Gravity Coherence Fall Coherence Defect

The coherence defect

The coherence defect is a number that measures how much a gravitational potential changes across the height of an extended object. In the Recognition Science framework, it is defined as the absolute difference in total potential between the object's head and its feet. If the potential is perfectly flat across the object, the defect is zero. If the object stands in a field that grows stronger with height, the defect is positive.

The definition is precise. An object is modeled as having a center of mass position and a positive extent, its half-height. A processing field supplies a potential function of position. The total potential in a frame accelerating with acceleration a is the gravitational potential plus an inertial term a times the height coordinate. The defect is then the absolute value of the potential at the head minus the potential at the feet, after both are evaluated in that accelerating frame. A machine-checked lemma simplifies this to the closed form: the defect equals the absolute value of two times the extent times the sum of the potential's derivative and the acceleration a.

The central result is a theorem: for any extended object in any smooth potential field, there exists a unique acceleration a that makes the defect zero. That unique acceleration is exactly the negative gradient of the potential, which is the gravitational acceleration g. In plain terms, falling with the field's local gradient cancels the defect. Standing still in a gravitational field leaves a nonzero defect; free fall removes it.

In Recognition Science, this is the formal content of why free fall feels like nothing. The framework models gravity not as a force pulling objects down, but as the requirement to accelerate in order to keep the local processing environment constant. The defect is the mismatch that appears when you resist that requirement. The theorem proves that there is exactly one acceleration that restores coherence, and it is the one free fall provides.

The declaration does not claim that this is a new prediction about gravity. It does not derive the value of G or any coupling constant. It does not say that physical objects actually experience a processing field. The framework models gravity this way; the theorem establishes the internal consistency of that model. The empirical content, if any, would be a separate claim about whether the real world behaves this way.

/-- 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)

/-- Closed form for the linearized coherence defect:
    `coherence_defect = | 2 * extent * (∂ϕ + a) |`. -/
lemma coherence_defect_simplify (field : ProcessingField) (obj : ExtendedObject) (a : ℝ) :
    coherence_defect field obj a =
      abs (2 * obj.extent * (deriv field.phi obj.h_cm + a)) := by
  rw [coherence_defect_expand]
  congr 1
  ring

/-- 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

The theorem does not claim that physical objects experience a processing field; that is a modeling choice. It does not derive the value of the gravitational constant G or any other coupling constant. It does not predict any new gravitational phenomenon beyond what the model already assumes.

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)

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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