Gravity Equivalence Principle Rs Consistent With Microscope

The zero prediction

The equivalence principle is the observation that a feather and a hammer, dropped in a vacuum, fall at the same rate. In physics this is expressed as the equality of inertial mass, the resistance to acceleration, and gravitational mass, the charge that couples to a gravitational field. Experiments have tested this sameness to extraordinary precision. The MICROSCOPE satellite mission, for instance, compared the accelerations of two different test masses in orbit and found any difference to be smaller than one part in 10^15.

Recognition Science (RS) is a framework that starts from a single forced cost function, a rule that assigns a price to any deviation from a balanced state. The framework's central theorem proves this cost function is unique. From that uniqueness, the framework derives the equivalence principle: both inertial and gravitational mass are defined as functionals of this same cost function, so for any body, at any strength of deviation, the two masses must be equal. The framework's library contains a machine-checked theorem, rs_consistent_with_microscope, that formalizes the consistency of this prediction with the MICROSCOPE result.

The theorem itself is a statement about a number. It takes the Eötvös parameter, a standard measure of equivalence-principle violation, sets it to zero, and proves that this zero is less than the MICROSCOPE bound of 10^-15. The proof is a direct computation: it uses the definition of the Eötvös parameter and the fact that zero divided by anything nonzero is zero. The declaration does not model the satellite, its instruments, or any measurement uncertainty. It is a formal statement that the framework's exact prediction of zero violation is compatible with the experimental upper bound.

The strength of the claim is that it is exact, not approximate. The framework's cost function has higher-order corrections beyond the simple quadratic form, but since both inertial and gravitational mass are derived from the same function, these corrections affect both equally and cancel in the ratio. The prediction is therefore zero violation at all orders, not just in the weak-field limit. This makes the prediction maximally falsifiable: any measured nonzero Eötvös parameter would require a modification of the framework's single-cost-function foundation.

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

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

/-- The MICROSCOPE experiment measures η < 10⁻¹⁵.
    RS predicts η = 0 exactly. This is consistent with experiment and
    makes the strongest possible prediction: any nonzero η falsifies RS. -/
def microscope_bound : ℝ := 1e-15

/-- For the EP, what matters is NOT the size of corrections, but whether
    they affect inertial and gravitational mass DIFFERENTLY.
    In SingleSourceMassTheory, they cannot differ: both use J_full. -/
theorem ep_exact_all_orders (T : SingleSourceMassTheory) (x : ℝ) (hx : 0 < x) :
    T.inertial_mass x = T.gravitational_mass x :=
  single_source_equivalence T x hx

What this page does not claim

The declaration does not model the MICROSCOPE satellite or its measurement process. The theorem does not prove that the framework's cost function is the correct description of physical reality. No claim is made that the framework derives the fine-structure constant or any other dimensionless coupling.

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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