Generative Structural Theory v1.20 — Where the 59 Cuts Come From: The Closure-Preserving Sub-relations of z_t, a Finite Machine Check of Their Characterisation, and a Conditional Connection with C_2

GST v1.19 recorded that the set C_2 used by the class C evaluation has 59 members, a value read from the evaluation output. This manuscript explains where that number comes from and checks the explanation by machine. Let z_t be the 12-pair cut of v1.19 (Y_9 without 4>1), and H the set of sub-relations w of z_t with the same reflexive transitive closure as z_t. A replacement lemma reduces membership in H to replacement paths; for z_t this gives a necessary and sufficient condition in four independent groups of pairs (five pairs always kept; at least one of 1>0, 1>3; 0>4, or both 0>3 and 3>4; 5>0 and 5>2 free), so |H| = 1 x 3 x 5 x 4 = 60 and |H - {z_t}| = 59. Every member of H is persistent and not closed: having the closure of z_t, it keeps the mutual reachability of 0 and 2, and it keeps the reachability of 2 from 1 while the pair 1>2 is absent, so it is not closed. A pre-registered finite check, run once against the fixed cone_lib code on all 4096 subsets of z_t, found the code's closure test and the stated condition in agreement on every subset, and the persistence and non-closedness properties on all 60 members, with its controls behaving as registered. Under the cited lemma D2 and the C-cone reservation kept from v1.19, H - {z_t} is C_2; by the code definitions, the same 59 cuts have letter 000 relative to z_t and 100 relative to Y_9, the only change being the new reachability from S = {0, 2, 3, 4} into node 1 through 4>1. The member list reported by the class C evaluation was not compared, and nothing is claimed about acceptance at a root or about why the stored answer was chosen.