Gravity Seven Gaps Causal Simplex4 D Is Timelike Three Two Eq Cross Slice

The (3,2) simplex

A 4-simplex is the four-dimensional analogue of a triangle: five points, with every pair joined by an edge, giving ten edges in total. In causal dynamical triangulations, a standard approach to quantum gravity, spacetime is built by gluing such simplices together. The recognition framework, which models how a discrete record of events can generate physical structure, formalizes the two basic building blocks of a four-dimensional spacetime: the (4,1) simplex, with four vertices on one spatial slice and one on the next, and the (3,2) simplex, with three on one slice and two on the next.

The declaration isTimelike_threeTwo_eq_crossSlice establishes a purely combinatorial fact about the (3,2) simplex. In this simplex, the edges that connect vertices on different time slices are called timelike, while those lying within a single slice are spacelike. The theorem proves that exactly six of the ten edges are timelike, and the remaining four are spacelike. This is not a physical assumption but a consequence of the chosen labeling: with three vertices on the earlier slice and two on the later one, there are 3 × 2 = 6 cross-slice pairs, and each such pair forms a timelike edge.

The result is verified by the machine-checked library of formal theorems using a finite case analysis: it checks all ten edges one by one and confirms the count. This combinatorial layer is a definitional choice, a model of the standard conventions used in causal dynamical triangulations, not a derivation from deeper principles. The framework's library proves the count exactly, with no gaps in the reasoning.

What the declaration does not claim is equally important. It does not establish that a (3,2) simplex with these edge lengths can actually be embedded in four-dimensional Euclidean space. That question, whether a positive Cayley-Menger determinant guarantees real geometric existence, remains open in this library for four dimensions. The theorem only certifies the edge counts, not the geometric realizability of the simplex as a whole.

/-- THEOREM (by `decide`): type (3,2) has exactly 6 timelike edges. -/
theorem timelike_count_threeTwo :
    (Finset.univ.filter fun e : Fin 10 =>
      isTimelike CausalPentType.threeTwo e = true).card = 6 := by
  decide

/-- THEOREM (by `decide`): type (3,2) has exactly 4 spacelike edges
(3 within the lower slice triangle plus 1 within the upper slice pair). -/
theorem spacelike_count_threeTwo :
    (Finset.univ.filter fun e : Fin 10 =>
      isTimelike CausalPentType.threeTwo e = false).card = 4 := by
  decide

What this page does not claim

The declaration does not prove that a (3,2) simplex with these edge lengths is geometrically realizable in four-dimensional Euclidean space. The declaration does not derive the edge counts from the recognition framework's cost function or forcing chain. The declaration does not address the action-level Wick rotation 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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