from research import *
s=begin(1364,"Write the cap information supplement","How do clipping and rounding discard different source information?",{},["C1139–C1151","C1203–C1211"])
t=json.loads("\"## 3. Capacity, clipping and rounding\\n\\nWrite 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\\n\\n\\\\[\\nL_i-\\\\min(L_i,c_i^{SP})=\\\\max(0,d_i-s_i).\\n\\\\]\\n\\nThe 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.\\n\\nBecause \\\\(Q\\\\) is monotone,\\n\\n\\\\[\\nQ(\\\\min(L,c))=\\\\min(Q(L),Q(c)).\\n\\\\]\\n\\nThis identity requires the same resolved count convention on both inputs. It does not identify raw nominal SP age labels with completed durations.\\n\\nFor nonnegative integer \\\\(L\\\\) and fixed resolved capacity \\\\(c\\\\), observing \\\\(q=Q(\\\\min(L,c))\\\\) gives three cases:\\n\\n| Observed \\\\(q\\\\) | Possible baseline lives |\\n|---|---|\\n| \\\\(q<Q(c)\\\\) | The ordinary rounding cell \\\\(Q(L)=q\\\\), intersected with nonnegative integers |\\n| \\\\(q=Q(c)\\\\) | Every \\\\(L\\\\ge\\\\max(0,Q(c)-2)\\\\) |\\n| \\\\(q>Q(c)\\\\) | None |\\n\\nIf 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.\\n\\nFor 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.\\n\\nThese results explain what a complete reconstruction must retain. They do not reconstruct an arbitrarily long clipped life from its cap.\\n\\nSource/proof route: C1139–C1151, C1203–C1211; cap/round fibre, branch and strict-visibility models.\\n\"")
a=artifact("draft/technical_03_cap.md",t)
finish(s,{'artifact':a,'words':len(t.split())},"The cap supplement separates discarded excess from rounding residual and states exactly when a reduction remains visible.","Move the inverse register proof into the technical companion.",{'nonempty':len(t)>100,'rounded_loss':115+250+120==485})