Foundation Rscoupled Axis Rs Primitive Count

Five primitives

In mathematics and computer science, a primitive is an irreducible building block, an object defined directly rather than assembled from other objects. The Recognition Science framework, which derives physical structure from a ledger of recognition events, uses five such primitives to tag the axes of its domain spaces. The five are the cost function, the phi ladder, the sigma charge, the q3 lattice, and the gap-45 ceiling. The framework's machine-checked library of formal theorems proves that this list is complete: the theorem rsPrimitive_count establishes that the number of recognition primitives is exactly five.

The proof is a direct enumeration, not a deep argument. The library defines the type RSPrimitive with five constructors, one for each primitive, and then verifies by computation that the type has exactly five elements. This is a theorem in the formal sense: it is checked by a computer, with no gaps and no additional axioms. The result is a counting fact about the framework's own vocabulary, not a statement about the physical world.

What the theorem does not claim is more interesting. It does not say that these five primitives are the only possible primitives, nor that they are the only ones needed to describe reality. It does not say that the five are independent of one another, nor that they are fundamental in any physical sense. The theorem only states a fact about the framework's internal definition: the type RSPrimitive, as defined, has five elements. The choice of which primitives to include is a definitional choice, not a derived result.

The five primitives are used to tag the axes of finite domain spaces. Two axes count as independent in the framework only when they are tagged by different primitives. This independence condition is what allows the framework to combine axes across domains, for example in the construction of triples of axes whose cardinalities multiply or add. The counting theorem is the foundation for these combination theorems, but it does not by itself establish any physical claim.

In the framework's own account, the five primitives are the vocabulary from which more complex structures are built. The theorem that there are exactly five of them is a small but necessary piece of that account. It is a fact about the framework's definitions, verified by computation, and it carries no physical content beyond what the definitions themselves supply.

theorem rsPrimitive_count : Fintype.card RSPrimitive = 5 := by
  decide

theorem rsPrimitive_count : Fintype.card RSPrimitive = 5 := by
  decide

/-- RS-independence means the axes are carried by different primitives. -/
def independent {n m : ℕ} (A : CoupledAxis n) (B : CoupledAxis m) : Prop :=
  A.primitive ≠ B.primitive

What this page does not claim

The theorem does not claim that the five primitives are physically fundamental or exhaustive. The theorem does not claim that the primitives are independent of one another. The theorem does not claim any physical content beyond the framework's own definitions.

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/Foundation/RSCoupledAxis.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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