Foundation Maximal Forcing Rsclosure Extension Extend Preserves Trichotomy

The closure guarantee

In the Recognition Science framework, a ledger, a discrete record of claims about reality, is organized by a classifier that sorts every claim into exactly one of three buckets: forced (derivable from the framework's axioms), independent (neither provable nor refutable, with explicit countermodels), or selected (adopted by a named principle). The theorem extend_preserves_trichotomy proves that this three-way sorting survives a specific operation: adding one new forced claim to the ledger.

The operation is called carrier extension. Given a complete classifier for a claim universe, and any claim that is forced over the universe's admissibility class, the theorem shows the extended universe still has a complete classifier. The new claim lands in the forced bucket, and every old claim keeps its original classification verbatim. This is a structural guarantee, not a cosmetic one: no forced invariant can ever be missing in a way that breaks the closure.

The proof is a direct specialization of a more general result. The library defines a certificate that a classifier is complete, then shows how to transport any classification along an extension. Because extension changes only the claim set, leaving the realization type and admissibility class fixed, every field is definitionally equal. The theorem extend_preserves_trichotomy then applies the trichotomy lemma to the extended certificate.

What this does not claim is just as important. The theorem does not say that every claim is forced, nor that the ledger is complete in an absolute sense. It says that if you have a complete classifier and add a forced claim, completeness is preserved. The only way a genuinely new claim adds content beyond the forced bucket is by carrying its own independence witness or named selection principle, each a real proof obligation. The register's incompleteness, if any, can only ever be an undiscovered independence or selection, never an undiscovered forced invariant.

The consequence is practical: the framework's curated claim set cannot be incomplete in a way the framework fails to absorb. If a new physically-real invariant is proposed, exactly one of three things happens, and all three are handled. The residual content of the maximal-forcing program is never finding a missing slot in the register; it is always producing the classification certificate for a specific proposed claim.

/-- **EXTENSION PRESERVES THE TRICHOTOMY (THEOREM).** Every claim in a universe
extended by a forced invariant is `Forced`, `Independent`, or `Selected`. The new
forced claim lands in `Forced`; everything else keeps its prior classification. -/
theorem extend_preserves_trichotomy {P : Primitive} {U : ClaimUniverse.{u}}
    (cert : MaximalClosureCert P U) {C0 : RealityClaim U.Realization}
    (hC0 : Forced U.admissibility.admissible C0)
    (C : RealityClaim U.Realization) (hC : InClosure P (U.extend C0) C) :
    Forced (U.extend C0).admissibility.admissible C ∨
    Independent (U.extend C0).admissibility.admissible C ∨
    Selected (U.extend C0).admissibility.admissible C :=
  maximal_forcing_closure_trichotomy (cert.extendForced hC0) C hC

/-- **No forced invariant can be missing (THEOREM).** For any claim `C₀` forced over
`U`'s admissibility, the extended universe still admits a complete classifier and
`C₀` itself is `Forced` there. This is the precise structural answer to the
carrier-completeness worry on the forced side: the closure absorbs any forced fact
without disruption. -/
theorem forced_invariant_absorbed {P : Primitive} {U : ClaimUniverse.{u}}
    (cert : MaximalClosureCert P U) {C0 : RealityClaim U.Realization}
    (hC0 : Forced U.admissibility.admissible C0) :
    (Nonempty (MaximalClosureCert P (U.extend C0))) ∧
    Forced (U.extend C0).admissibility.admissible C0 :=
  ⟨⟨cert.extendForced hC0⟩, hC0⟩

What this page does not claim

The theorem does not assert that every claim in reality is forced. It does not provide a method to discover new forced invariants. It does not claim the ledger is complete in an absolute sense, only closed under extension by classifiable claims.

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/RSClosureExtension.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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