technical 03 cap · 490d Research

3. Capacity, clipping and rounding

Write baseline slack as $s_i=c_i^{MT}-L_i$ , and capacity loss as $d_i=c_i^{MT}-c_i^{SP}$ . Then the clipped life reduction is

L_i-\min(L_i,c_i^{SP})=\max(0,d_i-s_i).

The SP capacity losses are 350 for the first six ancestors, 250 for Enoch and Methuselah, and 130 for Lamech. Baseline slack prevents six of these losses from clipping a life; the remaining reductions are 115, 249 and 124.

Because $Q$ is monotone,

Q(\min(L,c))=\min(Q(L),Q(c)).

This identity requires the same resolved count convention on both inputs. It does not identify raw nominal SP age labels with completed durations.

For nonnegative integer $L$ and fixed resolved capacity $c$ , observing $q=Q(\min(L,c))$ gives three cases:

Observed $q$	Possible baseline lives
$q<Q(c)$	The ordinary rounding cell $Q(L)=q$ , intersected with nonnegative integers
$q=Q(c)$	Every $L\ge\max(0,Q(c)-2)$
$q>Q(c)$	None

If the clipping branch is known, its residual is fixed by $c-Q(c)$ , and the missing information is the discarded excess. In the unclipped branch the excess is zero and the missing information is the within-cell residual. Forgetting which branch occurred combines those possibilities.

For Jared, Methuselah and Lamech, observing only the capped rounded values admits baseline thresholds 843, 718 and 653. A reduction is visibly strict after rounding only at baseline thresholds 848, 723 and 658. The actual baselines exceed them. The exact total reduction 488 becomes 485 in the rounded display: 115+250+120.

These results explain what a complete reconstruction must retain. They do not reconstruct an arbitrarily long clipped life from its cap.

Source/proof route: C1139–C1151, C1203–C1211; cap/round fibre, branch and strict-visibility models.