AdmissibilityClass
plain-language theorem explainer
An admissibility class packages a set of allowed realizations of an abstract type R together with an audit label. Maximal-forcing arguments cite it as the carrier for subset-order tightenings and for forced/independent claims. It is a bare structure definition: two fields, no proof obligations.
Claim. An admissibility class on a type $R$ is a pair $(S,\ell)$ where $S\subseteq R$ is the set of admissible realizations and $\ell$ is a string label used for audit. The carrier $R$ is left abstract so that later phases may take $R$ to be logical models, costed realizations, or physical structures.
background
Maximal forcing refuses free parameters. When a reality claim is not forced on the current admissible set, the next step is either to tighten that set by imposing a deeper law, or to exhibit an independent countermodel. This module supplies the bookkeeping for that loop.
An admissibility class is the basic unit: a named subset of realizations. The type parameter $R$ is intentionally free; the same machinery applies whether realizations are strict-logic models, cost functionals, or domain-specific physical structures. Downstream, a tightening from class $A$ to class $B$ is a proof that every $B$-admissible realization is already $A$-admissible (with an optional strictness witness kept separate so the order remains usable before strictness is known).
Related upstream notions include the Recognition Operator's trivial admissibility predicate on ledger states and the primitive-distinction bridge that reduces axiom load; those feed later legitimate-tightening witnesses rather than this structure itself.
proof idea
Pure structure definition. Two fields are declared: admissible : Set R (the allowed realizations) and label : String (human-readable audit tag). No constructors beyond the default, no axioms, no tactics.
why it matters
This is the carrier type for the entire maximal-forcing closure stack in the module. Every subsequent notion is indexed by admissibility classes: Tightening (subset order between classes), ForcedAfterTightening (claims forced only after a gate), LegitimateTightening (non-vacuous tightenings justified by a named deeper RS law), and the smart constructor that turns an independence-to-forcing flip into a legitimate gate.
Downstream, ForcingClosure.ClaimUniverse consumes these classes when assembling the closed claim set. In the Recognition Science forcing chain the pattern matches the T0–T8 style: never leave a degree of freedom open; either force it by a deeper law or prove independence. The abstract $R$ lets the same gate machinery run from pure logic through costed realizations into physical models without rewriting the order theory.
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