Gravity Seven Gaps Simplicial Class One Tet Complex Is Simplicial

The one-tetrahedron witness

In combinatorial geometry, a simplicial complex is a way to build a space from simple pieces: points, line segments, triangles, and tetrahedra, glued together so that the intersection of any two pieces is a face of both. The framework's Recognition Science works with a particular presentation of these objects, a ledger (a discrete record of events) that lists vertices, edges, and tetrahedra. The declaration oneTetComplex_isSimplicial proves that the simplest possible nonempty ledger, one tetrahedron with its four vertices and all six edges, meets the framework's four formal conditions for being simplicial.

The four conditions are the combinatorial content of being a simplicial 3-complex. First, no edge may have the same vertex at both ends. Second, no two distinct edges may connect the same pair of vertices. Third, each tetrahedron must have four distinct vertices. Fourth, every pair of vertices in a tetrahedron must be joined by an edge in the ledger. The proof is a finite check: the single tetrahedron has only four vertices and six edges, so a computer can verify each condition directly. The declaration is a theorem in the framework's machine-checked library of formal theorems, with zero unproved assumptions.

The result matters because it shows the simplicial subclass is not empty in a trivial way. The framework had already proved that the collection of all simplicial complexes has positive size, but that proof alone could have rested on the empty complex, the object with no vertices, edges, or tetrahedra. The one-tetrahedron witness supplies a concrete, nonempty example, so the positivity claim stands on real ground. It also opens the door to larger constructions: any bounded complex that contains this single tetrahedron as a piece is automatically simplicial as well.

What the declaration does not claim is just as important. It does not assert that every tetrahedron in a simplicial complex must have its full set of six edges present; the definition only requires that the edges between a tetrahedron's own vertices exist. It does not address triangles or higher-dimensional faces, because the ledger's data structure carries only vertices, edges, and tetrahedra. And it does not say anything about the physical meaning of these complexes in a theory of gravity. The theorem is purely combinatorial: it certifies that a specific small object belongs to a precisely defined class.

/-- **THEOREM (non-vacuous simplicial witness).**  The single-tetrahedron
complex is simplicial: distinct edge endpoints, no multi-edges, injective
corners, and every corner pair realized by one of the six skeleton edges.
Kernel-checked by `decide` on the finite index types (`Fin 4`, `Fin 6`,
`Fin 1`); no `native_decide`. -/
theorem oneTetComplex_isSimplicial : IsSimplicial oneTetComplex := by
  decide

/-- **The simplicial predicate** on a bounded incidence configuration:
no degenerate edges, no multi-edges, injective tetrahedron corners, and
skeleton closure (every corner pair of every tet is an edge of the
complex). -/
def IsSimplicial {B : ℕ} (K : BoundedComplex B) : Prop :=
  (∀ e : Fin K.nE, (K.edgeVerts e).1 ≠ (K.edgeVerts e).2) ∧
  (∀ e e' : Fin K.nE,
    sameUnorderedPair (K.edgeVerts e) (K.edgeVerts e') → e = e') ∧
  (∀ t : Fin K.nT, Function.Injective (K.tetVerts t)) ∧
  (∀ (t : Fin K.nT) (i j : Fin 4), i ≠ j →
    ∃ e : Fin K.nE,
      sameUnorderedPair (K.edgeVerts e) (K.tetVerts t i, K.tetVerts t j))

/-- **THEOREM (non-vacuous simplicial witness).**  The single-tetrahedron
complex is simplicial: distinct edge endpoints, no multi-edges, injective
corners, and every corner pair realized by one of the six skeleton edges.
Kernel-checked by `decide` on the finite index types (`Fin 4`, `Fin 6`,
`Fin 1`); no `native_decide`. -/
theorem oneTetComplex_isSimplicial : IsSimplicial oneTetComplex := by
  decide

/-- **THEOREM.**  Every cap `B ≥ 6` admits a genuinely 3-dimensional
simplicial configuration (one tetrahedron, full skeleton): the subclass
positivity is not carried by the empty complex alone. -/
theorem exists_simplicial_with_tet (B : ℕ) (hB : 6 ≤ B) :
    ∃ K : SimplicialComplex B, 0 < K.1.nT :=
  ⟨⟨relax hB oneTetComplex, relax_isSimplicial hB oneTetComplex_isSimplicial⟩,
    Nat.one_pos⟩

What this page does not claim

The declaration does not assert that every tetrahedron in a simplicial complex must have its full set of six edges present. The declaration does not address triangles or higher-dimensional faces, because the ledger's data structure carries only vertices, edges, and tetrahedra. The declaration does not say anything about the physical meaning of these complexes in a theory of gravity.

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/SimplicialClass.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.

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