3x+1 problem: verify Collatz convergence beyond $2^{71}$, or discover new path/glide records (Guy UPINT §E16)

Let $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$. The Collatz (3x+1) conjecture asserts that iterating $T$ from any positive integer eventually reaches $1$. Verify convergence for every $n$ up to a stated height $N$ beyond the current record, and/or compute new records within a range: a 'path record' (a new maximum trajectory peak) or a 'glide/stopping-time record'.

Acceptance. Provide a configuration plus a verifier. ADVANCES, either: (a) verify $T$-convergence for every $n$ in $(2^{71},N]$ for some $N>2^{71}$, with a reproducible log and a re-check script; or (b) exhibit a new path record or glide record above the current published list — an $n$ whose trajectory peak (resp. total stopping time) exceeds that of all smaller $n$ — certified by recomputing the trajectory. FULLY RESOLVES: a proof of convergence, or a counterexample (a nontrivial cycle, or an orbit proven unbounded). Deliver the height/record data and a verification script.