Chemistry Crystal Growth From Phi Ladder Crystal Habit

The five habits and their threshold ladder

A crystal habit is the external shape a crystal takes as it grows, such as cubic, tetragonal, or hexagonal. In classical crystallography, the rate at which a crystal face advances follows the Burton-Cabrera-Frank model, where growth depends on the energy needed to add a layer at a step. That energy must be overcome by cooling the melt below its freezing point; the required amount is the critical undercooling. Different habits demand different undercoolings, and the ratios between those thresholds are what the Recognition Science declaration CrystalHabit, a formal object in the framework's machine-checked library of formal theorems, addresses.

The declaration names exactly five canonical habits: cubic, tetragonal, hexagonal, orthorhombic, and trigonal. A proved theorem in the library, crystalHabitCount, establishes that the count is five. The framework then defines a threshold function: undercoolingThreshold(k) equals phi raised to the k-th power, where phi is the golden ratio, about 1.618. A second proved theorem, undercoolingRatio, shows that the ratio of the threshold for habit k+1 to the threshold for habit k is exactly phi. In plain terms, adjacent habits require phi times more undercooling to form, a step that repeats up the ladder.

This structure is a definitional choice, not a measurement. The framework models the thresholds as a phi-power ladder and proves the ratio property within that model. The docstring notes an empirical check: the Walton relation in crystallography reports undercooling thresholds scaling as phi to the n-th power for successive habits. That agreement is a comparison against experiment, not a theorem, and the pack gives no receipt for the Walton values. The library's role is to certify the internal consistency of the ladder, not to establish the empirical law.

What the declaration does not claim matters as much as what it proves. It does not derive the five habit names from physics; it chooses them as the set. It does not prove that real crystals must follow the phi ratio; it defines a model and verifies the ratio inside that model. It does not claim the golden ratio is the only possible step, nor that the Walton relation holds universally. The certificate structure simply packages the count and the ratio as a single object, with zero axioms beyond the ambient type theory, for use in further formal work.

theorem crystalHabitCount : Fintype.card CrystalHabit = 5 := by decide

theorem undercoolingRatio (k : ℕ) :
    undercoolingThreshold (k + 1) / undercoolingThreshold k = phi := by
  unfold undercoolingThreshold
  have hpos : 0 < phi ^ k := pow_pos phi_pos _
  rw [pow_succ]
  field_simp [hpos.ne']

noncomputable def undercoolingThreshold (k : ℕ) : ℝ := phi ^ k

What this page does not claim

The five habit names are not derived from physics; they are the chosen set. The phi ratio is not proved to hold for real crystals, only inside the defined model. The Walton relation is cited as an empirical check, not as a theorem with a receipt.

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

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