Gravity Coherence Fall Position

Position and the coherence defect

In the Recognition Science framework, Position is a plain real number, written as ℝ. It represents a height or location along a single vertical axis. The framework's machine-checked library of formal theorems defines it as an abbreviation, not as a new kind of object. A potential function Φ(h) maps each position to a real value, and an extended object carries a center-of-mass position plus a positive extent, its size. These are the raw ingredients for a statement about gravity.

The framework's library proves one theorem about falling: falling_restores_coherence. It says that for any potential field and any extended object, there exists a unique acceleration a that makes a quantity called the coherence defect equal to zero. That defect measures how much the potential varies across the object's extent. In a uniform field, the defect simplifies to |2 × extent × (∂Φ + a)|. Setting it to zero forces a = -∂Φ, which is exactly the local gravitational acceleration. The theorem is a formal statement about real numbers and derivatives; it is not a measurement of any actual falling body.

The interpretation section of the library reads the result as a picture of gravity. Standing still in a gravitational field leaves a nonzero coherence defect, a mismatch in the potential across your body. Free falling, with acceleration equal to the negative gradient of the potential, cancels that mismatch. The framework's account says this is why free fall feels like nothing: you are in the coherent state, where the potential is locally flat across you. This is a definitional choice about how to model gravity, not a claim about the cause of gravitational forces.

What the declaration does not claim is narrower than it looks. It does not claim that position has any special structure beyond being a real number; it does not claim that the theorem applies to curved spacetime or to fields with nonlinear gradients, since the proof uses a linear approximation around the center of mass; and it does not claim that the interpretation is the only way to read the mathematics. The theorem is a clean statement about a linearized model, and the interpretation is a gloss on that statement.

abbrev Position := ℝ

/-- 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

/-- 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

What this page does not claim

Position has no structure beyond being a real number; it is not a vector or a manifold point. The theorem applies only under the linear approximation around the center of mass, not to curved spacetime or strong fields. The interpretation of free fall as coherence restoration is a definitional choice, not an empirical discovery.

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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