Gravity Energy Processing Bridge Jcost Quadratic Ratio

The quadratic bridge

The declaration Jcost_quadratic_ratio is a theorem in the machine-checked library of formal theorems of Recognition Science. It states a precise inequality about a function called J-cost, which the framework defines as J(x) = (x + 1/x)/2 - 1 for positive x. The theorem says that when the input is written as 1 + ε, with ε a small positive number, the value of J-cost is at most ε²/2. In symbols: J(1 + ε) ≤ ε²/2.

The classical fact this leans on is an exact identity, also proved in the library: J(1 + ε) equals ε² divided by 2(1 + ε). For small ε, the denominator is close to 1, so the exact value is nearly ε²/2. The inequality then says the exact value never exceeds that simpler quadratic expression. This is the bridge to kinetic energy: in classical mechanics, kinetic energy for a slow object is approximately (1/2)mv², a quadratic form in speed. Here, the framework's cost of deviation from balance behaves like a quadratic energy for small deviations.

The theorem's conditions matter. It requires ε to be positive and greater than -1, so the input 1 + ε stays positive. The inequality is an upper bound, not an equality, and it holds only in this small-deviation regime. The framework uses this to model how an energy distribution creates a processing field, which it identifies with a gravitational source. But the theorem itself does not prove that gravity exists; it proves a bound on a cost function.

What the declaration does not claim is equally clear. It does not say J-cost equals kinetic energy exactly; it gives an upper bound. It does not extend to large deviations, where the exact identity would make the bound fail. And it does not, by itself, derive the fine-structure constant or any other physical constant from scratch. The theorem is a step in a larger framework, not a complete physical theory.

/-- For small ε, J(1+ε) ≈ ε²/2. Specifically, the ratio approaches 1. -/
theorem Jcost_quadratic_ratio (ε : ℝ) (hε_neg : -1 < ε) (hε_pos : 0 < ε) :
    Jcost (1 + ε) ≤ ε ^ 2 / 2 := by
  rw [Jcost_one_plus_exact ε hε_neg]
  apply div_le_div_of_nonneg_left (sq_nonneg ε) (by positivity) (by nlinarith)

/-- J-cost exact identity: J(1 + ε) = ε²/(2(1+ε)) for ε > -1.
    This is the bridge between J-cost and the Hamiltonian (kinetic energy ≈ ε²/2). -/
theorem Jcost_one_plus_exact (ε : ℝ) (hε : -1 < ε) :
    Jcost (1 + ε) = ε ^ 2 / (2 * (1 + ε)) := by
  unfold Jcost
  have h1ε : (0 : ℝ) < 1 + ε := by linarith
  have h1ε_ne : (1 + ε) ≠ 0 := ne_of_gt h1ε
  field_simp
  ring

/-- For small ε, J(1+ε) ≈ ε²/2. Specifically, the ratio approaches 1. -/
theorem Jcost_quadratic_ratio (ε : ℝ) (hε_neg : -1 < ε) (hε_pos : 0 < ε) :
    Jcost (1 + ε) ≤ ε ^ 2 / 2 := by
  rw [Jcost_one_plus_exact ε hε_neg]
  apply div_le_div_of_nonneg_left (sq_nonneg ε) (by positivity) (by nlinarith)

What this page does not claim

The theorem does not prove that J-cost equals kinetic energy exactly. The theorem does not extend to large deviations where the bound fails. The theorem does not derive the fine-structure constant or any other physical constant.

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/EnergyProcessingBridge.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: