Geometry Cofactor Derivatives Dihedral Cos3 Sq Closed Form Deriv Eq Generic

The derivative identity

In geometry, a tetrahedron is a solid with four triangular faces, six edges, and four vertices. A dihedral angle is the angle between two faces that share an edge. The Recognition Science theorem dihedralCos3SqClosedFormDeriv_eq_generic states that the derivative of the squared cosine of a dihedral angle, taken with respect to one of the six edge lengths, equals a specific rational expression involving the cofactors of the Cayley-Menger matrix. The theorem is a formal identity: it says the closed-form derivative, which is written as a quotient of polynomial-like terms, is exactly equal to the value obtained by applying the general quotient rule to the numerator and denominator of the cosine-squared function.

The statement is proved in the machine-checked library of formal theorems. It is a theorem, not a definition or a model. The proof relies on the fact that the denominator of the cosine-squared function is never zero for a non-degenerate tetrahedron, meaning one whose four vertices are not coplanar. The theorem also uses the fact that the product of the two diagonal cofactors is positive in that case. These conditions are stated explicitly in the theorem's hypotheses, so the identity holds for all non-degenerate tetrahedra and all choices of edge and vertex indices.

What the theorem does not claim is that the derivative has been simplified to a single polynomial or that the expression has been reduced to a fully expanded symbolic form. The docstring for the module says the hard symbolic derivative simplifications are still downstream. The theorem establishes the equality of two expressions, but it does not claim that either expression is the simplest possible form, nor does it claim that the derivative is continuous or differentiable at degenerate configurations. The theorem is a building block for later work, not the final simplification.

In the Recognition Science framework, this theorem is part of the calculus layer for Cayley-Menger cofactors, which are determinants of submatrices of a matrix whose entries are squared distances between points. The framework uses these cofactors to study the geometry of tetrahedra and, ultimately, to derive physical structure from geometric constraints. This particular theorem is a technical result that supports the larger project of understanding how geometric quantities change as edge lengths vary.

The practical consequence is that anyone working with the derivative of a dihedral angle in a tetrahedron can now rely on a verified closed form, rather than having to recompute the derivative from scratch or trust an unverified symbolic manipulation. The theorem provides a rigorous foundation for further derivations involving the geometry of tetrahedra, such as the forces that act on the edges of a simplex in a discrete spacetime model.

/-- The closed-form cosine derivative is exactly the generic quotient
derivative value after substituting the explicit cofactor partials. -/
theorem dihedralCos3SqClosedFormDeriv_eq_generic
    (a : SqEdges) (e : Fin 6) (k : Fin 6) :
    dihedralCos3SqClosedFormDeriv a e k =
      dihedralCos3SqDerivValue (fun _ : ℝ => a) 0
        (dihedralNumeratorClosedDeriv a e k)
        (dihedralDenom3ClosedDerivValue a e k) e := by
  unfold dihedralCos3SqClosedFormDeriv dihedralCos3SqDerivValue
  simp

/-- The closed-form cosine derivative is exactly the generic quotient
derivative value after substituting the explicit cofactor partials. -/
theorem dihedralCos3SqClosedFormDeriv_eq_generic
    (a : SqEdges) (e : Fin 6) (k : Fin 6) :
    dihedralCos3SqClosedFormDeriv a e k =
      dihedralCos3SqDerivValue (fun _ : ℝ => a) 0
        (dihedralNumeratorClosedDeriv a e k)
        (dihedralDenom3ClosedDerivValue a e k) e := by
  unfold dihedralCos3SqClosedFormDeriv dihedralCos3SqDerivValue
  simp

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)

/-- 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

What this page does not claim

The derivative has been simplified to a single polynomial or fully expanded symbolic form. The identity holds for degenerate tetrahedra where the denominator is zero. The theorem provides a physical interpretation of the derivative.

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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