Extremal character spectra of finite projective systems

arXiv:2610.42006v1 Announce Type: cross Abstract: Let $\mathcal M$ be a finite spanning multiset of points in ${\rm PG}(k-1,q)$, and let ${\rm Char}_{\mathcal M}$ be its character function. For $1\le r\le k$, we study the maximum number of distinct values attained by ${\rm Char}_{\mathcal M}$ on the codimension-$r$ flats under a total-multiplicity budget. The spectrum is subject to two natural upper bounds, whose extremal cases lead respectively to $r$-maximum character spectrum ($r$-MCS) and $r$-full character spectrum ($r$-FCS) systems. We establish the existence of $r$-FCS systems in a broad parameter range, and solve the size-constrained extremal problem completely when $r=k-1$, including the minimum-cost multiplicity profiles. For $1\le r