Foundation Maximal Forcing Reality Closure Maximal Forcing Closure Trichotomy

The closure trichotomy

A formal system that aims to describe reality faces a basic question: for any given claim, is it forced by the system's starting points, is it independent of them, or is it chosen by an explicit rule? The recognition framework, which treats reality as holding a discrete ledger of events, answers this question with a trichotomy theorem. The theorem states that, given a certificate that classifies every claim in a system's closure, every claim falls into exactly one of three categories: forced, independent, or selected.

The three categories carry distinct meanings. A forced claim is one that follows necessarily from the system's primitives and rules; it cannot be otherwise. An independent claim is one for which there exists an explicit countermodel witness, meaning a consistent scenario where the claim fails. A selected claim is one that is neither forced nor independent but is admitted through a named selection principle. The theorem is deliberately conditional: it does not assert that such a classification exists for any particular universe, but rather that if a classifier certificate exists, then the trichotomy holds for every claim in the closure.

The theorem is proved in the framework's machine-checked library of formal theorems. Its proof is a short case analysis: given the certificate's classification of a claim, the theorem maps each outcome to the corresponding disjunct. The significance is not in the proof's length but in its role as an interface. The framework's program is to build such a classifier certificate for the real claim universe, and this theorem states what that certificate would deliver: a complete triage of every claim into forced, independent, or selected.

The trichotomy is not a claim that the certificate exists. It is a conditional statement: if you have the certificate, then you get the triage. The framework's session protocol tracks progress toward that certificate, recording whether a session has landed a forced invariant, an independence witness, tightened admissibility, or updated the execution plan. The theorem itself concedes no contingency lazily: both independent and selected outcomes carry their own proof obligations, namely an explicit countermodel and a named selection principle respectively.

What the theorem does not claim is equally important. It does not assert that the real universe's claim set has been classified, nor that any particular claim is forced, independent, or selected. It does not assert that the certificate can be built, only what follows if it is. The trichotomy is a structural promise about the framework's target end state, not a report of progress toward it.

/-- Crown theorem in the exact disjunction form: given a classifier certificate,
every claim in the forcing closure is `Forced`, `Independent`, or `Selected`.
This is the literal "as forced as possible" statement; it concedes no contingency
lazily, because `Independent` and `Selected` are themselves proof obligations
(an explicit countermodel witness and a named selection principle, respectively).
-/
theorem maximal_forcing_closure_trichotomy
    {P : Primitive} {U : ClaimUniverse.{u}} (cert : MaximalClosureCert P U)
    (C : RealityClaim U.Realization) (hC : InClosure P U C) :
    Forced U.admissibility.admissible C ∨
    Independent U.admissibility.admissible C ∨
    Selected U.admissibility.admissible C := by
  rcases cert.classifies C hC with h | hw | hs
  · exact Or.inl h
  · exact Or.inr (Or.inl (independent_of_witness hw))
  · exact Or.inr (Or.inr hs)

/-- Crown theorem in the exact disjunction form: given a classifier certificate,
every claim in the forcing closure is `Forced`, `Independent`, or `Selected`.
This is the literal "as forced as possible" statement; it concedes no contingency
lazily, because `Independent` and `Selected` are themselves proof obligations
(an explicit countermodel witness and a named selection principle, respectively).
-/
theorem maximal_forcing_closure_trichotomy
    {P : Primitive} {U : ClaimUniverse.{u}} (cert : MaximalClosureCert P U)
    (C : RealityClaim U.Realization) (hC : InClosure P U C) :
    Forced U.admissibility.admissible C ∨
    Independent U.admissibility.admissible C ∨
    Selected U.admissibility.admissible C := by
  rcases cert.classifies C hC with h | hw | hs
  · exact Or.inl h
  · exact Or.inr (Or.inl (independent_of_witness hw))
  · exact Or.inr (Or.inr hs)

/-- Crown theorem in the exact disjunction form: given a classifier certificate,
every claim in the forcing closure is `Forced`, `Independent`, or `Selected`.
This is the literal "as forced as possible" statement; it concedes no contingency
lazily, because `Independent` and `Selected` are themselves proof obligations
(an explicit countermodel witness and a named selection principle, respectively).
-/
theorem maximal_forcing_closure_trichotomy
    {P : Primitive} {U : ClaimUniverse.{u}} (cert : MaximalClosureCert P U)
    (C : RealityClaim U.Realization) (hC : InClosure P U C) :
    Forced U.admissibility.admissible C ∨
    Independent U.admissibility.admissible C ∨
    Selected U.admissibility.admissible C := by
  rcases cert.classifies C hC with h | hw | hs
  · exact Or.inl h
  · exact Or.inr (Or.inl (independent_of_witness hw))
  · exact Or.inr (Or.inr hs)

/-- Conditional crown theorem: once a classifier certificate exists, every claim
in the forcing closure is forced, independent, or selected. This is deliberately
conditional; the program is to build `MaximalClosureCert` for the real universe,
not to postulate it. -/
theorem maximal_forcing_closure
    {P : Primitive} {U : ClaimUniverse.{u}} (cert : MaximalClosureCert P U) :
    forall C : RealityClaim U.Realization,
      InClosure P U C -> ClaimClassification U C :=
  cert.classifies

What this page does not claim

The certificate for the real universe has not been built; the theorem is conditional on its existence. No specific claim about the physical world is asserted to be forced, independent, or selected by this theorem. The theorem does not prove that the trichotomy is exhaustive for claims outside the closure.

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/RealityClosure.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.

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