Foundation Rscoupled Axis Disjoint Sum Card

The disjoint sum theorem

The theorem disjoint_sum_card is a counting fact about three finite collections. If each collection has exactly n items, and the collections are disjoint (no item belongs to two of them), then the total number of items across all three is 3n. This is the plain arithmetic that the declaration proves: add three copies of n and you get three times n.

The framework's contribution is the word independent. In Recognition Science, two finite axes of the same size do not automatically count as independent. They count as independent only when they are tagged by different recognition primitives, the five basic kinds of meaning the framework uses: jCost, phiLadder, sigmaCharge, q3Lattice, and gap45. The theorem applies to a structure called a pairwise independent disjoint sum of three same-size axes, which packages three axes of size n, each tagged differently, and asserts that their combined cardinality is 3n.

The proof is a direct rewrite: each axis has cardinality n by its defining property, so the sum of the three cardinalities is n + n + n, which the ring tactic simplifies to 3n. The declaration is a theorem in the machine-checked library of formal theorems, with no unproved assumptions and no axioms beyond the standard logical ones. It is a small, fully verified piece of infrastructure for cross-domain combination theorems.

What the theorem does not claim is just as important. It does not say that three axes of size n are independent, only that if they are independent, then their disjoint sum has size 3n. It does not say anything about the product of the axes, which is a separate theorem (triple_card) giving n^3. And it does not say that the five primitives are the only possible tags, only that these five are defined. The theorem is a counting identity under a specific independence condition, not a claim about the physical meaning of the axes themselves.

/-- Cardinality of the disjoint sum of three same-size RS-independent axes. -/
theorem disjoint_sum_card {n : ℕ} (S : RSDisjointSum3 n) :
    @Fintype.card S.axis1.Ix S.axis1.finite +
      @Fintype.card S.axis2.Ix S.axis2.finite +
      @Fintype.card S.axis3.Ix S.axis3.finite = 3 * n := by
  rw [S.axis1.card_eq, S.axis2.card_eq, S.axis3.card_eq]
  ring

/-- 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

/-- The five RS primitives used to tag domain axes. -/
inductive RSPrimitive where
  | jCost
  | phiLadder
  | sigmaCharge
  | q3Lattice
  | gap45
  deriving DecidableEq, Repr, Fintype

What this page does not claim

The theorem does not establish that three axes of size n are independent; it only states the cardinality of their disjoint sum under that condition. The theorem says nothing about the product of the axes, which is covered by a separate theorem giving n^3. The five primitives are defined, but the theorem does not claim they are exhaustive or that they carry any physical meaning beyond their role as tags.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: