Gravity Coherence Fall Processing Field

The processing field

A processing field, in the Recognition Science framework, is a mathematical structure that assigns a real number, called the potential, to every position in space. The framework models this with a single function Φ(h), where h is a position. The field itself is just that function; it carries no other data. The framework's library defines it as a structure with one component, the potential function, and nothing more.

The framework uses this field to study what happens to an object that has physical size, not just a point. It defines an extended object by its center of mass and a positive extent, a measure of its length. The key idea is the coherence defect: the difference in total potential between the object's head and feet, taken as an absolute value. If the potential is flat across the object, the defect is zero. If the potential varies, the defect is positive.

The framework's central theorem about this field, proved in its machine-checked library of formal theorems, is that there exists a unique acceleration a that makes the coherence defect zero. The theorem states this as an existence and uniqueness claim: there is exactly one such acceleration. The proof shows that this acceleration is exactly the negative derivative of the potential, which the framework identifies as the gravitational acceleration g = −∇Φ.

The physical interpretation the framework draws is that gravity is not a force but a requirement. Standing still in a gravitational field means experiencing a coherence defect, an inconsistency in the processing environment across your own body. Free-falling, accelerating at exactly the rate the field demands, cancels that defect and restores coherence. This is why free fall feels like nothing: you are in the coherent state. The framework's library states this interpretation explicitly, but it is an interpretation, not a theorem.

In Recognition Science, the declaration ProcessingField establishes the vocabulary for this account. It defines what a field is, what an extended object is, and what coherence means in this context. It does not claim that this is the only way to think about gravity, nor does it derive the value of any physical constant. The theorem it proves is about the mathematics of a given field, not about which field describes the actual universe.

structure ProcessingField where
  /-- Potential function Φ(h) -/
  phi : Position → ℝ

/-- Coherence Defect: Variance of the potential across the object.
    If Potential is flat, Defect is 0.
-/
def coherence_defect (field : ProcessingField) (obj : ExtendedObject) (a : ℝ) : ℝ :=
  -- Difference in potential between Head (z = +extent) and Feet (z = -extent)
  let pot_head := total_potential_in_frame field obj a obj.extent
  let pot_feet := total_potential_in_frame field obj a (-obj.extent)
  abs (pot_head - pot_feet)

/-- Falling (Acceleration) Restores Coherence.

    Theorem: There exists a unique acceleration `a` that reduces the
    linear Coherence Defect to zero.

    This `a` is exactly the gravitational acceleration `g = -∇Φ`.
-/
theorem falling_restores_coherence (field : ProcessingField) (obj : ExtendedObject) :
    ∃! a : ℝ, coherence_defect field obj a = 0 := by
  -- We want |2 * e * (ϕ' + a)| = 0 ⇒ a = -ϕ' (since e > 0).
  -- This is exactly "Falling with acceleration = -Gradient".
  use -(deriv field.phi obj.h_cm)
  constructor
  · -- Existence
    -- | 2 * e * (ϕ' + (-ϕ')) | = |0| = 0
    simp [coherence_defect_simplify]
  · -- Uniqueness
    intro a' h_zero
    -- Reduce to a product equals zero
    have h0 : 2 * obj.extent * (deriv field.phi obj.h_cm + a') = 0 := by
      simpa [coherence_defect_simplify, abs_eq_zero] using h_zero
    -- From |x| = 0 we get x = 0
    -- Since obj.extent > 0, we have 2 * obj.extent ≠ 0
    have h_extent_pos : (0 : ℝ) < 2 * obj.extent := by
      have htwo : (0 : ℝ) < 2 := by norm_num
      exact mul_pos htwo obj.extent_pos
    have h_extent_ne : 2 * obj.extent ≠ 0 := ne_of_gt h_extent_pos
    -- So (deriv field.phi obj.h_cm + a') = 0
    have h2 : deriv field.phi obj.h_cm + a' = 0 := by
      have := mul_eq_zero.mp h0
      cases this with
      | inl h => exact absurd h h_extent_ne
      | inr h => exact h
    -- Therefore a' = -(deriv field.phi obj.h_cm)
    linarith

/-- Falling (Acceleration) Restores Coherence.

    Theorem: There exists a unique acceleration `a` that reduces the
    linear Coherence Defect to zero.

    This `a` is exactly the gravitational acceleration `g = -∇Φ`.
-/
theorem falling_restores_coherence (field : ProcessingField) (obj : ExtendedObject) :
    ∃! a : ℝ, coherence_defect field obj a = 0 := by
  -- We want |2 * e * (ϕ' + a)| = 0 ⇒ a = -ϕ' (since e > 0).
  -- This is exactly "Falling with acceleration = -Gradient".
  use -(deriv field.phi obj.h_cm)
  constructor
  · -- Existence
    -- | 2 * e * (ϕ' + (-ϕ')) | = |0| = 0
    simp [coherence_defect_simplify]
  · -- Uniqueness
    intro a' h_zero
    -- Reduce to a product equals zero
    have h0 : 2 * obj.extent * (deriv field.phi obj.h_cm + a') = 0 := by
      simpa [coherence_defect_simplify, abs_eq_zero] using h_zero
    -- From |x| = 0 we get x = 0
    -- Since obj.extent > 0, we have 2 * obj.extent ≠ 0
    have h_extent_pos : (0 : ℝ) < 2 * obj.extent := by
      have htwo : (0 : ℝ) < 2 := by norm_num
      exact mul_pos htwo obj.extent_pos
    have h_extent_ne : 2 * obj.extent ≠ 0 := ne_of_gt h_extent_pos
    -- So (deriv field.phi obj.h_cm + a') = 0
    have h2 : deriv field.phi obj.h_cm + a' = 0 := by
      have := mul_eq_zero.mp h0
      cases this with
      | inl h => exact absurd h h_extent_ne
      | inr h => exact h
    -- Therefore a' = -(deriv field.phi obj.h_cm)
    linarith

What this page does not claim

This answer does not claim the processing field is a physical field in the sense of general relativity. This answer does not claim the framework derives the value of the gravitational constant G. This answer does not claim the theorem applies to any field other than the one given.

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

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