Geometry Cayley Menger Derivatives Cm3 Cubic Single Perturb

The cubic remainder

The Cayley-Menger polynomial is a formula that decides whether six lengths can form a tetrahedron in three-dimensional space. For a tetrahedron with vertices labeled 0 through 3, the polynomial CM_3 takes six squared edge lengths as input. It vanishes exactly when those lengths describe a flat, degenerate tetrahedron, and its sign separates realizable tetrahedra from impossible ones. The polynomial is cubic in the squared lengths, meaning that if you change one edge length, the polynomial changes by terms up to the third power of that change.

The declaration cm3_cubic_singlePerturb isolates one piece of that cubic behavior. It states that when you perturb a single edge length by an amount t, the cubic part of the polynomial's change is exactly the cube of t multiplied by a coefficient. That coefficient is zero for every edge. In other words, the cubic remainder vanishes identically under a single-coordinate perturbation. The theorem holds for all six edges and for every real value of t.

This result is not a numerical approximation. It is an exact polynomial identity, proved in the framework's machine-checked library of formal theorems. The proof proceeds by expanding the cubic term and simplifying each of the six cases, one per edge. The same library also proves the fuller Taylor expansion of cm3 under a single-coordinate update, which includes the linear, quadratic, and cubic contributions. The vanishing of the cubic term means that the Taylor expansion for a single-edge perturbation stops cleanly at the quadratic term.

In Recognition Science, this fact feeds into a larger geometric program. The framework models physical space through a discrete ledger of recognition events, and it derives three spatial dimensions from a forcing chain of theorems. The Cayley-Menger derivatives appear when the framework studies how geometric quantities change under edge-length variations. The single-perturbation result simplifies those calculations by removing the cubic term from the expansion, leaving a quadratic form that is easier to analyze.

The theorem does not claim that the full cubic term vanishes for arbitrary perturbations of multiple edges. It applies only when exactly one edge length changes. It also does not identify which tetrahedra are realizable; that is the job of the polynomial's sign, not of this derivative identity. And it makes no statement about physics beyond the algebraic fact itself. The framework's use of this result in physical modeling remains a separate step, one that the library has not yet formalized.

theorem cm3_cubic_singlePerturb (i : Fin 6) (t : ℝ) :
    cm3_cubic (singlePerturb i t) = cm3_cubic_coeff i * t ^ 3 := by
  unfold cm3_cubic singlePerturb cm3_cubic_coeff
  fin_cases i <;> simp

/-- Cubic-in-`t` coefficient of the single-coordinate correction.  In each
case there is no `t³` contribution because the cm3 polynomial is degree 2 in
each *individual* squared-edge coordinate. -/
def cm3_cubic_coeff : Fin 6 → ℝ := fun _ => 0

theorem cm3_cubic_singlePerturb (i : Fin 6) (t : ℝ) :
    cm3_cubic (singlePerturb i t) = cm3_cubic_coeff i * t ^ 3 := by
  unfold cm3_cubic singlePerturb cm3_cubic_coeff
  fin_cases i <;> simp

/-- Combined polynomial form of the Taylor expansion in single-coordinate
direction:

```
cm3 (a.update i (a i + t))
  = cm3 a + cm3_grad(a)(i) · t + cm3_quadratic_coeff i a · t² + cm3_cubic_coeff i · t³
```
-/
theorem cm3_update_polyform (a : SqEdges) (i : Fin 6) (t : ℝ) :
    cm3 (Function.update a i (a i + t)) =
      cm3 a + cm3_grad a i * t
        + cm3_quadratic_coeff i a * t ^ 2
        + cm3_cubic_coeff i * t ^ 3 := by
  rw [cm3_update_taylor a i t]
  rw [cm3_quadratic_singlePerturb, cm3_cubic_singlePerturb]

What this page does not claim

The full cubic term vanishes for arbitrary perturbations of multiple edges. The theorem identifies which tetrahedra are realizable from given edge lengths. The framework's physical modeling based on this result is formalized in the library.

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

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