Foundation Maximal Forcing Forced Invariant Claim Classification

Classifying claims

In Recognition Science, a claim about reality is a statement that could in principle be checked against the framework's model. The declaration ClaimClassification establishes that every such claim must eventually land in exactly one of three categories. A claim is forced when it holds in every admissible realization of the framework: no matter how the details are filled in, the claim comes out true. A claim is independent when two admissible realizations disagree on it, meaning the framework neither forces it nor rules it out. The third category, selected, is a temporary tag for claims that are not yet forced but are governed by a named selection principle, a specific rule that picks one option among several.

The key point is that selected is not a resting place. The framework requires each selected claim either to be promoted to forced by finding a deeper admissibility condition, or to be demoted to independent by producing a countermodel, an admissible realization where the claim fails. This keeps the classification honest: no claim gets to sit forever in an undefined middle state. The structure ForcedInvariant packages the useful end product: a forced invariant is a closure claim, one that belongs to the closure of the framework's primitive notions, together with a proof that it is forced. In plain terms, it is a statement about reality that the framework can certify as unavoidable.

What ClaimClassification does not do is decide which category any particular claim falls into. It provides the three bins and the rules for moving between them, but it does not itself prove that any specific claim is forced, independent, or selected. That work belongs to other declarations that supply the admissibility conditions, the selection principles, and the countermodels. The classification is a scaffold, not a verdict.

/-- A forced invariant is a closure claim with a proof of forcedness. -/
structure ForcedInvariant (P : Primitive) (U : ClaimUniverse.{u}) where
  claim : RealityClaim U.Realization
  in_closure : InClosure P U claim
  forced : Forced U.admissibility.admissible claim

/-- A forced invariant is a closure claim with a proof of forcedness. -/
structure ForcedInvariant (P : Primitive) (U : ClaimUniverse.{u}) where
  claim : RealityClaim U.Realization
  in_closure : InClosure P U claim
  forced : Forced U.admissibility.admissible claim

/-- Classification of one claim in one closure universe. -/
inductive ClaimClassification (U : ClaimUniverse.{u})
    (C : RealityClaim U.Realization) : Prop where
  /-- Holds in every admissible realization. -/
  | forced : Forced U.admissibility.admissible C -> ClaimClassification U C
  /-- Two admissible realizations disagree on the claim. -/
  | independent : IndependenceWitness U C -> ClaimClassification U C
  /-- Not currently forced, but governed by a named selection principle. -/
  | selected : Selected U.admissibility.admissible C -> ClaimClassification U C

/-- Classification of one claim in one closure universe. -/
inductive ClaimClassification (U : ClaimUniverse.{u})
    (C : RealityClaim U.Realization) : Prop where
  /-- Holds in every admissible realization. -/
  | forced : Forced U.admissibility.admissible C -> ClaimClassification U C
  /-- Two admissible realizations disagree on the claim. -/
  | independent : IndependenceWitness U C -> ClaimClassification U C
  /-- Not currently forced, but governed by a named selection principle. -/
  | selected : Selected U.admissibility.admissible C -> ClaimClassification U C

What this page does not claim

ClaimClassification does not itself prove that any particular claim is forced, independent, or selected. The declaration does not define what makes a realization admissible.

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

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