Foundation Maximal Forcing Rsclosure Extension Forced Invariant Absorbed

Closure under new facts

A claim universe, a discrete record of statements about reality, comes with a classification that sorts every claim into one of three buckets: forced, independent, or selected. A forced claim is one that follows necessarily from the framework's rules. The theorem forced_invariant_absorbed addresses a specific worry: if someone later names a new forced fact that was not in the original record, does the whole classification fall apart? The answer is no, and the proof is machine-checked.

The theorem states that given a complete classifier for a claim universe, and any claim that is forced over that universe's admissibility class, the extended universe (with the new claim added) still has a complete classifier. The new claim itself is classified as forced. Every old claim keeps its prior classification verbatim. The structural reason is that the classification depends on the universe only through its admissibility class and realization type, both of which are untouched by enlarging the claim set.

This is a closure property. It means the ledger of claims cannot be incomplete in the forced direction: any forced fact, once named, is absorbed without disruption. 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 residual incompleteness, if any, can only be an undiscovered independence or selection, never an undiscovered forced invariant.

The theorem is part of a broader result: the predicate "this universe has a complete classifier" is closed under extension by any classifiable claim. Forced, independent, and selected cases are all handled uniformly. The honest reading is that the curated carrier cannot be incomplete in any way the framework fails to absorb. The remaining work is always to produce a classification certificate for a specific proposed claim, not to find a missing slot in the recognition register.

/-- **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⟩

/-- **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⟩

/-- **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

What this page does not claim

The theorem does not prove that any particular forced invariant exists or is physically real. It does not show that the classification is complete for all possible claims, only that adding a forced claim preserves completeness. It does not address what happens when a new claim is neither forced, independent, nor selected under the framework's rules.

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)

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