Gravity Propagates at the Speed of Light

1. Setting

This is the simplest gravitational prediction in RS. Light and gravity do not propagate through two media. They ride the same recognition ledger, so their native speed is the same tick-per-voxel speed. The Lean theorem is a reflexivity proof because both speeds are defined as 1 in RS-native units; the empirical content is that nature never reveals a separate gravitational clock.

2. Equations

(E1)

$$ c_{\mathrm{grav}}^{\mathrm{RS}} = c_{\mathrm{EM}}^{\mathrm{RS}} = 1 $$

Native RS speed equality.

(E2)

$$ \left|\frac{c_{\mathrm{grav}}-c}{c}\right|\lesssim 10^{-15} $$

GW170817 observational bound.

3. Prediction or structural target

• c_grav/c_EM: predicted 1 (dimensionless); empirical 1 within ~10^-15. Source: GW170817 / GRB 170817A

GW170817 is the flagship test; it already supports equality to roughly one part in 10^15.

4. Formal anchor

The primary anchor is Gravity.PropagationSpeed..c_grav_eq_c_RS.

    GW170817 confirmed c_grav = c to 10⁻¹⁵. RS predicts exact equality. -/
theorem c_grav_eq_c_RS : c_grav_RS = c_RS := rfl

/-- Propagation-speed structural marker implies gravity/light equality in RS units. -/
theorem propagation_implies_equal_speed (h : c_grav_RS = c_RS) :
    c_grav_RS = c_RS :=
  h

/-- When both speeds are defined from the same tick rate, their ratio is 1. -/
theorem speed_ratio_unity : c_grav_RS / c_RS = 1 := by

5. What is inside the Lean module

Key theorems:

• c_grav_eq_c_RS
• propagation_implies_equal_speed
• speed_ratio_unity
• propagation_equality_forced

Key definitions:

• c_RS
• c_grav_RS

6. Derivation chain

• c_grav_eq_c_RS - c_grav = c_RS
• speed_ratio_unity - Speed ratio unity
• propagation_equality_forced - Equality forced
• propagation_implies_equal_speed - Propagation implies equal speed

7. Falsifier

Any reproducible measurement of gravitational waves propagating at a speed different from electromagnetic waves in the same local inertial frame refutes this derivation.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

10. Audit path

To audit gravity-equals-light, start with the primary Lean anchor Gravity.PropagationSpeed.c_grav_eq_c_RS. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.