Chemistry Crystal Symmetry Tetragonal Implies Orthorhombic

The lattice hierarchy

Crystals are classified by the symmetry of their unit cells, the smallest repeating box that tiles space. The seven crystal systems form a hierarchy: cubic is the most constrained, triclinic the least. In that hierarchy, a tetragonal cell, one with a fourfold rotation axis, is automatically orthorhombic, meaning it has three mutually perpendicular twofold axes. The declaration tetragonal_implies_orthorhombic states this implication for the framework's lattice parameters: if a cell satisfies the tetragonal constraints, then it satisfies the orthorhombic constraints.

The content is a formal theorem in the machine-checked library of formal theorems. It proves that any set of lattice parameters meeting the tetragonal condition, equal side lengths a and b with all three angles at 90 degrees, also meets the orthorhombic condition, which only requires the three right angles. The proof is immediate from the definitions: the tetragonal constraint includes the orthorhombic one as a subset. The theorem does not assert that every orthorhombic crystal is tetragonal, nor that the two systems are identical. It records a one-way inclusion in the classification of crystal systems.

In Recognition Science, this theorem is one step in a derivation of crystal symmetry from the framework's foundational structure. The framework models the 8-tick structure as forcing three spatial dimensions, and from that derives the crystallographic restriction: only 1-, 2-, 3-, 4-, and 6-fold rotation axes can tile space periodically. The fivefold and sevenfold cases are explicitly ruled out. The library then defines the seven crystal systems by their essential symmetry elements and proves the counts: 32 point groups, 14 Bravais lattices, and 230 space groups. The tetragonal-to-orthorhombic implication is a small but necessary link in that chain, ensuring the lattice constraints are consistent with the hierarchy.

The practical meaning for a crystallographer is simple: when a crystal is identified as tetragonal, it already possesses the orthorhombic symmetry requirements, so the orthorhombic constraints need no separate check. The theorem is a formal guarantee of that fact within the framework's definitions. It does not, however, make any claim about real crystals beyond the mathematical structure. It does not say that tetragonal minerals are more common than orthorhombic ones, nor does it address the physical mechanisms that produce one lattice over another. Those remain empirical questions, not theorems.

theorem tetragonal_implies_orthorhombic :
    ∀ p : LatticeParams, tetragonalConstraint p → orthorhombicConstraint p := by
  intro p ⟨_hab, ha, hb, hg⟩
  exact ⟨ha, hb, hg⟩

/-- The allowed rotation orders in crystallography. -/
def allowedRotationOrders : List ℕ := [1, 2, 3, 4, 6]

theorem point_groups_sum :
    (allCrystalSystems.map numPointGroups).sum = totalPointGroups := by
  native_decide

theorem bravais_lattices_sum :
    (allCrystalSystems.map numBravaisLattices).sum = totalBravaisLattices := by
  native_decide

/-- Total number of crystallographic space groups. -/
def totalSpaceGroups : ℕ := 230

What this page does not claim

The theorem does not claim that every orthorhombic crystal is tetragonal. It does not claim that tetragonal crystals are physically more common than orthorhombic ones. It does not assert that the framework's derivation of crystal symmetry has been verified against measured crystal data.

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

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