Relative perturbations and the growth of irrationality sequences

../

increasing_blocks: A recursive block choice makes every relative perturbation irrational while forcing the root liminf to be one.

---

The complete argument reconstructs the increasing-block idea recorded on the Price claim page with a complete denominator estimate. The reconstruction is author-recorded and gives no new acceptance to the uninspected manuscript or formal file.

The two demands on the blocks are compatible: their lengths greatly exceed the logarithm of their starting points, yet their reciprocal mass can be made arbitrarily small relative to all earlier denominators. The first gives root liminf one, and the second makes a rational tail, after multiplication by its denominator, a positive integer below one.

This resolves the growth implication at author scope. The separate question in Problem 263 about $a_n=2^{2^n}$ remains untouched: the construction chooses different blocks and does not specialize to that fixed sequence. No finite test or generic statement excluding only countably many bases identifies the behavior of the particular base $2$.