semiconductorCert
plain-language theorem explainer
This definition assembles a certificate confirming exactly five semiconductor types with band gaps forming a geometric sequence under the golden ratio and remaining strictly positive. Condensed-matter researchers applying Recognition Science to solid-state systems would reference the certificate when connecting the configDim parameter to observed band structures. The construction is a direct record instantiation that supplies the three required fields from the count, ratio, and positivity results.
Claim. A certificate asserting that the set of semiconductor types has cardinality 5, that successive band-gap energies satisfy $E_{k+1}/E_k = phi$ for every natural number $k$, and that every band gap satisfies $E_k > 0$.
background
The module treats semiconductor band structure as a direct consequence of setting configDim to 5, which produces five canonical types: intrinsic, n-type doped, p-type doped, compensated, and degenerate. Band-gap energies are required to lie on the phi-ladder, so each successive gap is larger than the previous by the golden ratio phi. The SemiconductorCert structure encodes exactly these three properties: the type count equals 5, the ratio property holds for all k, and all gaps are positive. Upstream, the phi_ratio definition supplies the numerical value 1/phi while the bandGap_ratio theorem proves the scaling relation and bandGap_pos proves strict positivity via the power rule for phi.
proof idea
The definition constructs the SemiconductorCert record by direct field assignment: five_types receives the result of the cardinality theorem, phi_ratio receives the scaling theorem, and bandGap_always_pos receives the positivity theorem. No tactics or intermediate lemmas are applied beyond these three assignments.
why it matters
The certificate closes the verification step inside the SemiconductorBandStructureFromConfigDim module, confirming that configDim = 5 yields precisely the five types with phi-ladder band gaps required by the Recognition framework. It supplies the concrete physical instance that downstream solid-state derivations would invoke when linking the forcing chain (T5 J-uniqueness through T8 D = 3) to observable semiconductor properties. No open questions are resolved here, but the definition removes the last scaffolding obligation for this domain.
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