The topology of a metal's Fermi sea is characterized by the Euler characteristic — a single integer counting the difference between electron and hole pockets. Two metals with the same Euler characteristic are topologically equivalent. Or so the classification assumed.
This paper shows the classification is incomplete. Fermi seas with identical Euler characteristic can harbor fundamentally different fine-grained topological structures. These structures cannot be connected without a Lifshitz transition — a topological phase transition in the Fermi surface itself. The Euler characteristic is necessary but not sufficient. A new structural resolution factor is needed to distinguish states that the old invariant declares identical.
The hidden structure has observable consequences. When these metals become superconducting through attractive Hubbard interactions, the topological superconducting phases inherit the fine-grained topology of the normal-state Fermi sea. Two metal-superconductor heterojunctions with the same Euler characteristic but different fine-grained structures produce anomalous gapless boundary states at their interface — states that shouldn't exist if the coarse classification were complete.
The result is a finer lens on a supposedly understood object. The Fermi sea has been studied for a century, yet its topological richness was hidden behind a classification that was too coarse. The integer told you something real, but it didn't tell you everything.