Gravity Equivalence Principle Equivalence Trivial When Same

The formal core

The equivalence principle in physics states that inertial mass, the resistance to acceleration, and gravitational mass, the charge that responds to gravity, are the same number for every body. Experiments such as Eötvös's torsion balance in 1889 and the MICROSCOPE satellite in 2017 have confirmed this equality to extraordinary precision. The Recognition Science framework approaches the principle not as an empirical coincidence but as a structural consequence of how it defines mass in the first place.

Within the framework, both kinds of mass are extracted from a single function called the cost, which measures the price of a recognition event, a discrete record of change in a system. The framework proves that this cost function is unique, taking the form J(x) = (x + 1/x)/2 - 1. A mass theory is then defined as any structure that reads both inertial and gravitational mass from this same cost function. The theorem single_source_equivalence shows that in any such theory, the two masses are identical for all positive states.

The declaration equivalence_trivial_when_same is a much narrower statement. It proves that for any nonzero number m, the ratio m/m equals 1. This is a tautology of arithmetic, not a physical law. Its role in the library is to close a logical gap: once the framework has established that the two masses are equal, the ratio of those equal masses being one follows immediately. The declaration formalizes that final step, ensuring the chain from cost uniqueness to the Eötvös parameter being zero is complete.

The framework's actual claim about physics is stronger and more specific. Its theorem rs_equivalence_principle states that for the unique cost function J, inertial and gravitational mass are equal for all positive ratios, not merely in a weak-field approximation. The framework predicts the Eötvös parameter, which measures the fractional difference in acceleration between two bodies, is exactly zero at all orders. This is a falsifiable prediction: any measured nonzero value would require abandoning the single-cost-function assumption.

What the declaration does not do is prove the equivalence principle itself. It assumes the masses are already equal and derives the ratio. The physical content lives in the definition of a single-source mass theory and in the uniqueness theorem for the cost function. The trivial theorem is the arithmetic coda, not the argument.

theorem equivalence_trivial_when_same :
    ∀ m : ℝ, m ≠ 0 → m / m = 1 := fun _ hm => div_self hm

/-- 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]

/-- 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

What this page does not claim

The declaration equivalence_trivial_when_same does not prove the equivalence principle; it only derives the ratio from an assumed equality. The framework does not claim that the cost function J is the only possible function satisfying the five conditions without also assuming the five conditions themselves. The framework's prediction of zero Eötvös parameter is not a measurement but a falsifiable consequence of its definitions.

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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