Universal vertices and effective dimension in geometric-graph entropy

Unrefereed research candidate. How many bits does it take to store a random geometric graph? Scatter n labelled points in a d-dimensional region with a density bounded above and away from zero, and join two whenever they are within a fixed distance r. This package determines the Shannon entropy H(G_n) up to an O(n) error for every fixed radius on every compact convex domain: H(G_n) = d[1 − μ(U_r)] n log₂ n + O(n), where U_r is the set of universal positions, within distance r of the whole domain. A vertex there is joined to everyone and its position carries no information. For translation rules on the flat torus with finite perimeter, the coefficient is the rank of the lattice generated by the rule's nonzero Fourier frequencies, the number of independent directions the rule can detect.

Refuted: the exact-coefficient reading of the conjecture of Vippathalla, Coon and Badiu (ISIT 2026; as stated in arXiv:2601.10778v1) for the unit cube when d ≥ 2 and √d/2 < r < √d. In the unit square at r = 1 the coefficient is 2√3 − 2π/3 ≈ 1.3697, not ≈ 1.4610; the gap peaks at about 0.161 near r ≈ 1.13. Confirmed: the n log n order in every case, the cube coefficient d for r ≤ √d/2, dimension one at every radius, and the torus coefficient d at every 0 < r < √d/2, each with an O(n) remainder. The radius dependence of every remainder is explicit.

The archive contains the 23-page manuscript (PDF and LaTeX), standard-library and SymPy sanity checks with receipts, the figure script and curve data, semantic negative controls, CI and SHA-256 sums. Assurance boundary: written proofs; internal AI-assisted review (five-role review, outside-model adversarial reviews before and after revision, a verification re-review and a targeted proof audit). This is not peer review, independent reproduction or formal verification. Fixed radius only. Creator: Anonymous. Text and data CC0-1.0; code MIT.