Gravity Coercive Projection Coercive Projection Cert

The certification

The coercive projection certificate is a formal object in the Recognition Science library: a machine-checked collection of four inequalities that together assert a certain energy functional is well-behaved. In plain terms, it certifies that the model's energy cannot run away to negative infinity, that a key constant exceeds one, that the operator is positive, and that a prefactor is positive. The certificate is a theorem, not a definition: the library proves that these four conditions hold for the specific constants chosen in the model.

The four conditions are concrete. The coercivity constant is 49/162, which lies between 0.30 and 0.31, and the certificate proves it is positive. The net constant K_net is (9/7)^2 = 81/49, which is greater than 1. The operator positivity condition states that for any function f and any weight w with w ≥ 1, the product w * f^2 is nonnegative. The prefactor C_ilg_prefactor is phi^(-3/2), where phi is the golden ratio, and the certificate proves it is positive. These are not arbitrary assumptions: each is a proved theorem in the library.

The certificate also bundles a pressure equivalence: the model's modified Poisson equation is equivalent to standard Poisson with an effective pressure source p = w * rho_b. This is a structural result, not a numerical prediction. The certificate does not claim that the model matches any particular galaxy rotation curve, nor that the pressure source is physical. It only certifies that the mathematical conditions hold.

What the certificate does not claim is as important as what it does. It does not prove that gravity is a coercive projection in the physical world; it proves that a specific formal model satisfies those inequalities. It does not claim that the constants are derived from first principles; the constants are chosen and then verified. It does not claim that the pressure equivalence is a physical pressure; it is a mathematical equivalence. The certificate is a foundation stone, not a complete theory.

theorem coercive_projection_cert : CoerciveProjectionCert where
  coercivity_positive := c_coercive_pos
  net_above_one := K_net_gt_one
  operator_positive := fun w f hw => energy_bounded_below w f hw (sq_nonneg f)
  prefactor_positive := C_ilg_prefactor_pos

theorem c_coercive_approx : (0.30 : ℚ) < c_coercive ∧ c_coercive < (0.31 : ℚ) := by
  unfold c_coercive; constructor <;> norm_num

theorem c_coercive_pos : (0 : ℚ) < c_coercive := by
  unfold c_coercive; norm_num

theorem K_net_value : K_net = 81 / 49 := by
  unfold K_net; norm_num

theorem K_net_gt_one : (1 : ℚ) < K_net := by
  unfold K_net; norm_num

/-- The ILG weight operator is positive: if w(x) >= 1 for all x,
    then <f, w*f> >= ||f||^2 (in L^2 inner product sense).

    We formalize this pointwise: w(x) * f(x)^2 >= f(x)^2. -/
theorem operator_positivity_pointwise (w_val f_val : ℝ) (hw : 1 ≤ w_val) :
    f_val ^ 2 ≤ w_val * f_val ^ 2 := by
  nlinarith [sq_nonneg f_val]

/-- 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 certificate does not prove that physical gravity is a coercive projection. The certificate does not claim that the constants are derived from first principles. The certificate does not claim that the effective pressure is a physical pressure.

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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Derived articles

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