A finite closed regular-faced polyhedral surface with a 5.5.4 vertex

This record gives an explicit finite closed polyhedral surface for M. Green's 554 problem (superliminal.com/geometry/acro): a polyhedron with regular faces having a vertex surrounded by exactly two regular pentagons and one square. The surface. Coordinates lie in the number field Q(√(1+√5)). It has 92 vertices, 264 edges and 164 faces (130 equilateral triangles, 32 squares, 2 regular pentagons, all of edge length 1). It is connected and orientable, of genus 5; every edge lies in exactly two faces and every vertex link is a single cycle. At the origin exactly one square and two pentagons meet, and no other part of the surface comes within distance 1/√18 of that vertex. Limitations, stated openly. The realization is self-intersecting (72 edge–face crossings), has genus 5, and has adjacent coplanar faces (124 flat edges; the square at the 5.5.4 vertex is one quarter of a flat 2×2 square). Under the conventions of the acrohedra catalogue (Flat, Co-P, S-Int allowed) it is a 5.5.4 acrohedron; under a convention that merges coplanar faces it would not qualify. No acceptance by the prize sponsor is claimed. Files. acro554_paper.pdf, acro554_paper.tex – the paper with full proofs; acro554_verify.py – Python 3, standard library only: builds the surface from the formulas of the paper and checks every claim in exact arithmetic (31 checks). Run python3 acro554_verify.py --check acro554.json; acro554.json – exact coordinates (coefficients in Q(h), h = √(1+√5)/2), decimal coordinates and faces; vertex 0 is the 5.5.4 vertex; acro554_figure.png – picture of the surface and of the 5.5.4 vertex. Use of AI. This solution was found with the help of artificial intelligence tools; the author checked the arguments and computations and takes responsibility for the content.