from research import *
s=begin(1337,"Draft the source-to-rectangle bridge","Which inputs make the two calendar gains equal?",{},["C1223–C1230"])
t=json.loads("\"The same located changes now supply the SP equal-gain rectangle. Starting with the selected standard MT frames and the 1446 Exodus anchor, the source construction gives two radii:\\n\\n\\\\[\\n\\\\begin{aligned}\\nR&=(4114-1446)+650-350-215+7=2760,\\\\\\\\\\nC&=(14004-1446)-488-120+10=11960.\\n\\\\end{aligned}\\n\\\\]\\n\\nThe Regular line retains the post-Flood additions, pre-Flood shortening, Sojourn choice and seven-year source opening. The Cumulative line retains the cap losses, post-Flood lifespan changes and ten-year source opening. Those openings and absolute frames remain source premises.\\n\\n| Mode | Source opening | Radius from 1446 | Appointed Key | Expanded radius | Resulting common anchor |\\n|---|---:|---:|---:|---:|---:|\\n| Regular | 4206 | 2760 | \\\\(70/69\\\\) | 2800 | 1406 |\\n| Cumulative | 13406 | 11960 | \\\\(300/299\\\\) | 12000 | 1406 |\\n\\nBoth radii gain forty. Their difference stays 9200; Priestly expansion then gives \\\\(9200\\\\times25/23=10000\\\\).\\n\\nThe equal-gain condition is \\\\(R/69=C/299\\\\), or \\\\(3C=13R\\\\). Before the source openings, the radii are 2753 and 11950 and their balance is \\\\(3C-13R=61\\\\). Adding seven and ten changes that balance by \\\\(3(10)-13(7)=-61\\\\). This identifies precisely how the source packet closes the rectangle.\\n\\nThe advance is an explanation of where its two inputs come from. The equality of gains follows once those inputs and Keys are fixed; it is one connected family of consequences. The cap mechanism does not independently select the source openings or the Exodus anchor.\\n\\nSource basis: Strategy §4.3; Report IV; C1223–C1230.\\n\"")
a=artifact("draft/04b_SP_rectangle.md",t)
finish(s,{'artifact':a,'words':len(t.split())},"The rectangle chapter connects its radii to source-row mechanisms and makes the wrapper condition explicit beside the positive result.","Prepare the five-block Moses table.",{'nonempty':len(t)>100,'radii':(4114-1446)+650-350-215+7==2760 and (14004-1446)-488-120+10==11960})