Chemistry Bond Angles Tetra Cos Eq

The tetrahedral cosine

The tetrahedral bond angle is the angle between any two bonds in a molecule where one central atom sits at the center of a regular tetrahedron and four identical bonds point to its corners. That angle is 109.47 degrees, and its cosine is exactly -1/3. The value is a classical result of geometry: a regular tetrahedron inscribed in a cube has corners whose coordinates are permutations of (1,1,1) and (-1,-1,-1), and the dot product of two such vectors gives cos(θ) = -1/3. It is the angle in methane (CH₄), where four hydrogen atoms surround a central carbon atom.

The framework's machine-checked library of formal theorems contains a declaration, tetra_cos_eq, that states this same fact: the cosine of the tetrahedral angle equals -1/3. The angle itself is defined as arccos(-1/3), so the theorem is a direct consequence of the definition of arccosine. The library also proves that this angle lies strictly between 90 degrees and 120 degrees, matching the geometric expectation. The declaration is a formal statement of a known geometric identity, not a new physical prediction.

In Recognition Science, the tetrahedral angle appears in a broader pattern. The framework defines a formula for the optimal cosine of the bond angle for n equivalent bonds: cos(θ) = -1/(n-1). For n=4, this gives -1/3; for n=3, it gives -1/2, the trigonal planar angle of 120 degrees; for n=2, it gives -1, the linear angle of 180 degrees. The framework also connects the tetrahedral angle to the golden ratio through the dodecahedron, and defines a positive bias proxy, 1 - 1/φ, as a dimensionless measure of tetrahedral preference.

The framework's claims about bond angles are falsifiable. The docstring lists explicit criteria: if the cosine of the tetrahedral angle for sp³ carbon were not -1/3, or if the bond angle did not decrease with lone pairs (CH₄ > NH₃ > H₂O), or if the formula cos(θ) = -1/(n-1) failed for other geometries, the derivation would be wrong. The library also notes a known limitation: the formula gives cos(θ) = -1/5 for octahedral geometry (n=6), but the real octahedral angle is 90 degrees.

/-- The tetrahedral cosine is -1/3. -/
theorem tetra_cos_eq : Real.cos tetrahedralAngleRadians = -1/3 := by
  rw [tetrahedralAngleRadians]
  apply Real.cos_arccos
  · norm_num
  · norm_num

/-- The tetrahedral angle is between 90° and 120° (in radians).
    90° = π/2 ≈ 1.571, 120° = 2π/3 ≈ 2.094
    arccos(-1/3) ≈ 1.911 -/
theorem tetra_angle_bounds :
    π/2 < tetrahedralAngleRadians ∧ tetrahedralAngleRadians < 2*π/3 := by
  constructor
  · -- θ > 90° because cos(θ) = -1/3 < 0 = cos(90°)
    rw [tetrahedralAngleRadians]
    have h_neg : (-1/3 : ℝ) < 0 := by norm_num
    -- arccos is strictly decreasing, so arccos(-1/3) > arccos(0) = π/2
    have h_zero_in : (0 : ℝ) ∈ Set.Icc (-1 : ℝ) 1 := by simp [Set.mem_Icc]
    have h_third_in : (-1/3 : ℝ) ∈ Set.Icc (-1 : ℝ) 1 := by simp [Set.mem_Icc]; norm_num
    have h_mono := Real.strictAntiOn_arccos h_third_in h_zero_in h_neg
    rwa [Real.arccos_zero] at h_mono
  · -- θ < 120° because cos(θ) = -1/3 > -1/2 = cos(120°)
    rw [tetrahedralAngleRadians]
    have h_neg_third_gt : (-1/3 : ℝ) > -1/2 := by norm_num
    have h_half_in : (-1/2 : ℝ) ∈ Set.Icc (-1 : ℝ) 1 := by simp [Set.mem_Icc]; norm_num
    have h_third_in : (-1/3 : ℝ) ∈ Set.Icc (-1 : ℝ) 1 := by simp [Set.mem_Icc]; norm_num
    have h_mono := Real.strictAntiOn_arccos h_half_in h_third_in h_neg_third_gt
    -- cos(2π/3) = -1/2, so arccos(-1/2) = 2π/3
    have h_in_range : 0 ≤ 2 * π / 3 ∧ 2 * π / 3 ≤ π := by
      constructor
      · positivity
      · have hp := Real.pi_pos
        linarith
    have h_arccos : Real.arccos (-1/2) = 2 * π / 3 := by
      rw [← cos_two_pi_div_three]
      exact Real.arccos_cos h_in_range.1 h_in_range.2
    rwa [h_arccos] at h_mono

/-- 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 declaration does not prove that real molecules adopt the tetrahedral angle; it states a geometric identity. The declaration does not derive the tetrahedral angle from the framework's cost function; it defines the angle as arccos(-1/3). The formula cos(θ) = -1/(n-1) is not claimed to hold for all geometries, as the octahedral case shows.

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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