Chemistry Semiconductor Band Gap2 Iiivband Gap Cert

The certificate's scope

In the Recognition Science framework, a ledger is a discrete record of events, and its cost measures how expensive a recognition event is. The declaration IIIVBandGapCert is a formal certificate, a machine-checked bundle of three proven facts about a specific cost function. That function, written domainCost, takes two real numbers m and e and returns a value based on their ratio.

The three facts are general properties of this cost function, not facts about any particular material. First, when the two inputs are equal, the cost is zero. Second, for positive inputs, the cost is never negative. Third, a certain constant called canonicalThreshold, defined as phi minus 1.5, is positive. These are proved in the machine-checked library of formal theorems. The certificate simply packages these three results together and shows that such a package exists.

The name of the certificate suggests a connection to III-V semiconductors, compounds made from elements in groups III and V of the periodic table, like gallium arsenide. The research note attached to the file records an intention: band gaps of such materials might cluster at values following a phi-power ladder, where phi is the golden ratio. But the formal proof itself does not define what m and e mean for a semiconductor. Without such a definition, the certificate proves nothing specific to GaAs, InP, or any other material.

What the certificate does establish is a small, solid piece of mathematics. It shows that a particular cost function has three desirable properties: it vanishes at equality, it stays nonnegative for positive inputs, and a threshold constant is positive. These are the building blocks for a future theory, but the bridge from these abstract facts to actual band gaps remains open. The certificate is a foundation stone, not a building.

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)

What this page does not claim

The certificate does not prove that any real semiconductor band gap follows a phi-power ladder. The certificate does not define what m and e are for a semiconductor. The certificate does not derive the existence of three spatial dimensions or any other physical structure.

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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