Geometry Dihedral Cofactor Formula Cm Cofactor3 Edge0 Diag Product Eq Sixteen De

The cofactor identity

A tetrahedron is a pyramid with four triangular faces. Pick one edge, the line where two faces meet. The dihedral angle is the angle between those two faces, measured as if you were looking straight down the edge. In three-dimensional space, you can compute that angle from the face normals, the vectors perpendicular to each face. The classic formula uses their dot product, the sum of products of matching coordinates, divided by the product of their lengths. That is the cosine of the dihedral angle.

There is a second, older way to get the same angle, through the Cayley-Menger determinant, a determinant built from the six squared edge lengths of the tetrahedron. The Berger cofactor formula says the geometric cosine equals a ratio of two such determinants. The declaration in question, cmCofactor3_edge0_diag_product_eq_sixteen_denomSq, is one algebraic step inside that proof. It states that for the first edge, the product of two diagonal cofactors of the Cayley-Menger matrix equals 16 times the square of the denominator used in the geometric cosine. The number 16 is not an approximation; it is exact, and it comes from the factor of 4 that appears when you take the square root of the denominator. The theorem is proved in a machine-checked library of formal theorems, meaning every step is verified by a computer.

The theorem does not claim that the geometric cosine equals the cofactor ratio for all six edges at once. It covers only the first edge, labeled 0. The other five edges have their own separate identities, each with its own algebraic form. Nor does it claim that the cofactor ratio is always between -1 and 1; that is a separate theorem, proved for all edges after the identification is made. The declaration is a lemma, not the final result. Its role is to supply the exact factor of 16 that lets the square root in the geometric formula match the cofactor ratio.

What the theorem changes is the proof strategy. Instead of expanding both sides of the Berger formula by hand, the library can reduce the problem to a single algebraic identity about determinants. The factor of 16 is the key that makes the square roots cancel. A reader who wants to verify the full formula for all six edges still needs the remaining lemmas, but the first edge is now closed, and the pattern for the rest is set.

theorem cmCofactor3_edge5_diag_product_eq_sixteen_denomSq (T : RealizedTet) :
    cmCofactor3 (sqEdgeOfPoints T) 1 1 *
      cmCofactor3 (sqEdgeOfPoints T) 2 2 =
        16 * geometricDihedralDenomSq T 5 := by
  rw [cmCofactor3_edge5_left_diag_eq_neg_four_normalSq,
    cmCofactor3_edge5_right_diag_eq_neg_four_normalSq]
  unfold geometricDihedralDenomSq
  simp [edgeVertices3, adjacentFaceOppositeVertices, ReggeRigorousFoundation.edgeVertices]
  ring

/-- The internal geometric dihedral cosine at an edge, computed from the
two face normals adjacent to the edge.  The sign is chosen to match the
internal Regge dihedral convention. -/
def geometricDihedralCos (T : RealizedTet) (e : Fin 6) : ℝ :=
  geometricDihedralNumerator T e / Real.sqrt (geometricDihedralDenomSq T e)

/-- The theorem target is discharged. -/
theorem bergerCofactorFormula3 : BergerCofactorFormula3 :=
  geometricDihedralCos_eq_cmCofactorRatio

What this page does not claim

The theorem does not cover the other five edges of the tetrahedron. The theorem does not assert that the cofactor ratio lies between -1 and 1. The theorem does not relate the dihedral angle to any physical constant or recognition cost.

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/DihedralCofactorFormula.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: