Materials Magnetism Types From Config Dim Magnetism Type

MagnetismType

Magnetism in solids is classically divided into five orderings: diamagnetism, paramagnetism, ferromagnetism, antiferromagnetism, and ferrimagnetism. In a diamagnet, applied fields induce a weak opposing response; a paramagnet aligns weakly with an applied field; a ferromagnet retains magnetization after the field is removed; an antiferromagnet has opposing sublattices that cancel; a ferrimagnet has opposing sublattices that only partially cancel. These five categories organize most of solid-state magnetism and appear in every standard textbook.

The Recognition Science framework's machine-checked library of formal theorems records this classification in a declaration named MagnetismType, a discrete record of the five canonical orderings. The declaration is an inductive type with exactly those five constructors, and a theorem proves that its cardinality is five. A certificate structure bundles that count, and a definition supplies the certificate. The entire file is checked with zero unproved assumptions and zero axioms beyond the ambient type theory.

What the declaration does not do is more interesting than what it does. It does not define the physical mechanism by which any material becomes magnetic. It does not assign a specific material to a type, and it does not predict which orderings exist in nature. The five names are a definitional choice, a taxonomy, not a derivation. The framework's forcing chain, which derives the golden ratio and three spatial dimensions from a cost function, does not here produce magnetism; this file merely records the standard classification as a finite enumerated set.

The practical consequence is modest but real: any later framework theorem that needs to quantify over magnetic orderings can do so soundly, because the list is closed and machine-checked. A proof that says "for all five types" means all five, not five plus an overlooked sixth. The declaration is a foundation stone for future materials work in the framework, not a result about physics itself.

inductive MagnetismType where
  | diamagnetism
  | paramagnetism
  | ferromagnetism
  | antiferromagnetism
  | ferrimagnetism
  deriving DecidableEq, Repr, BEq, Fintype

theorem magnetismType_count : Fintype.card MagnetismType = 5 := by decide

structure MagnetismTypesCert where
  five_types : Fintype.card MagnetismType = 5

inductive MagnetismType where
  | diamagnetism
  | paramagnetism
  | ferromagnetism
  | antiferromagnetism
  | ferrimagnetism
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The declaration does not define the physical mechanism by which any material becomes magnetic. The declaration does not assign a specific material to a magnetic type. The five orderings are not derived from the framework's cost function; they are a recorded taxonomy.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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