Gravity Parameterization Bridge Accel Mul Tdyn Sq

The orbit bridge

For an object moving in a circle at constant speed, two quantities describe the motion: the acceleration that keeps it turning and the time for one full revolution. Classical mechanics connects them through the radius of the circle. The relation is exact and independent of the object's mass: acceleration multiplied by the square of the orbital period equals four times π squared times the radius. In symbols, if a is the acceleration, T the period, and r the radius, then a·T² = 4π²r.

This identity follows directly from two standard definitions. The acceleration for circular motion is a = v²/r, where v is the speed. The orbital period is T = 2πr/v, the time to complete one revolution. Substituting the second into the first and simplifying gives the relation above. The result holds for any circle, any speed, and any object, provided the speed and radius are nonzero. It is a purely kinematic fact; it does not depend on gravity, mass, or any particular force law.

The same relation appears in the framework's machine-checked library of formal theorems as the declaration accel_mul_Tdyn_sq. The framework models the identity as a proved theorem about real numbers, with the speed and radius assumed nonzero. The proof unfolds the two definitions and simplifies algebraically. This is not a new physical discovery; it is a formal confirmation that the classical relation follows from the definitions without hidden assumptions.

In Recognition Science, the identity is a bridge between two ways of parameterizing gravity. One can describe a system by its acceleration or by its dynamical time, the time for one revolution. The framework's library derives a related identity for a characteristic time T₀ = 2π√(r₀/a₀) and shows how ratios of acceleration and time connect. At a chosen characteristic radius, an exponent on the acceleration ratio maps to twice that exponent on the time ratio. This bridge lets models written in either parameterization be translated exactly.

The theorem does not claim anything about actual orbits beyond the kinematics. It assumes circular motion at constant speed; it does not cover elliptical orbits, changing speeds, or non-circular paths. It says nothing about the cause of the acceleration, whether gravity, a string, or an electric field. The identity is a tool for translating between descriptions, not a statement about which forces exist or how they behave.

What the theorem gives is a clean, checkable relation that ties together two common ways of describing circular motion. Any model that uses acceleration and any model that uses orbital time can be compared through this bridge, as long as the motion is circular. The framework's contribution is not the identity itself but the formal guarantee that it follows from the definitions with no unstated assumptions.

/-- Core identity: \(a\,T_{\rm dyn}^2 = 4\pi^2 r\). -/
theorem accel_mul_Tdyn_sq (v r : ℝ) (hv : v ≠ 0) (hr : r ≠ 0) :
    accel v r * (Tdyn v r)^2 = 4 * (Real.pi ^ 2) * r := by
  unfold accel Tdyn
  field_simp [hv, hr]
  ring

/-- Core identity: \(a\,T_{\rm dyn}^2 = 4\pi^2 r\). -/
theorem accel_mul_Tdyn_sq (v r : ℝ) (hv : v ≠ 0) (hr : r ≠ 0) :
    accel v r * (Tdyn v r)^2 = 4 * (Real.pi ^ 2) * r := by
  unfold accel Tdyn
  field_simp [hv, hr]
  ring

What this page does not claim

The theorem does not cover elliptical orbits, changing speeds, or non-circular paths. The identity says nothing about the cause of the acceleration, whether gravity, a string, or an electric field.

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/ParameterizationBridge.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: