# Verification of the octahedron minimum slab theorem

Author: Alper Ferudun. Version 1.0, 5 October 2026.

The manuscript proves a complete analytic theorem for the minimum volume
of central slabs in the three-dimensional cross-polytope with vertices
+/-e_i. It includes every half-width, the exact transition between the
diagonal and coordinate normals, and every equality case. It does not
solve the full four-part AIM source record, higher dimensions or maxima.

The originating researcher checked the simplex normalization, the
three chamber formulas, the continuity and first-derivative matching
at chamber boundaries, the fixed-smallest-coordinate endpoint reduction,
both equal-coordinate families, the connected critical curve and its
resultant sign, and the exact radical comparisons. No mathematical gap
is currently identified in the stated theorem. This is a self-audit,
not independent peer review or formal verification.

The supplied SymPy program passed 31 exact checks. These include
derivative identities, the repeated-knot tail, polynomial endpoint
certificates, the resultant, radical signs and root isolation. The
transition is
tau = (96 - 36 sqrt(3) - 8 sqrt(33) + 9 sqrt(11))/37
= 0.47403494238958883233084958351549569....

The supplementary floating formula check agreed with 200 independently
clipped-polytope volumes to below 4.45e-16. That numerical diagnostic
and the optimization scan are not proof certificates. Three attempted
generic Z3 queries returned UNKNOWN and are not used in the proof.
Their outputs and the unsuccessful initial cube search remain in the
research archive for future work.

The PDF was compiled successfully with the desktop LaTeX compiler and
exported with Tectonic. Its final seven pages were checked visually,
including formula layout, references and the author footnote. The
final export has no TeX warnings. The source ZIP is integrity-checked.

The primary-source review is bounded. The known large-width result of
Liu and Tkocz and the standard simplex formula are credited. No claim
of absolute priority, exhaustive literature coverage, independent
review or formal proof-assistant certification is made. AI assistance
is disclosed in an ordinary paragraph of the manuscript.