LanguageAcquisitionCert
plain-language theorem explainer
Language acquisition in Recognition Science is certified by a structure requiring exactly five phonological features and a canonical J-band threshold for the critical period. Researchers modeling the closure of the critical period via J-cost would reference this when connecting config dimension to acquisition limits. The definition is a direct packaging of the feature cardinality and the upstream CanonicalCert properties with no additional proof steps.
Claim. A language acquisition certificate consists of the assertion that the set of phonological features has cardinality 5 together with a canonical J-band certificate $C$ satisfying $J(1)=0$, $J(x)=J(1/x)$ for $x≠0$, $J(ϕ)>0$, $0.11<J(ϕ)<0.13$, and $J(ϕ^{-2})>0$.
background
The module models the critical period for language acquisition using the J-cost function from Recognition Science, where plasticity ratio r yields J(r) with J(1)=0 at peak and rising cost as r drops below 1/ϕ. PhonologicalFeature is the inductive type with constructors vowel, consonant, tone, stress, prosody, giving five classes that align with configDim D=5. The CanonicalCert upstream structure supplies the six clauses that any domain certificate must satisfy, including the phi-band bounds on J(phi). This local setting treats each feature class as independently subject to the same critical period threshold.
proof idea
The declaration is a structure definition. It directly records the Fintype cardinality of the five-constructor PhonologicalFeature inductive and embeds an instance of the CanonicalCert structure. No tactics or lemmas are applied; the definition serves as a type-level package for the two required properties.
why it matters
This certificate is used to construct the concrete languageAcquisitionCert instance in the same module, providing a reusable object for J-cost based models of language acquisition. It fills the role of linking the five-dimensional phonological system to the T5 J-uniqueness and T6 phi fixed point in the forcing chain, with the critical period threshold tied to the phi-band. It touches the open question of how the same J-cost law scales from physics to cognitive processes without additional axioms.
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