Some more ideas to get more mileage out of the single-atom system. Please tell me whether this is getting too weird or far-fetched!

The original g ('Grasshopper') modifier is a restriction of the p ('Pao') modifier, where the platform must be adjacent to the destination square. One can also imagine a piece that is the 'time-reversed Grasshopper', where the platform needs to be adjacent to the square of origin. Another way of viewing the Grasshopper is like a Cannon where the slider move downgrades to the corresponding leaper move at the platform. The time-reversed Grasshopper would upgrade the leaper move that managed to reach the platform to the corresponding slider. So if we assign the g modifier this general two-way slider<->leaper conversion property, it would be natural to write the time-reversed Grasshopper as gK (which originally made no sense).

Unfortunately this is not yet enough to describe 'skip-sliders', which skip the adjacent square before starting to slide. Tenjiku Shogi has such a piece, the Heavenly Tetrarchs. Apart from the time-reversed Grasshopper, it would also require a hop-less version of the move, when the adjacent square is empty. So spontaneous upgrading from leaper to slider. Now it is a co-incidence that the letters d and u are both still available! So we could use the 'again' operator to specify a two-leg mode, and explicitly write a u or d on the later leg to specify it uses an upgraded or downgraded version of the atom that was written. This is a kludge to use the same atom as both a slider and leaper move: we can write uW to mean R in any context (like WW, W7 or W0 wasn't already enough...).

So we could write mpafuW to describe the Skip-Rook: make a Wazir step to an empty or occupied square without touching it, and then make again a Wazir step, an unlimited number of times. Of course the original Betza 'then' notation could also do this, as t[D,R], glueing a Rook move to the D leap that skipped the square. I never liked that notation much, however; it seems quite out of line with the normal Betza syntax.

Now the 'then' notation was originally developed for describing bent riders. The 'a' notation also allowed bent trajectories, as in the example of hook movers, by putting a directional modifier on the second leg. And the choice is especially large on the pseudo-atom K.

Now originally it made little sense to have directional selection from an 'atom' that in itself already joined different directions. It would almost always be much simpler to just put them on the W or F components. But in a continuation leg it is very useful to put f or v on them, to enforce a linear path without having to write the orthogonal and diagonal paths separately. So we already have crossed that bridge. Now the K has non-degenerate 8-fold symmetry, non-degenerate because, unlike oblique true atoms there is a unique forward move, in stead of a pair. So we need a different system of direction specification.

Some analysis shows we could require here that the sideway direction always precedes the vertical direction, as there are only 4 directions that have components along both dimensions. The other 4 can be written by single direction specifiers. With this convention, rf would mean the single rightt-forward direction, while fr would be non-combinable, and describe a pair of directions, pure forward or pure right. This way it becomes possible to write svK = (l+r)(f+b)K = (lf+rf+lb+rb)K, all combinable and thus indicating diagonal directions. I.e. svK = F. OTOH, vsK would have a non-combinable v+s, meaning joining their direction sets, so f+b+r+l, all indicating orthogonal direction. Thus vsK = W!

Now we could write vsmasfK. In the K directional system sf is combinable, (l+r)f = lf + rf, i.e. continuation with 45-degree deflection, while the direction specification vs for the first leg was the Ferz. In other words, we have described the Mao as a two-leg King move without hopping ability. The Moa would similarly be svmasfK, and the Moo masfK. The latter was the fully 8-fold pseudo-symmetric bent-lame-leaper move, and the vs or sv prefixes were just needed to pick the W and F starters from it.

Now combining this with the 'upgrade' modifier, we get vsmasfuK for the Gryphon, which is essentially a Mao with an elongated second leg. Similarly the Ancaa is svmasfuK. (Good thing they were not capture-only... ;-) ) The F or W moves will have to put on them separately (if we want them to have those).

We can do bifurcators too, as these are similar to bent riders, except that they don't bend spontanously, but at the platform. So the ultimate bifurcator is gasfQ: move as Q to the nearest occupied square, and then again as Q, but in the relative sideway-forward direction (i.e. at 45 degrees). With the vs or sv prefix you could select the parts of this that start in the Rook or Bishop direction. Of course by giving different directional specs behind the 'a' you could also turn 135-degree angles. For 90-degree angles it would of course be more clear to simply write gmasR or gmasB, as there the trajectories are orthogonal or diagonal all the way, and R reads a lot clearer than vsQ! Of course the compound of these is gmasQ.

So with the u modifier, and abusing the pseudo-atom K, we can actually achieve a lot of quite useful stuff within the framework of the single-atom + modifiers system!