Geometry Cofactor Derivatives Dihedral Cofactor Product Poly Pos Of Non Degenera

A positivity guarantee

A tetrahedron is a pyramid with four triangular faces. It is non-degenerate when its four corner points are not all in one plane, so it has genuine volume. For such a tetrahedron, geometry attaches to each edge a number called a Cayley-Menger cofactor, built from the squared lengths of all six edges. The declaration dihedralCofactorProductPoly_pos_of_nonDegenerate proves that for any non-degenerate tetrahedron and any chosen edge, the product of the two cofactors belonging to the two vertices opposite that edge is strictly greater than zero.

The classical background: Cayley-Menger determinants generalize Heron's formula for triangle area to higher dimensions. For a tetrahedron, they encode volume and dihedral angles. A dihedral angle is the angle between two faces meeting along an edge. The cofactor product in question is a polynomial in the six squared edge lengths. The theorem states that this polynomial, evaluated at any non-degenerate tetrahedron, is positive. This is not a numerical approximation; it is an exact algebraic inequality.

In the framework's machine-checked library of formal theorems, this positivity result is a ledger, a discrete record of proven statements, entry. It is used to show that a related denominator, the square root of that product, is non-zero. That non-zero condition is what makes derivative formulas for dihedral angles meaningful: dividing by zero would be undefined. The theorem therefore underpins the calculus of how dihedral angles change as edge lengths vary.

The declaration does not claim that the cofactor product is positive for degenerate tetrahedra, where volume is zero. It does not compute the value of the product, only its sign. It also does not assert that the individual cofactors are positive; only their product is guaranteed. Finally, it says nothing about the physical meaning of these quantities, which is a separate question from the algebraic fact proved here.

/-- Nondegenerate tetrahedra have positive polynomial cofactor denominator
products for every dihedral edge. -/
theorem dihedralCofactorProductPoly_pos_of_nonDegenerate
    (T : ReggeRigorousFoundation.NonDegenerateTet) (e : Fin 6) :
    0 < dihedralCofactorProductPoly T.sqEdge e := by
  have hdisc := dihedralCofactorPoly_discriminant_eq T.sqEdge e
  have hdisc' :
      dihedralCofactorProductPoly T.sqEdge e -
          dihedralCofactorNumeratorPoly T.sqEdge e ^ 2 =
        2 * cm3 T.sqEdge * T.sqEdge e := by
    simpa [dihedralCofactorProductPoly, dihedralCofactorNumeratorPoly] using hdisc
  have hpos : 0 < 2 * cm3 T.sqEdge * T.sqEdge e := by
    nlinarith [T.cm_pos, T.sqEdge_pos e]
  have hsq : 0 ≤ dihedralCofactorNumeratorPoly T.sqEdge e ^ 2 := sq_nonneg _
  nlinarith

theorem dihedralDenom3Poly_ne_zero_of_nonDegenerate
    (T : ReggeRigorousFoundation.NonDegenerateTet) (e : Fin 6) :
    dihedralDenom3Poly T.sqEdge e ≠ 0 :=
  ne_of_gt (dihedralDenom3Poly_pos_of_nonDegenerate T e)

What this page does not claim

The cofactor product is positive for degenerate tetrahedra. The individual cofactors are each positive. The theorem computes the numerical value of the cofactor product.

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

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