This sentence has forty-five (45) characters.
This sentence has three as, one b, two cs, two ds, thirty-six es, three fs, three gs, eleven hs, nine is, one j, one k, three ls, one m, eighteen ns, twelve os, one p, one q, eight rs, twenty-six ss, twenty ts, two us, five vs, seven ws, three xs, four ys and one z.
I eventually found the solution by iteratively updating the approximate distribution of each letter and finally sampling [1] - not sure if there is a better way!
[1] solve_2 in tsh.py at https://bitbucket.org/akxlr/tsh/src
I bet you can construct a language where a solution exists, but all of its neighbors' errors (using the topology implied by your gd function) are local maxima.
The basic idea instead is to treat each letter count as a random variable, and ask what the distribution of each count is. In particular, each letter count can be expressed as a sum of (a mapping of) the other letter counts, so if you know the distribution of all other letter counts you can improve the distribution of the letter of interest. Initially assume uniform for all counts, then iterate until convergence. The images in that repo show the distribution of some letters at various stages. After doing this for about 10 iterations you stop seeing any improvement (the letter distributions are as 'peaked' as they are going to get), at which point I drew samples until I found a solution.
The update function you wrote on the distribution space is continuous, and distribution space is compact (since there's no way to have more than N of any letter), so there is necessarily at least one fixed point.
Fixed points could still be sources though, right? It clearly wasn't, but I'm curious to know if you got lucky, or if you merely didn't get unlucky.
I wonder how effective your code would be on the following self descriptive sentences (base j).
For instance when j=2, "100 1,11 0" has 4 ones and 3 zeros as described.
Does your code consistently find solutions as you increase j? At what point does the computation become unfeasible?
I will endeavour to try your problem, is it related in some way to a well-known problem?