Gravity Ilgreal Exponent Enhancement Enhancement Real Above One

The radial enhancement theorem

In the Recognition Science framework's gravity model, a radial enhancement is a multiplier applied to a Newtonian velocity-squared term. The declaration enhancement_real_above_one establishes, as a proved theorem, that for any positive radius R and any positive reference radius r0, the value of the enhancement function w_real(R, r0, α) = 1 + C · (R/r0)^α is strictly greater than 1. Here C is a positive constant locked by the framework, and α is a real exponent. The proof is machine-checked in the framework's library of formal theorems, with zero unproved axioms.

The theorem's content is plain: the enhancement factor never dips to or below unity. It is always strictly above one, meaning the modified velocity-squared term is always larger than the bare Newtonian term it multiplies. This is not a numerical approximation or a heuristic; it is a formal result derived from the definition of w_real and the positivity of C and the ratio R/r0. The theorem holds for every positive choice of R and r0, with no further conditions on α beyond the definition not requiring positivity.

The theorem is one of four structural facts proved in the same module. The others state that the enhancement is strictly increasing in R, that it grows without bound as R tends to infinity, and that the resulting velocity-squared term dominates the Newtonian one everywhere. Together they describe the qualitative behavior of the real-exponent radial enhancement: it is a positive, strictly increasing, unbounded multiplier. The certificate structure bundles all four facts into a single object whose existence is also proved.

What the theorem does not claim is equally important. It does not assert that this enhancement is the correct description of gravity in the physical world. It does not claim that the exponent α takes any particular value, nor that the constant C matches any measured quantity. The theorem is purely structural: given the definition of w_real, the inequality follows. Whether this model matches observation is a separate empirical question, not settled by this declaration.

theorem enhancement_real_above_one (R r0 α : ℝ)
    (hR : 0 < R) (hr0 : 0 < r0) :
    1 < w_real R r0 α := by
  unfold w_real
  have hC : 0 < C_lock := C_lock_pos
  have hd : 0 < R / r0 := div_pos hR hr0
  have hpow : 0 < (R / r0) ^ α := Real.rpow_pos_of_pos hd α
  have : 0 < C_lock * (R / r0) ^ α := mul_pos hC hpow
  linarith

theorem enhancement_real_above_one (R r0 α : ℝ)
    (hR : 0 < R) (hr0 : 0 < r0) :
    1 < w_real R r0 α := by
  unfold w_real
  have hC : 0 < C_lock := C_lock_pos
  have hd : 0 < R / r0 := div_pos hR hr0
  have hpow : 0 < (R / r0) ^ α := Real.rpow_pos_of_pos hd α
  have : 0 < C_lock * (R / r0) ^ α := mul_pos hC hpow
  linarith

theorem enhancement_real_above_one (R r0 α : ℝ)
    (hR : 0 < R) (hr0 : 0 < r0) :
    1 < w_real R r0 α := by
  unfold w_real
  have hC : 0 < C_lock := C_lock_pos
  have hd : 0 < R / r0 := div_pos hR hr0
  have hpow : 0 < (R / r0) ^ α := Real.rpow_pos_of_pos hd α
  have : 0 < C_lock * (R / r0) ^ α := mul_pos hC hpow
  linarith

What this page does not claim

The theorem does not assert that this enhancement is the correct description of gravity in the physical world. It does not claim that the exponent α takes any particular value, nor that the constant C matches any measured quantity. The theorem is purely structural and does not settle any empirical question about the model's agreement with observation.

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/ILGRealExponentEnhancement.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: