# Technical companion: the common chronological construction

## 1. Row changes and ordered measurement

For ordinary compatible rows, \(R_d(b,r,L)=(b+d,r-d,L)\) and \(A_e(b,r,L)=(b,r+e,L+e)\). Any compatible row difference has the unique parameters \(d=\Delta b,e=\Delta L\). These algebraic operations commute on unrestricted triples. When nonnegative ages or source-selected states are required, each intermediate row must also be admissible.

The nineteen matched MT/LXX rows have the following nonzero supports:

| Source change | Rows | Amplitude |
|---|---|---:|
| Begetting change | Adam, Seth, Enosh, Kenan, Mahalalel, Enoch; Arphaxad, Shelah, Eber, Peleg, Reu, Serug | \(u=100\) |
| Begetting change | Nahor | \(u/2=50\) |
| Lifespan change | Lamech | \(\lambda=-24\) |
| Lifespan change | Arphaxad, Shelah | \(a=27\) |
| Lifespan change | Eber | \(e=40\) |
| Lifespan change | Peleg, Reu, Serug | \(u=100\) |
| Lifespan change | Nahor | \(n=60\) |

The 38-output change matrix has rank five. Imposing the fitted relation \(e+n=u\) reduces the parameter count to four; neither description claims to generate the baseline source rows. The matched head measurements are

\[
\Delta_{\rm Reg}=12.5u,\qquad
\Delta_{\rm Cum}=3u+\lambda+2a+e+n.
\]

At the source values these are 1250 and 430. Native Cainan contributes a labelled insertion with measures 130 and 460.

Let \(U\) be the suffix-sum matrix for a fixed ordered path. With common terminal, \(X'-X=U(w'-w)\); with changed terminal, add its difference to each boundary. Adjacent differences recover the weights. Thus the complete field retains location, while its head alone cannot recover the distribution.

Source/proof route: complete source-row packet; whole-genealogy five-amplitude matrix; C1162–C1189; Regular staircase C1253–C1264.