Physics Piezoelectric Coupling From Jcost

A formal placeholder

Piezoelectricity is the property of certain crystals, like quartz and PZT ceramics, to generate an electric voltage when squeezed or struck, and conversely to change shape when a voltage is applied. The strength of this effect in a material is summarized by a coupling coefficient, k², a number between 0 and 1 that represents the fraction of mechanical energy converted into electrical energy. In the 1880s, Jacques and Pierre Curie discovered the effect in quartz; today it drives sensors, actuators, and ultrasonic transducers.

The framework's library contains a module that defines a candidate formula for this coefficient: k² is set equal to J(m/e), where J is a specific cost function and m and e are two real numbers. The module proves three general facts about this formula, but it does not prove that this formula describes any real piezoelectric material. It proves that the cost vanishes when m equals e, that the cost is never negative when both inputs are positive, and that the golden ratio minus 1.5 is a positive number. These are properties of the cost function itself, not of quartz or PZT.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and the cost of recognition, a forced penalty for each event. The cost function J(x) = (x + 1/x)/2 - 1 is the unique function satisfying five plain conditions, a theorem proved in the framework's machine-checked library of formal theorems. For piezoelectricity, the idea was to identify m and e as two physical quantities in the crystal's own terms, making k² a derived prediction rather than a free parameter.

That identification is missing. The module defines m and e only as real numbers with no physical meaning attached, so the formula remains a formal template, not a physical law. The library itself states this plainly: the module proves nothing specific to piezoelectricity. The research note records the intended direction, but the definition of m and e in the subject's own terms is what would turn this placeholder into a theorem about the effect.

What the module does establish is a checkable skeleton. Any future claim that k² equals J(m/e) for a real material must first supply the physical definitions of m and e, and then the three proved properties will apply. Until then, the coupling coefficient remains an empirical quantity measured in laboratories, and the framework's contribution is a formal shape waiting for content.

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)

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

What this page does not claim

This module proves that any real piezoelectric material has coupling coefficient equal to J(phi). The definition of m and e in terms of crystal properties is absent from the module. The framework's cost function J is derived from the five conditions, not from piezoelectric measurements.

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

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