Foundation Maximal Forcing Admissible Realization Forced Of Forced Under Tighten

The tightening lemma

The declaration forced_of_forced_under_tightening establishes a monotonicity property for the framework's forcing relation. In plain language: if a claim is forced across a broad class of admissible realizations, and you then tighten that class by adding more constraints, the claim remains forced. The proof is immediate: every realization in the narrower class is also in the wider one, so if the claim holds for all wider realizations, it holds for all narrower ones too.

The framework models reality as a collection of admissible realizations, each a possible way the world could be structured. A claim is forced when it holds for every realization in a given class. Tightening means moving to a smaller class, one that excludes some previously allowed realizations. The lemma says that forcing survives this restriction: it is a one-way street, from broader to narrower, and never the reverse.

The declaration does not claim that tightening is always justified. A separate structure, LegitimateTightening, requires that a tightening do real work (exclude at least one previously admissible realization) and be justified by a deeper law, not by free choice. The lemma itself is silent on whether the tightening is legitimate; it only states the logical consequence of applying it.

In practice, this lemma is the backbone of the framework's closure operator. When a claim is forced on a wide class and a deeper law narrows the class, the claim carries over without re-proving it. It is a bookkeeping guarantee: adding rules never breaks what was already established.

/-- If a claim is forced on a wider admissible class, it remains forced after
tightening. -/
theorem forced_of_forced_under_tightening {R : Type u}
    {A B : AdmissibilityClass R} {C : RealityClaim R}
    (hT : Tightening A B) (hA : Forced A.admissible C) :
    Forced B.admissible C := by
  intro r hr
  exact hA r (hT.subset r hr)

What this page does not claim

The lemma does not claim that any particular tightening is justified or legitimate. It does not claim that a claim forced on a narrower class is forced on a wider one.

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

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