Materials Piezoelectric3 From Config Dim

A template, not a result

Piezoelectric materials convert mechanical stress into electric charge and vice versa. The piezoelectric tensor d_ijk describes this coupling. For a tetragonal crystal, symmetry reduces the tensor's independent components from 27 to 6, and in full three dimensions the tensor has 18 components. The number 18 also equals D^(D-1) * 2 for D = 3, a coincidence the module notes.

The file materials piezoelectric3 from config dim defines a cost function domainCost as J(m/e), where J is the framework's recognition cost. The Lean code proves three general facts: the cost vanishes when m equals e, the cost is nonnegative for positive inputs, and the number phi minus 3/2 is positive. These are properties of the cost function itself, not of any piezoelectric material.

In Recognition Science, the module is a template. Its docstring states that the Lean code proves nothing specific to piezoelectricity because domainCost is defined without reference to the subject. The module is shared verbatim with 2383 sibling modules. What would make it a theorem about piezoelectricity is a definition of m and e in that subject's own terms, such as stress and strain components.

The module does establish that the framework's cost function has the basic properties one would expect: zero at equality, nonnegative values, and a positive threshold. These facts are proved for any positive real inputs. The structure PiezoTensor3Cert packages these three theorems, and the certificate is inhabited, meaning the facts hold.

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]

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

What this page does not claim

This module derives any property of piezoelectric materials. The number 18 being equal to D^(D-1) * 2 is a proved theorem, not an observation. The framework's cost function J is defined or proved here; it is imported from elsewhere.

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/Piezoelectric3_FromConfigDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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