Geometry Cayley Menger Matrix Det Regular Unit Off Diag Minor Matrix14

A specific cofactor value

The Cayley-Menger matrix is a standard tool in geometry that encodes the distances between points. For a tetrahedron, it is a 5 by 5 matrix whose entries are 0 on the diagonal, 1 in the first row and column, and the six squared edge lengths elsewhere. Its determinant, called the Cayley-Menger determinant, vanishes exactly when the six lengths can form a tetrahedron in ordinary three-dimensional space.

The declaration det_regularUnitOffDiagMinorMatrix14 concerns a specific 4 by 4 matrix obtained by deleting one row and one column from the 5 by 5 Cayley-Menger matrix of a regular tetrahedron with all edge lengths equal to 1. The matrix in question is:

[[0, 1, 1, 1], [1, 1, 0, 1], [1, 1, 1, 0], [1, 1, 1, 1]]

The declaration proves, by a machine-checked calculation in the Recognition Science library, that the determinant of this matrix is -1. This value is a cofactor of the full 5 by 5 Cayley-Menger matrix. Cofactors appear in formulas for the dihedral angles of a tetrahedron, the angles between its faces, and this particular cofactor is one of the six off-diagonal cofactors that enter such formulas.

The declaration does not claim that the tetrahedron exists, that the determinant has geometric meaning by itself, or that the cofactor relates to any physical quantity. It establishes only the algebraic fact that this particular matrix, built from a regular tetrahedron with unit edges, has determinant -1. The Recognition Science framework uses this value as a building block in its geometry library, but the declaration itself is a pure statement of linear algebra.

theorem det_regularUnitOffDiagMinorMatrix14 :
    Matrix.det regularUnitOffDiagMinorMatrix14 = -1 := by
  unfold regularUnitOffDiagMinorMatrix14
  rw [Matrix.det_succ_row_zero]
  simp [Fin.sum_univ_succ, Matrix.det_fin_three, Fin.succAbove]

What this page does not claim

The declaration does not assert that the tetrahedron with these edge lengths exists in space. The declaration does not assign any physical meaning to the determinant value. The declaration does not relate this cofactor to any other geometric property beyond its algebraic definition.

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

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