Geometry Dihedral Angle Deficit Eq Zero Of Flat

The flat hinge

A dihedral angle is the angle between two flat faces that meet along a shared edge. Picture an open book: the angle between the two pages is the dihedral angle at the spine. In a regular tetrahedron, the angle between any two faces is about 70.53 degrees, the arccosine of one third. In a cube, the angle between adjacent faces is exactly 90 degrees. These are classical facts of solid geometry, known since antiquity and formalized here without any unproved assumptions.

When several solid pieces meet along a single edge, the sum of their dihedral angles around that edge is called the total angle at the hinge. In a flat, or Euclidean, space, that total must be a full 360 degrees, or 2π radians. Imagine four cubes meeting at a common edge: each contributes a right angle, and four right angles make the full circle. The difference between 2π and the actual sum is the angular deficit. If the space is genuinely flat at that hinge, the deficit is zero. This is the piecewise-flat way of saying there is no curvature concentrated along that edge.

The machine-checked theorem deficit_eq_zero_of_flat proves exactly this: whenever the flat-sum condition holds, meaning the angles sum to 2π, the deficit is zero. The proof is a direct algebraic rewrite, from the definition of deficit as 2π minus the sum, using the hypothesis that the sum equals 2π. It is a tautology in the best sense: it unpacks what the words mean. The same file also proves the concrete instance that four cube right angles sum to 2π, so the cubic lattice is flat at every edge.

In Recognition Science, this lemma is a building block, not a destination. The framework's program aims to discharge a hypothesis about Regge calculus, a discrete approach to general relativity where spacetime is approximated by flat pieces glued together. The flat-sum condition is the piecewise-flat analog of zero curvature, and this theorem certifies that the definition of deficit behaves correctly under that condition. The certificate structure bundles the key results, including the cube right angle and the tetrahedron range, into one machine-checked package for later phases.

What the theorem does not claim is just as important. It does not prove that any particular physical space is flat, nor that the flat-sum condition actually holds for any real configuration. It only says that if the condition holds, the deficit follows. It does not address curved spaces, where the deficit is nonzero and measures the concentrated curvature. The theorem is a statement about the internal consistency of the definitions, not about the geometry of the universe.

/-- At a flat hinge, the deficit is zero. -/
theorem deficit_eq_zero_of_flat (ds : List DihedralAngleData)
    (h : FlatSumCondition ds) : deficit ds = 0 := by
  unfold deficit
  rw [h]; ring

/-- The regular-tetrahedron dihedral angle is `arccos(1/3) ≈ 70.53°`. -/
theorem regular_tet_dihedral_theta :
    regular_tet_dihedral.theta = Real.arccos (1/3) := rfl

/-- The cube dihedral is `π / 2` exactly. -/
theorem cube_dihedral_theta : cube_dihedral.theta = Real.pi / 2 := by
  unfold DihedralAngleData.theta cube_dihedral
  simp only
  exact Real.arccos_zero

/-- Four cube-dihedral angles (`π/2 + π/2 + π/2 + π/2 = 2π`) sum to `2π`:
    the classical "Z³ lattice is flat" statement. -/
theorem cubic_lattice_flatSum :
    FlatSumCondition [cube_dihedral, cube_dihedral, cube_dihedral, cube_dihedral] := by
  unfold FlatSumCondition sumThetas
  simp only [List.map, List.sum_cons, List.sum_nil, cube_dihedral_theta]
  ring

What this page does not claim

The theorem does not prove that any physical space is flat. The theorem does not show that the flat-sum condition holds for any real configuration of simplices. The theorem does not address curved spaces, where the deficit is nonzero.

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

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