Chemistry Bond Angles Trigonal Cosine

The trigonal cosine

A trigonal planar molecule, such as boron trifluoride, places three identical bonds around a central atom in one flat plane. Electrostatic repulsion pushes the bonds as far apart as possible, which puts 120 degrees between any two of them. The cosine of 120 degrees is exactly minus one half. That value is not an approximation; it follows from the geometry of a circle divided into three equal arcs.

The framework's machine-checked library of formal theorems records this classical fact as a small proof. The declaration trigonal_cosine states that the optimal bond cosine for three equivalent bonds, as defined by the framework's own cost-minimization formula, equals minus one half. The formula in question, cos(θ) = -1/(n-1) for n equivalent bonds, gives -1/2 when n is 3. The library proves this by direct computation: substituting 3 into the definition and simplifying yields the result. The same library also proves the neighboring cases, linear with cosine minus one and tetrahedral with cosine minus one third.

In Recognition Science, the framework models bond angles as the outcome of minimizing a forced cost, not as a separate assumption. The cost function J(x) = (x + 1/x)/2 - 1, proved unique in the framework, drives the preference for certain geometries. For three bonds, the minimum lands exactly on the trigonal planar arrangement. The framework's contribution is not a new measurement of the angle; it is a derivation of the same classical cosine from its own cost principle. The proof itself is a theorem in the library, checked by a machine, with no unverified steps.

What the declaration does not claim is broader. It does not claim that all trigonal molecules are perfectly planar, nor that real bond angles never deviate from 120 degrees. Lone pairs, differing atomic sizes, and other effects bend real molecules. The framework's own notes list falsification criteria: if a measured trigonal cosine differs from minus one half, or if the angle trend with lone pairs fails, the derivation would be wrong. The theorem is about the ideal symmetric case, not about every molecule that roughly fits the shape.

/-- Trigonal planar (n=3) has angle ≈ 120° (cos = -1/2). -/
theorem trigonal_cosine : optimalBondCosine 3 = -1/2 := 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

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

What this page does not claim

The declaration does not claim that real trigonal molecules always have exactly 120 degree angles. It does not claim that the framework's cost formula is the only way to derive the trigonal cosine. It does not claim that the framework has measured any bond angle; the theorem is a derivation, not a measurement.

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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