technical 02 rounding · 490d Research

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.