Geometry Cayley Menger Matrix Det Regular Unit Off Diag Minor Matrix24

A determinant in the tetrahedron's bookkeeping

The Cayley-Menger matrix is a standard tool in geometry that encodes the distances between points. For a tetrahedron, the matrix is 5 by 5, with zeros on the diagonal and the squared lengths of the six edges filling the remaining entries. Its determinant, up to a constant factor, equals the square of the tetrahedron's volume, so it vanishes exactly when the four points are coplanar. The framework's machine-checked library of formal theorems works with this matrix and its submatrices.

One submatrix, called regularUnitOffDiagMinorMatrix24, comes from a regular tetrahedron whose edges all have length 1. It is a 4 by 4 matrix that keeps the first two rows and columns while deleting row 3 and column 4 of the original. The entries are 0 on the diagonal and 1 elsewhere, except for one extra 1 in the bottom row. The declaration det_regularUnitOffDiagMinorMatrix24 establishes that the determinant of this particular matrix equals 1. The proof is a direct computation, expanding along the first row and simplifying the resulting 3 by 3 determinants.

This determinant is not an isolated curiosity. The library uses it to compute the cofactor of the original Cayley-Menger matrix at position (2,4), which is also 1. Cofactors appear in the formula for the dihedral angle between two faces of the tetrahedron. The value 1 here means that this particular minor contributes a clean, nonzero term to that formula, without any sign or scale complications.

In Recognition Science, this result is a small piece of the framework's larger project of deriving geometry from a discrete ledger of recognition events. The framework models space as emerging from a cost function on such a ledger, and it proves that three spatial dimensions are forced by the structure of recognition. The Cayley-Menger determinant is the bridge between that abstract derivation and the ordinary geometry of triangles and tetrahedra that working mathematicians use.

The declaration does not claim anything about the volume of the tetrahedron, about any other minor, or about the general formula for cofactors. It is a single, exact computation for one specific matrix. The value 1 is a fact about that matrix alone, not a general theorem about all tetrahedra or all Cayley-Menger matrices.

theorem det_regularUnitOffDiagMinorMatrix24 :
    Matrix.det regularUnitOffDiagMinorMatrix24 = 1 := by
  unfold regularUnitOffDiagMinorMatrix24
  rw [Matrix.det_succ_row_zero]
  simp [Fin.sum_univ_succ, Matrix.det_fin_three, Fin.succAbove]

theorem regularUnit_cofactor_24 :
    cmCofactor3 regularUnitSqEdges 2 4 = 1 := by
  unfold cmCofactor3 cmCofactorSign3 cmMinor3
  simp [show Even (6 : Nat) by decide]
  rw [regularUnit_minor_24_eq_offDiag, det_regularUnitOffDiagMinorMatrix24]

What this page does not claim

This declaration says nothing about the volume of a tetrahedron. The value 1 is specific to this one matrix, not a general property of all Cayley-Menger minors. The framework's derivation of three dimensions is not established by this determinant computation.

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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