Foundation Primitive Recognition Calculus Objecthood Registry Classify Observabl

The observable contract

In the Recognition Science framework, the declaration classify_observable establishes a precise contract about what an observable is. The framework models a physical system as a set of states, and an observable as a function that assigns a value to each state. The theorem states that every admissible observable descends to the physical quotient: if two states are identified by the quotient, then every admissible observable assigns them the same value. In plain terms, the quotient loses no observable information; the observable family is exactly the set of probes that survive the identification.

The theorem is proved in the framework's machine-checked library of formal theorems. Its statement is: for any type X of states, any collection F of functions from X to a value type C, and any function f in F, there exists a function g on the physical quotient such that for every state x, g applied to the quotient projection of x equals f applied to x. This is the formal way of saying that every admissible observable factors through the quotient. The proof is a single step, deriving from the definition of the quotient as the identification of states that no observable distinguishes.

The declaration is one of seven commitments in the framework's objecthood registry, a classification of the kinds of mathematical objects the framework builds. The registry assigns each object a commitment type: forced, permitted, quotient, completion, display, observable, or convention. The observable commitment is distinct from the others. For example, the rational numbers are forced into every carrier, while the cost-scale unit is a convention, a gauge choice. The observable commitment is specifically about the relationship between a quotient and the functions that respect it.

What classify_observable does not claim is equally important. It does not say which functions are admissible observables; that is a separate modeling choice. It does not say that observables are the only way to access a system. It does not say that the quotient is unique; the theorem holds for any quotient defined by a family F. The theorem is a structural fact about quotients and functions, not a physical law about what can be measured.

In the framework, this theorem supports the broader claim that the physical quotient is forced by indistinguishability. The observable family is the set of probes that define that indistinguishability, and the theorem guarantees that the quotient respects them. This is a step toward the framework's goal of building physical structure from a ledger of recognition events, where the cost of recognition is forced, not chosen.

/-- `observable`: every admissible observable descends to the physical quotient;
the quotient loses no observable information. -/
theorem classify_observable :
    ∀ {X C : Type} (F : Set (X → C)) (f : X → C), f ∈ F →
      ∃ g : QuotientSelection.PhysicalQuotient F → C, ∀ x, g (QuotientSelection.proj F x) = f x :=
  fun F f hf => QuotientSelection.observable_descends F f hf

/-- `observable`: every admissible observable descends to the physical quotient;
the quotient loses no observable information. -/
theorem classify_observable :
    ∀ {X C : Type} (F : Set (X → C)) (f : X → C), f ∈ F →
      ∃ g : QuotientSelection.PhysicalQuotient F → C, ∀ x, g (QuotientSelection.proj F x) = f x :=
  fun F f hf => QuotientSelection.observable_descends F f hf

/-- **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 theorem does not specify which functions are admissible observables. The theorem does not say that observables are the only way to access a physical system. The theorem does not claim the physical quotient is unique.

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)

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: