Chemistry Crystal Growth From Phi Ladder Undercooling Threshold

The undercooling threshold

In crystal growth, undercooling is the amount by which a melt must be cooled below its freezing point before a crystal habit, a characteristic shape, can form. The Burton-Cabrera-Frank model describes growth rate as proportional to exp(-E_step/kT), where E_step is the energy of a step on the crystal surface. The Recognition Science framework models the critical undercooling threshold for each of five canonical crystal habits (cubic, tetragonal, hexagonal, orthorhombic, trigonal) as a power of the golden ratio phi, about 1.618. The declaration undercoolingThreshold is a definition: it sets the threshold for the k-th habit as phi^k.

What the declaration proves is a ratio. The theorem undercoolingRatio shows that the threshold for the next habit divided by the current threshold equals phi, for every natural number k. The certificate CrystalGrowthCert bundles this ratio with the fact that there are exactly five habits. These are formal, machine-checked statements: the library reports zero axioms and zero unproved obligations. In plain terms, the framework defines a ladder where each rung of crystal habit requires phi times more undercooling than the one before.

The declaration does not claim that real crystals follow this ladder. It defines a mathematical object and proves its internal ratio. Whether the empirical Walton relation, where undercooling thresholds scale as phi^n, actually holds for measured crystals is a separate empirical check. The docstring notes that the phi-ladder matches the Walton relation, but that match is not part of the formal theorem. The declaration also does not derive the five habit count from physics; it simply lists five canonical habits and proves the count is five by enumeration.

For a reader, the useful fact is this: within the framework, if you accept the phi-ladder as a model, then the ratio between successive undercooling thresholds is exactly the golden ratio, not approximately. The proof is a simple algebraic consequence of the definition, and it is machine-checked. What remains open is whether the model describes real crystal growth; that is a question for experiment, not for the formal library.

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

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']

structure CrystalGrowthCert where
  five_habits : Fintype.card CrystalHabit = 5
  phi_ratio : ∀ k, undercoolingThreshold (k + 1) / undercoolingThreshold k = phi

noncomputable def crystalGrowthCert : CrystalGrowthCert where
  five_habits := crystalHabitCount
  phi_ratio := undercoolingRatio

What this page does not claim

The declaration does not claim that real crystal growth follows the phi-ladder. The declaration does not derive the five habit count from physical principles. The declaration does not prove the Walton relation; it only notes a match in the docstring.

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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