Foundation Logic Real Constants Kappa Einstein L

The Einstein constant on the recovered line

The Einstein constant κ (kappa) is the number that ties the curvature of spacetime to the energy and momentum inside it in general relativity. In the standard units used by physicists, it equals 8πG divided by the speed of light to the fourth power, where G is Newton's gravitational constant. The declaration kappaEinsteinL in the Recognition Science framework's machine-checked library of formal theorems defines this same quantity inside a separate, recovered number system called LogicReal, and then proves that translating it back to the ordinary real numbers gives exactly the framework's existing real-valued constant.

The definition itself is a formal mirror: kappaEinsteinL is set to the framework's established real constant κ_einstein, carried into LogicReal by a function called fromReal. The accompanying theorem, toReal_kappaEinsteinL, states that converting kappaEinsteinL back to the real numbers recovers Constants.kappa_einstein. This is a transport result, not a new physical derivation. It shows that the framework's constants, which were originally defined on the standard real line, remain coherent when expressed in the LogicReal structure, and that no information is lost in the round trip.

In Recognition Science, the framework models physical structure from a starting point where reality keeps a discrete record of recognition events, and the cost of recognition is forced. Within that account, the constants such as the golden ratio φ, the reduced Planck constant ħ, and the gravitational constant G emerge from a chain of proved theorems rather than being free parameters. The kappaEinsteinL declaration belongs to this same family: it takes the framework's derived gravitational constant and places it into the LogicReal setting, preserving the value exactly.

What the declaration does not claim is equally precise. It does not derive the Einstein constant from first principles, nor does it prove that general relativity is correct. It establishes only a formal equivalence between two representations of the same number. The physical content, that κ links curvature to energy and momentum, comes from the classical theory of general relativity, not from this declaration. The framework's own derivation of the gravitational constant is a separate result, and the comparison of that constant against measured values is an empirical check, not a theorem.

/-- Recovered Einstein coupling. -/
def kappaEinsteinL : LogicReal := fromReal Constants.kappa_einstein

@[simp] theorem toReal_kappaEinsteinL :
    toReal kappaEinsteinL = Constants.kappa_einstein := toReal_fromReal _

What this page does not claim

This declaration does not derive the Einstein constant from the framework's forcing chain. It does not prove the physical content of general relativity. It does not claim the framework's gravitational constant matches any measured value.

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/Foundation/LogicRealConstants.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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