claimroutineinferred_dep
Approximants chosen with center-independent weight thresholds
Picks approximants with weights whose margin is below
last parameter choice; distinct centers for interpolation
Claim·§4 Choice of parameters and proof of Theorem~thm:main·inside thmIrrationality exponent of pi is 2
Statement (paper)
With , , , fixed, one can choose approximations satisfying , with , successively sufficiently large for Theorem and such that holds, i.e. with . The thresholds for weight separation are independent of the centers, and for sufficiently large one has , so that , , are distinct across in each coordinate, with no independence between different coordinates required.
In plain English
Once , , , are fixed, one can pick approximations of with , each successively large enough for the interpolation theorem and satisfying the weight-margin inequality . The weight-separation thresholds do not depend on the centers, and for large we have , so the centers are distinct in each coordinate.
Why it's here
This is the last parameter choice: it picks the approximants and weights after , , , , are fixed. Because the interpolation theorem's weight thresholds do not depend on the points or centers, the can be taken large in advance, and makes the centers pairwise distinct in each coordinate, which is exactly the hypothesis that theorem needs. The weight-margin bound uses the last third of in the combined error budget used by the next claim.
Classified routine:
This is a parameter-choice step: under the assumed existence of arbitrarily large good approximants, their weights can be made successively large enough to meet the center-independent thresholds of geom-interpolation and the stated error margin. The distinctness of the centers follows directly from p_i ≠ 0.