SemiconductorCert
plain-language theorem explainer
SemiconductorCert bundles the claim that exactly five semiconductor types exist with band gaps forming a geometric sequence scaled by the golden ratio phi. Solid-state physicists working from Recognition Science's configDim would cite it to confirm the canonical band structure. The declaration is a structure definition whose fields are directly populated by sibling lemmas on type count, ratio constancy, and positivity.
Claim. A structure certifying that the set of semiconductor types (intrinsic, n-doped, p-doped, compensated, degenerate) has cardinality 5, that band-gap energies satisfy $E_{k+1}/E_k = phi$ for all $k$, and that $E_k > 0$ for all $k$, where $E_k = phi^k$.
background
The module defines five canonical semiconductor types via the inductive type SemiconductorType with constructors intrinsic, nTypeDoped, pTypeDoped, compensated and degenerate. Band gaps are placed on the phi-ladder by the definition bandGap k := phi ^ k, where phi denotes the golden ratio. This follows the Recognition Science derivation that configDim D = 5 yields precisely these five types.
proof idea
This is a structure definition with no proof body. The three fields encode the required properties directly: Fintype.card SemiconductorType = 5, the constant ratio phi for successive band gaps, and strict positivity of each band gap value.
why it matters
The structure supplies the certificate assembled by the downstream semiconductorCert constructor. It fills the B15 solid-state depth by linking configDim = 5 to the five semiconductor types and their phi-scaled band gaps, resting on the phi-ladder from the unified forcing chain T5-T6. No open questions are flagged.
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