Materials Magnetism Types From Config Dim Magnetism Type Count

Five magnetic orderings

Magnetism in solid materials appears in five classical orderings: diamagnetism, paramagnetism, ferromagnetism, antiferromagnetism, and ferrimagnetism. Diamagnetic materials weakly repel an applied field; paramagnetic ones weakly attract it. Ferromagnets, like iron, retain a strong internal field; antiferromagnets align neighboring moments in opposition; ferrimagnets, like certain oxides, have opposing moments that do not fully cancel, leaving a net magnetization.

These five are the standard categories in solid-state physics, taught in every introductory course and used by engineers to classify permanent magnets, transformer cores, and memory devices. The list is not exhaustive of all magnetic phenomena, but it is the canonical set of orderings that appears in textbooks and in the design of magnetic materials.

In the Recognition Science framework, the declaration magnetismType_count (a theorem in the framework's machine-checked library of formal theorems) establishes that the inductive type MagnetismType, which names exactly these five orderings, has cardinality five. The proof is by computation: the type has five constructors, and the theorem states that the number of distinct elements is five. The library checks this with no unproved assumptions and no axioms beyond the standard logical ones.

The declaration does not claim that these five are the only possible magnetic orderings in nature, nor that the framework derives their physical existence from first principles. It is a counting statement about a defined type, not a physical law. The framework's library also packages this result in a certificate structure, but that structure adds no new content; it merely records the same counting fact.

What the theorem gives a reader is a precise, machine-verified statement that the canonical list has five entries. That is useful as a foundation for further formal work on magnetism, but it is a small step, not a theory of magnetism. The physical content, the classification of real materials, comes from experiment and from the classical theory of condensed matter, not from this declaration.

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

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

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

What this page does not claim

The declaration does not claim that these five are the only possible magnetic orderings in nature. It does not derive the physical existence of these orderings from first principles. It does not state anything about the relative strength or temperature dependence of these orderings.

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.

Derived articles

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