Foundation Seam Bridge Bridge Op Bridge

The bridge that carries theorems

A bridge, in mathematics, is a way of showing that two structures behave alike. If you have a map from one set to another that respects the operations on both sides, then anything you prove about the first structure's operations tells you something about the second. The Recognition Science framework calls this a derivation bridge: a single-operation map from a delta-rooted object to a theory carrier, where the map preserves the operation. The framework's library, a machine-checked collection of formal theorems, defines this as OpBridge.

OpBridge is a structure with four parts. First, a carrier map from a source type to a target type. Second, a source operation built from distinction, the framework's primitive notion. Third, a target operation, which must be the genuine theory operation, not a trivial one. Fourth, a preservation condition: applying the source operation then the carrier map gives the same result as applying the carrier map then the target operation. In symbols, for every s, carrier (srcOp s) = tgtOp (carrier s). This is the homomorphism condition, the heart of the bridge.

From this single preservation condition, the framework proves two theorems. The first, transport_iterate, says that the bridge carries every iterate of the source operation to the matching iterate of the target operation. So if a theory theorem is about repeated dynamics, it transports to a statement about the delta side. The second, transport_fixedPoint, says that a fixed point of the source operation maps to a fixed point of the target operation. If the source operation leaves an object unchanged, its image under the carrier map is unchanged by the theory operation. These are proved in the machine-checked library, with no framework-specific axioms.

What OpBridge does not claim is just as important. It does not assert that any particular bridge exists; it only defines what a bridge is and what it would do. It does not claim that the target operation is the true physics operation, only that the bridge must use it if it is to prove anything. And it does not claim that the source operation is the only possible one built from distinction. The structure is a tool, not a result: it sets up the condition under which theorems can travel, and leaves the actual construction of bridges to other declarations.

/-- A single-operation derivation bridge: a map from a delta-side carrier `Src`
to a theory carrier `Tgt` that intertwines a source operation `srcOp` with a
target (theory) operation `tgtOp`. This is the homomorphism whose existence lets
the theory's dynamical theorems transport from the delta side. A faithful bridge
must use the genuine theory operation as `tgtOp`; mapping to a trivial operation
proves nothing and is forbidden by the commitment. -/
structure OpBridge (Src Tgt : Type*) where
  /-- The carrier map from the delta-rooted object to the theory carrier. -/
  carrier : Src → Tgt
  /-- The delta-side operation (built from distinction). -/
  srcOp : Src → Src
  /-- The genuine theory operation the bridge must preserve. -/
  tgtOp : Tgt → Tgt
  /-- Primitive preservation: the bridge intertwines the two operations. -/
  preserves : ∀ s, carrier (srcOp s) = tgtOp (carrier s)

/-- The transport core. A derivation bridge carries every iterate of the
delta-side operation to the matching iterate of the theory operation. Any theorem
of the theory phrased about iterated dynamics therefore transports to a statement
about the delta-rooted object: it suffices to read the theory's `tgtOp^[n]`
behaviour off the delta side's `srcOp^[n]`. -/
theorem transport_iterate (s : Src) :
    ∀ n : ℕ, B.carrier (B.srcOp^[n] s) = B.tgtOp^[n] (B.carrier s) := by
  intro n
  induction n generalizing s with
  | zero => simp
  | succ k ih =>
    rw [Function.iterate_succ_apply, Function.iterate_succ_apply, ih (B.srcOp s),
        B.preserves s]

/-- 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

OpBridge does not assert that any particular bridge exists. OpBridge does not claim that the source operation is the only one built from distinction. OpBridge does not prove that the target operation is the true physics operation.

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)

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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