nanoScienceCert
plain-language theorem explainer
nanoScienceCert supplies the two cardinality facts needed to inhabit the NanoScienceCert structure. Materials physicists deriving nanoscale behavior from the Recognition Science lattice would cite it to anchor the claim that configDim equals 5 at Q3 spacing. The definition succeeds by direct substitution of the two decide-proven count theorems into the structure fields.
Claim. The NanoScienceCert record is the pair of equalities $Fintype.card(NanoscalePhenomenon)=5$ and $Fintype.card(NanostructureType)=5$.
background
The module Nanoscience from RS treats nanoscale phenomena as recognition events at the Q3 lattice spacing with unit cell a0. Five canonical phenomena (quantum confinement, surface plasmon resonance, van der Waals forces, quantum tunneling, size-dependent catalysis) are asserted to match configDim D=5. Five canonical nanostructure types (nanoparticle, nanowire, nanosheet, nanotube, quantum dot) are likewise asserted to match the same dimension. The upstream theorems nanoscalePhenomenonCount and nanostructureTypeCount each establish the required cardinality by exhaustive decision on the finite inductive enumerations of the respective types.
proof idea
The definition is a one-line record construction that assigns the theorem nanoscalePhenomenonCount to the five_phenomena field and the theorem nanostructureTypeCount to the five_structures field.
why it matters
This definition supplies the concrete inhabitant of the NanoScienceCert structure, completing the Lean formalization of the E2/B10 nanoscience section. It directly encodes the module claim that five phenomena plus five structures realize configDim D=5, consistent with the Recognition Science forcing chain in which spatial dimension D=3 emerges at larger scales while configDim remains free at the nanoscale. The module reports zero sorry and zero axiom, so the definition closes the local formalization without open scaffolding.
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