Gravity Zero Parameter Gravity Kappa Ne Zero

The nonzero coupling

In the Recognition Science framework, gravity is not treated as a fundamental force but as a large-scale curvature of a discrete structure called the ledger, a record of recognition events. The framework's library of machine-checked theorems contains a declaration, kappa_ne_zero, which establishes that the Einstein gravitational constant, written κ, is not zero. This constant appears in the Einstein field equations, the standard equations of general relativity that describe how matter curves spacetime. The declaration proves κ ≠ 0, a fact that follows directly from a companion theorem showing κ is positive.

The value of κ in this framework is not a free parameter. It is derived as κ = 8φ⁵, where φ is the golden ratio, approximately 1.618. This expression is a definition in the framework's library, and the theorem kappa_ne_zero confirms it is nonzero. The proof is short: it relies on the positivity of the number 8 and the positivity of φ raised to the fifth power. The framework also provides numerical bounds, showing κ lies between 85.6 and 90.4, based on bounds for φ⁵.

This declaration is a small but necessary piece of a larger claim within the framework: that gravity emerges from the ledger's curvature, and that the Einstein field equations appear as a continuum limit. The framework further asserts that the equivalence principle, the idea that inertial and gravitational mass are the same, follows automatically because all mass comes from the same cost function. The nonzero coupling is a consistency check, ensuring the derived equations of gravity are not degenerate.

What kappa_ne_zero does not do is establish that gravity itself exists, that the Einstein field equations are correct, or that the framework's derivation of gravity is physically valid. The theorem only proves a mathematical property of a defined constant within the framework. It does not compare κ to any measured value, nor does it claim the framework's value for κ matches experiment. The declaration is a formal statement about a definition, not an empirical result.

/-- The derived Einstein coupling cannot vanish. -/
theorem kappa_ne_zero : kappa_rs ≠ 0 := ne_of_gt kappa_pos

/-- κ > 0. -/
theorem kappa_pos : 0 < kappa_rs := by
  unfold kappa_rs
  apply mul_pos (by norm_num : (0:ℝ) < 8)
  exact pow_pos phi_pos 5

/-- The RS prediction for the Einstein gravitational coupling: κ = 8φ⁵.
    This is derived, not assumed. -/
noncomputable def kappa_rs : ℝ := 8 * phi ^ 5

/-- Einstein coupling is explicitly the derived `8*phi^5` factor. -/
theorem kappa_rs_closed_form : kappa_rs = 8 * phi ^ 5 := rfl

/-- Numerical bounds on κ = 8φ⁵.
    From 10.7 < φ⁵ < 11.3 and κ = 8φ⁵: 85.6 < κ < 90.4. -/
theorem kappa_bounds : (85.6 : ℝ) < kappa_rs ∧ kappa_rs < 90.4 := by
  unfold kappa_rs
  have h1 := phi_fifth_bounds.1
  have h2 := phi_fifth_bounds.2
  constructor <;> nlinarith

What this page does not claim

The declaration does not prove gravity exists or that the Einstein field equations are physically correct. The declaration does not compare κ to any measured value from experiments. The declaration does not establish the framework's derivation of gravity is valid.

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/ZeroParameterGravity.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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