Materials Semiconductor Dopant Types From Config Dim Dopant Type Count

The five dopant categories

Semiconductor doping is the deliberate introduction of impurities to change how easily a material conducts electricity. Silicon, the workhorse of modern electronics, is typically doped with atoms from group V of the periodic table, such as phosphorus, arsenic, or antimony, which donate an extra electron and are called donors. Atoms from group III, such as boron, aluminum, or gallium, accept an electron and are called acceptors. These two families are the classic dopant types taught in every solid-state physics course.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration, dopantType_count, that establishes a precise count: there are exactly five canonical dopant categories for silicon-type semiconductors. The five are group-V donor, group-III acceptor, deep-level impurity, compensating, and transition-metal scattering center. The theorem states that the number of these categories is five, and the proof is a direct computation: the Lean kernel checks the finite enumeration and confirms the cardinality. The declaration is accompanied by a certificate structure that packages the count as a reusable fact.

The five categories are not arbitrary. The framework derives the number from a configurational dimension, a structural parameter that the framework associates with the material system. In this account, the count of five is forced by the same kind of dimensional reasoning that the framework applies elsewhere, though the physical bridge from recognition cost to semiconductor band structure is not part of this declaration. The theorem itself only certifies the arithmetic: given the five named categories, their number is five.

What the declaration does not claim is just as important. It does not assign specific chemical elements to the categories, beyond the illustrative examples in the docstring. It does not assert that these five categories are the only possible dopant types in real materials, nor that the framework's configurational dimension is the correct physical explanation for why silicon doping has this structure. The theorem is a formal counting result, not a materials-science discovery. It says that if you accept the five categories as the canonical list, then the count is five, and that is all it says.

For a reader, the practical takeaway is modest but solid: the framework has a machine-checked proof that its five dopant categories number five. The declaration is a small piece of a larger formal structure, and its value lies in the certainty of the count, not in any new physics. The categories themselves come from the standard textbook treatment of semiconductor doping, and the framework's contribution is to formalize that count in a way that can be checked mechanically.

theorem dopantType_count : Fintype.card DopantType = 5 := by decide

theorem dopantType_count : Fintype.card DopantType = 5 := by decide

inductive DopantType where
  | groupVDonor
  | groupIIIAcceptor
  | deepLevel
  | compensating
  | transitionMetal
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The declaration does not assign specific chemical elements to the five categories. The declaration does not prove that these five categories are exhaustive for all real semiconductor materials. The declaration does not establish the physical bridge from recognition cost to semiconductor band 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/Materials/SemiconductorDopantTypesFromConfigDim.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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