Chemistry Metallic Bond Transition Cohesive Gt Alkali

The cohesive energy ordering

In chemistry, cohesive energy is the energy needed to separate a solid into isolated atoms. It is a direct measure of how strongly a material holds itself together. For metals, this energy is tied to the metallic bond: the sharing of delocalized valence electrons across a lattice of positive ions, often pictured as an electron sea. The general trend is well established experimentally: transition metals, with their partially filled d-orbitals and multiple available bonding electrons, have higher cohesive energies than alkali metals, which contribute only a single valence electron per atom.

The machine-checked library of formal theorems in Recognition Science (RS) encodes this trend as transition_cohesive_gt_alkali. The framework models a metal's cohesive energy with a proxy: a simplified scalar value, not the full physical quantity. In this model, the proxy equals the golden ratio φ for transition metals, 1/φ for alkaline earth metals, and 1/φ² for alkali metals. The theorem states that for any transition metal and any alkali metal, the transition metal's proxy exceeds the alkali's. The proof is a direct computation from these assigned values, checked by the machine.

The declaration does not claim that real measured cohesive energies of transition metals are always greater than those of alkali metals in every case. It establishes an ordering of a definitional proxy within the framework's model. The proxy is a modeling choice, not a derived physical constant. The theorem's strength lies in its formal correctness relative to that model, not in empirical prediction. The framework's library also defines related notions: lists of transition, alkali, and alkaline earth atomic numbers, a predicate for membership, free electron counts, lattice types with coordination numbers, and packing efficiencies.

Within the framework, the ordering is a consequence of the model's assignment of φ-scaled values. The framework's broader claim is that recognition cost, the forced cost of maintaining a discrete record of events, drives physical structure. Here, that structure is expressed as a simple hierarchy of bond strengths. The theorem is a small, fully formalized piece of that larger picture. It shows how a macroscopic trend can be represented in the framework's terms, but it does not derive the trend from first principles or from the recognition cost function itself.

What the declaration does not claim is equally important. It does not claim that the proxy values are measured or empirically fitted to real cohesive energies. It does not claim that the theorem predicts the exact numerical values of cohesive energies for any specific element. It does not claim that the ordering holds for real materials in all conditions, such as alloys or high-pressure phases. It establishes a formal ordering of a model quantity, nothing more. The gap between the proxy and the physical observable remains a modeling assumption, not a proved equivalence.

/-- Transition metals have higher cohesive energy than alkali metals. -/
theorem transition_cohesive_gt_alkali (Z_trans Z_alkali : ℕ)
    (h_trans : Z_trans ∈ transitionMetalZ) (h_alkali : Z_alkali ∈ alkaliMetalZ) :
    cohesiveEnergyProxy Z_trans > cohesiveEnergyProxy Z_alkali := by
  simp only [cohesiveEnergyProxy]
  -- Need to show: cohesiveEnergyProxy Z_trans > cohesiveEnergyProxy Z_alkali
  -- Transition metals: Z_trans ∈ transitionMetalZ → φ
  -- Alkali metals: Z_alkali ∈ alkaliMetalZ → 1/φ²
  -- First show that sets are disjoint
  have h_trans_not_alkali : Z_trans ∉ alkaliMetalZ := by
    simp only [transitionMetalZ, alkaliMetalZ] at h_trans h_alkali ⊢
    fin_cases h_trans <;> simp
  have h_trans_not_alk_earth : Z_trans ∉ alkalineEarthZ := by
    simp only [transitionMetalZ, alkalineEarthZ] at h_trans ⊢
    fin_cases h_trans <;> simp
  have h_alkali_not_trans : Z_alkali ∉ transitionMetalZ := by
    simp only [transitionMetalZ, alkaliMetalZ] at h_trans h_alkali ⊢
    fin_cases h_alkali <;> simp
  have h_alkali_not_alk_earth : Z_alkali ∉ alkalineEarthZ := by
    simp only [alkaliMetalZ, alkalineEarthZ] at h_alkali ⊢
    fin_cases h_alkali <;> simp
  simp only [h_trans, h_trans_not_alkali, h_trans_not_alk_earth,
             h_alkali_not_trans, h_alkali, h_alkali_not_alk_earth, ite_true, ite_false]
  -- Now need: φ > 1/φ² (which is φ³ > 1)
  have h_phi_pos := Constants.phi_pos
  have h_phi_gt_1 : Constants.phi > 1 := by
    have := Constants.phi_gt_onePointFive
    linarith
  have h_phi_cubed_gt_1 : Constants.phi^3 > 1 := by
    have : Constants.phi^3 > 1^3 := by
      apply pow_lt_pow_left₀ h_phi_gt_1 (by norm_num) (by norm_num)
    simpa using this
  calc Constants.phi = Constants.phi^3 / Constants.phi^2 := by field_simp
    _ > 1 / Constants.phi^2 := by
      apply div_lt_div_of_pos_right h_phi_cubed_gt_1
      apply pow_pos h_phi_pos

/-- The metallic bond strength proxy is related to cohesive energy.
    Transition metals have higher cohesive energy than alkali metals. -/
def cohesiveEnergyProxy (Z : ℕ) : ℝ :=
  if Z ∈ transitionMetalZ then Constants.phi  -- Higher for transition metals
  else if Z ∈ alkalineEarthZ then 1 / Constants.phi  -- Medium
  else if Z ∈ alkaliMetalZ then 1 / Constants.phi ^ 2  -- Lower for alkali
  else 0

/-- The metallic bond strength proxy is related to cohesive energy.
    Transition metals have higher cohesive energy than alkali metals. -/
def cohesiveEnergyProxy (Z : ℕ) : ℝ :=
  if Z ∈ transitionMetalZ then Constants.phi  -- Higher for transition metals
  else if Z ∈ alkalineEarthZ then 1 / Constants.phi  -- Medium
  else if Z ∈ alkaliMetalZ then 1 / Constants.phi ^ 2  -- Lower for alkali
  else 0

What this page does not claim

The theorem does not prove that measured cohesive energies of transition metals always exceed those of alkali metals. The proxy values are not derived from first principles or from the recognition cost function. The theorem does not predict exact numerical cohesive energies for any specific element.

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

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