Adalogical Ænigma #7

Greetings, gentle patron!
If you are encountering difficulty making progress on my seventh ænigma, perhaps you'll find relief in one or both of the following pedagogical aids.
First, I have collected a small assortment of advisory notes, or ‘tips’ as one might say, which are enumerated a bit below on this page.
Second, concerning the example puzzle shown on the ænigma paper, I've penned a detailed description of how one might go about solving it.
I trust that these resources will suffice to lift your mental ship off the shoals of perplexity and allow you happily to continue your journey down the river of clarity and decipherment, but if you find that they do not, I invite you to contact my associate, Pavel Curtis, directly for more individualized furtherance.
With my very best wishes for your imminent enlightenment,

General advice for solving the current ænigma

  1. As with so many ænigmas, it is ever so much easier to proceed when one has as much contextual information as possible. In this case, right at the beginning and from time to time thereafter, you will encounter regions in which all but a single square have already been assigned their numbers; at the very start, this includes any regions that comprise exactly one square. In such circumstances, you may instantly deduce which number is missing from that region. Pray do not hesitate to fill in such numbers immediately, as soon as you discover them.
  2. The great majority of deductions you will make in solving this ænigma will be situations in which you can eliminate all but one square in a region from the candidate locations in which some particular number may legally be placed. For example, you might find that every unoccupied square but one in some region R is within 3 squares of a square labelled “3” in a surrounding region. By the rules of this ænigma, then, none of those squares may contain the “3” in region R, so the “3” must be placed in that one remaining square.
  3. Less frequently, but oftentimes quite critically, you will encounter circumstances in which you can eliminate all but one of the available numbers that might be placed in some particular square. For example, you might discover a position within a five-square region that is within 1 square of a square labelled “1” in some surrounding region, within 2 spaces of a nearby “2”, within 4 of a “4”, and within 5 of a “5”. That position, therefore, cannot contain any of “1”, “2”, “4”, or “5”, and so must contain a “3”.
  4. Sometimes, you will be able to reject a potential placement for some number because that placement would leave some other nearby region with no legal placements for that number. For example, consider this snippet from a hypothetical ænigma grid: If a “3” were placed in the square labelled with a question mark, then there would be no legal placements for a “3” in the three-square region shown. We may therefore conclude that the question-mark square does not contain a “3”.
  5. As always with my ænigmas, progress toward the solution here will come almost entirely in the form of small, incremental steps, often adding but a single number to the grid. Do not endeavour to rush ahead, attempting to guess or otherwise ascertain the positions of several numbers in one fell swoop. The whole of your solution will come as the sum of many steps; “patience” and “perseverence” must be your watchwords.
All of the above notes are illustrated in my detailed walkthrough of the example ænigma, should you feel the need to observe them being put into action.
Wishing you the best of solving luck,