NanoScienceCert
plain-language theorem explainer
NanoScienceCert is a structure that asserts the finite sets of nanoscale phenomena and nanostructure types each have cardinality exactly five. Materials physicists applying Recognition Science to E2/B10 materials would cite it to record the match between enumerated effects and configDim D = 5. The declaration is a direct structure whose fields invoke the derived Fintype.card on two inductive enumerations with no further computation.
Claim. Let $P$ be the finite set whose elements are quantum confinement, surface plasmon resonance, van der Waals forces, quantum tunneling, and size-dependent catalysis. Let $S$ be the finite set whose elements are nanoparticle, nanowire, nanosheet, nanotube, and quantum dot. NanoScienceCert is the structure whose fields assert $|P|=5$ and $|S|=5$.
background
The NanoScienceFromRS module treats nanoscale physics as recognition occurring at Q₃ lattice spacing, with lattice parameter a₀ serving as the recognition unit cell. NanoscalePhenomenon is the inductive type whose five constructors are quantumConfinement, surfacePlasmon, vanDerWaals, quantumTunneling, and sizeCatalysis; it derives DecidableEq, Repr, BEq, and Fintype. NanostructureType is the parallel inductive type with constructors nanoparticle, nanowire, nanosheet, nanotube, and quantumDot, likewise deriving Fintype. The module equates both cardinalities to configDim D = 5.
proof idea
The declaration is a structure definition. Its two fields are literal statements of Fintype.card equality; the equalities hold by the Fintype instances automatically derived from the finite inductive enumerations of NanoscalePhenomenon and NanostructureType. No lemmas or tactics are invoked.
why it matters
NanoScienceCert supplies the type required by the downstream definition nanoScienceCert, which constructs the explicit instance. It fills the E2/B10 Materials slot in the Recognition Science framework by recording that five phenomena and five structures realize configDim D = 5 at the nanoscale. The construction sits alongside the forcing chain (T0–T8) in which spatial dimension is fixed at D = 3 while configuration dimension at this scale reaches 5.
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