jcost_energy x hx := by
 156  intro x hx
 157  exact jcost_energy_min_at_ground x hx
 158
 159/-! ## §IV. Storage Density Hierarchy -/
 160
 161/-- Energy density ratio between two rungs: φ^(n - m). -/
 162def storage_density_ratio (n m : ℤ) : ℝ := phi ^ (n - m)
 163
 164/-- **THEOREM EN-004.12**: Storage density ratios are positive. -/
 165theorem storage_density_ratio_pos (n m : ℤ) : 0 < storage_density_ratio n m := by
 166  unfold storage_density_ratio
 167  exact zpow_pos phi_pos _
 168
 169/-- **THEOREM EN-004.13**: Higher rung → higher storage density. -/
 170theorem higher_rung_denser (n m : ℤ) (h : m < n) :
 171    1 < storage_density_ratio n m := by
 172  unfold storage_density_ratio
 173  exact one_lt_zpow₀ one_lt_phi (by omega)
 174
 175/-- **THEOREM EN-004.14**: The three-level energy hierarchy.
 176    Mechanical < Chemical < Nuclear (in RS rungs). -/
 177theorem energy_storage_density_hierarchy :