Foundation Primitive Recognition Calculus Frscarrier Has Protocol Display

The protocol display

A protocol display is a formal bridge between two ways of describing a number. On one side sits a finite expression, a term built from rational numbers and a short list of named constants using only addition, multiplication, and division. On the other side sits a protocol, which the framework models as a basic process that yields a real number as its output. The declaration has_protocol_display states that for every such finite expression, there exists a protocol whose output equals the expression's value. It is a theorem in the machine-checked library of formal theorems, proved directly from the fact that every real number is the value of some protocol.

The statement matters because it closes a gap between syntax and process. The framework's carrier, its working vocabulary of numbers, is a countable set: only countably many finite terms exist, so the carrier values form a proper subset of the real line. Yet every one of those values, however it is written, can be rendered as the output of a protocol. The declaration does not say the protocol is unique, or that it is efficient, or that it is physically realizable. It says only that the display exists, for every term in the carrier.

In Recognition Science, this is part of a larger claim about what the framework computes on. The carrier is finite generation over a fixed inventory, not the uncountable continuum. The protocol display shows that this finite vocabulary is not cut off from the framework's process layer: each term has a protocol that produces it. The theorem is a completeness statement in one direction, from expression to process, and it is silent on the reverse direction, from arbitrary process back to expression.

What the declaration does not claim is just as precise as what it does. It does not claim that every protocol output is a carrier term; the carrier is a proper subset of the reals, so most real numbers have no finite expression in this syntax. It does not claim that the protocol display is constructive, in the sense of giving an algorithm to find the protocol. It does not claim that the protocol has any physical meaning, only that it exists in the formal model. The theorem is a bridge of existence, not a bridge of construction or of physics.

/-- **Protocol display.** Every carrier term value is the value of a Delta-real
protocol, so the finite-description carrier renders into the `ℝδ` interface. -/
theorem has_protocol_display (ex : FRSExpr) :
    ∃ x : DeltaReal.Protocol, x.value = eval ex :=
  DeltaReal.Protocol.value_surjective (eval ex)

/-- The carrier values are a proper subset of ℝ. -/
theorem carrierValues_proper : carrierValues ≠ Set.univ := by
  intro h
  exact Cardinal.not_countable_real (h ▸ carrierValues_countable)

/-- The carrier value set is countable: only countably many finite terms exist. -/
theorem carrierValues_countable : carrierValues.Countable :=
  (MinimalField.rsField_countable).mono carrierValues_subset

What this page does not claim

The theorem does not claim that every protocol output is a carrier term. The theorem does not claim the protocol display is constructive or provides an algorithm. The theorem does not claim the protocol has physical meaning, only formal existence.

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

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