from research import *
s=begin(1363,"Write the exact rounding supplement","What information makes row rounding compatible with later source changes?",{},["C1190–C1202","C1212–C1222","C1277–C1279"])
t=json.loads("\"## 2. Rounded rows, residuals and recovery\\n\\nFor 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\\\\}\\\\).\\n\\nRetaining \\\\(y=Q(x)\\\\) and \\\\(\\\\rho\\\\) allows an exact source change \\\\(d\\\\):\\n\\n\\\\[\\nT_d(y,\\\\rho)=\\\\bigl(y+Q_{\\\\mathbb Z}(\\\\rho+d),\\\\\\n\\\\rho+d-Q_{\\\\mathbb Z}(\\\\rho+d)\\\\bigr).\\n\\\\]\\n\\nSince \\\\(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.\\n\\nFor 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\\n\\n\\\\[\\nu+v=z,\\\\qquad -2\\\\le u,v\\\\le2.\\n\\\\]\\n\\nIn 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.\\n\\nIf 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\\\\).\\n\\nAccumulation 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.\\n\\nSource/proof route: C1190–C1202, C1212–C1222, C1277–C1279; joint-row and residual-lift models.\\n\"")
a=artifact("draft/technical_02_rounding.md",t)
finish(s,{'artifact':a,'words':len(t.split())},"The rounding companion gives exact recovery, composition and ambiguity domains, with the Moses path as its concrete application.","Write the cap information-loss supplement.",{'nonempty':len(t)>100,'lift_bound':(ROOT/'model/operation_rounded_residual_lift.json').exists()})