Chemistry Polarization Catastrophe From Jcost

The polarization catastrophe

In solid-state chemistry, a polarization catastrophe is the runaway growth of electric polarization in a crystal as its dielectric response diverges. The classic textbook picture is a perovskite: as temperature approaches a critical value, the material's susceptibility ε climbs without bound, and the lattice becomes unstable against a spontaneous shift of ions. That instability is what makes a ferroelectric, a material with a permanent, switchable electric polarization, lose its ordered state at the Curie temperature T_C.

Recognition Science (RS) proposes to see this through its own lens. The framework's central object is a cost, a forced price of recognition events, and its proved form is J(x) = (x + 1/x)/2 − 1. The idea for chemistry is to model the instability as a cost crossing: at the Curie temperature, the ratio of a domain scale m to a charge scale e hits a threshold where J(m/e) equals J(φ), with φ the golden ratio. That threshold, φ − 3/2, is a positive number, about 0.118. The physical picture is that when the cost of maintaining the ordered polarization exceeds a critical value, the material gives way.

What the machine-checked library actually proves is narrower. Its module defines domainCost(m, e) = J(m/e), and proves three general facts: the cost is zero when m equals e, it is never negative for positive inputs, and the threshold φ − 3/2 is positive. These are true for any positive real numbers m and e. The library itself states plainly that this proves nothing specific to ferroelectrics, because the module never defines m and e in terms of a perovskite's ions, lattice, or temperature. The connection to T_C and the dielectric susceptibility is a research note, a plan for where the idea was meant to go, not a result.

In plain terms: the framework has a universal cost function with nice arithmetic properties, and a chemist can choose to call m/e the ratio that drives a polarization catastrophe. What would make that choice a theorem about perovskites is a definition of m and e from the material's own physics, for instance the ionic displacement and the charge that responds to it. Until such a definition exists, the module is a template, not an explanation of any real crystal.

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)

What this page does not claim

This module proves any statement about actual ferroelectrics or perovskites. The polarization catastrophe is derived from the cost function; it is only modeled by it. The threshold φ − 3/2 is not shown to match any measured Curie temperature.

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

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