Geometry Cofactor Polynomial Has Deriv At Cm Cofactor3 34 Along Coord

A derivative along one edge

In the geometry of a tetrahedron, a Cayley-Menger cofactor is a number built from the six edge lengths that appears in formulas for volume and dihedral angles. The declaration hasDerivAt_cmCofactor3_34_along_coord is a formal statement about how one specific cofactor, the one in row 3 and column 4 of a 5 by 5 matrix, responds when a single edge length is varied while the other five stay fixed.

The theorem says that this cofactor, viewed as a function of one edge length, has a derivative at every point. The derivative is itself a polynomial in the six squared edge lengths, written out explicitly as cmCofactorPartial 3 4 k a. In plain terms: if you nudge one edge of the tetrahedron by a tiny amount, the cofactor changes by an amount proportional to that nudge, and the proportionality constant is a known polynomial expression. The statement is proved for every choice of which edge is varied, indexed by k, and for every assignment of squared edge lengths a.

This is a technical but useful result. It lets downstream calculations about dihedral angles refer to named polynomial partial derivatives instead of opaque abstract derivative terms. The machine-checked library of formal theorems records the proof, so the derivative statement is not an assumption but a verified consequence of the definitions of the cofactor and the polynomial.

The declaration does not claim that the cofactor itself has any particular geometric meaning, nor that it is positive, nor that it relates to any physical quantity. It only establishes the existence and explicit form of the derivative along one coordinate direction. It also does not claim that the derivative is continuous or that higher derivatives exist; those would be separate statements.

/-- Closed-form coordinate derivative of the `(3,4)` cofactor polynomial. -/
theorem hasDerivAt_cmCofactor3Poly_34_along_coord
    (k : Fin 6) (a : SqEdges) :
    HasDerivAt (fun t : ℝ => cmCofactor3Poly 3 4 (Function.update a k t))
      (cmCofactorPartial 3 4 k a) (a k) := by
  have hfun :
      (fun t : ℝ => cmCofactor3Poly 3 4 (Function.update a k t)) =
        (fun t : ℝ => cmCofactor3Poly 3 4 a
          + cmCofactorPartial 3 4 k a * (t - a k)
          + (if k = 0 then -1 else 0) * (t - a k) ^ 2
          + 0 * (t - a k) ^ 3) := by
    funext t
    have h := cmCofactor3Poly_34_update_polyform a k (t - a k)
    have hbase : a k + (t - a k) = t := by ring
    rw [hbase] at h
    simpa using h
  rw [hfun]
  exact hasDerivAt_shifted_cubic (cmCofactor3Poly 3 4 a)
    (cmCofactorPartial 3 4 k a) (if k = 0 then -1 else 0) 0 (a k)

/-- Closed-form coordinate derivative of the `(3,4)` cofactor polynomial. -/
theorem hasDerivAt_cmCofactor3Poly_34_along_coord
    (k : Fin 6) (a : SqEdges) :
    HasDerivAt (fun t : ℝ => cmCofactor3Poly 3 4 (Function.update a k t))
      (cmCofactorPartial 3 4 k a) (a k) := by
  have hfun :
      (fun t : ℝ => cmCofactor3Poly 3 4 (Function.update a k t)) =
        (fun t : ℝ => cmCofactor3Poly 3 4 a
          + cmCofactorPartial 3 4 k a * (t - a k)
          + (if k = 0 then -1 else 0) * (t - a k) ^ 2
          + 0 * (t - a k) ^ 3) := by
    funext t
    have h := cmCofactor3Poly_34_update_polyform a k (t - a k)
    have hbase : a k + (t - a k) = t := by ring
    rw [hbase] at h
    simpa using h
  rw [hfun]
  exact hasDerivAt_shifted_cubic (cmCofactor3Poly 3 4 a)
    (cmCofactorPartial 3 4 k a) (if k = 0 then -1 else 0) 0 (a k)

What this page does not claim

The cofactor has a specific geometric meaning or physical interpretation. The derivative is continuous or that higher derivatives exist. The result applies to cofactors of matrices other than the Cayley-Menger matrix for a tetrahedron.

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

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