# One row grammar, three post-Flood families

The complete post-Flood comparison shows why Regular and Cumulative chronology can agree in one measurement and differ in another. A source row contains a begetting interval b and a lifespan L; its calculated remainder is L−b. Two simple changes describe the whole shared field: R moves years from remainder to begetting while keeping life fixed; A changes remainder and life together while keeping begetting fixed.

After Cainan is leveled for comparison, SP and LXX share every begetting value from Shem through Terah. Their lifespans differ in a located pattern:

| Source block | SP→LXX lifespan change | Begetting change |
|---|---:|---:|
| Shem | 0 | 0 |
| Arphaxad and Shelah | +27 each | 0 |
| Eber, Peleg, Reu and Serug | +100 each | 0 |
| Nahor and Terah | +60 each | 0 |

The shared birth path therefore coexists with a Cumulative difference. Ordered accumulation of the life changes produces the complete comparison field. Its outer574 is generated by4×100+2×60+2×27; the interior boundaries retain the placement of those contributions.

For Peleg, Reu and Serug, the SP source row is also a concrete intermediate between MT and LXX. MT→SP increases begetting by100 while keeping lifespan fixed. SP→LXX then increases lifespan by100 at the same begetting age. The combined change preserves the original MT remainder:209,207 or200. The early Arphaxad/Shelah block instead preserves equal remainders within each tradition:403 inMT,303 inSP and330 inLXX.

Eber, Nahor and Terah retain their explicit roles. With u=100,t=60,a=27, the full MT→SP/LXX packet is generated by three amplitudes and their named source supports. Regular measurements seeu; SP life differences seet; the full LXX life field seesu,t,a. Outer totals alone lose some of this information.

Native LXX Cainan is then restored as a distinct labelled row between Arphaxad andShelah. Its130/330/460 vector equalsShelah’s numerical vector and extends the330 block. Numeric sharing does not derive the existence or position of the inserted person.

This is a complete source-conditioned reconstruction, not a historical sequence of edits. The baseline values, the located supports, the amplitudes, the Cainan insertion and the source states all remain part of its descriptive cost. The21-parameter joint model includes18 MT baseline values; the rank3 change model does not erase them. Four selected baseline equalities reduce that joint model to17 dimensions, but low rank alone is not an explanation: a rank1 fit can simply hide every output number in its coefficients.

The shared grammar is clearest in the conserved quantities. Lifespan, begetting and remaining years are preserved by different row moves. Their different projections explain how related genealogies generate shared Regular paths and distinct Cumulative fields.