Chemistry Crystal Structure Stability Tradeoff

Stability tradeoff

In crystallography, a metal's crystal structure is a tradeoff. Body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed (HCP) lattices each offer different coordination numbers and packing efficiencies. BCC has coordination 8 and a packing efficiency near 0.68; FCC and HCP both reach coordination 12 and the maximum packing efficiency of π/(3√2), about 0.74. Dense packing lowers cohesive energy, which is why FCC and HCP usually win on that axis. But a structure can also win on another axis: how well its coordination number matches a preferred internal period.

In Recognition Science, the framework models that second axis as eight-tick coherence, a measure of how well a structure's coordination number aligns with the framework's eight-tick recognition cycle. BCC's coordination of 8 is a perfect match, scored 1.0; FCC and HCP, with coordination 12, score 2/3. The framework's library defines a stability score as a weighted sum: packing efficiency times a packing weight, plus eight-tick coherence times a coherence weight. The theorem stability_tradeoff proves that with weights 0.3 for packing and 0.7 for coherence, BCC outscores FCC; with weights 0.9 and 0.1, FCC outscores BCC. The result is a formal statement that no single structure dominates on both criteria.

The framework's library also proves supporting facts. BCC has strictly lower packing efficiency than FCC, and FCC and HCP have equal packing efficiency. BCC's eight-tick coherence strictly exceeds both close-packed structures. The HCP ideal c/a ratio is defined as √(8/3), which the library proves lies between 1.63 and 1.64 and within 0.03 of the golden ratio φ. These are machine-checked theorems about the framework's definitions, not empirical measurements of real metals.

The tradeoff theorem does not claim that real metals follow these scores. It proves an inequality about the framework's own stabilityScore definition with two chosen weight pairs. The framework's lists of which elements prefer which structure, such as alkali metals preferring BCC, are definitions, not derived theorems. The framework models crystal stability as this two-factor tradeoff; it does not prove that this model matches measured cohesive energies or observed crystal structures.

/-- BCC is favored when 8-tick coherence dominates.
    The coordination number 8 directly reflects ledger periodicity. -/
def eightTickCoherence : Structure → ℝ
| .BCC => 1.0      -- Perfect 8-tick match
| .FCC => 2/3      -- 8/12 = 2/3 match
| .HCP => 2/3      -- Same as FCC

/-- With high coherence weight, BCC wins; with very high packing weight, FCC wins.
    For FCC to beat BCC: 0.74p + 0.667c > 0.68p + 1.0c → 0.06p > 0.333c → p/c > 5.5
    So we need packing weight over 5× the coherence weight. -/
theorem stability_tradeoff :
    stabilityScore .BCC 0.3 0.7 > stabilityScore .FCC 0.3 0.7 ∧
    stabilityScore .FCC 0.9 0.1 > stabilityScore .BCC 0.9 0.1 := by
  simp only [stabilityScore, packingEfficiencyApprox, eightTickCoherence]
  -- BCC (0.3, 0.7): 0.3 * 0.68 + 0.7 * 1.0 = 0.204 + 0.7 = 0.904
  -- FCC (0.3, 0.7): 0.3 * 0.74 + 0.7 * (2/3) ≈ 0.222 + 0.467 = 0.689
  -- So BCC > FCC with high coherence weight ✓
  -- BCC (0.9, 0.1): 0.9 * 0.68 + 0.1 * 1.0 = 0.612 + 0.1 = 0.712
  -- FCC (0.9, 0.1): 0.9 * 0.74 + 0.1 * (2/3) = 0.666 + 0.067 ≈ 0.733
  -- So FCC > BCC with very high packing weight ✓
  constructor <;> norm_num

/-- BCC has lower packing than FCC/HCP. -/
theorem bcc_packing_lt_fcc : packingEfficiencyApprox .BCC < packingEfficiencyApprox .FCC := by
  simp only [packingEfficiencyApprox]
  norm_num

/-- FCC and HCP have same packing. -/
theorem fcc_hcp_same_packing : packingEfficiencyApprox .FCC = packingEfficiencyApprox .HCP := rfl

/-- The ideal HCP c/a ratio is approximately 1.633. -/
theorem ideal_hcp_ratio_value : 1.63 < idealHCPRatio ∧ idealHCPRatio < 1.64 := by
  -- √(8/3) ≈ 1.6329931..., so 1.63 < √(8/3) < 1.64
  -- We verify: 1.63² = 2.6569 < 8/3 ≈ 2.6667 < 2.6896 = 1.64²
  simp only [idealHCPRatio]
  have h_sq_lo : (1.63 : ℝ)^2 < 8/3 := by norm_num
  have h_sq_hi : (8 : ℝ)/3 < (1.64 : ℝ)^2 := by norm_num
  constructor
  · -- 1.63 < √(8/3) ⟺ 1.63² < 8/3 (for positive values)
    rw [Real.lt_sqrt (by norm_num : (0 : ℝ) ≤ 1.63)]
    exact h_sq_lo
  · -- √(8/3) < 1.64 ⟺ 8/3 < 1.64² (for positive values)
    rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1.64)]
    exact h_sq_hi

/-- The ideal HCP ratio is close to φ ≈ 1.618.
    √(8/3) ≈ 1.633, φ ≈ 1.618, difference ≈ 0.015.
    Using available bounds: 1.63 < √(8/3) < 1.64, 1.61 < φ < 1.62.
    This gives |√(8/3) - φ| < 1.64 - 1.61 = 0.03. -/
theorem hcp_ratio_near_phi : |idealHCPRatio - phi| < 0.03 := by
  simp only [idealHCPRatio]
  -- First establish that √(8/3) > φ, so |√(8/3) - φ| = √(8/3) - φ
  have h_phi_lt : phi < 1.62 := phi_lt_onePointSixTwo
  have h_163_lt_sqrt : (1.63 : ℝ) < Real.sqrt (8/3) := by
    rw [Real.lt_sqrt (by norm_num : (0 : ℝ) ≤ 1.63)]
    norm_num
  have h_sqrt_gt_phi : Real.sqrt (8/3) > phi := by linarith
  rw [abs_of_pos (by linarith : Real.sqrt (8/3) - phi > 0)]
  -- Now show √(8/3) - φ < 0.03
  -- √(8/3) < 1.64 and φ > 1.61, so √(8/3) - φ < 1.64 - 1.61 = 0.03
  have h_sqrt_lt : Real.sqrt (8/3) < 1.64 := by
    rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1.64)]
    norm_num
  have h_phi_gt : phi > 1.61 := phi_gt_onePointSixOne
  linarith

What this page does not claim

The stability_tradeoff theorem does not claim that real metals follow these scores. The framework's lists of which elements prefer which structure are definitions, not derived theorems. The framework does not prove that its two-factor model matches measured cohesive energies or observed crystal structures.

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