What Rounded rows remember

Regular and Cumulative chronology measure different parts of the same genealogical row. Rounding each part provides a further comparison, but loses exact residuals. If B and R are the nearest-five values of begetting age b and remaining years r, and exact lifespan L is retained, then the hidden residuals satisfy u+v=L−B−R, with u and v each between−2 and2.

This bounded relation explains both recovery and disagreement between Rounded paths. It has5−|L−B−R| possible decompositions in an ordinary interior cell. Across the55 ordinary source rows, five have only one: MT and LXX Methuselah, and Reu in all three traditions. Fifty remain ambiguous. The three inclusive SP cap rows are outside this ordinary-row test.

The source rows were used to calculate these rounded observations. Their conditional recovery is not independent testimony for those same rows. The result explains what an observer could recover if both rounded measurements and exact life were separately supplied.

Source relations can narrow the possibilities further. Equal remaining years for Arphaxad and Shelah reduce their paired alternatives, but do not uniquely generate their ages. The shared century operations also leave four common Peleg/Serug choices even after Reu is fixed.

The same hidden residual sum determines the rounding carry: rounded lifespan minus the sum of rounded parts is−5,0 or+5. Exact residuals locate the source within a rounded cell; the carry reports only which branch it occupies. This connects the Regular death construction and Cumulative contribution through one local mechanism.