Proper classes (in the project's sense: circular self-embedding structures) have their mathematical home in terminal coalgebras νF ≅ F(νF).

Under the Anti-Foundation Axiom, every (flat) system of equations has a unique solution (Aczel's Solution Lemma); primordial equations replace axiomatic propositions on this basis.

Bisimulation serves as the identity criterion; in the finite case it is decidable via Paige–Tarjan partition refinement.

5 is the least integer admitting, on one element set, two mutually irreducible directed Hamiltonian cycles (steps 1 and 2); hence the "five" of the Five Phases is fixed by number theory.

Restraint = generation²: on ℤ₅ the overcoming map is the square of the generating map.

5 ∤ 2ᵏ: the Five-Phase track cannot be reduced to a binary (yin-yang track) representation.

For the logistic map, all period-2ᵏ orbits appear below the accumulation point r∞≈3.570; period 5 appears only beyond it.

Sharkovskii's theorem (external anchor): periods of continuous interval maps obey the Sharkovskii ordering.

Nuclear-hexagram basin spectrum: the nuclear map is the only irreversible inter-hexagram relation; the basin spectrum of the 64 hexagrams under its iteration (three attractor basins) is exhaustively determined.

The transition semigroup of the full 32-state rotor cell (coupled phase and magnitude) is solvable with group component ℤ₈; an S₈ counterexample marks the boundary.

The rotor sequent calculus RSC₀ is sound and complete.

Coding threshold 16 < 20 ≤ 64: doublet capacity cannot encode 20 amino acids; triplet capacity can.

C₅ is solvable and A₅ is not (external anchor); this marks the outer boundary of the generative theory of molecular genetics.

64-group isomorphism: 64 = 4³ = 2⁶; codon space and hexagrams are isomorphic as (ℤ₂)⁶.