Geometry Schlaefli Tetrahedron Proof Has Deriv At Sq Edge Coordinate From Edge L

The edge-length derivative

In the geometry of a tetrahedron, each of the six edges has both a length and a squared length. The declaration hasDerivAt_sqEdgeCoordinate_from_edgeLength records a basic calculus fact: as a function of the edge length L, the squared length L² has derivative 2L at every positive L. In the framework's notation, when the edge length equals the square root of the squared-edge coordinate, the derivative is 2 times that square root. This is the familiar chain-rule step that connects changes measured in length units to changes measured in squared-length units.

The theorem is proved in the machine-checked library, a collection of formal theorems verified by a proof assistant. Its proof is short: it applies the power rule to the identity function and simplifies. The statement holds for any non-degenerate tetrahedron, meaning any tetrahedron with positive volume, and for any of its six edges. It does not require the tetrahedron to be regular or to have any special symmetry.

This fact matters because the framework's tetrahedral Schläfli work expresses dihedral angle and volume derivatives in terms of squared-edge coordinates. Converting those derivatives to length coordinates requires exactly this chain-rule step. The theorem supplies that conversion cleanly, so later results can switch between the two coordinate systems without re-deriving the calculus.

What the theorem does not claim is equally precise. It says nothing about how the volume or dihedral angles change; it only relates the edge length coordinate to the squared-edge coordinate. It does not assert that any particular edge length is optimal or that the tetrahedron satisfies any geometric constraint beyond being non-degenerate. It is a local statement about one edge at a time, not a global statement about the whole tetrahedron.

/-- The square map derivative at the positive edge length `sqrt (a k)`.
This is the scalar chain-rule factor behind `d/dL = 2L d/da`. -/
theorem hasDerivAt_sqEdgeCoordinate_from_edgeLength
    (T : NonDegenerateTet) (k : Fin 6) :
    HasDerivAt (fun L : ℝ => L ^ 2)
      (2 * Real.sqrt (T.sqEdge k)) (Real.sqrt (T.sqEdge k)) := by
  have h := (hasDerivAt_id (Real.sqrt (T.sqEdge k))).pow 2
  simpa [pow_succ, two_mul, mul_comm, mul_left_comm, mul_assoc] using h

What this page does not claim

This theorem does not describe how volume or dihedral angles change with edge length. This theorem does not impose any symmetry or regularity condition on the tetrahedron beyond non-degeneracy. This theorem does not establish any global property of the tetrahedron; it is local to one edge.

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

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