How can we prove that a determinant is not merely large—but as large as mathematically possible?
Unlocking Maximal Determinants over Third Roots of Unity transforms Daoyu Dong’s order-15 proof into a clear, step-by-step journey through one of computational mathematics’ most intricate determinant problems.
At its center is a remarkable exact result: among all order-15 matrices whose entries are third roots of unity, the maximal determinant is $120,932,352\sqrt{19}$. But discovering this value is only part of the story. Proving that no matrix can do better requires a powerful combination of Eisenstein arithmetic, Gram matrices, spectral inequalities, support graphs, Schur complements, and carefully organized computational elimination.
This companion unpacks the entire argument without hiding the difficult steps behind “a computation shows.” Instead, it shows what is being computed, why each reduction is valid, and how the result can be independently verified.
Inside, you will learn how to:
- Master the arithmetic of Eisenstein integers and their norms, units, associates, and prime structure.
- Translate determinant maximization into the language of Gram matrices and off-diagonal energy.
- Reduce an enormous search space to just 23 admissible energy shells.
- Use support graphs, independence bounds, and spectral projections to exclude impossible configurations.
- Understand the spectral intertwining that connects matrix symmetries with algebraic constraints.
- Reconstruct the extremal matrix and its connection to the Fano plane.
- Verify determinants exactly using fraction-free Bareiss elimination.
- Certify polynomial root counts with exact Sturm sequences and directed rational enclosures.
- Replay the computations through both pure-Python and SymPy/SageMath tracks.
Worked numerical examples, matrix diagrams, proof maps, and the complete 136-entry inner-product catalogue make the argument accessible without sacrificing rigor. Every major reduction is explained, every computational certificate has a mathematical purpose, and the final extremizer is examined from both algebraic and spectral perspectives.
Whether you are a graduate student, researcher, educator, or computational mathematics enthusiast, this book offers more than a guided reading of a single proof. It provides a reusable model for turning advanced mathematical arguments into **understandable, reproducible, and verifiable knowledge**.
An independent educational study companion to Daoyu Dong’s “The Maximal Determinant of Order 15 over the Third Roots of Unity,” arXiv:2610.40679v1.