Geometry Affine Indep Interior Adjacent Face Normals Independent Iff Cross Ne Ze

The normal test

A tetrahedron has four triangular faces. Any two faces that share an edge each have a normal vector, the arrow perpendicular to the face. The theorem adjacentFaceNormalsIndependent_iff_cross_ne_zero states that these two normals are linearly independent, meaning neither is a scalar multiple of the other, exactly when their cross product is not the zero vector. This is not a new geometric fact but a formal restatement of a standard linear algebra equivalence: two vectors in three-dimensional space are independent if and only if their cross product is nonzero.

The statement lives in the framework's machine-checked library of formal theorems, specifically in the file AffineIndepInterior.lean. The library proves this equivalence as a theorem, meaning it is a derived result, not an assumption. The proof is short: it unfolds the definition of AdjacentFaceNormalsIndependent and applies the existing lemma crossProduct_ne_zero_iff_linearIndependent. The declaration is named adjacentFaceNormalsIndependent_iff_cross_ne_zero and is anchored at lines 130-135 of that file.

This equivalence is a tool, not a conclusion about physics. It does not claim that any particular tetrahedron has independent face normals, nor that any physical tetrahedron exists. It only says that if you have a tetrahedron and you want to know whether two of its adjacent face normals are independent, you can check whether their cross product is zero. The theorem is a bridge between two ways of stating the same condition, useful for later steps in the framework's Regge closure program, but it proves no geometric fact about any specific tetrahedron.

What the theorem does not claim is equally precise. It does not assert that the cross product of the two face normals is nonzero for every tetrahedron. A degenerate tetrahedron, one where four points lie in a plane, can have dependent normals and a zero cross product. The theorem also does not say anything about the angle between the faces; independence means the normals are not parallel, but the angle could be anything from 0 to 180 degrees, excluding the endpoints. The theorem is a pure statement about linear algebra, not about the geometry of any particular shape.

/-- Adjacent face-normal independence is equivalent to the cross product of
the two adjacent face normals being nonzero. -/
theorem adjacentFaceNormalsIndependent_iff_cross_ne_zero
    (T : RealizedTet) (e : Fin 6) :
    AdjacentFaceNormalsIndependent T e ↔
      (adjacentFaceNormals T e).1 ⨯₃ (adjacentFaceNormals T e).2 ≠ 0 := by
  unfold AdjacentFaceNormalsIndependent
  exact (crossProduct_ne_zero_iff_linearIndependent).symm

/-- Adjacent face-normal independence is equivalent to the cross product of
the two adjacent face normals being nonzero. -/
theorem adjacentFaceNormalsIndependent_iff_cross_ne_zero
    (T : RealizedTet) (e : Fin 6) :
    AdjacentFaceNormalsIndependent T e ↔
      (adjacentFaceNormals T e).1 ⨯₃ (adjacentFaceNormals T e).2 ≠ 0 := by
  unfold AdjacentFaceNormalsIndependent
  exact (crossProduct_ne_zero_iff_linearIndependent).symm

What this page does not claim

The theorem does not assert that any particular tetrahedron has independent face normals. The theorem does not claim that the cross product of adjacent face normals is always nonzero. The theorem does not say anything about the size of the angle between two faces.

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/Geometry/AffineIndepInterior.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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