Chemistry Semiconductor Band Gap2

The band gap ladder

A band gap is the energy a bound electron needs to jump into a conducting state, a quantity that decides whether a material acts as an insulator, a semiconductor, or a metal. In the III-V family, gallium arsenide has a gap near 1.42 electron volts, indium phosphide near 1.35, indium arsenide near 0.36, and gallium nitride near 3.4. The ratio of the gallium nitride gap to the gallium arsenide gap is about 2.39, which sits close to phi squared, roughly 2.618, where phi is the golden ratio, about 1.618. That proximity is the seed of an idea: band gaps might cluster at phi to a whole-number power times a base energy.

The module named chemistry-semiconductor-band-gap2 in the Recognition Science framework does not prove that idea. Its machine-checked library of formal theorems defines a cost function on a ratio of two positive numbers, and proves three facts about that function: it returns zero when the two numbers are equal, it never returns a negative value for positive inputs, and a certain threshold constant, phi minus 1.5, is positive. These are general properties of the cost function, stated once in a shared template and reused verbatim by 2383 sibling modules. The module itself defines no meaning for the two numbers in semiconductor terms, so it proves nothing specific to gallium arsenide or any other material.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and a forced cost of recognition. The cost function J(x) = (x + 1/x)/2 - 1 measures how far a ratio sits from unity, and the framework proves this is the only cost function satisfying five plain conditions. The semiconductor module applies that cost to a ratio of two masses, but the masses are left abstract. The research note attached to the module records where the idea was meant to go: a ladder of band gaps at phi-power steps. The gap between the note and the proof is exactly the missing definition of the two masses in terms of the materials themselves.

What the module does establish, in plain language, is a certificate: a packaged collection of the three proved facts, assembled so that any future work can rely on them. That certificate is real and machine-checked. What it does not establish is any connection between the golden ratio and measured semiconductor band gaps. The measured values are empirical facts, and the proposed phi-power ladder is a hypothesis awaiting a definition that ties the abstract cost function to actual materials.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

structure IIIVBandGapCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The module does not prove that any measured semiconductor band gap equals a phi-power times a base energy. The module does not derive the band gaps of gallium arsenide, indium phosphide, indium arsenide, or gallium nitride. The research note about III-V alloys is not a proved theorem; it is a statement of intent.

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/Chemistry/Semiconductor_Band_Gap2.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.

Derived articles

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