Gravity Coercive Projection Pressure Equiv From W

The pressure equivalence

In classical physics, gravity is a force between masses. Some alternative theories recast it as a pressure effect, where the presence of matter alters the local pressure of some background medium. The declaration pressure_equiv_from_w in the framework's machine-checked library of formal theorems establishes a precise mathematical identity in one such model: for any given weight function w, density rho, and auxiliary field delta, there exists a pressure function p such that at every point x, the pressure equals the product w(x) * rho(x) * delta(x). In plain terms, the theorem proves that the model can always express its gravitational effect as a local pressure term.

The theorem is a formal existence result. It constructs the pressure function explicitly: define p(x) as w(x) * rho(x) * delta(x), and the identity holds by definition. The proof is a single step of unfolding definitions, tagged THEOREM in the framework's library. This is not an empirical claim about the universe; it is a statement about the internal consistency of a particular mathematical model. The model itself, called the coercive projection law of gravity, is a definitional choice, not a derived physical law.

The theorem does not claim that gravity actually is a pressure phenomenon in the physical world. It does not assert that the weight function w is the true dark matter profile, nor that the auxiliary field delta has a physical interpretation. It only shows that within this model, the mathematical language of pressure is rich enough to describe any configuration. The existence of the pressure function is guaranteed, but its physical meaning is left entirely open.

What the theorem changes is the expressive power of the model. A reader can now take any density distribution in this framework and know, without further calculation, that an equivalent pressure description exists. This is a useful bridge between two formulations of the same idea. The cost is that the bridge is purely formal: it carries no information about which pressure function is physically correct, only that one exists.

/-- Any ILG kernel with w >= 1 defines a valid pressure equivalence. -/
theorem pressure_equiv_from_w (w rho delta : ℝ → ℝ) :
    ∃ p : ℝ → ℝ, ∀ x, p x = w x * rho x * delta x :=
  ⟨fun x => w x * rho x * delta x, fun _ => rfl⟩

/-- Any ILG kernel with w >= 1 defines a valid pressure equivalence. -/
theorem pressure_equiv_from_w (w rho delta : ℝ → ℝ) :
    ∃ p : ℝ → ℝ, ∀ x, p x = w x * rho x * delta x :=
  ⟨fun x => w x * rho x * delta x, fun _ => rfl⟩

What this page does not claim

The theorem does not claim that gravity is physically a pressure phenomenon. The theorem does not assign a physical meaning to the weight function w or the auxiliary field delta. The theorem does not prove that the coercive projection model itself is a correct description of real gravity.

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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