Generative Structural Theory v1.15 — A Conditional Fibre Characterisation of a Provenance-Population Condition and a Recorded Verdict

GST v1.14 found, on a fixed single-arc census record, that a provenance-population condition Q_D selects a non-empty set strictly inside the set of candidates whose recorded verdict is the E1 word (E_rec). This note describes what the two conditions require. For each candidate in a stated range, the full and prefix chain-tuple sets F_a and P_a are finite and componentwise truncation is a map rho_a : F_a -> P_a; with b_a the surplus of the fibres over the image and h_a the number of empty fibres, |F_a| - |P_a| = b_a - h_a, a statement about finite sets and maps. Assuming the provenance premise PR (that the record lines were produced by the specified code on the specified inputs and were not changed after generation) and the correspondence between these objects and the census implementation, established by drafting-side static matching and reading, Q_D holds on the domain D exactly when b_a > h_a, and E_rec holds exactly when the code's prefix condition prefix_ok holds (the code's test, not the census rule's whole prefix clause), some letter index has an extra level, and F_a is non-empty. This gives a conditional characterisation of the difference between the two conditions by the fibres of rho_a: under Q_D, E_rec reduces to prefix_ok; E_rec without Q_D falls into exactly one of two cases (rho_a injective, or non-injective with at least as many empty fibres as the surplus). Which concrete fibre structure produced the strict inclusion on the fixed record is not determined. The result is conditional on PR, which is not verified; it does not resolve single-arc eligibility; and it does not identify the mechanism by which the difference arises on the record.