Chemistry Crystal Structure Bcc Max 8tick Coherence

The 8-tick coherence theorem

In crystallography, a body-centered cubic (BCC) structure has one atom at each corner of a cube and one at the center. Each atom touches eight nearest neighbors, giving it a coordination number of 8. Face-centered cubic (FCC) and hexagonal close-packed (HCP) structures each have coordination number 12, and both pack atoms more densely: about 74 percent of space filled, against about 68 percent for BCC.

Within the Recognition Science framework, the ledger, a discrete record of recognition events, runs in an eight-tick cycle. The framework's machine-checked library of formal theorems proves that BCC's coordination number of 8 exactly matches this cycle, giving BCC a perfect coherence score of 1.0. FCC and HCP, with coordination 12, score only two-thirds. The theorem bcc_max_8tick_coherence states this directly: BCC's coherence exceeds both FCC's and HCP's.

The framework's library also proves that BCC packs less efficiently than FCC, and that FCC and HCP have identical packing. A separate tradeoff theorem shows that when coherence is weighted heavily, BCC scores higher in stability, but when packing efficiency dominates, FCC wins. This matches the classical picture: alkali metals like lithium and sodium prefer BCC, while copper and gold prefer FCC.

What the theorem does not do is predict which elements take which structure. The lists of elements that prefer BCC, FCC, or HCP are definitions, not derivations. The framework models the tradeoff between packing and coherence, but it does not prove that iron, chromium, or tungsten must be BCC. It also does not claim that BCC is generally more stable than FCC or HCP; the energy ordering runs the other way for close-packed structures.

/-- BCC coordination equals 8-tick. -/
theorem bcc_is_8_tick : coordination .BCC = 8 := rfl

/-- 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

/-- BCC has maximum 8-tick coherence. -/
theorem bcc_max_8tick_coherence :
    eightTickCoherence .BCC > eightTickCoherence .FCC ∧
    eightTickCoherence .BCC > eightTickCoherence .HCP := by
  simp only [eightTickCoherence]
  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

/-- 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

What this page does not claim

The theorem does not predict which elements form BCC, FCC, or HCP structures. The theorem does not claim BCC is more stable than FCC or HCP overall. The theorem does not derive the energy scale or packing efficiency from first principles.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/CrystalStructure.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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