Materials Piezoelectric Constant From Jcost

The piezoelectric constant

The piezoelectric constant d33 is a material property that quantifies the strain, the fractional change in length, a piezoelectric crystal undergoes per unit of applied electric field. For a common actuator ceramic like lead zirconate titanate, known as PZT, manufacturers list d33 values between 400 and 800 picometers per volt; a higher value means the material deforms more for the same voltage, which is why transducer and actuator designers seek high-d33 compositions. The constant is anisotropic, meaning its value depends on the direction of the applied field relative to the crystal axes, and the subscript 33 denotes the standard measurement direction along the poling axis.

In Recognition Science, the framework models the piezoelectric constant through its cost function. The framework's core object is the recognition cost, a discrete record of the price the universe pays to distinguish one state from another; the cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions and is proved in the framework's machine-checked library of formal theorems. The piezoelectric module defines a domain-specific cost as J applied to the ratio of two material parameters, m over e. The module proves three general facts about this cost: it vanishes when the two parameters are equal, it is never negative for positive inputs, and the golden-ratio threshold phi minus 3/2 is positive. These are theorems about the cost function itself, not about any particular material.

The framework's research note goes further: it proposes that the piezoelectric constant d33 divided by a reference value of about 1 picometer per volt equals J(phi) to the power -2 times the recognition dimensionality D. With D equal to 3, the spatial dimension count, this gives d33 approximately 8.47 squared times 3, about 215 picometers per volt, which the note labels a lower bound. That number sits below the PZT range of 400 to 800, but within the broader family of piezoelectric ceramics; for example, some quartz and other crystals have d33 values in the tens of picometers per volt. The framework's estimate is a model prediction, not a theorem, because the definition of m and e in material terms is not yet formalized.

What the module actually establishes in plain language is narrower and fully rigorous: the cost function used for this domain has the three properties above, and these properties are assembled into a certificate structure that is provably inhabited. The certificate is a formal object that packages the three facts, and the theorem cert_inhabited proves such a certificate exists. This is a solid foundation, but it is not a derivation of any specific material's d33. The connection from the abstract cost to a real piezoelectric constant remains open.

The practical takeaway is that Recognition Science offers a route from a universal cost function to a material property, with the piezoelectric constant as a test case. The framework's estimate of about 215 pm/V is a falsifiable prediction: if a PZT sample measured below that value, the model's lower bound would be violated in that regime. No such measurement is presented here, so the prediction remains a target for experiment, not a confirmed result.

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]

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

theorem cert_inhabited : Nonempty PiezoStressCert := ⟨cert⟩

What this page does not claim

The module derives the piezoelectric constant for any specific material. The value 215 pm/V is a measured or confirmed result. The framework's estimate matches the PZT range of 400 to 800 pm/V.

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

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