Chemistry Metallic Bond Lorenz Positive

The Lorenz number's positivity

The Lorenz number (symbol L) is a physical constant that links a metal's thermal conductivity to its electrical conductivity. The Wiedemann-Franz law, named after Gustav Wiedemann and Rudolf Franz (1853), states that the ratio of thermal conductivity to the product of electrical conductivity and temperature is roughly constant for metals. That constant, L, is approximately 2.44 × 10⁻⁸ watt-ohm per kelvin squared. The law holds well at ordinary temperatures for many metals, though it fails at very low temperatures or in alloys.

The standard derivation of L begins with the free electron model of metals, where conduction electrons behave like a gas. In that model, the ratio of conductivities can be calculated from the electron's charge and the Boltzmann constant. The result is L = (π²/3)(k_B/e)², where k_B is the Boltzmann constant and e is the elementary charge. Plugging in the exact SI values for these constants, k_B = 1.380649 × 10⁻²³ J/K and e = 1.602176634 × 10⁻¹⁹ C, gives the numerical value above. The positivity of L is then a direct consequence of the definitions: it is a ratio of squares of positive quantities.

In Recognition Science, the framework's machine-checked library of formal theorems records this fact as the declaration lorenz_positive. The theorem states that the Lorenz number, as defined in the library, is greater than zero. The proof is a short chain of arithmetic steps: it shows that the number is a product of two positive factors. The first factor is π²/3, which is positive because π is positive. The second factor is the square of the ratio (k_B/e)², which is positive because it is a square of a nonzero real number. The library's definition of lorenzNumber uses the same formula as the classical derivation, so the theorem is a formal confirmation of a known physical fact.

The theorem does not claim that the Wiedemann-Franz law is universally valid, nor that the Lorenz number is constant across all temperatures or materials. It only asserts that the number defined by the standard formula is positive. The framework's derivation of the Lorenz number from fundamental constants is a definitional choice, not a proved result. The positivity theorem is a small but necessary step in the framework's larger project of deriving physical constants from its core cost function, but it does not by itself establish the Wiedemann-Franz law or any other physical prediction.

What the theorem shows is that the framework's formal library can reproduce a basic, well-known fact about a physical constant. This is a consistency check, not a new discovery. It confirms that the framework's definitions align with standard physics at least at this level. For a reader, the practical takeaway is that the Lorenz number's positivity is not an assumption in the framework; it is a proved theorem, verified by machine. This is the kind of small, exact result that gives confidence in the framework's formal foundations.

/-- The Lorenz number is positive. -/
theorem lorenz_positive : lorenzNumber > 0 := by
  simp only [lorenzNumber]
  apply mul_pos
  · apply div_pos
    · exact sq_pos_of_pos Real.pi_pos
    · norm_num
  · apply sq_pos_of_pos
    apply div_pos <;> norm_num

/-- The Lorenz number is positive. -/
theorem lorenz_positive : lorenzNumber > 0 := by
  simp only [lorenzNumber]
  apply mul_pos
  · apply div_pos
    · exact sq_pos_of_pos Real.pi_pos
    · norm_num
  · apply sq_pos_of_pos
    apply div_pos <;> norm_num

What this page does not claim

The theorem does not prove the Wiedemann-Franz law is universally valid. The definition of lorenzNumber is a choice, not a derived result. No claim is made about the Lorenz number's value beyond its positivity.

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expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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