Gravity Seven Gaps Causal Simplex4 D Cm Matrix N Lorentzian Three Two

The Lorentzian 3-2 simplex

Causal dynamical triangulations (CDT) is an approach to quantum gravity that builds spacetime by gluing together simple flat pieces, called simplices, much as a geodesic dome is built from triangles. In four dimensions, the pieces are 4-simplices, the 4D analogue of a tetrahedron. A standard convention, from the work of Ambjorn, Jurkiewicz, and Loll, distinguishes two types based on how many vertices lie on successive time slices. The type (3,2) has three vertices on one slice and two on the next.

For any simplex, the Cayley-Menger determinant is a number computed from its edge lengths. In Euclidean geometry, a positive value means the edge lengths can be realized as an actual simplex in flat space, while a zero value means the simplex is degenerate, collapsed flat. The Recognition Science framework's machine-checked library of formal theorems proves that for the (3,2) type with a timelike squared edge length of -alpha * a^2 (where alpha > 0), the Cayley-Menger determinant equals -(12 * alpha + 7) * a^8. This expression is strictly negative for all alpha >= 0.

This negative result is not a failure but a feature. It means the Lorentzian (3,2) simplex, with its timelike edges, cannot be embedded in ordinary four-dimensional Euclidean space. This is precisely why the framework introduces the Wick rotation, a map that flips the sign of the timelike squared lengths, turning the Lorentzian simplex into a Euclidean one. The library proves this rotation is an involution, meaning applying it twice returns the original configuration, and that it acts as the algebraic continuation alpha to -alpha.

After this rotation, the determinant becomes positive exactly when alpha > 7/12. This threshold is the exact point where the Euclideanized simplex becomes non-degenerate. At the physical point alpha = 1, both types reduce to the regular 4-simplex, with a determinant of 5 * a^8. The framework proves these facts with zero unproven axioms, but it does not claim that a negative determinant proves non-realizability; that classical equivalence, cm4 > 0 if and only if embeddable in R^4, is not formalized in this library for four dimensions.

/-- THEOREM: type (3,2) Lorentzian Cayley-Menger determinant,
`cm4 = -((12*alpha + 7) * a^8)`. -/
theorem cm4_lorentzian_threeTwo (a alpha : ℝ) :
    cm4 (lorentzianSqEdges CausalPentType.threeTwo a alpha)
      = -((12 * alpha + 7) * a ^ 8) := by
  unfold cm4 cmDetN
  rw [cmMatrixN_lorentzian_threeTwo, det_pentMatrix32]
  ring

/-- THEOREM: type (3,2) Lorentzian Cayley-Menger determinant,
`cm4 = -((12*alpha + 7) * a^8)`. -/
theorem cm4_lorentzian_threeTwo (a alpha : ℝ) :
    cm4 (lorentzianSqEdges CausalPentType.threeTwo a alpha)
      = -((12 * alpha + 7) * a ^ 8) := by
  unfold cm4 cmDetN
  rw [cmMatrixN_lorentzian_threeTwo, det_pentMatrix32]
  ring

/-- THEOREM: the Wick map is involutive (Mathlib `Function.Involutive`
packaging of `wick_wick`). -/
theorem wick_involutive (ty : CausalPentType) :
    Function.Involutive (wick ty) :=
  fun x => wick_wick ty x

/-- THEOREM (core, exact range): for `0 < a`, the Euclideanized causal
4-simplex satisfies the CM positivity criterion `cm4 > 0` if and only if
`alpha > alphaMin ty`.  The threshold is exact in both directions.
Reading note: `cm4 > 0` is `9216 * simplexVolumeSqN > 0` (proved above);
its classical equivalence to embeddability in R^4 is not formalized in
this repo for n = 4 (3D analog: `Geometry/TetrahedronRealization.lean`). -/
theorem cm4_euclidean_pos_iff (ty : CausalPentType) (a alpha : ℝ)
    (ha : 0 < a) :
    0 < cm4 (euclideanSqEdges ty a alpha) ↔ alphaMin ty < alpha := by
  have h8 : 0 < a ^ 8 := pow_pos ha 8
  cases ty
  · rw [cm4_euclidean_fourOne, alphaMin_fourOne]
    constructor
    · intro h
      by_contra hle
      push_neg at hle
      have hprod : 0 ≤ (3 - 8 * alpha) * a ^ 8 :=
        mul_nonneg (by linarith) h8.le
      linarith
    · intro h
      have hprod : 0 < (8 * alpha - 3) * a ^ 8 :=
        mul_pos (by linarith) h8
      linarith
  · rw [cm4_euclidean_threeTwo, alphaMin_threeTwo]
    constructor
    · intro h
      by_contra hle
      push_neg at hle
      have hprod : 0 ≤ (7 - 12 * alpha) * a ^ 8 :=
        mul_nonneg (by linarith) h8.le
      linarith
    · intro h
      have hprod : 0 < (12 * alpha - 7) * a ^ 8 :=
        mul_pos (by linarith) h8
      linarith

What this page does not claim

The theorem does not prove that a negative Cayley-Menger determinant implies the simplex cannot exist in spacetime. The library does not formalize the classical equivalence between a positive determinant and embeddability in R^4 for n = 4. The framework does not claim to have derived the action-level Lorentzian continuation in four dimensions.

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/CausalSimplex4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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