Foundation Primitive Recognition Calculus Objecthood Registry Classify Forced Ra

Why rationals are unavoidable

The rational numbers are the fractions, numbers like 1/2, 3/4, or 22/7, formed by dividing one whole number by another. Any system of numbers that deserves to be called a field, meaning it supports addition, subtraction, multiplication, and division, must contain the rationals. This is a classical fact of algebra: the rationals form the smallest field, the prime subfield, inside any field of characteristic zero.

In the Recognition Science framework, which builds mathematical objects from a discrete record of recognition events, this classical fact takes on a specific meaning. The framework classifies every object it uses by the kind of commitment that produces it. The classification is called the objecthood registry, and it assigns each object one of seven tags: forced, permitted, quotient, completion, display, observable, or convention. A forced object is one uniquely determined by the framework's law, with no freedom left over.

A theorem in the framework's machine-checked library of formal theorems, named classify_forced_rationals, proves that the rationals are forced. The theorem states that for every subfield K of the real numbers and every rational number q, q belongs to K. In plain language, no carrier of the framework can omit the rationals; they are the unique prime subfield, present in every number system the framework can build. The proof is a direct application of the standard algebraic fact that the rationals embed in any field of characteristic zero.

The framework's periodic table of objecthood records this result. The table assigns the rationals the forced commitment, and the theorem objecthood_periodic_table confirms this assignment. This matters because it distinguishes the rationals from other objects in the registry. The real numbers, for example, are classified as a completion, requiring an independent axiom to add limit points. The rationals need no such extra axiom; they are forced by the mere structure of a field.

What the theorem does not claim is that the rationals are the only forced object, or that the framework derives the rationals from nothing. The proof relies on the ambient theory of fields and the real numbers, which the framework takes as given. The classification is a statement about the rationals' status within the framework, not a derivation of arithmetic from first principles.

/-- `forced`: the rationals are forced into every carrier; they are the unique
prime subfield. No carrier of the framework can omit them. -/
theorem classify_forced_rationals :
    ∀ (K : Subfield ℝ) (q : ℚ), (q : ℝ) ∈ K :=
  fun K q => SubfieldClass.ratCast_mem K q

/-- The seven commitments that produce a mathematical object. -/
inductive Commitment where
  | forced       -- uniquely determined by the law; no freedom
  | permitted    -- an admissible free choice
  | quotient     -- identification under an equivalence
  | completion   -- closure adding limit points; an independent axiom
  | display      -- a rendering / instrument, not a native ingredient
  | observable   -- defined by what can be measured
  | convention   -- a gauge / labeling choice
  deriving DecidableEq, Repr

/-- **Phase 8 headline: the periodic table of objecthood.** Each catalogued object
carries its commitment, and the assignment is exactly the evidence above:
distinction-forced (rationals, the coherence scale), display (ℝδ value map),
completion (ℝ), convention (cost unit), quotient and observable (the gauge
quotient and its probes), permitted (the generable inventory). Objecthood is not
flat: each object is produced by a specific kind of commitment, and the kind is
now a typed, proved attribute. -/
theorem objecthood_periodic_table :
    commitmentOf RSObject.deltaRationals = Commitment.forced
      ∧ commitmentOf RSObject.protocolReals = Commitment.display
      ∧ commitmentOf RSObject.classicalReals = Commitment.completion
      ∧ commitmentOf RSObject.calibrationUnit = Commitment.convention
      ∧ commitmentOf RSObject.physicalQuotient = Commitment.quotient
      ∧ commitmentOf RSObject.observableFamily = Commitment.observable
      ∧ commitmentOf RSObject.generableCarrier = Commitment.permitted
      ∧ commitmentOf RSObject.primeScale = Commitment.forced :=
  ⟨rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl⟩

What this page does not claim

The rationals are the only forced object in the registry. The framework derives the rationals from nothing; it assumes the ambient theory of fields and real numbers. The theorem says anything about the physical interpretation of rationals as measurements.

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

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