# Exact Line Packings from Numerical Solutions

## TL;DR

This paper introduces a reproducible numerical-to-exact method for line packings in the real setting, avoiding deep algebraic conjectures, and successfully converts a numerical packing into an exact solution.

## Contribution

The authors present a novel, conjecture-free technique to transform numerical line packings into exact solutions using cylindrical algebraic decomposition.

## Key findings

- Successfully promoted an 8-point packing to an exact solution
- Method matches existing numerical packings in Sloane's database
- Establishes a foundation for exact solutions in line packing problems

## Abstract

Recent progress in Zauner's conjecture has leveraged deep conjectures in algebraic number theory to promote numerical line packings to exact and verifiable solutions to the line packing problem. We introduce a numerical-to-exact technique in the real setting that does not require such conjectures. Our approach is completely reproducible, matching Sloane's database of putatively optimal numerical line packings with Mathematica's built-in implementation of cylindrical algebraic decomposition. As a proof of concept, we promote a putatively optimal numerical packing of eight points in the real projective plane to an exact packing, whose optimality we establish in a forthcoming paper.