C.40:5.13 - Separate an inexpressible figure from an unreachable one

A small icon generator draws four horizontal bars in fixed rows. Their integer lengths must lie from 0 to 3, and bars 2 and 4 must remain equal. The current figure is (1,2,1,2); the designer asks for (1,2,3,2). These stipulated conditions make the full construction small enough to inspect directly.

With four independently stored lengths, the target is representable and admissible. Suppose, however, the only available change adds 1 modulo 4 to every length at once. Repeating it visits just four tuples and always preserves equality of bars 1 and 3. More repeats cannot reach the target. Keep the representation and add the needed local operation: change bar 3 to 3 while leaving the others fixed. For general variations, change bars 2 and 4 together and keep each length within its bound. The repair concerns the changes, not missing expressive capacity.

Now the editor instead stores two parameters and draws D(a,b)=(a,b,a,b), with a,b in {0,1,2,3}. Every produced figure has equal bars 1 and 3. No change of those two parameters can express the target. Extend the construction to D′(a,b,c)=(a,b,a+c,b), with integer parameters satisfying 0≤a≤3, 0≤b≤3 and 0≤a+c≤3. The old value (a,b) transfers to (a,b,0), preserving every old figure. The target becomes (1,2,2). Changing b still moves bars 2 and 4 together; changing c can alter bar 3 alone. Changes to a and c must respect their joint bound.

The old generator produces 16 figures; the extended one produces all 64 admissible tuples, since a selects the first length, b the equal pair and a+c the third. This finite coverage is a property of the stipulated construction. It does not show that arbitrary random changes sample those figures equally or find them cheaply. Because the desired tuple is known, direct construction already finishes the task.

Change the circumstances once more: c may change only by one unit per step, and selection discards every intermediate with c=1 before it can be changed again. The target c=2 remains expressible and reachable by the permitted operations, but this continuation policy never retains the necessary intermediate. Keep c=1 for one justified further step or provide a direct c=2 change; enlarging the representation is unnecessary. If the new requested third length is 4 while the hard upper bound stays 3, the request itself conflicts with admissibility. Return that conflict instead of treating it as another failure of search.