bandGap
Band gap energies for the five semiconductor types are placed on the phi-ladder by defining the gap at rung k as phi to the power k. Solid-state modelers working from configDim D equals 5 would cite this when establishing ratios or positivity in band structure calculations. The definition is a direct power assignment imported from the solid-state physics module.
claimThe band-gap energy at rung $k$ is $E_g(k) = phi^k$, where $phi$ denotes the self-similar fixed point of the Recognition Composition Law.
background
The module defines five canonical semiconductor types (intrinsic, n-type, p-type, compensated, degenerate) corresponding to configDim D equals 5. Band-gap energies sit on the phi-ladder, the same construction appearing upstream in SolidStatePhysicsFromRS.bandGap. Recognition Science places all such energies via the self-similar fixed point phi forced at T6, with the Recognition Composition Law supplying the functional equation that fixes phi.
proof idea
One-line definition that directly sets bandGap k to phi raised to the power k.
why it matters in Recognition Science
This supplies the explicit values used by SemiconductorCert to enforce the constant ratio phi and strict positivity for every rung, thereby certifying the five semiconductor types. It anchors the phi-ladder segment of the mass formula and connects to T6 forcing of phi as the self-similar fixed point. The same definition is re-exported from SolidStatePhysicsFromRS, closing the link between solid-state and semiconductor layers.
scope and limits
- Does not assign numerical values in electron-volts or other physical units.
- Does not derive the numerical value of phi from the forcing chain T0-T8.
- Does not map specific rungs to particular semiconductor types.
- Does not incorporate temperature, doping concentration, or external fields.
Lean usage
theorem bandGap_ratio (k : ℕ) : bandGap (k + 1) / bandGap k = phi := by unfold bandGap; rw [pow_succ]; ringformal statement (Lean)
29noncomputable def bandGap (k : ℕ) : ℝ := phi ^ k
proof body
Definition body.
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