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It is well known that the compactifications of the canonical contact systemsliving on real jet spaces $J^{\,r}(1,1)$, $r \ge 2$, are locally universalGoursat distributions, $\Delta^r$, living on compact manifolds (called Goursatmonsters) having open dense jet-like ($J^{\,r}(1,1)$-like) parts.  By virtue of the results of Jean (1996), one was able, for each $r \ge 2$, torecursively compute the {\it small growth vector\,} of \,$\Delta^r$ at anypoint of the $r$-th monster. The result was got by performing series of $r$operations taken, in function of the local geometry of \,$\Delta^r$ inquestion, from the set of fixed recursive rules $\{\,1,\,2,\,3\,\}$ (called inthe present text S,\,T,\,G, respectively). By the local universality of\,$\Delta^r$ one was thus able to compute {\it all\,} small growth vectors ofall existing Goursat distributions.  In the work of Mormul (2004) proposed were explicit solutions to the seriesof Jean's recurrences. The solutions uncovered a surprisingly involvedunderlying arithmetics -- a kind of G{\"o}del-like encoding of words over athree-letter alphabet $\{$G,\,S,\,T$\}$ by neat sequences of positive integers.Yet, those formulas, though characterizing really existing objects, appeared asif from thin air, and proofs were postponed to another publication.  In the present contribution we submit proofs of our formulas from 2004. It isnot, however, a plain check that our candidates satisfy Jean's recurrences.Under the way we retrieve and re-produce those surprising formulas.

Small growth vectors of the compactifications of the contact systems on $J^{\,r}(1,1)$