A One-Step Cascade Symmetric Model: Rank-$1$ Packets, Binary Shielding, and the Even Exact-Cardinality Profile

Abstract

We introduce a one-step cascade symmetric system whose local symmetry geometry is organized by finite $\rho$-closed windows and one-step stars rather than by rowwise-independent toggles. The resulting symmetric model isolates a new $ZF + DC + \neg \mathrm{BPI}$ geometry in which rank-$1$ hereditarily symmetric reals admit a packet normalization theorem over countable $\rho$-closed supports. The technical center of the paper is the finite star-span lemma and the associated rank-$1$ packet calculus. From this we obtain a normalization theorem and a two-layer coding consequence for rank-$1$ reals (in the metatheory, via a well-orderable base of packets). We then apply the same binary fresh-support shielding pattern to prove $\neg C_2$, hence $\neg AC_{\mathrm{fin}}$, and therefore the failure of every even $C_n$ (where $C_n$ denotes the principle that every family of nonempty $n$-element sets admits a choice function). On the odd side, the present bounded packet calculus remains dyadic: support-fixed local actions factor through finite $2$-groups, bounded support-equivariant quotients of finite local orbits have power-of-two size, and trace-separated bounded rigid ternary families admit canonical selectors within a fixed finite trace window. Accordingly, the odd exact-cardinality profile remains open beyond the current local binary machinery.