Chemistry Ionic Bond Born Exponent In Range

The Born exponent bound

The Born exponent is a number used in the classical theory of ionic crystals. It appears in the Born-Landé equation, which estimates the lattice energy of a crystal, the energy released when oppositely charged ions assemble into a solid. The exponent measures how sharply the repulsive force between ions rises as they are pushed together. For many real crystals, fitting the equation to measured compressibility gives values between about 6 and 12, with sodium chloride near 9 and some zinc blende structures higher.

The framework's machine-checked library of formal theorems contains a declaration called born_exponent_in_range. It proves that a proxy value, defined as the golden ratio raised to the fifth power, lies strictly between 10 and 12. The golden ratio, about 1.618, is the number satisfying r² = r + 1. Its fifth power is about 11.09. The theorem is a pure inequality, checked by the kernel of the proof assistant, with no empirical input.

In Recognition Science, the framework models the cost of recognition events, and from that cost it derives the golden ratio as a natural scaling constant. The ionic bond page applies that constant to chemistry. The proxy value φ⁵ is offered as a candidate for the Born exponent, and the theorem establishes only that this candidate falls in the same numerical band as many measured exponents. It does not prove that any real crystal has an exponent of exactly 11.09, nor that the Born-Landé equation with this exponent reproduces any measured lattice energy.

The gap between the theorem and experiment is the honest boundary. The theorem is a statement about a defined number, not about sodium chloride or any other substance. The framework's own documentation describes the φ-stability idea as a prediction, not a measured result. So the declaration earns its place as a consistency check: the framework's preferred constant lands in a plausible range for a real physical parameter. What it does not do is identify which crystal, if any, realizes that value, or show that the Born-Landé equation with this exponent beats the standard fitted values.

/-- The Born exponent proxy is between 10 and 12.
    φ^5 ≈ 11.09, which matches empirical Born exponents of 9-12. -/
theorem born_exponent_in_range : 10 < bornExponentProxy ∧ bornExponentProxy < 12 := by
  dsimp [bornExponentProxy]
  -- Use phi_fifth_bounds: 10.7 < φ^5 < 11.3
  constructor
  · linarith [Constants.phi_fifth_bounds.1]
  · linarith [Constants.phi_fifth_bounds.2]

/-- The Born exponent proxy is between 10 and 12.
    φ^5 ≈ 11.09, which matches empirical Born exponents of 9-12. -/
theorem born_exponent_in_range : 10 < bornExponentProxy ∧ bornExponentProxy < 12 := by
  dsimp [bornExponentProxy]
  -- Use phi_fifth_bounds: 10.7 < φ^5 < 11.3
  constructor
  · linarith [Constants.phi_fifth_bounds.1]
  · linarith [Constants.phi_fifth_bounds.2]

/-- The Born-Mayer repulsion exponent is close to φ.
    Empirically, n ≈ 9-12 for most ions. This can be connected to φ via φ^5 ≈ 11.09. -/
def bornExponentProxy : ℝ := Constants.phi ^ 5

What this page does not claim

No claim that any real crystal has a Born exponent equal to φ⁵. No claim that the Born-Landé equation with this exponent reproduces any measured lattice energy. No claim that the theorem relates to a specific substance such as sodium chloride.

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/IonicBond.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.

Derived articles

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