Materials Anisotropic Etching From Jcost

Anisotropic etching and the cost ratio

Anisotropic etching is a process used in microfabrication where a chemical etchant attacks a crystalline material at different rates along different crystal planes. In silicon, for example, potassium hydroxide (KOH) etches the (100) plane much faster than the (110) plane, with measured rate ratios often around 30:1 to 50:1. This directional preference is what lets engineers carve precise channels, pits, and membranes into wafers for sensors and microfluidic devices.

The classical explanation for this anisotropy lies in atomic packing density: planes with more atoms per unit area tend to etch more slowly because each atom requires more energy to remove. The ratio of etch rates between planes is therefore a measurable materials property that depends on crystal structure, etchant chemistry, and temperature. A predictive formula for this ratio from first principles has long been a target in process engineering, since it would let designers choose crystal orientations without empirical trial runs.

In Recognition Science, the framework models this ratio through its cost function. The framework's cost, a forced measure of the price of recognition events, takes the ratio of two rates as input. The framework defines domainCost as the cost of dividing one rate by another. The machine-checked library of formal theorems proves three general facts about this cost: it is zero when the two rates are equal, it is never negative for positive rates, and a threshold constant built from the golden ratio is positive. These are properties of the cost function itself, not of silicon or any specific etchant.

The framework also records a research note comparing the cost expression to measured silicon data. The note reports that the ratio J(φ)^(-2) ≈ 71.7 lands within an order of magnitude of the measured 30:1 to 50:1 range for the (110)/(100) plane pair. This is a numerical comparison, not a derived result: the definition of domainCost does not reference silicon, so the framework proves nothing specific to anisotropic etching. The note records where the idea was meant to go, not a theorem about materials.

What the framework establishes in plain language is a template: if someone later defines what m and e mean for a specific crystal and etchant, then the cost function's proven properties would apply. Until that definition exists, the etching claim remains a research aspiration, not a framework result. The general facts about the cost function are proved; their application to anisotropic etching is not.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

This framework proves any specific etch rate ratio for silicon or any other material. The numerical comparison to silicon data is a derived result rather than a research note. Anisotropic etching itself is derived from the cost function; only the template for such a derivation is formalized.

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

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