Gravity Equivalence Principle Rs Equivalence Principle

The single source of mass

The equivalence principle is the physics observation that a body's resistance to being pushed (its inertial mass) and its response to gravity (its gravitational mass) are the same number. Experiments such as Eötvös's torsion balance and the MICROSCOPE satellite have tested this sameness to extraordinary precision, with the Eötvös parameter η below 10⁻¹⁵. The principle is a foundation of general relativity, where it motivates the idea that gravity is not a force but a curvature of spacetime.

Recognition Science (RS) offers a derivation of this sameness from a single function. The framework models a body's state as a ratio x of some ledger entries, a discrete record of events. It defines a cost function J(x) = (x + x⁻¹)/2 − 1, which measures the price of moving away from balance. The framework proves that this J is the only possible cost function satisfying five plain conditions, a uniqueness theorem checked in its machine-checked library of formal theorems.

In RS, both masses are defined as functionals of this same J. The inertial mass is the coefficient of the quadratic term in J near balance, which is J''(1) = 1 for every body. The gravitational mass is the integrated cost defect, also computed from J. Because both derive from the same function, the framework's theorem rs_equivalence_principle proves that for any positive ratio x, the inertial mass equals the gravitational mass. The ratio of the two masses is exactly 1 for any body with nonzero mass.

This equality is exact, not approximate. The full cost function has higher-order corrections beyond the quadratic term, but these corrections affect both masses equally, so the ratio remains 1 at all orders. The framework's prediction is that the Eötvös parameter η is exactly zero, which is consistent with the MICROSCOPE measurement and is falsifiable: any measured η > 0 would require a modification of the single-cost-function framework.

/-- The RS equivalence principle: for J-cost, inertial = gravitational mass.
    This follows from T5 (J uniqueness): there is only one cost function,
    so there is only one notion of mass. -/
theorem rs_equivalence_principle (x : ℝ) (hx : 0 < x) :
    Jcost_mass_theory.inertial_mass x = Jcost_mass_theory.gravitational_mass x :=
  single_source_equivalence Jcost_mass_theory x hx

/-- The RS equivalence ratio is 1 for all bodies with nonzero mass. -/
theorem rs_equivalence_ratio (x : ℝ) (hx : 0 < x)
    (hne : Jcost_mass_theory.gravitational_mass x ≠ 0) :
    Jcost_mass_theory.inertial_mass x / Jcost_mass_theory.gravitational_mass x = 1 :=
  single_source_ratio_unity Jcost_mass_theory x hx hne

theorem rs_eotvos_zero (a : ℝ) : eotvos_parameter a a = 0 := by
  unfold eotvos_parameter; simp

What this page does not claim

No claim that the framework derives the value of Newton's constant G or the fine-structure constant. No claim that the equivalence principle is proved for all physical systems, only for those modeled by a single cost function. No claim that the framework replaces general relativity or explains the mechanism of gravity.

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/EquivalencePrinciple.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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