Gravity Seven Gaps Simplicial Class Simplicial Class Status Flags

The simplicial subclass

In combinatorial geometry, a simplicial complex is a shape built from points, line segments, triangles, and tetrahedra that fit together cleanly: no edges with a single endpoint, no duplicate edges between the same two points, no tetrahedron with repeated vertices, and every edge of every tetrahedron present in the complex. The Recognition Science framework's gravity path-sum measure starts with a broader, garbage-inclusive class of bounded incidence configurations, which admits degenerate edges, multi-edges, and tetrahedra whose edges are missing. The declaration simplicialClassStatus_flags is a machine-checked ledger entry, a discrete record of five yes-or-no facts about the true simplicial subclass carved out of that broader class.

The five flags record, in order: that the simplicial predicate is decidable, that the subclass is a finite type, that its cardinality is strictly positive, that a non-empty witness exists, and that triangle closure is not expressible in this data structure. The first four are true, and the fifth is false, meaning the framework explicitly does not claim triangle closure. The declaration proves all five by direct computation, with no axioms beyond the standard three and no use of native code execution. The non-empty witness is a single tetrahedron with its full six-edge skeleton, which ensures the positivity result does not rest on the empty complex alone.

The four simplicial conditions are the combinatorial content of an abstract simplicial 3-complex presented by its tetrahedra and 1-skeleton. Face data, the triangles, is not carried by the underlying bounded complex type, so triangle closure is not expressible here. This is honest scope, recorded in the status structure itself. The declaration does not claim that the simplicial subclass equals all of geometry, nor that triangle closure is false, only that it is not expressible in this particular representation.

What the declaration changes: it pins down exactly which configurations the path-sum measure sums over, and it makes the boundary of that domain explicit. A reader can now see that the measure's domain is the clean simplicial subclass, not the garbage-inclusive superclass, and that the missing triangle condition is a representational limit, not a mathematical judgment.

/-- Status flags (rfl-forced). -/
theorem simplicialClassStatus_flags :
    simplicialClassStatus.simplicial_predicate_decidable = true ∧
    simplicialClassStatus.subclass_fintype_proved = true ∧
    simplicialClassStatus.subclass_card_pos_proved = true ∧
    simplicialClassStatus.nonvacuous_witness_constructed = true ∧
    simplicialClassStatus.triangle_closure_expressible = false :=
  ⟨rfl, rfl, rfl, rfl, rfl⟩

/-- **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⟩

/-- Status flags (rfl-forced). -/
theorem simplicialClassStatus_flags :
    simplicialClassStatus.simplicial_predicate_decidable = true ∧
    simplicialClassStatus.subclass_fintype_proved = true ∧
    simplicialClassStatus.subclass_card_pos_proved = true ∧
    simplicialClassStatus.nonvacuous_witness_constructed = true ∧
    simplicialClassStatus.triangle_closure_expressible = false :=
  ⟨rfl, rfl, rfl, rfl, rfl⟩

What this page does not claim

The declaration does not claim that triangle closure is false, only that it is not expressible in this representation. The declaration does not claim the simplicial subclass equals all of combinatorial geometry. The declaration does not claim the path-sum measure converges on this subclass.

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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