The structural role of 529
The number 529 has a specific role in the Priestly expansion family: it is 23², the denominator needed for two applications of E=25/23 to produce integral duration widths. A width 529n therefore follows the ladder 529n→575n→625n. The inherited examples 1058→1150→1250 and 10580→11500→12500 share that duration grammar, with the second ten times the first. Their agreement follows from the same operation and scale relation.
Widths alone do not determine endpoints. Holding the same integer pivot for both stages requires each starting point’s offset from that pivot to be divisible by 529. With different pivots, the allowed residue changes. Two fields can therefore share a 529-multiple width while only one has integral endpoints under its declared pivots. The successful inherited small example and the fixed-12026 comparison exhibit exactly this distinction.
The Rounded macro family also supplies a particular endpoint realization: with AD 12026 held, AD 1446→AD 526→476 BC accompanies the widths 10580→11500→12500. Its last crossing uses the declared Rounded coordinate convention; substituting ordinary civil counting changes the final display.
Thus 529 connects these families through a reusable two-stage duration structure. It does not license a universal date conversion, supply a missing pivot, or establish independent evidence merely because another multiple follows the same inherited ladder.
Sources: File_52c.Rounded_Whole_Span_Inverse_Detailed_Study_Draft (2)(1).md §§3.8–3.9; coordinate convention in§1 and§6; c482_c531/journal.json C505–C507: duration ladder and placement; C510–C511: Rounded endpoint realization; c932_c1131/journal.json C1001: inherited width/placement distinction.
Research record: C505, C506, C507, C510, C511, C998, C1001, C1007.