languageAcquisitionCert
plain-language theorem explainer
This definition constructs the language acquisition certificate by populating the feature count field with the decided value 5 and assigning the critical period threshold to the canonical cert. Researchers modeling language acquisition via J-cost in Recognition Science would cite it to instantiate the structure that encodes the five phonological feature classes and the plasticity threshold. The construction proceeds by direct field assignment from the upstream count theorem.
Claim. The language acquisition certificate is the structure satisfying $Fintype.card(PhonologicalFeature)=5$ together with a critical period threshold that meets the canonical certificate condition.
background
The module treats language acquisition through the J-cost formalism, where the plasticity ratio r determines acquisition difficulty: r=1 yields J(r)=0 at peak childhood plasticity, while r entering the J(phi) band marks the critical period boundary and r<1/phi renders near-native fluency impossible. The structure LanguageAcquisitionCert requires exactly five independent phonological feature classes (vowels, consonants, tone, stress, prosody) and a canonical threshold. The upstream phonologicalFeatureCount theorem establishes the cardinality by direct decision.
proof idea
The definition is a direct construction that assigns the feature_count field to the result of the phonologicalFeatureCount theorem and sets the critical_period_threshold field to the canonical cert.
why it matters
This definition supplies the concrete instance of LanguageAcquisitionCert needed to close the linguistics tier in the Recognition Science framework. It links the J-cost function to the empirical critical period for language acquisition and provides the explicit certificate required by downstream models. The construction touches the open question of how the five-feature count interacts with the eight-tick octave and D=3 spatial embedding in the broader forcing chain.
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