Geometry Realisability Cone Right Angle Unit Mem Realisable Tet Cone

A right-angle tetrahedron in the cone

A tetrahedron is a pyramid with four triangular faces. A right-angle unit tetrahedron is one where three edges meeting at a single vertex are all length 1 and meet at right angles, like the corner of a room where three walls and the floor come together. This shape is a familiar object in solid geometry, often used as the simplest three-dimensional analogue of a right triangle.

For a tetrahedron to exist in ordinary three-dimensional space, its six edge lengths cannot be arbitrary. They must satisfy a condition named after the mathematicians Arthur Cayley and Karl Menger: the Cayley-Menger determinant, computed from the squared edge lengths, must be positive. The recognition ledger, a discrete record of geometric events the framework studies, defines a realisable tetrahedral cone as the set of all squared-edge-length tuples with positive entries and a positive Cayley-Menger determinant. This cone is the domain where the tetrahedral formulas are valid.

In Recognition Science, the machine-checked library of formal theorems proves that the right-angle unit tetrahedron lies in this cone. The declaration rightAngleUnit_mem_realisableTetCone establishes exactly that: its six squared edge lengths are all positive, and its Cayley-Menger determinant is positive. The proof is direct, checking each of the six edges and then evaluating the determinant for this specific shape.

This result is a sanity check on the framework's definitions. It confirms that the realisability cone is not empty of familiar shapes: the corner-of-a-room tetrahedron is inside it. The same library also proves the regular unit tetrahedron, with all six edges equal to 1, lies in the cone. These two theorems anchor the cone's definition to concrete, recognisable geometry.

What the declaration does not claim is broader. It does not say that every positive-edge-length tuple with a positive Cayley-Menger determinant corresponds to a tetrahedron; that full characterisation remains a target. It does not assert anything about dihedral angles, face areas, or the volume of this tetrahedron. It only certifies membership in the basic cone, leaving stronger geometric properties for later phases of the framework's development.

/-- The right-angle unit tetrahedron lies in the basic realisability cone. -/
theorem rightAngleUnit_mem_realisableTetCone :
    rightAngleUnitSqEdges ∈ RealisableTetCone := by
  constructor
  · intro i
    unfold rightAngleUnitSqEdges
    fin_cases i <;> norm_num
  · rw [cm3_rightAngle_unit]
    norm_num

/-- Basic open tetrahedral cone: positive squared edge lengths and positive
Cayley-Menger determinant.  Later phases strengthen this with face-minor
positivity as needed by cofactor denominators. -/
def RealisableTetCone : Set SqEdges :=
  {a | (∀ i : Fin 6, 0 < a i) ∧ 0 < cm3 a}

What this page does not claim

The declaration does not prove that every tuple in the cone corresponds to a real tetrahedron. It does not compute the volume, face areas, or dihedral angles of the right-angle unit tetrahedron. It does not establish that the cone is non-empty beyond the two unit tetrahedra named.

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

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