Gravity Coherence Collapse Born Weight Is Sin Sq

The sine-squared weight

The Born rule is one of quantum mechanics' most famous formulas: the probability of a measurement outcome equals the square of the amplitude's magnitude, P = |α|². The declaration born_weight_is_sin_sq in the Recognition Science framework's machine-checked library of formal theorems derives this rule from a specific geometric setup. It shows that when a quantum state splits into two branches separated by an angle θ, the weight of one branch is exactly sin²θ, which is the square of the sine of that angle.

The derivation starts with a cost function, the recognition cost, a measure of how much effort reality's ledger, a discrete record of events, must spend to register a change. Along a path where two branches rotate apart by angle θ, this cost accumulates as C = -2 ln(sin θ). The framework also defines a residual action A = -ln(sin θ), the leftover rate after the main cost is paid. The central identity, established as a theorem, is that C = 2A: the total recognition cost is exactly twice the residual action for any rotation angle.

From this identity, the Born weight emerges. The weight of an outcome is defined as exp(-C), an exponential decay of the cost. Substituting C = 2A gives exp(-2A) = exp(2 ln sin θ) = sin²θ. The theorem born_weight_is_sin_sq formalizes this chain: for any angle with positive sine, the weight equals sin²θ. This is not a numerical approximation; it is an identity that holds for all such angles, and the machine-checked proof confirms every algebraic step.

In Recognition Science, this result connects two domains. On one side, it links to gravitational collapse: the residual action A is tied to geodesic separation, the bending of paths in curved spacetime. On the other, it produces the quantum measurement rule. The framework models the collapse of a quantum superposition as a recognition event whose cost determines probabilities. The threshold where this becomes significant is around 0.2 nanograms, a mass accessible to optomechanical experiments, where the framework predicts a plateau in collapse rate that differs from the Penrose-Diósi model.

The declaration does not claim to explain why the universe uses this rule, nor does it derive the full Born rule for arbitrary multi-outcome measurements. It establishes a precise equivalence for the two-branch, sine-squared case within the framework's definitions. The physical bridge from recognition cost to actual gravitational collapse remains an open target, not a proven theorem.

/-- 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)]

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

/-- The recognition action C along a geodesic rotation is TWICE the
    residual rate action A. This is the central identity connecting
    quantum measurement (Born weights from C) to gravitational
    collapse (rate from A).

    Derivation: For a geodesic rotation by angle θ_s,
    - The recognition cost accumulated is C = -2 ln(sin θ_s)
    - The residual action is A = -ln(sin θ_s)
    - Therefore C = 2A identically.

    This holds for ALL geodesic rotations, not just specific angles. -/
noncomputable def recognition_action (theta_s : ℝ) : ℝ := 2 * rate_action theta_s

/-- The residual rate action A for a two-branch geodesic rotation
    with separation angle θ_s. A = -ln(sin θ_s) for 0 < θ_s < π/2. -/
noncomputable def rate_action (theta_s : ℝ) : ℝ := -Real.log (Real.sin theta_s)

/-- 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

What this page does not claim

The full Born rule for arbitrary multi-outcome measurements is not derived here. The physical bridge from recognition cost to gravitational collapse is not proven; it remains an open target. The declaration does not explain why the universe follows this rule, only that it follows from the framework's definitions.

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.

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