Gravity Seven Gaps Causal Simplex Wick Dihedral Angle3 Physical Mem Ioo

The angle at the physical point

In the framework's discrete gravity program, spacetime is built from tetrahedra, the four-sided solids that tile a three-dimensional space. Recognition, the framework's term for a discrete record of events, enters here through the causal structure: each tetrahedron's edges are labeled spacelike or timelike depending on whether they lie within a spatial slice or connect one slice to the next. The declaration dihedralAngle3_physical_mem_Ioo is a theorem about the angle between two faces of such a tetrahedron at a special setting.

The setting is the physical point, where the parameter alpha, which controls the squared length of timelike edges, is set to 1. At this point, the Euclideanized tetrahedron, the one obtained by flipping the sign of timelike squared lengths, becomes the regular tetrahedron, whose six edges all have the same length. The theorem states that for either of the two allowed tetrahedron types, the cosine of any dihedral angle, the angle between two faces, equals exactly 1/3. The angle itself is therefore arccos(1/3), approximately 70.53 degrees, the same value a regular tetrahedron has in ordinary Euclidean geometry.

The proof is machine-checked in the framework's library of formal theorems. It relies on a chain of earlier results: the Euclideanized squared edges at alpha = 1 equal the regular unit tetrahedron's edges, the cosine of the dihedral angle at that point is 1/3, and that value lies strictly between -1 and 1, so the arccos is a real angle, not an artifact of a degenerate configuration. The theorem also confirms that this angle is real for both tetrahedron types, the (3,1) type with three vertices on one slice and one on the next, and the (2,2) type with two on each slice.

The declaration does not claim that the angle is real for every value of alpha. The symbolic range of alpha for which the arccos arguments stay inside (-1,1) is left open; the cofactor denominators are square roots of degree-2 minors, and controlling their signs symbolically is a separate problem. The theorem proves the reality only at the concrete physical point alpha = 1. It also does not address the action-level Lorentzian continuation, the complex dihedral angles and boost-angle assignments at timelike hinges, which the framework lists as open. What the theorem does establish is a precise, certified fact about the geometry of spacetime tetrahedra at the physical point, a fixed landmark in a largely unexplored Lorentzian sector.

/-- THEOREM: the Euclidean dihedral angle at every edge of the physical-point
causal tetrahedron is `arccos (1/3)`, hence strictly inside `(0, π)`:
deficit angles at all edges are real and well-defined. -/
theorem dihedralAngle3_physical (ty : CausalTetType) (e : Fin 6) :
    dihedralAngle3 (physicalCausalTet ty) e = Real.arccos (1 / 3) :=
  congrArg Real.arccos (dihedralCos3Sq_alpha_one ty e)

/-- THEOREM (deficit-angle reality, physical point): the arccos argument
lies strictly inside `(-1, 1)` at every edge, for both causal types. -/
theorem dihedralCos3Sq_alpha_one_mem_Ioo (ty : CausalTetType) (e : Fin 6) :
    -1 < dihedralCos3Sq (euclideanSqEdges ty 1 1) e
      ∧ dihedralCos3Sq (euclideanSqEdges ty 1 1) e < 1 := by
  rw [dihedralCos3Sq_alpha_one ty e]
  norm_num

/-- THEOREM: at `alpha = 1`, `a = 1`, both causal types Euclideanize to the
regular unit tetrahedron tuple. -/
theorem euclideanSqEdges_alpha_one (ty : CausalTetType) :
    euclideanSqEdges ty 1 1 = regularUnitSqEdges := by
  funext e
  simp only [euclideanSqEdges, regularUnitSqEdges]
  by_cases h : isTimelike ty e = true
  · rw [if_pos h]
    norm_num
  · rw [if_neg h]
    norm_num

What this page does not claim

The theorem does not prove the dihedral angle is real for any alpha other than 1. It does not address the Lorentzian action-level continuation or complex dihedral angles. It does not claim the causal tetrahedron classes are physically realized in spacetime.

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

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