# Verification of the cube section minimum theorem

The proof covers every real normal in every dimension n>=3. Dimensions
three and four use the graph, clipping and one-variable concavity proofs.
For n>=7, ordered facet coefficients and a global cubic Taylor lower bound
give analytic estimates, supplemented near coordinate directions by
variance and fourth-moment tail estimates. Continuity follows from a
quantitative graph comparison, not an assumption about facet formulas at
degenerate normals. Equality is checked throughout.

The n=5 cover has 756600 nodes, 6213 analytic leaves and 698775 exact-center
leaves, with maximum depth six. Python unbounded integers independently
check every leaf and complete coverage. The n=6 cover has 21993217 nodes,
328808 analytic leaves and 20884943 exact-center leaves, with maximum
depth seven. A separate C++ verifier uses Boost arbitrary-precision
comparisons, integer square-root bisection and a different signed-subset
construction. Both covers have zero unresolved leaves. All checks passed
on 5 October 2026, and their exact inputs are bound by SHA-256.

The continuum implication uses the written neighboring-section comparison
and the first-clipping-chamber lemma. It does not follow from a grid alone.
Analytic endpoint constants are certified by rational enclosures with
integer square roots. Earlier finite rational checks, 480 facet-order
checks and geometric spot checks are supplementary, not universal proofs.

The original research computation receipts retain their historical wording
that the covers depend on the geometric lemmas; those lemmas are now proved
in the manuscript. Earlier failed concavity arguments and test repairs are
preserved in the dossier and are not treated as source counterexamples.

The acceptance is the originating researcher's self-audit. No independent
human review, outside-model approval or proof-assistant formalization is
claimed. The complete theorem concerns the minimum subquestion only.
The whole bundled source remains partially solved and novelty undetermined.