On two conjectures of Wu on consecutive integers of the same height in the Collatz problem

The height of a positive integer is the number of steps of the Collatz map (n ↦ n/2 for even n, n ↦ 3n+1 for odd n) needed to reach 1. Wu Jiabang (1992) computed, for N ≤ 24, the proportion d(2^N) of the integers in [1, 2^N) that belong to a run of at least two consecutive integers of the same height, found that it increases with N, and proposed two conjectures. As read from the full text through the AI reading assistant of the CNKI database, they state that d(M) → 1 and that the length L(M) of the longest run of equal height in [1, M) tends to infinity. We prove the second conjecture: it follows from the existence of arbitrarily long runs of equal height, proved in a companion paper. For the first, we show that d(M) → 1 if and only if the positive integers that reach 1 have natural density one. One direction is immediate. The other uses a theorem of a second companion paper, on a conjecture of Gao, by which the proportion of n < x such that n and n+1 coalesce tends to 1, together with the fact that coalescing integers that reach 1 have the same height. Wu's first conjecture is therefore equivalent to a density form of the 3x+1 conjecture, which is open: the best known lower bound for the number of n ≤ x that reach 1 is x^0.84. We extend Wu's table to N = 26, and note that the monotonicity he observed, if it held for all N, would already give a positive lower density of integers reaching 1. Both results are formalized in Lean 4 with Mathlib. The wording of the conjectures has still to be checked against the printed article.