Chemistry Crystal Structure Close Packed Coordination

Close-packed coordination

In crystallography, the coordination number of an atom in a crystal structure is the number of its nearest neighbors. For the two close-packed structures, face-centered cubic (FCC) and hexagonal close-packed (HCP), that number is 12: each atom touches twelve others. This is a classical fact, taught in every introductory materials science course, and it is the content of the declaration close_packed_coordination in the framework's machine-checked library of formal theorems.

The declaration is a theorem, proved by direct computation from the framework's definitions. It states that the coordination number of FCC is 12 and the coordination number of HCP is 12. The proof is trivial in the formal sense: the definitions of coordination for these structures are literally the numbers 12, so the theorem holds by reflexivity. The framework's contribution is not the number itself, which has been known for over a century, but that it records the fact as a formal, checkable statement within its system.

In Recognition Science, this coordination number is interpreted as a reflection of the framework's eight-tick recognition cycle. The framework models a discrete record of events, called a ledger, with a forced period of eight ticks. It then proposes that body-centered cubic (BCC), with coordination 8, directly matches this period, while FCC and HCP, with coordination 12, represent a different balance. The framework's model also assigns FCC and HCP the same maximum packing efficiency, about 74 percent, and places their energy scale below BCC's, which it connects to the classical observation that close-packed metals are often softer and more ductile than BCC metals.

The framework's library proves several related formal statements: that BCC has coordination 8, that FCC and HCP have equal packing efficiency, that BCC's packing efficiency is less than FCC's, and that the ideal HCP axial ratio, sqrt(8/3) ≈ 1.633, is within 0.03 of the golden ratio. These are all theorems about the framework's own definitions, not new measurements of real crystals. The framework also defines a stability score that trades off packing efficiency against eight-tick coherence, and proves that different weights favor different structures, but this is a modeling choice, not a derivation from first principles.

What the declaration does not claim is equally important. It does not claim that real FCC or HCP crystals always have coordination exactly 12; real crystals have defects, surfaces, and impurities that change local environments. It does not claim that the framework's model predicts which elements will form which structure; the lists of elements that prefer BCC, FCC, or HCP are definitions, not theorems. And it does not claim that the golden ratio appearance in the HCP ratio is a proof of the framework's core cost function; it is a numerical proximity, stated as a theorem about the framework's own ideal ratio, not as a law of nature.

/-- FCC and HCP have coordination 12. -/
theorem close_packed_coordination : coordination .FCC = 12 ∧ coordination .HCP = 12 := by
  constructor <;> rfl

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

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

/-- 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 declaration does not claim that real crystals always have coordination exactly 12, ignoring defects and surfaces. The declaration does not claim that the framework's model predicts which elements form which crystal structure. The declaration does not claim that the golden ratio proximity in the HCP ratio proves the framework's core cost function.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: