minPhonemes
plain-language theorem explainer
minPhonemes fixes the lower bound on natural-language phoneme inventory size at the integer 8. Quantitative linguists testing universal grammar constraints against Recognition Science would cite this constant when checking empirical inventories against the derived band. The declaration is a direct constant assignment that imports the vertex count of Q₃ from the upstream SpectralEmergence structure.
Claim. The minimum phoneme inventory size is defined to be the integer 8, equal to the cardinality of the vector space $F_2^3$ (equivalently, the number of vertices of the 3-cube $Q_3$).
background
Recognition Science forces three spatial dimensions (T8) from the phi fixed point and eight-tick octave, yielding the discrete hypercube $Q_3$ whose vertices number $2^3=8$. The module imports five upstream structures: NucleosynthesisTiers.of (nuclear density tiers), LedgerFactorization.of (J-cost on the positive reals), PhiForcingDerived.of (J-cost properties), SpectralEmergence.of (Q₃ forcing of gauge groups, three generations, and 24 chiral flavors), and PhysicsComplexityStructure.of (convexity of J-cost minimization). SpectralEmergence.of states that Q₃ simultaneously forces SU(3)×SU(2)×U(1) gauge content and exactly three particle generations from face-pair count.
proof idea
The definition is a direct constant assignment of 8. It inherits the identification of the Q₃ vertex count from SpectralEmergence.of without applying any lemmas or tactics inside the declaration itself.
why it matters
This definition supplies the lower endpoint for the phoneme inventory band used by human_language_phonemes_in_band, phoneme_bound, minPhonemes_eq_F2cube, and the PhonemeInventoryCert structure. It translates the T8 landmark (D=3) into a concrete linguistic prediction by equating the minimum inventory to |F₂³|. PhonemeInventoryCert packages the bound with the upper limit of 45 (gap-45 orbits) and the strict inequality minPhonemes < maxPhonemes. The declaration closes the link from general Q₃ spectral emergence to testable phoneme-count constraints.
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