= Rank certificate from rounded canonical heights
The <Gram matrix> of the <canonical height pairing> of rational points gives a robust independence certificate. For two points, set $a=\widehat h(P)$, $b=\widehat h(Q)$ and $c=(\widehat h(P+Q)-a-b)/2$. If bounds from rounded measurements imply $a>0$ and $ab-c^2>0$, the matrix is <positive-definite>. Any integer relation $rP+sQ=O$ would give $ar^2+2crs+bs^2=0$, forcing $r=s=0$. Thus the points generate a free <abelian group> of rank two. Checking an interval lower bound for the determinant, rather than just its approximate central value, makes the certificate insensitive to rounding. It certifies independence, not that the points are a basis of the entire <Mordell-Weil group>.
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