We study a family of directed graphs on n = 2k+ 5 vertices whose adjacency matrices are{0, 1}-matrices with zero diagonal, uniform column sum 2, and a rigid Jordan structure: thenon-zero spectrum {2, 1, −1, −1, −1} is constant in k, while a single nilpotent block J2k(0)grows with k. We call this the Wen class Uk and define it through five axioms D1—D5. The main result of the paper settles the existence question. We construct an explicit infinite subfamily {Am}m≥1 with ord(Am) = 2m + 7, obtained by iterating a chain-insertionoperator T on a fixed 7 × 7 core C. A Schur-complement identity for T, combined with aresolvent computation that reduces to three cofactors of the single 7 × 7 matrix zI − CT, shows that T multiplies the characteristic polynomial by z 2 ; the exact determination ofdim ker Am = 1 and dim ker(Am + I) = 2 then upgrades this spectral statement to the fullJordan axiom D4. Consequently Uk 6= ∅ for every integer k ≥ 1: the class is infinite, andno order is empty. Every step of the proof is a finite exact computation of order at most 7,independent of m. We prove the closed-walk counting formula NL = 2L + 1 + 3(−1)L, the rank formula rank(Am) = max(n − m, 5), the minimal polynomial x 2k (x − 2)(x − 1)(x + 1)2 of degree n − 1, the commutant dimension 7 + 2k, the Newton-identity recurrence of order 5,and the doubler-chain power-of-two property. A computer search at seven orders (k =1, . . . , 7, i.e. n = 7, . . . , 19) produced 233 pairwise non-isomorphic members, distributed as 34, 50, 39, 22, 68, 13, 7; these numbers are lower bounds for the number of isomorphism classes at each order, and ten representatives of each order are listed in Appendix 18. We disprove two natural conjectures (the "minimum-one integer Perron vector" property and the "core-template" hypothesis) and organise the observed structure into a three-layer architecture separating rigid invariants, semi-invariants and flexible realisation.