Generative Structural Theory v1.23 — Connecting the Class C Evaluation with Closure-Preserving Sub-relations

GST v1.19 stored, for the fixed cut z_t of twelve pairs, a list of 59 sub-relations as the set C_2 of its class C evaluation; GST v1.20-v1.22 described, counted and analysed the family H_lt(z_t) of the proper sub-relations of z_t that keep its reflexive transitive closure, which also has 59 members. This manuscript connects the two by members, not by counts, and keeps three objects apart: the set E_alg that the evaluator's loop and acceptance condition define, the set E_stored of the elements of the stored list, and the set C_2 of the paper's definition. (T2a) E_alg = H_lt(z_t), by an argument based on existing computation results (the registered B1 check of v1.20) and on the drafting side's static reading of the evaluator, with no new enumeration. (T2b) E_stored = H_lt(z_t), and the stored list has no repeated element, confirmed by a pre-registered, LOCKed machine comparison (a single run, no automatic correction, no rerun; word EQUAL), which builds the reference set in two ways that must agree and checks four controls first. Hence E_stored = H_lt(z_t) = E_alg as sets, relations being identified with their integer codes. (T2c) Every member of C_2, where its definition applies, lies in H_lt(z_t), by the definition alone, without D2 or a check of condition C-cone; the converse inclusion holds if the recorded certified set V of the v1.18 runs is the V of those runs. So C_2 = H_lt(z_t) holds under that assumption; it is not established without it, and the assumption is undetermined. (T3) Through the equalities of T2a and T2b, the v1.22 results for z_t hold for E_alg and E_stored: five pairs lie in every member, there are four inclusion-minimal members, of 7, 7, 8 and 8 pairs, the least number of pairs is 7, and the stored answer y = z_t - {1>3} is one of the seven maximal members and is not inclusion-minimal. A short lemma says why removing the maximum z_t does not change the essential pairs, the inclusion-minimal members or the members of minimum cardinality. These are structural consequences; they do not explain why the procedure stored y. Nothing is claimed about the provenance of the stored file, other cuts, or the classification inside components.