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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.25950 |
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| _version_ | 1866910077070344192 |
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| author | Gilson, Frank |
| author_facet | Gilson, Frank |
| contents | We introduce a one-step cascade symmetric system whose local symmetry geometry is organized by finite $ρ$-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 $ρ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25950 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A One-Step Cascade Symmetric Model: Rank-$1$ Packets, Binary Shielding, and the Even Exact-Cardinality Profile Gilson, Frank Logic 03E25 We introduce a one-step cascade symmetric system whose local symmetry geometry is organized by finite $ρ$-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 $ρ$-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. |
| title | A One-Step Cascade Symmetric Model: Rank-$1$ Packets, Binary Shielding, and the Even Exact-Cardinality Profile |
| topic | Logic 03E25 |
| url | https://arxiv.org/abs/2603.25950 |