Gravity Regge Calculus Deficit Pos Of Angle Deficit

The deficit angle

In geometry, the deficit angle measures how much curvature is concentrated at a point or along a line. Four squares meeting at a corner on a flat table have angles that sum to a full 360 degrees. When the same four squares are forced into a pyramid, the angles at the apex no longer fill the full circle; the shortfall is the deficit angle. The larger the shortfall, the sharper the curvature. This idea, formalized by Tullio Regge in 1961, lets physicists build curved spacetime from flat building blocks, with all the bending concentrated along the edges where the blocks meet.

The ledger, a discrete record of events, enters here because this geometric picture is what the Recognition Science framework uses to describe gravity. Its machine-checked library of formal theorems contains a statement, deficit_pos_of_angle_deficit, that captures the most basic property of this construction. The theorem says: if the sum of the dihedral angles around a hinge is less than a full turn of 2π, then the deficit angle is greater than zero. This is a purely arithmetic consequence of the definition, verified by the library's checker. It establishes that a positive deficit angle corresponds exactly to a shortfall in the surrounding angles, nothing more.

The theorem's companion, deficit_neg_of_angle_excess, states the mirror case: if the angles sum to more than 2π, the deficit angle is negative. Together they confirm that the sign of the deficit angle tracks whether the local geometry is curved inward or outward. The framework also proves that a flat configuration, where the angles sum exactly to 2π, has a zero deficit, and that the total deficit over a closed surface relates to its topology through a Gauss-Bonnet identity. These results are the scaffolding for the Regge action, the sum over all hinges of area times deficit angle, which the framework uses as its discrete version of the Einstein-Hilbert action.

What the declaration does not claim is equally precise. It does not assert that any particular physical configuration has a positive deficit. It only states the conditional: if the angles sum to less than 2π, then the deficit is positive. It also does not claim that the deficit angle formula itself is the correct description of gravity in the framework; that is a modeling choice, not a theorem. The declaration is a lemma about the definition, not a physical law. It is the kind of small, exact step that a machine-checked library can certify without ambiguity, and it is the foundation on which the framework's discrete gravity programme is built.

/-- Deficit angle is positive when total angle < 2*pi (positive curvature). -/
theorem deficit_pos_of_angle_deficit (h : HingeData)
    (h_less : h.dihedral_angles.sum < 2 * Real.pi) :
    0 < deficit_angle h := by
  unfold deficit_angle; linarith

/-- Deficit angle is negative when total angle > 2*pi (negative curvature). -/
theorem deficit_neg_of_angle_excess (h : HingeData)
    (h_more : 2 * Real.pi < h.dihedral_angles.sum) :
    deficit_angle h < 0 := by
  unfold deficit_angle; linarith

/-- For a flat configuration, all hinges have zero deficit. -/
theorem flat_deficit_zero (h : HingeData)
    (h_flat : h.dihedral_angles.sum = 2 * Real.pi) :
    deficit_angle h = 0 := by
  unfold deficit_angle; linarith

What this page does not claim

The declaration does not assert that any particular physical configuration has a positive deficit angle. The declaration does not claim that the deficit angle formula is the correct description of gravity; it is a modeling choice. The declaration does not establish the full Regge calculus framework, only a single lemma about the definition of deficit angle.

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

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