Chronological families: checkpoint C771

Forty steps have developed one connected explanation. Ordered source rows generate regular and cumulative boundary fields through different selectors. The gap field is the suffix accumulation of lifespan minus effective begetting interval.

Rounded acts on these rows before accumulation. Its local residual e=Q(x)−x generates the full boundary field D by suffix summation. Adjacent differences recover e, and Actual=Rounded−D. The rounded mode bridge changes by D_cumulative−D_regular.

This mechanism reproduces all26 literal MT cumulative boundaries. Three changed residual rows generate the entire LXX/MT comparison, and two generate SP/MT. It also explains why date-grid rounding and whole-subpath rounding can fail to reproduce the source construction.

Two source qualifications matter: Methuselah and Reu both exhibit the MT parts-versus-total lifespan defect; SP inclusive lifespans remain ledger counts. LXX/SP transferred rounded columns are derived comparisons.

A useful Strategy connection is conditional:13396+2+3+5=13406, separating completion-member selection, SP rounding, and the rounded Aaron branch. This reconstructs the supplied cumulative rectangle head but does not establish the controlling report’s intended derivation.

Next: finite source variants, their local supports, their distinct regular/cumulative realizations, and the whole BJ paths. No second decimal inversion, new Gear transport, or primer reopening.