Ascending Run
You don't need to climb the whole way — one burst of three is enough.
Anywhere in the digit string, three or more consecutive digits each exactly +1 — like 1012, 912340 or 4567. If the whole string qualifies, it also earns Fully Ascending.
Your record
Ascending Run is the local edition of Fully Ascending: the global version has only 22 members in range, while the local one has 28,848 — because only one burst of three needs to land, and the rest of the digits can do whatever they like. Relaxing "locked everywhere" to "three in a row somewhere" inflates the count by three orders of magnitude.
By length: three-digit numbers contribute just 7 (with three digits there is no "local" — it degenerates to Fully Ascending), four-digit 136, five-digit 2,017 and six-digit 26,688. The longer the number, the more likely a 234 or 789 is hiding in some corner.
It and the Straight are two intensity levels of one family: a Straight demands five in a row — the extended deluxe edition — and every Straight is automatically an Ascending Run.