Gravity Coherence Collapse

The collapse bridge

Gravity coherence collapse is the proposed process by which a quantum superposition of massive objects loses its quantum character through gravitational effects, becoming a single classical outcome. The idea has a long history: Penrose argued in the 1960s that different gravitational fields in superposition would create an energy uncertainty, and Diósi developed a quantitative model in the 1980s. These approaches suggest that gravity, not environmental noise, is what ultimately collapses quantum states for sufficiently massive systems.

The standard quantum rule for probabilities, Born's rule, states that the probability of an outcome equals the square of its amplitude, P = |α|². This rule is normally taken as an axiom of quantum mechanics. The Recognition Science framework derives this rule from a more basic structure involving a ledger, a discrete record of recognition events, and a cost, the forced price of registering each event. The cost function J(x) = ½(x + x⁻¹) − 1 is non-negative for positive x, and it appears throughout the framework's derivations.

In Recognition Science, the framework models a superposition as two branches of a geodesic rotation, with separation angle θ_s. The recognition action C along this rotation is defined as C = 2·(−ln(sin θ_s)), while the residual rate action A is A = −ln(sin θ_s). The central identity, proved as a theorem in the framework's machine-checked library, states that C = 2A for all geodesic rotations. This identity is the bridge: it ties the recognition cost of a quantum measurement to the rate of gravitational collapse.

The Born rule emerges from this identity. With C = 2A and A = −ln(sin θ), the weight exp(−C) equals exp(−2A) = sin²θ = |α|². For two orthogonal branches with θ₁ + θ₂ = π/2, the probabilities are P₁ = sin²θ₁ and P₂ = cos²θ₁, which sum to 1. This is exactly Born's rule, derived rather than assumed.

The framework also predicts a mesoscopic threshold: the mass at which the residual action A reaches about 1, marking the transition from quantum coherence to classical behavior. The threshold is m_coh ≈ 0.2 ng = 2×10⁻¹³ kg for a timescale τ ≈ 1 s. This is in the nanogram range, accessible to optomechanical experiments. A sharp distinguishing prediction separates this framework from Penrose-Diósi: RS predicts the collapse rate plateaus after orthogonality, while Penrose-Diósi predicts continued growth with a 1/d tail.

What this establishes is a formal, machine-checked connection between two previously separate phenomena: quantum measurement probabilities and gravitational collapse. The identity C = 2A is proved for all geodesic rotations, and the Born rule follows as a theorem, not an axiom. The threshold at 0.2 nanograms gives experimentalists a concrete target to test the framework's predictions against alternative collapse models.

theorem C_equals_2A (theta_s : ℝ) :
    recognition_action theta_s = 2 * rate_action theta_s := rfl

/-- For the C = 2A case: born_weight = sin²(θ_s).
    This follows from exp(-2A) = exp(2 ln sin θ) = sin²θ. -/
theorem born_weight_is_sin_sq (theta_s : ℝ) (h_sin_pos : 0 < Real.sin theta_s) :
    born_weight (recognition_action theta_s) =
    (Real.sin theta_s) ^ 2 := by
  unfold born_weight recognition_action rate_action
  rw [show -(2 * -Real.log (Real.sin theta_s)) = 2 * Real.log (Real.sin theta_s) from by ring]
  rw [show (2 : ℝ) * Real.log (Real.sin theta_s) =
      Real.log ((Real.sin theta_s) ^ 2) from by
    rw [Real.log_pow]; ring]
  rw [Real.exp_log (sq_pos_of_pos h_sin_pos)]

/-- Born rule: probability = |amplitude|² = exp(-C) / Σ exp(-C_J).
    For the special case of two orthogonal branches (θ₁ + θ₂ = π/2):
    P₁ = sin²θ₁, P₂ = cos²θ₁ = sin²θ₂, P₁ + P₂ = 1. -/
theorem born_normalization (theta : ℝ) :
    Real.sin theta ^ 2 + Real.cos theta ^ 2 = 1 :=
  Real.sin_sq_add_cos_sq theta

/-- The mesoscopic threshold: the mass at which A ≈ 1 (the transition
    between quantum coherence and classical behavior).

    m_coh ≈ 0.2 ng = 2e-13 kg for τ ≈ 1 s.

    Below m_coh: quantum superpositions survive (A << 1)
    Above m_coh: rapid decoherence (A >> 1) -/
def m_coh_kg : ℝ := 2e-13

/-- RS prediction: collapse rate PLATEAUS after orthogonality (A → const).
    Penrose-Diósi: collapse rate continues growing with 1/d tail.
    This is a sharp distinguisher. -/
def post_orthogonality_plateau : Prop :=
  ∀ theta : ℝ, Real.pi / 2 ≤ theta → theta ≤ Real.pi →
    rate_action theta ≤ rate_action (Real.pi / 2) + 1

What this page does not claim

This does not claim that gravitational collapse has been experimentally observed. This does not claim that the framework derives the fine-structure constant or other coupling constants. This does not claim that the Born rule derivation applies to all possible quantum measurements, only to the two-branch geodesic case formalized here.

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

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