Geometry Cofactor Derivatives Dihedral Cofactor Product Poly Nonneg Of Non Degen

The positivity guarantee

A tetrahedron is the simplest solid shape: four triangular faces, six edges. In the geometry library of the Recognition Science framework, a tetrahedron is called non-degenerate when its four corner points do not collapse into a flat or smaller shape. For any such tetrahedron, and for any one of its six edges, the framework's machine-checked library of formal theorems proves that a particular algebraic expression, the dihedral cofactor product polynomial, is greater than or equal to zero. The theorem is named dihedralCofactorProductPoly_nonneg_of_nonDegenerate.

The expression itself is a product of two cofactors, each derived from the tetrahedron's edge lengths. A cofactor, in matrix algebra, is a signed number attached to an entry of a square matrix; here, the two cofactors come from the diagonal entries of a 3 by 3 matrix built from the squared edge lengths. The theorem states that their product is never negative, meaning it is either positive or exactly zero, whenever the tetrahedron is non-degenerate. This is a purely algebraic fact about the edge lengths; it does not depend on any physical law or on the Recognition Science framework's broader claims about recognition or cost.

The proof is short and relies on a stronger result: for a non-degenerate tetrahedron, the same product is strictly positive, not merely non-negative. The non-negativity statement follows directly from that stronger positivity result. This chain of reasoning is fully checked by the machine, so the claim carries the weight of a proved theorem, not a conjecture or a numerical observation.

What this theorem does not claim is equally important. It does not say that the product is positive for degenerate tetrahedra, those where the points collapse; for those, the product could be zero. It does not provide a formula for the cofactors or for the product itself, only a sign guarantee. It also does not connect this algebraic fact to any physical quantity, such as an angle or a force, within the Recognition Science framework. The theorem is a standalone piece of geometry, useful as a building block for later results about derivatives of dihedral angles, but it makes no statement about the physical meaning of those angles.

theorem dihedralCofactorProductPoly_nonneg_of_nonDegenerate
    (T : ReggeRigorousFoundation.NonDegenerateTet) (e : Fin 6) :
    0 ≤ dihedralCofactorProductPoly T.sqEdge e :=
  le_of_lt (dihedralCofactorProductPoly_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 theorem does not apply to degenerate tetrahedra, where the product may be zero. The theorem does not give a formula for the cofactors or the product, only a sign guarantee. The theorem does not connect this algebraic fact to any physical quantity within the Recognition Science framework.

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)

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: