Cost Classical Results Theta Min Spec Inequality

A bound on the minimum angle

The theorem theta_min_spec_inequality is a statement about a sphere and a logarithm. It says: if you have a positive number A, and an angle θ between zero and a right angle, and the quantity −log(sin θ) is no larger than A, then θ must be at least arcsin(e^−A). In plainer terms, it gives a lower bound on an angle once you know an upper bound on a related logarithmic quantity. The angle is measured in radians, and the logarithm is the natural logarithm.

The statement is a piece of classical mathematics, not a physical claim. It is a theorem in real analysis: the sine function, the logarithm, and the arcsine function are all standard objects, and the inequality follows from their elementary properties. The proof uses the fact that sine is increasing on the interval from zero to a right angle, and that the logarithm is increasing everywhere. The result is a useful technical lemma: it converts a bound on a logarithmic expression into a bound on an angle, which is a common maneuver in geometry and analysis.

In the Recognition Science framework, this lemma appears as a supporting result. The framework models recognition events as points on a sphere, and the angle between two such points is a measure of how different the two recognitions are. The quantity −log(sin θ) is related to the cost of distinguishing two events. The theorem then says: if the cost of distinguishing two events is at most A, then the angle between them cannot be too small. This is a precise, formal statement of a trade-off: a limited budget for distinguishing events forces a minimum separation between them.

The theorem is proved in the framework's machine-checked library of formal theorems, and the proof is complete. It does not, however, say anything about what A should be, or what the actual minimum angle is for any physical process. It is a conditional statement: given a bound A, here is a bound on θ. The framework may later supply a value for A from other principles, but this theorem does not do so. It is a tool, not a conclusion.

theorem theta_min_spec_inequality :
    ∀ (Amax θ : ℝ), 0 < Amax → 0 < θ → θ ≤ π/2 →
      (- Real.log (Real.sin θ) ≤ Amax) →
      θ ≥ Real.arcsin (Real.exp (-Amax)) := by
  intro Amax θ _hAmax hθpos hθle hlog
  have h1 : Real.log (Real.sin θ) ≥ -Amax := by linarith
  have hsin_pos : 0 < Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθpos (by linarith [Real.pi_pos])
  have h2 : Real.sin θ ≥ Real.exp (-Amax) := by
    have := Real.exp_log hsin_pos
    rw [← this]
    exact Real.exp_le_exp.mpr h1
  have h3 : Real.arcsin (Real.sin θ) = θ := by
    apply Real.arcsin_sin
    · linarith
    · linarith [Real.pi_pos]
  rw [← h3]
  exact Real.arcsin_le_arcsin h2

theorem theta_min_spec_inequality :
    ∀ (Amax θ : ℝ), 0 < Amax → 0 < θ → θ ≤ π/2 →
      (- Real.log (Real.sin θ) ≤ Amax) →
      θ ≥ Real.arcsin (Real.exp (-Amax)) := by
  intro Amax θ _hAmax hθpos hθle hlog
  have h1 : Real.log (Real.sin θ) ≥ -Amax := by linarith
  have hsin_pos : 0 < Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθpos (by linarith [Real.pi_pos])
  have h2 : Real.sin θ ≥ Real.exp (-Amax) := by
    have := Real.exp_log hsin_pos
    rw [← this]
    exact Real.exp_le_exp.mpr h1
  have h3 : Real.arcsin (Real.sin θ) = θ := by
    apply Real.arcsin_sin
    · linarith
    · linarith [Real.pi_pos]
  rw [← h3]
  exact Real.arcsin_le_arcsin h2

theorem theta_min_spec_inequality :
    ∀ (Amax θ : ℝ), 0 < Amax → 0 < θ → θ ≤ π/2 →
      (- Real.log (Real.sin θ) ≤ Amax) →
      θ ≥ Real.arcsin (Real.exp (-Amax)) := by
  intro Amax θ _hAmax hθpos hθle hlog
  have h1 : Real.log (Real.sin θ) ≥ -Amax := by linarith
  have hsin_pos : 0 < Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθpos (by linarith [Real.pi_pos])
  have h2 : Real.sin θ ≥ Real.exp (-Amax) := by
    have := Real.exp_log hsin_pos
    rw [← this]
    exact Real.exp_le_exp.mpr h1
  have h3 : Real.arcsin (Real.sin θ) = θ := by
    apply Real.arcsin_sin
    · linarith
    · linarith [Real.pi_pos]
  rw [← h3]
  exact Real.arcsin_le_arcsin h2

What this page does not claim

The theorem does not assign a numerical value to A or to the minimum angle. The theorem does not state that recognition events are always separated by at least this angle. The theorem does not depend on any Recognition Science-specific axiom; it is a classical result.

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