Foundation Maximal Forcing Rsgravity Universe

The gravity layer

In general relativity, the strength of gravity is set by the Einstein coupling constant, written κ = 8πG/c⁴. It is a number that has to be measured, not derived from theory. The Recognition Science framework's ledger, a discrete record of recognition events, takes a different route: it treats this coupling as a quantity that the framework's own logic forces to a specific value.

The gravity layer is the fifth in a chain of single-constant instantiations. Where an earlier layer pins the electromagnetic coupling, this layer pins the gravitational one. The framework works in its native units, where the recognition length λ_rec and the speed of light c are set to 1, and Planck's constant ħ equals φ⁻⁵, with φ the golden ratio (1 + √5)/2. Under those units, the Einstein coupling κ is forced to the pure number 8·φ⁵, about 47.0. The theorem forced_kappa in the machine-checked library of formal theorems proves this: any candidate value that meets the framework's gate condition must equal 8·φ⁵.

The forcing is not an assumption. The library defines a loose class of every real number as a candidate coupling, and a tightened class where the candidate must equal the framework's native Einstein coupling. Over the loose class, the claim is independent: the value 8·φ⁵ satisfies it, but the number 0 does not. Over the tightened class, the claim is forced. The derivation of G from λ_rec, c, and ħ does the work; the value 8·φ⁵ is not put in by hand. The theorem tightening_Lgrav0_LgravRS_effective records both facts.

This is the gravitational analogue of the alpha layer: a derived physical coupling forced to a parameter-free golden-ratio expression. The library also proves the value is strictly positive, a sanity check that gravity is attractive, and that the entire gravity-layer universe is fully classified: every claim in it is either forced or independent, with no gaps. The result is a concrete prediction: in the framework's units, the Einstein coupling is not a free parameter but a fixed number.

/-- **Einstein coupling as a forced invariant.** Over the RS-native gate, the
coupling equals `8·φ⁵`. Wraps `Constants.kappa_einstein_eq`; the only content is
the parameter-free RS derivation `G = λ_rec²c³/(πℏ)` with `ℏ = φ⁻⁵`. -/
theorem forced_kappa : Forced LgravRS.admissible isKappaClaim := by
  intro k hk
  have hk' : k = kappa_einstein := hk
  show k = 8 * phi ^ (5 : ℝ)
  rw [hk', kappa_einstein_eq]

/-- Over the loose class `Lgrav0`, the value claim is independent: the RS coupling
satisfies it, and `0` does not. -/
theorem kappa_independent_over_Lgrav0 :
    Independent Lgrav0.admissible isKappaClaim := by
  refine ⟨kappa_einstein, 0, ?_, ?_, ?_, ?_⟩
  · trivial
  · trivial
  · show kappa_einstein = 8 * phi ^ (5 : ℝ); exact kappa_einstein_eq
  · intro h
    have h0 : (0 : ℝ) = 8 * phi ^ (5 : ℝ) := h
    have hp := kappa_value_pos
    linarith

/-- The forced value is strictly positive: `8φ⁵ > 0`. -/
theorem kappa_value_pos : 0 < 8 * phi ^ (5 : ℝ) := by
  rw [← kappa_einstein_eq]; exact kappa_einstein_pos

/-- The gravity-layer universe is fully classified. -/
theorem gravUniverse_classifier :
    ∀ C : RealityClaim gravUniverse.Realization,
      InClosure Primitive.lawOfLogic gravUniverse C → ClaimClassification gravUniverse C := by
  intro C hC
  have hCeq : C = isKappaClaim := Set.mem_singleton_iff.mp hC
  subst hCeq
  exact ClaimClassification.forced forced_kappa

What this page does not claim

This page does not claim the framework derives the measured value of G in SI units. This page does not claim the Einstein coupling is forced without the framework's native unit convention. This page does not claim the gravity layer is the last or complete instantiation of the framework.

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/MaximalForcing/RSGravityUniverse.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.

Derived articles

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