# Verification of the broom counterexample

The general result is the analytic proof in the five-page manuscript. For every fixed V,sigma>0, the broom B_N with ceil(sqrt(log N)) trunk vertices has normalized worst-case squared error at most (3V^2+12sigma^2)sqrt(log N)/N. The hub-rooted star has error at least sigma^2 log(N-1)/(8N) for sufficiently large N, already at the zero signal. Their risk ratio therefore tends to zero. Both trees have exactly N vertices and use full-cone least squares.

The originating researcher's detailed self audit checks cone orthogonality, the quadratic projection identity, the uniform signal-energy estimate, the Gaussian maximum bounds, equal vertex counts, and the order of the supremum and lower bound. No mathematical gap was found within that stated scope. This is not an independent review or a proof-assistant certification.

## Exact diagnostics

The included standard-library checker performs 90,995 enabled rational checks in each of normal and optimized Python:

- 492 cone projections, certified by active-set equations, primal nonnegativity, polar residual inequalities and complementarity;
- 20,460 exhaustive nonnegative parallel-arm witnesses;
- 22,500 coefficient comparisons;
- three negative controls, including rejection of signed-leaf witnesses and distinction between full-cone and budget-constrained projections.

The two reports agree except for the optimization flag. The code does not verify Gaussian integration, asymptotic probability limits or the full general theorem. These are supplied in the written argument.

## Source and publication scope

The source question was checked on the complete relevant page of the 2016 Oberwolfach report. Three primary PDFs were retained, and eight complete relevant pages were visually inspected. The known parallel-arm obstruction and the path risk bound are credited to Chatterjee and Lafferty. Wang's September 2026 preprint concerns a fixed-root estimator and minimax comparison; no theorem from that preprint is used here.

The five final manuscript pages were visually inspected. The portable source package is rebuilt and both checker modes are replayed before release. No third-party PDF is redistributed. The manuscript is AI-assisted, self-audited and unrefereed. The bounded literature search does not certify novelty or absolute priority.