Materials Ferroelectric Piezo From Jcost

Ferroelectricity and the cost ledger

Ferroelectricity is the property of certain crystals, such as lead zirconate titanate (PZT), to hold a spontaneous electric polarization that can be reversed by an applied electric field. The field strength needed to flip the polarization is the coercive field. In PZT, measured coercive fields run from about 10 to 50 kV/cm, while the saturation field, the level where all domains align, reaches roughly 100 to 200 kV/cm. The ratio of the two, about 0.1 to 0.25, is the empirical fact this framework aims to frame.

In Recognition Science, the framework keeps a discrete record of events called a ledger, and every recognition event carries a forced cost. The cost function J is the unique function satisfying five plain conditions, and it equals (x + 1/x)/2 - 1. The framework defines a domain cost as J(measured / expected), so the cost is zero when the measured value equals the expected one. On positive inputs the cost is never negative, and the golden ratio phi, about 1.618, appears as a canonical threshold because phi - 3/2 is positive.

The framework proves three general facts: the cost vanishes at equilibrium, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive. It proves nothing specific to ferroelectrics, because the definition of domain cost uses the ratio m/e without tying m or e to any physical quantity. The coercive field relation, E_c = J(phi) * E_saturation, is a research note in the documentation, not a theorem. What would make this framework a theorem about ferroelectricity is a definition of m and e in the material's own terms, such as coercive and saturation fields.

In the framework's account, the empirical ratio of coercive to saturation field in PZT, about 0.1 to 0.25, lands near J(phi), which is about 0.118. That agreement is a measured observation, not a derived result. The framework models the threshold as a cost ratio, and the match is suggestive but unproven. The framework's value is that it names the gap: the general cost machinery is in place, and the missing piece is a physical definition of measured and expected for a ferroelectric domain.

What the framework establishes, in plain language, is that the cost function has the right formal properties to serve as a domain-switching cost, and that the golden ratio threshold is positive. It does not establish that ferroelectrics obey the framework. The research note records where the idea was meant to go, not a result. A reader can see the shape of a future derivation, but the derivation itself remains open.

theorem domainCost_at_equilibrium (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 (measured expected : ℝ) : ℝ := Jcost (measured / expected)

What this page does not claim

The framework does not prove that ferroelectric coercive fields follow the cost function. The empirical ratio of coercive to saturation field in PZT is a measured observation, not a derived result. No claim is made that the cost function is the only possible model for ferroelectric switching.

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

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