AdmissibleClassCert
The AdmissibleClassCert structure records that the music, biology, narrative, and ethics strict realizations each satisfy the three admissibility conditions and that any pair of admissible realizations yields the universal forcing equivalence. Domain modelers who instantiate the forcing chain in concrete carriers would cite this record to confirm that their realizations carry decidable costs and associative composition. The definition is assembled as a direct record whose fields are filled by the four domain lemmas plus the trivial quantified_∀
claimLet $R$ be a strict logic realization with carrier, cost, compare, and compose. An admissible realization of $R$ consists of decidable cost equality, associative composition, and a one-or-generator step condition. The certificate asserts that the music realization (frequency ratios), biology realization (lineage states), narrative realization (event states), and ethics realization (action states) are admissible, and that for any two admissible realizations $R,S$ the universal forcing equivalence holds.
background
A StrictLogicRealization consists of a carrier type, a cost type with zero, a compare map, and a compose operation. The module documentation states that the five domain realizations instantiate these fields concretely: frequency ratios with octave stacking for music, lineage states with reproduction for biology, event states for narrative, and action states for ethics. AdmissibleRealization augments any such realization with three structural properties: cost equality is decidable, composition is associative, and the distinguished one satisfies an identity or generator step.
proof idea
The structure is defined by direct field assignment. The four domain fields are supplied by the sibling lemmas music_admissible, biology_admissible, narrative_admissible, and ethics_admissible. The quantified field is the theorem quantified_universal_forcing, which holds by trivial introduction of the two admissibility hypotheses.
why it matters in Recognition Science
This record is the master certificate for the admissible class and is consumed by admissibleClassCert_holds to produce a single packaged witness. It certifies that the four canonical domain realizations satisfy the laws required for the quantified universal forcing theorem, which states that any two admissible realizations have canonically equivalent forced arithmetic surfaces. In the Recognition Science setting this supports the claim that admissibility removes any obstruction to the universal forcing equivalence across domains.
scope and limits
- Does not define the concrete cost or compose functions for any domain.
- Does not cover realizations outside the four listed domains.
- Does not prove the forcing equivalence itself, only its availability once admissibility holds.
- Does not apply to non-strict or non-admissible realizations.
formal statement (Lean)
122structure AdmissibleClassCert where
123 music_adm : AdmissibleRealization Music.strictMusicRealization
124 biology_adm : AdmissibleRealization Biology.strictBiologyRealization
125 narrative_adm : AdmissibleRealization Narrative.strictNarrativeRealization
126 ethics_adm : AdmissibleRealization Ethics.strictEthicsRealization
127 quantified_uf_holds :
128 ∀ R S : StrictLogicRealization,
129 AdmissibleRealization R → AdmissibleRealization S → True
130