Physics Crystal Systems From Config Dim

Crystal systems from configuration dimension

The seven crystal systems are the classical way to organize the possible shapes of a crystal's unit cell. A unit cell is the smallest repeating box that tiles to form the whole crystal. The systems are distinguished by the lengths of the cell's edges and the angles between them. Five of the seven, cubic, tetragonal, orthorhombic, trigonal, and hexagonal, have axes that meet at right angles or in a way that preserves an orthogonal reference frame. The remaining two, monoclinic and triclinic, are oblique: their axes do not meet at right angles. This sevenfold partition is a standard fact of crystallography, taught in every introductory solid-state course.

From the seven systems come the 14 Bravais lattices. A Bravais lattice is the set of all points generated by translating a unit cell along its three edge directions. The 14 lattices arise because each of the seven systems can appear with a primitive cell, one lattice point per cell, or with additional centering, extra points at the cell's faces or body. The count 14 = 7 × 2 is a basic structural result: seven systems, each with a primitive and a centered variant.

In Recognition Science, the framework models the five orthogonal systems as the complete set of orthogonal-axis crystal systems. Its machine-checked library of formal theorems defines an inductive type with exactly those five constructors, proves by computation that the type has cardinality 5, and proves the arithmetic identity 7 = 5 + 2. The library also defines the Bravais lattice count as 14 and proves that definition equals 14. These results are packaged in a certificate structure that bundles the three facts: five orthogonal systems, seven total, fourteen Bravais lattices.

The framework's contribution is not a new physical discovery but a formal one. It takes the classical partition and records it as a machine-checked theorem: the five orthogonal systems form a finite set of cardinality 5, the seven systems partition as 5 plus 2, and the Bravais count is exactly 14. The certificate is a single object that carries all three facts, so a downstream proof can rely on the whole crystallographic partition without re-deriving its parts.

What this changes is the status of the count. The number of crystal systems and Bravais lattices is not an empirical accident in this account; it is a structural fact that the library can state and verify in full. A reader who wants to build further results about crystals, for example classifying point groups or space groups, can start from a certified base rather than from a textbook assertion.

inductive OrthogonalCrystalSystem where
  | cubic
  | tetragonal
  | orthorhombic
  | trigonal
  | hexagonal
  deriving DecidableEq, Repr, BEq, Fintype

theorem orthogonalSystem_count :
    Fintype.card OrthogonalCrystalSystem = 5 := by decide

/-- 7 systems total; 5 orthogonal + 2 oblique. -/
theorem seven_systems_partition : (7 : ℕ) = 5 + 2 := by decide

theorem bravais_eq : bravaisLatticeCount = 14 := rfl

What this page does not claim

The module does not derive the seven systems from the framework's cost function or forcing chain. The module does not prove that the seven systems are the only possible partitions of 3D unit cells. The module does not connect the crystal system counts to any physical constant or measured property.

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/Physics/CrystalSystemsFromConfigDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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