= Uniqueness of Chebyshev derivative-norming nodes
{title2=$q^{(k)}(1)>T_n^{(k)}(1),\quad1\le k\le n$}
For $n\ge1$, any distinct node set in $[-1,1]$ different from the $n+1$ <Chebyshev polynomial> extrema admits one degree-$n$ <polynomial> $q$, bounded by one at all those nodes, whose every nonconstant derivative violates the corresponding sharp endpoint bound. In decreasing order set $q(t_i)=(-1)^i$. The <endpoint derivative signs of Lagrange cardinal polynomials> give $q^{(k)}(1)=\sum_i|L_i^{(k)}(1)|$. Interpolating $T_n$ on the same nodes shows
$$
q^{(k)}(1)-T_n^{(k)}(1)=\sum_i|L_i^{(k)}(1)|\bigl[1-(-1)^iT_n(t_i)\bigr]>0.
$$
Each term is nonnegative and at least one is positive, because the other node set includes a nonextremal point. Uniqueness is a claim about node sets: permuting the same nodes does not change their norming property.
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