Chemistry Metallic Bond Lattice Type

The lattice catalog

In crystallography, a metal's lattice type describes how its atoms arrange themselves in a repeating three-dimensional pattern. The three common metallic lattices are body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed (HCP). Each has a coordination number, the count of nearest neighbors touching any given atom: BCC atoms touch 8 neighbors, while FCC and HCP atoms each touch 12. This neighbor count determines how tightly the atoms pack together. BCC packs atoms at 68 percent efficiency, while FCC and HCP both reach 74 percent, the theoretical maximum for equal spheres. The German physicist August Bravais classified these repeating patterns in the mid-19th century, and the close-packed structures appear in metals like copper, aluminum, and zinc.

The Recognition Science framework's machine-checked library of formal theorems, a collection of statements verified by computer, defines a LatticeType with exactly these three cases: BCC, FCC, and HCP. The definition is a modeling choice, not a discovery. The framework then proves three facts about this model. First, the coordination number of BCC is exactly 8, matching the framework's eight-tick recognition cycle, a repeating rhythm of eight steps that the framework derives from its cost function. Second, FCC and HCP both have coordination number 12. Third, FCC and HCP are denser than BCC, meaning their packing efficiency is higher. These proofs are simple arithmetic checks on the chosen definitions, not empirical measurements or physical derivations.

The framework also defines a freeElectrons function that assigns each element a count of valence electrons contributing to the electron sea, the delocalized pool of electrons that characterizes metallic bonding. Alkali metals get 1, alkaline earth metals get 2, and transition metals get a simplified average of 2. A conductivityProxy simply returns this free electron count, so it is a rough stand-in for electrical conductivity, not a physical calculation. The framework proves that alkali metals have one valence electron and that transition metals have higher cohesive energy than alkali metals, using a proxy that assigns transition metals the golden ratio phi, a constant the framework derives elsewhere, as their cohesive energy value.

What the declaration does not claim is as important as what it proves. LatticeType does not prove that real metals actually form these lattices; that is an empirical fact established by X-ray diffraction experiments. The framework's coordination numbers and packing efficiencies are definitions chosen to match known crystallography, not theorems derived from first principles. The eight-tick connection to BCC is an observation about the number 8, not a proof that the recognition cycle causes BCC structure. The free electron count for transition metals is an admitted simplification, averaging 1 to 3 electrons to a flat 2, so it cannot distinguish iron from copper. The conductivityProxy is not a theory of electrical conduction; it is a monotone ranking that ignores temperature, impurities, and band structure.

What the reader gains is a clear boundary. The framework has a precise, machine-checked vocabulary for talking about metallic lattices, and it proves the arithmetic relationships among its chosen definitions. What remains open is the physical bridge: whether the framework's recognition cost, the forced expense of retaining a discrete record of events, actually explains why metals delocalize electrons or why specific elements choose specific lattices. That connection is not made in this declaration.

/-- Coordination number in metallic lattices.
    BCC: 8, FCC/HCP: 12 -/
inductive LatticeType
| BCC  -- Body-centered cubic (coordination 8)
| FCC  -- Face-centered cubic (coordination 12)
| HCP  -- Hexagonal close-packed (coordination 12)

/-- BCC has coordination number 8 (matching 8-tick). -/
theorem bcc_8tick : coordinationNumber .BCC = 8 := rfl

/-- Close-packed structures (FCC, HCP) have coordination 12. -/
theorem close_packed_12 : coordinationNumber .FCC = 12 ∧ coordinationNumber .HCP = 12 := by
  constructor <;> rfl

/-- FCC/HCP have higher packing efficiency than BCC. -/
theorem fcc_hcp_denser_than_bcc :
    packingEfficiency .BCC < packingEfficiency .FCC ∧
    packingEfficiency .BCC < packingEfficiency .HCP := by
  constructor <;> { simp only [packingEfficiency]; norm_num }

/-- Metals have low ionization energy (alkali metals are easiest to ionize). -/
theorem alkali_low_ionization (Z : ℕ) (h : Z ∈ alkaliMetalZ) :
    valenceElectrons Z = 1 := by
  simp only [alkaliMetalZ] at h
  -- alkaliMetalZ = [3, 11, 19, 37, 55, 87]
  -- Each alkali metal: Z - prevClosure Z = 1
  fin_cases h <;> native_decide

What this page does not claim

The framework does not prove that real metals form these lattices; that is empirical. The free electron count for transition metals is a simplified average, not a per-element derivation. The conductivityProxy is a ranking, not a physical theory of electrical conduction. The eight-tick connection to BCC is an observation, not a causal proof.

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/MetallicBond.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

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