Gravity Coercive Projection Energy Bounded Below

The bounded energy

The statement energy_bounded_below is a theorem in the framework's machine-checked library of formal theorems. It says: for any two real numbers w and f, if w is at least 1 and f squared is nonnegative (which it always is), then the product w times f squared is nonnegative. In plainer terms, if you scale a nonnegative quantity by a factor of at least one, the result cannot be negative. This is a basic fact about multiplication, and the proof in the library is a direct application of the rules of ordered arithmetic.

The theorem's importance comes from what it is used for. In the framework's model of gravity, the energy of a configuration is written as an integral of w times f squared, where f represents a field and w is a weight operator. The theorem energy_bounded_below (a lower bound on the energy) proves that this energy cannot go below zero, because every point in the integral is nonnegative. This is the first step toward showing that the energy has a minimum, a state of lowest possible energy. The library then combines this with other results, such as a coercivity constant of 49/162, to argue that this minimum is unique and that no per-galaxy retuning is needed.

The theorem is a formal statement about real numbers, not a physical law. It does not claim that gravity is actually described by this energy functional. It does not state that the minimum exists, only that the energy is bounded below. It does not say anything about the value of the minimum, only that it is not negative. The theorem is a piece of mathematics, a lemma that supports a larger argument.

/-- Operator positivity implies the energy functional is bounded below. -/
theorem energy_bounded_below (w_val f_val : ℝ) (hw : 1 ≤ w_val) (hf : 0 ≤ f_val ^ 2) :
    0 ≤ w_val * f_val ^ 2 := by
  exact mul_nonneg (by linarith) hf

/-- The ILG modified Poisson equation is EQUIVALENT to the standard Poisson
    equation with an effective pressure source:

    ILG:      nabla^2 Phi = 4*pi*G * a^2 * (w * rho_b * delta_b)
    Standard: nabla^2 Phi = 4*pi*G * a^2 * p

    where p = w * rho_b * delta_b is the "effective pressure."

    This equivalence means ILG is NOT a modification of GR's field equations
    but rather a modification of the SOURCE SIDE only. -/
structure PressureEquivalence where
  w_kernel : ℝ → ℝ
  rho_b : ℝ → ℝ
  delta_b : ℝ → ℝ
  effective_pressure : ℝ → ℝ
  equiv : ∀ x, effective_pressure x = w_kernel x * rho_b x * delta_b x

theorem c_coercive_value : c_coercive = 49 / 162 := rfl

What this page does not claim

The theorem does not prove that gravity is described by this energy functional. The theorem does not show that the energy minimum exists, only that it is bounded below. The theorem does not specify the value of the minimum energy.

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/CoerciveProjection.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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