Ordered simplex caps and Stringer bounds

Ordered simplex caps and the three-observation Stringer bound (LaTeX source).

For a uniform tetrahedron and any nondecreasing barycentric probability vector, the largest cap over ordered normals occurs at a step normal. A section-centroid argument reduces the result to one exact rational polynomial inequality and a separate repeated-upper-knot identity.

Consequently, for three independent observations from any distribution on ([0,1]), the binomial Stringer bound has coverage at least the nominal level whenever

[ 0<\alpha\le\left(\frac{19+\sqrt{21}}{34}\right)^3 =0.3336852118672717\ldots . ]

This endpoint is sharp for pointwise domination of Gaffke's valid upper mean bound, not asserted to be a coverage threshold. A two-knot perturbation gives an obstruction to pointwise domination in every sample size; its critical tail levels converge to (0.28466813704083846\ldots).

The article contains the geometric proof, its statistical application, and the obstruction. It does not combine the separate Poisson, finite-frame, or larger fixed-dimension calculations into the article.

• Proved with exact rational computation: the four-coordinate cap theorem and the three-observation comparison above.
• Conjectural: the global five- and six-coordinate monotone cap extensions and the dependent full confidence ranges at sample sizes four and five. Existing local calculations bound a first-coordinate beta cap; they do not establish the required active-prefix bound on repeated-lowest knot faces. See the precise remaining inequality.
• Separate results: the two-observation proof, the independent fixed-level certificates through sample size seven, and the valid modified procedures remain in the supporting materials. They are not proofs of the conjectural higher-dimensional extensions.
• Open: general finite-sample coverage of ordinary Stringer at conventional confidence levels under independent sampling.

The distinct fixed-population sampling problem is treated in Exact finite-frame inference under one-start systematic PPS (source). Its random-start inversion argument is separate from the independent-observation cap theorem.

Python 3.12, uv, and Tectonic are required. Dependencies are pinned in uv.lock.

make sync
make tetrahedral-vertex-barrier-check terminal-edge-bifurcation-check
make claim-manifest-check public-corpus-check
make paper

The tetrahedral check reconstructs the cap and centroid formulas, verifies the rational multiplier identity, and checks all Bernstein coefficient signs, including the repeated-upper boundary. The terminal-edge check verifies the derivative identities and numerical enclosures; the all-dimensional limiting argument is proved in the text. No proof-assistant formalization is asserted. make test runs the wider repository regression suite, including mutation tests for the public-text policy.

See the supporting guide for exact file locations and the other computations. The claim manifest distinguishes results in the article from results and conjectures in the research notes.

This is v1.1.0. The GitHub release contains the manuscript PDFs and the exact source package. The Zenodo series DOI resolves to the latest archived version; CITATION.cff cites this version using that stable series identifier.

Archived v1.0.0 predates the current article and is preserved unchanged.