 121/-! ## Structure Stability Energetics -/
 122
 123/-- Energy scale (dimensionless) inversely related to packing efficiency. -/
 124def energyScale : Structure → ℝ
 125| .BCC => 1.0
 126| .FCC => 0.917  -- ~68/74 ratio
 127| .HCP => 0.917
 128
 129/-- Close-packed structures have lower energy scale. -/
 130theorem close_packed_lower_energy :
 131    energyScale .FCC < energyScale .BCC ∧
 132    energyScale .HCP < energyScale .BCC := by
 133  simp only [energyScale]
 134  constructor <;> norm_num
 135
 136/-! ## 8-Tick Stability Explanation -/
 137
 138/-- BCC is favored when 8-tick coherence dominates.
 139    The coordination number 8 directly reflects ledger periodicity. -/
 140def eightTickCoherence : Structure → ℝ
 141| .BCC => 1.0      -- Perfect 8-tick match
 142| .FCC => 2/3      -- 8/12 = 2/3 match
 143| .HCP => 2/3      -- Same as FCC
 144
 145/-- BCC has maximum 8-tick coherence. -/
 146theorem bcc_max_8tick_coherence :
 147    eightTickCoherence .BCC > eightTickCoherence .FCC ∧
 148    eightTickCoherence .BCC > eightTickCoherence .HCP := by
 149  simp only [eightTickCoherence]
 150  constructor <;> norm_num
 151
 152/-- Stability is a balance of packing and 8-tick coherence. -/
 153def stabilityScore (s : Structure) (packing_weight coherence_weight : ℝ) : ℝ :=
 154  packing_weight * packingEfficiencyApprox s + coherence_weight * eightTickCoherence s

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.