Materials Piezo Pvdf From Jcost

A placeholder for a real material

Polyvinylidene fluoride, or PVDF, is a common polymer with a useful property: when you squeeze it, it generates a voltage. That effect, called piezoelectricity, is why PVDF appears in sensors, microphones, and medical imaging devices. Its piezoelectric constant d33, a number measuring how strongly strain produces charge, is roughly -24 picometers per volt in the beta phase. That is a real, measured property of a real material.

The module named materials-piezo-pvdf-from-jcost does not prove anything about that measured property. Its own docstring says so plainly: the Lean code below proves three general facts about a cost function, and nothing specific to PVDF. The paragraph above the code, with the d33 estimate, is a research note recording where the idea was meant to go, not a result.

What the code actually proves is about cost, a measure of recognition effort in the framework. The cost function J(x) = (x + 1/x)/2 - 1 vanishes when its two inputs are equal, is never negative for positive inputs, and the golden-ratio constant phi exceeds 1.5. These are true for any two positive real numbers, not for PVDF specifically.

The module defines a certificate structure that packages those three facts, and proves the certificate exists. That is a valid piece of formal mathematics, but it is a template. The docstring explains the situation: the same body is shared verbatim with 2383 sibling modules. To make this a theorem about PVDF, someone would need to define what m and e mean in the material's own terms, connecting the cost function to the polymer's actual physics. That definition is absent, so the module remains a placeholder.

In plain language: the machine-checked library proves the cost function behaves well, and the research note guesses that a scaled version of an inorganic reference value might land near PVDF's measured constant. The guess is not a proof, and the module does not claim it is. The honest summary is that this file establishes generic cost properties, not piezoelectric facts.

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]

theorem cert_inhabited : Nonempty PVDFPiezoCert := ⟨cert⟩

What this page does not claim

This module proves the PVDF d33 value or any piezoelectric property. The research note's estimate of 16 pm/V is a derived result. The cost function has been connected to any specific material physics.

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

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