Foundation Seam Bridge Bridge Transport Fixed Point

Transporting fixed points

A fixed point of an operation is a value that the operation leaves unchanged: apply the operation and you get the same value back. Fixed points matter because they are the stable outcomes of repeated application, the states a process settles into. In mathematics, proving that a system has a fixed point is often the first step toward showing that it has a meaningful long-term behavior.

The declaration transport_fixedPoint in the framework's machine-checked library of formal theorems establishes a precise transfer rule. One can have two systems, a source and a target, and a bridge between them: a function that maps elements of the source to elements of the target, in a way that respects the operations of both systems. The theorem states that if an element of the source is a fixed point of the source operation, then its image under the bridge is a fixed point of the target operation. In symbols, if the source operation applied to s gives back s, then the target operation applied to the bridge's image of s gives back that same image.

The proof is short and direct. The bridge's defining property says that applying the target operation to the image of s equals the image of the source operation applied to s. Since s is a fixed point, the source operation applied to s is just s, so the target operation applied to the image of s equals the image of s. That is exactly what it means for the image to be a fixed point of the target operation.

In Recognition Science, the source side is built from distinction, the primitive act of telling things apart, and the target side is a theory with its own dynamics. The bridge is a derivation bridge, a map that intertwines the source operation with the genuine theory operation. The framework uses this theorem to transport stationary-state results: if the delta-rooted object has a fixed point, then the theory has a corresponding fixed point in the image of the bridge. This is a structural transfer, not a numerical computation.

The theorem does not claim that every fixed point of the target arises this way, nor that the bridge is surjective. It establishes a one-way implication: a fixed point on the source side guarantees a fixed point on the target side, in the image of the carrier map. It also does not assert that the target operation is the only one the bridge could preserve; the bridge's definition requires it to use the genuine theory operation, but the theorem itself does not prove that this condition holds for any particular theory.

What this gives a reader is a clean, reusable lemma: whenever a theory's dynamics are linked to a simpler underlying structure by a faithful bridge, the theory inherits the stationary states of that structure. The transfer is formal and machine-checked, so the reasoning is not a matter of analogy but of proof.

/-- Corollary: a fixed point of the delta-side operation maps to a fixed point of
the theory operation. (A representative transported theorem: any "stationary
state" result of the theory holds for the delta image of a delta-side fixed
point.) -/
theorem transport_fixedPoint (s : Src) (hs : B.srcOp s = s) :
    B.tgtOp (B.carrier s) = B.carrier s := by
  have := B.preserves s
  rw [hs] at this
  exact this.symm

/-- Corollary: a fixed point of the delta-side operation maps to a fixed point of
the theory operation. (A representative transported theorem: any "stationary
state" result of the theory holds for the delta image of a delta-side fixed
point.) -/
theorem transport_fixedPoint (s : Src) (hs : B.srcOp s = s) :
    B.tgtOp (B.carrier s) = B.carrier s := by
  have := B.preserves s
  rw [hs] at this
  exact this.symm

/-- Corollary: a fixed point of the delta-side operation maps to a fixed point of
the theory operation. (A representative transported theorem: any "stationary
state" result of the theory holds for the delta image of a delta-side fixed
point.) -/
theorem transport_fixedPoint (s : Src) (hs : B.srcOp s = s) :
    B.tgtOp (B.carrier s) = B.carrier s := by
  have := B.preserves s
  rw [hs] at this
  exact this.symm

What this page does not claim

The theorem does not prove that any particular theory has a derivation bridge. It does not claim the bridge is surjective or that all target fixed points are images of source fixed points. It does not assert that the source operation is the only operation the bridge could preserve.

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

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Derived articles

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