Physics Semiconductor Band Structure From Config Dim

Semiconductor types from a count

Semiconductors are materials whose electrical conductivity sits between conductors and insulators, controlled by a band gap: the energy range no electron can occupy. Doping, the deliberate addition of impurities, shifts this behavior into distinct classes. The standard practical families are intrinsic (pure), n-type (electron-rich), p-type (hole-rich), compensated (both donors and acceptors), and degenerate (heavily doped to near-metallic).

In Recognition Science, the framework models these five classes as the complete set of a counted configuration type. Its machine-checked library of formal theorems proves that exactly five semiconductor types exist, matching the five canonical families above. The framework then places band-gap energies on a ladder where each step multiplies the previous one by the golden ratio φ ≈ 1.618, the same proportion found in pentagons and Fibonacci sequences. The library proves that every band gap is positive and that consecutive gaps maintain this φ ratio exactly.

The framework's contribution is a structural derivation: it does not measure silicon or gallium arsenide, but shows that a discrete count of five types, combined with the golden-ratio scaling forced by its recognition cost function, reproduces the coarse taxonomy and energy spacing that solid-state physics observes. The classical band-structure theory, with its Bloch waves and Brillouin zones, remains the detailed predictive machinery; the framework supplies a compact origin story for why five families and φ-spaced gaps are natural.

theorem semiconductorType_count :
    Fintype.card SemiconductorType = 5 := by decide

theorem bandGap_ratio (k : ℕ) : bandGap (k + 1) / bandGap k = phi := by
  unfold bandGap
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem bandGap_pos (k : ℕ) : 0 < bandGap k := pow_pos phi_pos k

What this page does not claim

This module does not derive the detailed quantum mechanics of band structure, such as Bloch waves or effective masses. The framework does not measure or fit any experimental band-gap value to silicon, germanium, or gallium arsenide. The five-type count is a theorem about the framework's configuration type, not a proof that nature must contain exactly five semiconductor classes.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/SemiconductorBandStructureFromConfigDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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