Foundation Primitive Recognition Calculus Formal System Prcembedding Into

The embedding bridge

A formal system is a set of rules for writing and comparing expressions. The declaration PRCEmbeddingInto defines a bridge from the primitive recognition calculus (PRC), the framework's minimal ledger of discrete events, into any such system. The bridge has two parts: it maps the two endpoints of a primitive distinction to tokens in the target system, and it maps finite traces of events to expressions. The bridge is required to preserve two facts: the target system must be able to tell the two endpoints apart, and if one trace extends another, the corresponding expressions must stand in the target system's own extension relation.

This is a definition, not a discovery. It says what it means for the PRC to be embedded, and it supplies the canonical construction: any formal system that can already distinguish the two endpoints gets a bridge by using its own interpretation fields. The machine-checked library proves that this construction always works, so the embedding target, the claim that every expressive formal system admits such a bridge, is a proved theorem. It also proves that the PRC itself, taken as a formal system, is expressive and therefore embeds into itself.

The honest limit is explicit in the source. The certificate that closes this step records that the formal-system surface and embedding theorem are done. The broader claim, that every external foundation satisfies this interface, is named as the next layer of work, not hidden inside this result. The bridge is a tool for connecting the calculus to other systems, not a proof that any particular external system exists or is consistent.

What this changes for a reader is the shape of the framework's ambition. The PRC is not sealed off; it is built to be portable. The embedding theorem is the mechanism that lets the calculus appear inside another formal system's own vocabulary, provided that system can express the primitive distinction. That is a precise, checkable condition, and the theorem guarantees the bridge exists whenever the condition holds.

/-- A PRC embedding into a formal system preserves the primitive endpoint
distinction and finite trace extension. -/
structure PRCEmbeddingInto (F : FormalSystem) where
  endpointMap : Endpoint → F.Token
  traceMap : Trace → F.Expr
  preserves_distinction :
    F.distinguishes (endpointMap Endpoint.left) (endpointMap Endpoint.right)
  preserves_trace_extension :
    ∀ {T U : Trace}, Trace.Extends T U → F.exprExtends (traceMap T) (traceMap U)

theorem FormalSystemEmbeddingTarget_proved :
    FormalSystemEmbeddingTarget := by
  intro F hF
  exact ⟨PRCEmbeddingInto.ofExpressive F hF⟩

theorem PRCFormalSystem_expressive :
    PRCFormalSystem.Expressive := by
  exact Endpoint.left_ne_right

theorem PRCFormalSystem_embedding :
    Nonempty (PRCEmbeddingInto PRCFormalSystem) :=
  FormalSystemEmbeddingTarget_proved PRCFormalSystem PRCFormalSystem_expressive

What this page does not claim

This does not claim that any particular external formal system exists or is consistent. This does not claim that the embedding theorem derives any physical constant or law. This does not claim that the PRC is the only formal system satisfying the interface.

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

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