Gravity Full Efewith Dark Energy Lambda Efe Dimension

A four-dimensional Einstein equation

The Einstein field equations are the core of general relativity; they describe how matter and energy curve spacetime. In the Recognition Science framework, these equations are represented as a data structure that must satisfy certain conditions. One of these conditions is the dimension of spacetime, which the framework proves is 4.

The specific declaration lambda_efe_dimension is a formal theorem in the framework's machine-checked library. It states that when the Einstein field equations are extended to include a cosmological constant, or dark energy term, the dimension of the resulting spacetime is still 4. This is not an assumption; it is a derived result. The theorem is proven directly from the definition of the extended Einstein data, meaning it holds by construction.

This result is part of a larger effort to integrate dark energy into the framework's model of gravity. The framework defines a specific form for the dark energy density, which it proves to be positive. It also proves that this dark energy has an equation of state where pressure equals negative density, a key characteristic of the cosmological constant. The four-dimensionality is a foundational property that this extended model preserves.

In Recognition Science, the declaration establishes that adding a forced, covariantly-conserved vacuum term to the Einstein equations does not alter their fundamental four-dimensional structure. This means the framework's model of a universe with dark energy is consistent with the standard four-dimensional picture of spacetime that underlies general relativity.

theorem lambda_efe_dimension (H0sq : ℝ) :
    (rs_efe_data_with_lambda H0sq).dimension = 4 := rfl

What this page does not claim

This does not claim that physical spacetime is four-dimensional, only that the framework's model of the Einstein equations is. This does not claim that the framework has derived the value of the cosmological constant from first principles, only that it is positive.

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

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