Powers of Four: The Depth Arithmetic of the Recurring Four-Gon

The counts 4, 16, 64, and 256 are the dimensions of the graded components of the iterated Hamiltonian generator within the path algebra constructed on the endomorphism's directed graph. The 4ⁿ count rests on three structural properties of Q acting together: γ₁ is Hamiltonian, so every vertex lies on the cycle and can base a traversal; Q is strongly connected, so no terminal vertex is stranded at the handoff; and the recursion-joint edge a₄ returns C to L, closing the cycle so the next iteration begins. The three together make all four vertices eligible at each handoff, and that eligibility is what produces the multiplicative factor of 4 at each iteration. The address route and the graph route are two independent derivations of one count, on operands that share nothing: the address route derives 4ⁿ from two binary distinctions iterated, and the graph route derives it from Q's Hamiltonian cycle, strong connectivity, and recursion joint. The distinction between iterated-Hamiltonian grading and walk-length grading matters. Walk-length grading counts every directed walk of length n through Q, and the resulting graded dimensions are 4, 6, 8, 12, 18 at lengths 0 through 4. Iterated-Four-gon (Hamiltonian) grading counts complete Hamiltonian cycles iterated n times, and the resulting graded dimensions are 4, 16, 64, 256 at depths 1 through 4. Depth-n positions are not an undifferentiated state space of 4ⁿ elements. They are strictly nested. The recursion is a containment hierarchy, not a list of independent state spaces. The scale invariance the counting derivation takes as input is derived in *On Occurrence: The Four-Gon* (Stewart, 2026g). These counts are structural consequences of scale invariance applied to a 2² architecture. This paper proposes the recurrence of these numbers across independent domains as convergent evidence for the endomorphism. Two filters specify what would settle the question empirically: a depth-3 instantiation with a flat 64-element state space lacking the 4 × 4 × 4 nested partition, or a depth-4 instantiation whose 256 positions index a graded component other than the iterated Hamiltonian. Either would show the counts belong to generic binary recursion rather than to Q. **Keywords:** scale invariance, recursion depth, powers of four, path algebra, Hamiltonian cycle, nested partition, binary encoding, genetic code, byte