Chemistry Crystal Structure Close Packed Lower Energy

Close-packed stability

Crystals of the same metal can often form in more than one structure. The three most common are body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed (HCP). The classical fact at stake is simple: FCC and HCP pack atoms more tightly than BCC does. Each FCC or HCP atom touches 12 neighbors, while a BCC atom touches only 8. The packing efficiency, the fraction of space actually filled by spheres, is about 0.74 for FCC and HCP and about 0.68 for BCC. Tighter packing usually means lower energy, because more neighbors means more bonding contacts per atom, so the close-packed structures tend to be favored when bonding is not strongly directional.

That classical ordering is what the declaration close_packed_lower_energy pins down inside the Recognition Science (RS) framework. The framework's machine-checked library of formal theorems defines a numerical energy scale for each structure: BCC gets 1.0, and both FCC and HCP get 0.917, a value chosen as roughly the ratio of the packing efficiencies, 68 over 74. The declaration then proves, by direct computation, that the energy scale for FCC is less than the scale for BCC, and the same holds for HCP versus BCC. In plain language, the declaration establishes that within this model, the close-packed structures sit at a lower energy than BCC does. The proof is a ledger, a discrete record of formal steps, checked by the machine, with no gaps left to human inspection.

The declaration does not claim that FCC or HCP is always the stable structure for a real metal. It works on a deliberately simplified energy scale that only reflects packing. Real stability also depends on the electronic structure, temperature, and pressure, which this scale ignores. The framework itself acknowledges the tradeoff elsewhere: a separate declaration shows that when an eight-tick coherence term, a measure of how well a structure matches the framework's recognition cycle, is weighted heavily, BCC scores higher than FCC despite its looser packing. So the declaration is a precise statement about one term in a model, not a universal law of metallurgy.

What the declaration does give a reader is a clean, checkable anchor for a familiar intuition. If you want to know whether the framework's account of crystal structure even respects the classical fact that close packing lowers energy, this declaration answers yes, and it does so with a proof the machine verified rather than a hand wave. That makes it a useful reference point for anyone comparing the framework's predictions against textbook chemistry, even though the model's simplifications mean the comparison stops at the level of relative packing, not absolute stability.

/-- Close-packed structures have lower energy scale. -/
theorem close_packed_lower_energy :
    energyScale .FCC < energyScale .BCC ∧
    energyScale .HCP < energyScale .BCC := by
  simp only [energyScale]
  constructor <;> norm_num

/-- Energy scale (dimensionless) inversely related to packing efficiency. -/
def energyScale : Structure → ℝ
| .BCC => 1.0
| .FCC => 0.917  -- ~68/74 ratio
| .HCP => 0.917

/-- Energy scale (dimensionless) inversely related to packing efficiency. -/
def energyScale : Structure → ℝ
| .BCC => 1.0
| .FCC => 0.917  -- ~68/74 ratio
| .HCP => 0.917

What this page does not claim

The declaration does not claim that FCC or HCP is always the stable structure for a real metal. The declaration does not account for electronic structure, temperature, or pressure in its energy comparison. The framework does not derive the energy scale values from first principles; they are a definitional choice.

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: