Chemistry Bond Angles Octahedral Formula Cosine

The octahedral cosine

In chemistry, an octahedral molecule places six bonds around a central atom, and the bonds point toward the corners of a regular octahedron. The angle between any two adjacent bonds is 90 degrees, and the cosine of that angle is 0. This is a textbook fact taught in every introductory chemistry course, and it holds for molecules such as sulfur hexafluoride (SF₆).

The Recognition Science framework, a system that derives physical structure from a forced cost function, contains a formula for the optimal bond angle when n equivalent bonds surround a central atom. The formula, cos(θ) = -1/(n-1), reproduces the familiar angles for linear (n=2, 180°), trigonal planar (n=3, 120°), and tetrahedral (n=4, 109.47°) geometries. The framework's machine-checked library of formal theorems proves that for n=6, this formula gives cos(θ) = -1/5, which corresponds to an angle of about 101.54 degrees.

That result, established by the declaration octahedral_formula_cosine, is a purely algebraic theorem about the framework's own definition. It shows that the general cosine formula, when applied to six bonds, yields the value -1/5. The theorem does not claim that real octahedral molecules have a 101.54 degree bond angle. In fact, the framework's own documentation explicitly notes that real octahedral geometry uses 90 degrees, and that the formula's output for n=6 is a known mismatch.

Within the framework, this mismatch is not treated as a failure of the framework itself, but as an indication that the simple formula -1/(n-1) does not capture the full physics of octahedral coordination. The framework treats conventional chemistry as itself derivable, so reproducing a known value like 109.47° is a derivation, not a defeat. The octahedral case, where the formula diverges from observation, remains a target for a more complete derivation that accounts for the actual geometry.

The practical lesson is that the framework's formula works for the first three geometries but breaks down at n=6. A reader should not take the octahedral cosine theorem as a prediction about real molecules. It is a statement about a definitional choice within the framework, and the gap between -1/5 and 0 marks a boundary of the current derivation.

/-- Octahedral (n=6) gives cos = -1/5 from formula, but real octahedral uses 90°. -/
theorem octahedral_formula_cosine : optimalBondCosine 6 = -1/5 := by
  simp only [optimalBondCosine]
  norm_num

/-- Linear geometry (n=2) has angle = 180° (cos = -1). -/
theorem linear_cosine : optimalBondCosine 2 = -1 := by
  simp only [optimalBondCosine]
  norm_num

/-- Trigonal planar (n=3) has angle ≈ 120° (cos = -1/2). -/
theorem trigonal_cosine : optimalBondCosine 3 = -1/2 := by
  simp only [optimalBondCosine]
  norm_num

/-- Tetrahedral (n=4) has angle ≈ 109.47° (cos = -1/3). -/
theorem tetrahedral_cosine : optimalBondCosine 4 = -1/3 := by
  simp only [optimalBondCosine]
  norm_num

/-- Optimal cosine of bond angle for n equivalent bonds.
    cos(θ_opt) = -1/(n-1) for n ≥ 2. -/
def optimalBondCosine (n : ℕ) : ℝ :=
  if n ≤ 1 then 0 else -1 / (n - 1 : ℝ)

What this page does not claim

The theorem does not claim that real octahedral molecules have a 101.54 degree bond angle. The framework does not claim to have derived the correct octahedral angle from its first principles. The formula cos(θ) = -1/(n-1) is not claimed to hold for all coordination numbers, only for n = 2, 3, and 4.

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/Chemistry/BondAngles.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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