Gravity Coherence Collapse Coherence Collapse Cert

The certificate

Quantum mechanics and gravity are usually treated as separate realms. This certificate, a formal object in the Recognition Science framework, packages a chain of definitions and proofs that connect the two. It is a statement in the framework's machine-checked library of formal theorems, and it asserts four facts: a geometric identity, a positivity condition, a normalization condition, and a numerical threshold.

The central identity is C = 2A. Here C is the recognition action, a cost accumulated along a path, and A is the residual rate action, a quantity built from the angle between two branches of a geodesic rotation. For any separation angle θ, the recognition action equals twice the residual rate action. The proof is a definitional equality: recognition_action is defined as twice rate_action, so the identity holds by unfolding definitions. This is not an approximation; it is exact for all angles.

The certificate also records that the Born weight, exp(-C), is always positive, and that for two orthogonal branches the probabilities sin²θ and cos²θ sum to one. The fourth fact is a numerical threshold: the coherence mass m_coh is defined as 2e-13 kg, about 0.2 nanograms, and the certificate proves this is less than 1e-9 kg, placing it in the nanogram range. The framework interprets this as the mass scale where quantum superpositions give way to classical behavior, a transition that experiments with optomechanical systems might probe.

What the certificate does not claim is equally important. It does not prove that gravitational collapse actually happens, nor that the Born rule is the only possible probability rule. It establishes a formal identity and some positivity and normalization facts within the framework's definitions. The physical interpretation, that this identity explains quantum measurement or gravitational collapse, is a hypothesis, not a theorem. The certificate itself is a formal object; its physical significance is an open question.

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

/-- Born weight is positive. -/
theorem born_weight_pos (C_I : ℝ) : 0 < born_weight C_I := Real.exp_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 threshold is in the nanogram range — accessible to
    optomechanical experiments (Aspelmeyer, Romero-Isart). -/
theorem m_coh_nanogram_range : m_coh_kg < 1e-9 ∧ m_coh_kg > 1e-15 := by
  unfold m_coh_kg; constructor <;> norm_num

What this page does not claim

The certificate does not prove that gravitational collapse occurs in nature. The certificate does not derive the Born rule from first principles; it shows a formal identity. The certificate does not establish the physical significance of the nanogram threshold.

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)

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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