Mathematics Boolean Algebra From Rs Atoms Eq 2cube D

The atom count

A Boolean algebra is a structure of true and false values combined by operations like AND, OR, and NOT. The simplest nontrivial example has two elements, true and false, and is written {0,1}. Stack three independent copies of this two-element algebra, and you get the algebra {0,1}³, whose elements are triples like (true, false, true). Each element of this algebra is an atom, a minimal nonzero piece, and there are exactly 2³ = 8 of them. This is the classical fact at the center of the question: a three-bit Boolean algebra has eight atoms.

The Recognition Science framework's machine-checked library of formal theorems records this fact in a declaration named atoms_eq_2cubeD. The declaration states that the atom count, defined as 2 raised to the power 3, equals 8. The proof is immediate by definition, since 2³ is 8 by the usual meaning of exponentiation. The library also records that the five canonical Boolean operations (AND, OR, NOT, NAND, NOR) number exactly five, and it packages both facts into a certificate structure. These are small, verified steps, not deep results.

In Recognition Science, the framework models a discrete record of events, called a ledger, and the cost of recognizing an event. The three-bit algebra appears when the framework counts the atoms available in a recognition space of dimension three. The declaration atoms_eq_2cubeD is a supporting lemma in that larger story: it pins down the size of the atom set. It does not, by itself, derive the number three from anything deeper, nor does it prove that the Boolean algebra structure is the only one possible.

What the declaration does not claim is just as important as what it proves. It does not claim that three dimensions of physical space follow from Boolean algebra. It does not claim that the five operations are the only possible ones, only that these five are recognized. It does not claim that the atom count 8 has any special arithmetic significance beyond being 2³. The declaration is a precise, narrow statement: the atom count is 2 to the third power. Everything else in the framework's larger claims rests on other theorems, not on this one.

theorem atoms_eq_2cubeD : atomCount = 2 ^ 3 := rfl

theorem boolOpCount : Fintype.card BoolOp = 5 := by decide

What this page does not claim

The declaration does not derive the number three from any deeper principle. The declaration does not claim the five Boolean operations are the only possible ones. The declaration does not connect the atom count to physical three-dimensional space.

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/Mathematics/BooleanAlgebraFromRS.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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