SolidStatePhenomenon
An inductive type enumerates the five canonical solid-state phenomena that Recognition Science associates with configuration dimension D = 5. A physicist working on the phi-ladder derivation of band gaps or Brillouin zones would cite this enumeration to anchor the materials sector. The definition is a direct inductive construction listing band structure, phonons, magnetism, superconductivity, and topology with no proof obligations.
claimLet $S$ be the inductive type whose constructors are band structure, phonons, magnetism, superconductivity, and topology.
background
In the Recognition Science framework, solid-state physics is obtained from the eight-vertex cube lattice $Q_3$ in three spatial dimensions. The module states that five canonical phenomena equal the configuration dimension $D = 5$, with band gaps given by $ΔE = φ^k × ℏω$ on the phi-ladder and the first Brillouin zone containing $2^3 = 8$ k-points. This inductive definition supplies the finite set of phenomena used to certify the solid-state sector.
proof idea
This declaration is an inductive definition that directly enumerates the five phenomena without invoking any lemmas or tactics.
why it matters in Recognition Science
This enumeration is used by the theorem that establishes cardinality five and by the structure that pairs the five phenomena with the eight k-points. It fills the B10 Materials step by linking the forcing chain (T8: $D = 3$) to observable solid-state effects. The parent result closes the certification that five phenomena and eight k-points follow from the RS lattice.
scope and limits
- Does not derive explicit energy spectra for each phenomenon.
- Does not connect the phenomena to specific rungs on the phi-ladder.
- Does not prove equivalence to standard solid-state classifications.
formal statement (Lean)
23inductive SolidStatePhenomenon where
24 | bandStructure | phonons | magnetism | superconductivity | topology
25 deriving DecidableEq, Repr, BEq, Fintype
26