Mathematical Definitions

Sets
P = universal position set, |P| = 133
T = {Shijūhatte, Kāmasūtra, Fángzhōngshù, Perfumed Garden, Greco-Roman}
M = {facing, rear, sitting, standing, side, inverted, oral, special}
P_t = positions belonging to tradition t

Functions
μ: P → M — mechanical category assignment
v: T → ℤ&sup8; — tradition vector (category counts)
σ: P × P → [0,1] — similarity metric
d(p,q) = 1 − σ(p,q) — pseudometric

Relations
p ~_ε q ⇔ σ(p,q) ≥ ε — equivalence relation
[p]_ε = equivalence class of p at threshold ε

Group G = ⟨α, β, γ⟩ ≤ S₈
α: facing ↔ rear (orientation reversal)
β: sitting ↔ standing (vertical inversion)
γ: facing ↔ inverted (role exchange)
|G| depends on generator closure
Orbit–stabilizer: |G| = |Orb(x)| × |Stab(x)| for all x ∈ M