from research import *
s=begin(1212,'Write the cap-and-round information bridge','How can the two operations agree forward while remaining non-invertible?',{},['C1202–C1211','C1143'])
body="""# The cap and rounding lose different information

The SP Flood cap selects the smaller of baseline lifespan and capacity to the Flood. Nearest-five rounding selects a coarser representative. Both preserve order, so rounding the capped count equals capping the rounded lifespan at the rounded capacity. This forward agreement does not make either operation reversible.

For an uncapped row, only the small rounding residual is lost. For a known clipped row, the capacity fixes its exact capped count and rounding residual; the unknown is the discarded excess lifespan. These are branch-dependent losses, not two independent missing quantities on every row.

| SP row | Exact capacity | Rounded output | Any baseline at least this large gives that output |
|---|---:|---:|---:|
| Jared | 847 | 845 | 843 |
| Methuselah | 720 | 720 | 718 |
| Lamech | 653 | 655 | 653 |

The output therefore does not identify the MT baseline962/969/777. Nor does it always reveal whether a cap was applied: an uncapped value within the same rounded cell can look identical. Source branch information and the appropriate loss register restore the original row.

The exact reductions115,249,124 sum to488. Their rounded counterparts115,250,120 sum to485. The difference follows from the rounding errors of the baseline and capped lives. It needs no new cap mechanism.

Keeping the exact capacity on one route while rounding it on another changes the construction. The valid comparison rounds both sides consistently. Inclusive SP counts must likewise be resolved before rounding; separately rounded date labels do not replace those counts.

This clarifies the larger family connection. Regular birth intervals determine capacities; capacities select Cumulative lives; Rounded chronology records coarser versions of the resolved row measures. Each stage has its own retained information and stated source premises.
"""
a=artifact('deliverables/Cap_and_Round_Information.md',body)
finish(s,{'explanation':a},'The cap-to-Rounded interface now has both a forward rule and an exact account of its lost information. This completes the bounded information analysis and returns attention to source mechanisms feeding larger chronology relations.','Check which whole-genealogy row operations are preserved by nearest-five rounding.',{'capacity_table':'843' in body and '718' in body,'source_counts':'Inclusive SP' in body,'stages':'Regular birth intervals determine capacities' in body})