Gravity Coercive Projection

Coercive projection

Coercive projection is a mathematical property of an energy functional, a rule that assigns a number to a possible state of a system. A functional is coercive when its values grow without bound as the state moves away from a central region, which forces the existence of a state with the lowest possible energy. The classic picture is a bowl: a ball rolling to the bottom finds a unique resting point because the bowl's walls rise steeply enough. The framework applies this idea to gravity, showing that a specific energy picture has exactly one lowest-energy state, not a family of equally good ones.

The classical starting point is the Poisson equation, which describes how a gravitational potential responds to a mass density. In the standard formulation, the potential's second derivative is proportional to the density. The framework builds a modified version of this equation, where the density is multiplied by a weight function. The key result is that this modified equation is equivalent to the standard one if the weight is absorbed into an effective pressure term. This pressure, defined as the weight times the density, is the mechanism that lets the modified equation reproduce standard gravity.

In Recognition Science, the weight function is not free. The framework derives it from a deeper principle: reality keeps a ledger, a discrete record of recognition events, and the cost of recognition is forced by a proved theorem. This cost function, J(x) = (x + 1/x)/2 - 1, leads to a chain of results that fix the weight operator. The framework's theorems certify that this operator is positive, meaning it never reduces the energy contribution of any state, and that the energy functional is bounded below. Together, these facts guarantee the existence of a unique minimizer, the state of lowest energy.

The framework also formalizes a constant, K_net = (9/7)^2, which is greater than one. This constant appears in a bound on how much the energy can deviate from its minimum. The framework's claim is that no per-galaxy retuning of the weight function is consistent with the energy minimization principle. The definition of no_retuning states that any different weight function that is pointwise larger would not be a valid alternative, because the original one already achieves the minimum. This is a formal statement, not an empirical fit.

The practical consequence is a testable prediction. The framework's energy minimization principle predicts that the same weight function applies to all galaxies, with no free parameters tuned to individual observations. This is a strong claim that can be falsified by finding a galaxy whose rotation curve requires a different weight. The framework does not prove that the prediction matches any particular galaxy's data; it establishes the mathematical consistency of the principle, leaving the comparison to measurements as an empirical check.

theorem c_coercive_value : c_coercive = 49 / 162 := rfl

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

/-- 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⟩

/-- 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]

/-- 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 no-retuning condition is consistent with operator positivity:
    if w_global(x) >= 1 for all x, the energy at w_global is bounded
    but finite, so a unique minimizer exists. -/
theorem no_retuning_consistent (w : ℝ → ℝ) (hw : ∀ x, 1 ≤ w x) :
    ∀ x f : ℝ, 0 ≤ w x * f ^ 2 :=
  fun x f => energy_bounded_below (w x) f (hw x) (sq_nonneg f)

What this page does not claim

This module does not prove that the predicted weight function matches any measured galaxy rotation curve. The uniqueness of the minimizer is a mathematical property, not a statement about the physical stability of the universe. The coercivity constant c = 49/162 is a derived constant, not a free parameter fitted to data.

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)

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

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