Proper classes (in the project's sense: circular self-embedding structures) have their mathematical home in terminal coalgebras νF ≅ F(νF).
Mathematics
The discipline entrances are dictionary-level correspondences: a ladder built in the other party's language, withdrawn once climbed.
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.
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.
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.
The Feigenbaum accumulation point r∞≈3.570 can be read as a dynamical realization of the Axiom-of-Foundation boundary.
Period-doubling 2ᵏ corresponds to the nested yin-yang trigram system; period 5 corresponds to the Five Phases.
Mutual arising ↔ adjunction (category theory).
Grand unification theorem: the necessity of the science of generation as a whole (in progress).
There is a unique hexagram–object assignment (e.g. one hexagram = one specific codon).
The order 60 of A₅ is the 60 of the sexagenary cycle.