phonologicalFeatureCount
Recognition Science models language acquisition difficulty via J-cost on the plasticity ratio. This declaration fixes the number of independent phonological feature classes at five. Linguists citing the critical period would reference this count when bounding total acquisition cost. The proof is a one-line decision procedure that enumerates the constructors of the feature type.
claimThe set of phonological features has cardinality $5$, consisting of the classes vowels, consonants, tone, stress, and prosody.
background
The module formalizes the critical period for language acquisition using the J-cost function on the synaptic plasticity ratio r. Peak plasticity gives J(r) = 0; the boundary occurs when r enters the J(phi) band, and near-native fluency becomes impossible for r < 1/phi. Each of the five feature classes is independently subject to this threshold, matching configuration dimension D = 5.
proof idea
The proof is a one-line wrapper that invokes the decide tactic. The tactic evaluates the cardinality equality by enumerating the five constructors of the inductive feature type and confirming the result is five.
why it matters in Recognition Science
This cardinality supplies the feature_count field inside the languageAcquisitionCert definition. It realizes the configDim = 5 for phonological features in the J-cost model of the critical period. The result closes the enumeration step without addressing the explicit threshold value or connections to the eight-tick octave.
scope and limits
- Does not derive the J-cost formula from the plasticity ratio.
- Does not prove the numerical location of the critical period boundary.
- Does not relate the feature classes to spatial dimension D = 3.
- Does not establish independence of the classes under the Recognition Composition Law.
formal statement (Lean)
30theorem phonologicalFeatureCount : Fintype.card PhonologicalFeature = 5 := by decide
proof body
31