Gravity Equivalence Principle Equivalence Ratio Unity Structural

The structural ratio

The equivalence principle is the physics observation that a body's inertial mass, its resistance to acceleration, equals its gravitational mass, its response to a gravitational field. Experiments by Loránd Eötvös in the 1880s and the MICROSCOPE satellite in 2017 have confirmed this equality with extraordinary precision. The Recognition Science framework encodes this equality as a formal theorem, and the declaration equivalence_ratio_unity_structural is the final step in that encoding.

The theorem establishes a simple mathematical fact: if two real numbers are equal and the second is nonzero, then their ratio equals one. The proof is a direct rewrite: given m_inertial = m_grav and m_grav ≠ 0, the ratio m_inertial / m_grav becomes m_grav / m_grav, which is 1. This is the structural core of the framework's equivalence principle, but it is important to see what it does and does not do.

In the framework, both masses are defined as functionals of a single cost function J(x) = ½(x + x⁻¹) − 1. The inertial mass is the quadratic coefficient of the restoring cost near balance, and the gravitational mass is the same cost function evaluated at the same point. Because both derive from the same J, they are equal by construction. The theorem single_source_equivalence proves this equality for any theory with a single cost source, and ratio_one_when_equal proves the ratio is one. The declaration equivalence_ratio_unity_structural packages these into a single statement.

What the declaration does not claim is that the framework has derived the physical equivalence principle from first principles. The theorem assumes a SingleSourceMassTheory, a structure where both masses are already defined as the same function. It proves that within such a theory, the ratio is one. The framework's broader claim, that any physical mass theory must have this single-source form because J is unique, is a separate derivation that the pack documents but does not formalize in this theorem.

The declaration also does not claim that the equivalence principle is exact in the sense of matching all experimental data. The framework predicts the Eötvös parameter η = 0 exactly, which is consistent with the MICROSCOPE bound η < 10⁻¹⁵, but this is a prediction, not a measured value. Any nonzero η would falsify the framework, but the theorem itself only establishes the ratio identity within the defined structure.

theorem equivalence_ratio_unity_structural : equivalence_ratio_unity := by
  intro m_i m_g hg heq
  exact ratio_one_when_equal m_i m_g heq hg

/-- In a single-source mass theory, inertial and gravitational mass are
    identical for all positive-ratio states. This is the equivalence
    principle derived from cost uniqueness. -/
theorem single_source_equivalence (T : SingleSourceMassTheory)
    (x : ℝ) (hx : 0 < x) :
    T.inertial_mass x = T.gravitational_mass x := by
  rw [T.inertial_from_cost x hx, T.gravitational_from_cost x hx]

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

theorem rs_consistent_with_microscope :
    eotvos_parameter 9.80665 9.80665 < microscope_bound := by
  rw [rs_eotvos_zero]; unfold microscope_bound; norm_num

What this page does not claim

The theorem does not prove that any physical mass theory must have the single-source form. The theorem does not claim the equivalence principle holds as a matter of empirical measurement. The declaration does not establish that the framework has derived the equivalence principle from more basic postulates.

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)

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Derived articles

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