phi_rung_energy_pos

formal source

  63def phi_rung_energy (n : ℤ) : ℝ := E_coh_storage * phi ^ n
  64
  65/-- **THEOREM EN-004.2**: Energy at each φ-rung is positive. -/
  66theorem phi_rung_energy_pos (n : ℤ) : 0 < phi_rung_energy n := by
  67  unfold phi_rung_energy
  68  exact mul_pos E_coh_storage_pos (zpow_pos phi_pos _)
  69
  70/-- **THEOREM EN-004.3**: Consecutive φ-rung energies differ by exactly φ. -/
  71theorem phi_rung_energy_ratio (n : ℤ) :
  72    phi_rung_energy (n + 1) / phi_rung_energy n = phi := by
  73  unfold phi_rung_energy
  74  rw [zpow_add_one₀ phi_ne_zero]
  75  have hpow_pos : 0 < phi ^ n := zpow_pos phi_pos n
  76  field_simp [phi_pos.ne', E_coh_storage_pos.ne', hpow_pos.ne']
  77
  78/-! ## §II. Energy Hierarchy -/
  79
  80/-- Chemical bonding rung index (n = 0, one coherence quantum). -/
  81def chemical_rung : ℤ := 0
  82
  83/-- Nuclear binding rung index (n = 45, matches nuclear/chemical ratio ≈ 10⁹). -/
  84def nuclear_rung : ℤ := 45
  85
  86/-- Chemical energy scale = E_coh · φ^0 = E_coh. -/
  87def E_chemical : ℝ := phi_rung_energy chemical_rung
  88
  89/-- Nuclear energy scale = E_coh · φ^45. -/
  90def E_nuclear : ℝ := phi_rung_energy nuclear_rung
  91
  92/-- **THEOREM EN-004.4**: Nuclear energy exceeds chemical energy. -/
  93theorem nuclear_exceeds_chemical : E_chemical < E_nuclear := by
  94  unfold E_chemical E_nuclear phi_rung_energy chemical_rung nuclear_rung
  95  simp only [zpow_zero, mul_one]
  96  apply lt_mul_of_one_lt_right E_coh_storage_pos