bandGap_pos
plain-language theorem explainer
The theorem establishes that the band gap energy remains strictly positive for every natural number rung k on the phi-ladder. Solid-state physicists working within Recognition Science would cite it when verifying that semiconductor models never predict unphysical negative gaps. The proof is a one-line application of the positivity of phi raised to any natural power.
Claim. $0 < phi^k$ for every natural number $k$.
background
The module places band-gap energies on the phi-ladder for the five canonical semiconductor types (intrinsic, n-type, p-type, compensated, degenerate) that correspond to configDim D = 5. The definition bandGap k := phi ^ k is identical to the one appearing in SolidStatePhysicsFromRS, so semiconductor band gaps inherit the same scaling as all other RS-native energies. This construction follows directly from the Recognition Composition Law and the forcing chain that fixes phi as the self-similar fixed point.
proof idea
One-line wrapper that applies pow_pos to phi_pos.
why it matters
The result supplies the bandGap_always_pos field inside the SemiconductorCert record that certifies the five semiconductor types. It anchors positivity of all band gaps to the positivity of phi, consistent with the phi-ladder construction that runs through the Recognition Science framework from T5 J-uniqueness onward.
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