Foundation Maximal Forcing Reality Closure Maximal Forcing Closure

The closure certificate

In Recognition Science, the framework that derives physical structure from the cost of recognition events, a central goal is to know which claims about reality are forced by the framework's own principles. The declaration maximal_forcing_closure in the machine-checked library of formal theorems states a conditional result: if a certain certificate exists, then every claim in the forcing closure is classified as forced, independent, or selected. The certificate is a function that takes any claim in the closure and returns its classification. The theorem itself is a projection from that certificate, not an assertion that the certificate exists.

The trichotomy theorem sharpens this. Given the certificate, any claim in the closure falls into exactly one of three categories: Forced, meaning the framework's principles entail it; Independent, meaning there is an explicit countermodel witness showing the framework does not decide it; or Selected, meaning a named selection principle chooses it. The key point is that the first two categories carry proof obligations. An independent claim is not a lazy concession; it comes with a concrete witness. A selected claim names its principle. The theorem concedes no contingency without evidence.

The program is to build the certificate for the real claim universe, not to postulate it. The docstring is explicit: the crown theorem is not asserted, the certificate is the target. This is a session protocol, not a completed result. Closing a session on this program means adding a new forced invariant, landing an independence witness, tightening admissibility, or updating the execution plan with the exact remaining blocker. The nonempty_progress field in the protocol structure guarantees that every session makes at least one of these moves.

What this means in plain language is that the framework has defined a precise sense in which it could be complete, and has proved that if it ever achieves that completeness, every claim it can ask about is settled with evidence. It does not claim that completeness has been achieved, and it does not claim that any particular physical claim is forced. The certificate is the missing piece, and the theorem is the bridge from that piece to the full trichotomy.

/-- 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

/-- 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)

What this page does not claim

No certificate has been built for the real claim universe, so no physical claim is asserted to be forced by this theorem. The theorem does not establish that any particular claim is independent or selected; it only states the conditional trichotomy given a certificate. The framework's completeness is not achieved; the certificate construction remains an open target.

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.

Derived articles

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