## 2. Rounded rows, residuals and recovery

For signed integer arguments use \(Q_{\mathbb Z}(x)=5\lfloor(x+2)/5\rfloor\). On source ages this is nearest-five rounding with residual \(\rho=x-Q(x)\in\{-2,-1,0,1,2\}\).

Retaining \(y=Q(x)\) and \(\rho\) allows an exact source change \(d\):

\[
T_d(y,\rho)=\bigl(y+Q_{\mathbb Z}(\rho+d),\
\rho+d-Q_{\mathbb Z}(\rho+d)\bigr).
\]

Since \(y\) is a multiple of five, this equals the encoding of \(x+d\). Consequently \(T_eT_d=T_{d+e}\), whenever the declared source domains allow the intermediate changes. A universal translation action on rounded values alone exists for integer changes precisely when \(5\mid d\); other changes need the residual.

For an ordinary row retain rounded begetting and remainder values \(B,R\), together with exact lifespan \(L\). Put \(z=L-B-R\). The possible residual pairs satisfy

\[
u+v=z,\qquad -2\le u,v\le2.
\]

In the interior nonnegative domain their number is \(5-|z|\), for \(|z|\le4\). Boundary restrictions can reduce this count. The selected 55 ordinary source rows contain five uniquely determined pairs and fifty ambiguous pairs under these retained observations. For LXX Lamech, \(L=753,B=180,R=570\) still permits 181/572 or 182/571; the selected source pair is needed.

If only the three rounded measures are retained, the carry \(Q(L)-B-R\) is −5, 0 or +5, with 3, 19 or 3 residual pairs respectively in the unrestricted interior cell. This is a different information question from retaining exact \(L\).

Accumulation acts before any optional re-rounding: \(Uw=UQ(w)+U\rho\). A block residual is a sum of row residuals and need not lie in one cell. The Moses blocks retain residuals \(0,2,4,-1,1\); their six-year total recovers every strict Actual coordinate without altering the Rounded field.

Source/proof route: C1190–C1202, C1212–C1222, C1277–C1279; joint-row and residual-lift models.