One-Weight Colorings, the Symmetric Class, and Lower Bounds for Hales--Jewett Numbers

A coloring of the Hales–Jewett cube $[t{]}^n$ is symmetric if it is invariant under all coordinate permutations, and one-weight if it reads only an integer-weighted count of the letters. We prove that the two classes coincide – a radix weight realizes every symmetric coloring – so the symmetric lower-bound problem for the Hales–Jewett numbers is exactly a one-dimensional coloring problem about homothetic copies of a $t$-point set, the case $d=1$ of Gallai’s theorem. Optimizing the weight yields $\mathrm{H}\mathrm{J}(3,3)\ge 22$ and $\mathrm{H}\mathrm{J}(4,2)\ge 14$, the latter in closed form from the new Gallai homothety numbers $G_2(\{0,2,3,5\})=67$ and $G_2(\{0,1,5,6\})=80$; new values at three colors – $G_3(\{0,1,3\})=42$, $G_3(\{0,1,4\})=57$ and $G_3(\{0,2,5\})\ge 77$ – give $\mathrm{H}\mathrm{J}(3,3)\ge 16$ from a one-line certificate. An anatomy of the $(4,2)$ palette locates the source of its compression: it is an extremal object of the bracket regime plus a single boundary scale. An exhaustive census shows how thin the class is: of the $1644$ line-free $2$-colorings of $[3{]}^3$, exactly $36$ are symmetric. For lines with at most $K$ active coordinates the same machinery gives infinite bracket numbers, ${\mathrm{H}\mathrm{J}}^{[12]}(3,3)={\mathrm{H}\mathrm{J}}^{[12]}(4,2)=\mathrm{\infty }$, strictly beyond the sum-type ceilings ${\kappa }_{\mathrm{s}\mathrm{u}\mathrm{m}}(3,3)=11$ and ${\kappa }_{\mathrm{s}\mathrm{u}\mathrm{m}}(4,2)=10$; for lines whose active set is an interval the machinery is provably blind, the interval ceiling $\lambda (3,r)$ is settled for every $r$ by assembling the known bounds, and a SAT computation gives the exact value ${\mathrm{H}\mathrm{J}}^{(1)}(3)=5>4=\mathrm{H}\mathrm{J}(3)$. We close with the Collapse, diagonal-only, and symmetric-extremality conjectures and with open problems on optimal weights. Every certificate displayed in this note has been re-verified by direct enumeration, independently of any solver.