Mathematics Combinatorics From Rs Choose84 Eq 70

The binomial count

The binomial coefficient C(8,4) counts the number of ways to select 4 objects from a set of 8 without regard to order. Its value is 70. This is a standard result in combinatorics, taught in introductory courses, and it follows directly from the definition of the binomial coefficient: 8! / (4! × 4!) = 40320 / (24 × 24) = 70. The Recognition Science framework, a formal system for deriving structure from recognition costs, records this identity as a proved theorem.

In the framework's machine-checked library of formal theorems, the declaration choose84_eq_70 states that Nat.choose 8 4 = 70, and the proof is by the decide tactic, meaning the computation is verified directly by the proof assistant. The framework also proves two related facts: that C(8,4) is greater than 45, and that C(8,4) equals 2 × C(7,3), which is 2 × 35. These are simple arithmetic identities, each verified without any unproved assumptions or axioms.

The framework's interest in this number is structural. It identifies five canonical combinatorial families: permutations, combinations, partitions, paths, and Young tableaux. The framework proves that this collection has exactly 5 members. It then notes the numerical coincidence that C(8,4) = 70 = 45 + 25, where 45 is a quantity the framework calls gap45 and 25 equals 5^(5-1), a power related to the number of families. The framework presents this as a combinatorial identity, not as a derived law of nature.

What the theorem does not claim is broader significance. It does not assert that 70 is a fundamental constant of physics, nor that the number 5 families has physical meaning. The theorem is purely about natural numbers: it verifies an arithmetic equality. The framework's own documentation describes it as a key combinatorial identity, but the proof itself establishes only the numerical fact, nothing about recognition costs, dimensions, or physical constants. The connection between this binomial value and the framework's other structures, such as the eight-tick cycle or three spatial dimensions, is not part of this theorem's content.

The practical upshot is that the framework has a verified foundation for one small piece of discrete mathematics. Anyone reading the library can trust that C(8,4) = 70 without rechecking the arithmetic, because the proof is machine-checked. The identity itself is unremarkable in classical combinatorics, but its presence in the framework's library shows that the system can handle standard combinatorial facts cleanly. The larger claims about why this number matters, if any, belong to other parts of the framework, not to this declaration.

/-- C(8,4) = 70. -/
theorem choose84_eq_70 : Nat.choose 8 4 = 70 := by decide

/-- C(8,4) = 70 > gap45 = 45. -/
theorem choose84_gt_gap45 : Nat.choose 8 4 > 45 := by decide

/-- C(8,4) = 2 × 35 = 2 × C(7,3). -/
theorem choose84_doubled : Nat.choose 8 4 = 2 * Nat.choose 7 3 := by decide

theorem combinatoricsFamilyCount : Fintype.card CombinatoricsFamily = 5 := by decide

What this page does not claim

This theorem does not claim that 70 is a physically meaningful constant. The proof does not establish any connection between C(8,4) and recognition costs or dimensions. The identity is a purely arithmetic fact, not a statement about the framework's other structures.

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

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