Mathematics Differential Geometry From Rs Rs Spacetime Dim Eq 4

The four-dimensional statement

In the mathematics of recognition, the framework's discrete record of events, the central object is a smooth three-dimensional manifold, a space that locally resembles ordinary three-dimensional Euclidean space. The declaration rsSpacetimeDim_eq_4 is a theorem in the framework's machine-checked library of formal theorems. It states that the dimension of this recognition manifold, called rsDimension and set to 3, plus one, equals 4.

The plus one is the time coordinate. The framework models spacetime as a pseudo-Riemannian manifold, a space where distances can be negative as well as positive, which is the mathematical setting for Lorentzian spacetime, the kind used in general relativity. The theorem rsSpacetimeDim_eq_4 proves that this spacetime dimension is exactly 4, written as 4 = 3 + 1. A companion theorem, rsSpacetimeDim_lorentzian, restates the same equality.

The library also counts five canonical differential geometric structures: smooth manifold, Riemannian, pseudo-Riemannian, Kähler, and symplectic. The theorem diffGeoStructureCount proves this count equals 5. The framework's certificate, diffGeoCert, bundles these two facts, the five structures and the four-dimensional spacetime, into a single object.

What the theorem does not claim is that the physical universe has four dimensions. The framework's model is a definitional choice, not a measurement of the cosmos. The theorem proves an internal consistency: if you accept the framework's starting points, the arithmetic 3 + 1 = 4 follows. It does not prove that space has three dimensions or that time is one-dimensional. Those are inputs to the model, not outputs of the theorem.

The practical upshot is that the framework's geometric language is compatible with the standard four-dimensional spacetime of physics. A reader who wants to use the framework's structures for relativity can do so without a dimensional mismatch. The theorem is a small but necessary bridge between the framework's discrete ledger and the continuous geometry of the physical world.

theorem rsSpacetimeDim_eq_4 : rsSpacetimeDim = 4 := by decide

theorem diffGeoStructureCount : Fintype.card DiffGeoStructure = 5 := by decide

def diffGeoCert : DiffGeoCert where
  five_structures := diffGeoStructureCount
  spacetime_4 := rsSpacetimeDim_eq_4

What this page does not claim

The theorem does not prove that physical space has three dimensions. The theorem does not establish that time is one-dimensional. The theorem does not derive the metric tensor or the field equations of general relativity.

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

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