Gravity Seven Gaps Causal Simplex Wick Dihedral Cos3 Sq Alpha One Mem Ioo

The regular tetrahedron's angles

A regular tetrahedron is the three-dimensional analogue of an equilateral triangle: four triangular faces, six edges of equal length, and every vertex connected to every other. Its dihedral angle, the angle between two adjacent faces, is a classic quantity. This page concerns a specific machine-checked theorem about that angle, proved within the Recognition Science framework's library of formal theorems.

The theorem, named dihedralCos3Sq_alpha_one_mem_Ioo, establishes a simple fact: for a regular tetrahedron with unit edge length, the cosine of the dihedral angle lies strictly between -1 and 1. In symbols, if the squared edge lengths are all 1, then the cosine of the angle between any two faces is exactly 1/3, a value comfortably inside the open interval (-1, 1). This means the angle is real and well-defined, not a degenerate or imaginary quantity.

The proof is a short chain of reasoning. It first shows that when the parameter alpha, which controls the squared length of timelike edges in the causal tetrahedron construction, is set to 1, the Euclideanized edge lengths all become 1. The theorem then invokes a separate result that the cosine of the dihedral angle for a regular unit tetrahedron is 1/3. The final step is a simple numerical check that 1/3 lies strictly between -1 and 1.

This result matters because it is a certified piece of a larger program. The framework is building a discrete model of spacetime from causal tetrahedra, where edges can be spacelike or timelike. A key step is the Wick rotation, a mathematical map that converts Lorentzian geometry (with its timelike directions) into Euclidean geometry, where standard geometric tools apply. The theorem here confirms that at the physical point alpha = 1, the Euclideanized tetrahedra are regular and their angles are real, a necessary condition for the geometry to be meaningful.

The theorem does not claim anything about the Lorentzian sector itself. The Lorentzian tetrahedra, with negative squared lengths on timelike edges, are never Euclidean-realizable, as a separate theorem shows. The reality of the angles is proved only for the Euclideanized version. Also, the theorem says nothing about the action-level continuation, the complex dihedral angles, or the sinh-action sector, which remain open targets in the framework's program.

/-- THEOREM: at the physical point the cofactor dihedral cosine is `1/3` at
every edge (both causal types), via the proved regular-tetrahedron cofactor
evaluation. -/
theorem dihedralCos3Sq_alpha_one (ty : CausalTetType) (e : Fin 6) :
    dihedralCos3Sq (euclideanSqEdges ty 1 1) e = 1 / 3 := by
  rw [euclideanSqEdges_alpha_one ty]
  exact dihedralCos3_regularUnit 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: type (3,1) Lorentzian Cayley-Menger determinant,
`cm3 = -(2 * (3*alpha + 1) * a^6)`: strictly negative for `alpha ≥ 0`,
`a ≠ 0` (see `lorentzian_cm3_neg_threeOne`). -/
theorem cm3_lorentzian_threeOne (a alpha : ℝ) :
    cm3 (lorentzianSqEdges CausalTetType.threeOne a alpha)
      = -(2 * (3 * alpha + 1) * a ^ 6) := by
  have h0 : lorentzianSqEdges CausalTetType.threeOne a alpha 0 = a ^ 2 := rfl
  have h1 : lorentzianSqEdges CausalTetType.threeOne a alpha 1 = a ^ 2 := rfl
  have h2 : lorentzianSqEdges CausalTetType.threeOne a alpha 2
      = -(alpha * a ^ 2) := rfl
  have h3 : lorentzianSqEdges CausalTetType.threeOne a alpha 3 = a ^ 2 := rfl
  have h4 : lorentzianSqEdges CausalTetType.threeOne a alpha 4
      = -(alpha * a ^ 2) := rfl
  have h5 : lorentzianSqEdges CausalTetType.threeOne a alpha 5
      = -(alpha * a ^ 2) := rfl
  unfold cm3
  rw [h0, h1, h2, h3, h4, h5]
  ring

What this page does not claim

The theorem does not establish the reality of dihedral angles for any alpha other than 1. The theorem does not address the Lorentzian action continuation or the sinh-action sector. The theorem does not imply that the Lorentzian tetrahedra themselves have real angles.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: