The Hodge conjecture for Fermat fourfolds of odd degree at most 199

Abstract

Let $X_m^4=\{x_0^m+\cdots +x_5^m=0\}\subset {\mathbb{P}}^5$ be the Fermat fourfold of degree $m$. We give a computer-assisted proof of the Hodge conjecture for $X_m^4$ for every odd $m\le 199$: three geometric closure criteria combined with an exhaustive machine census of the Hodge $(2,2)$-orbits, with a completeness proof and re-verifiable certificates. Criteria: (1) if the character multiset splits into two zero-sum triples, the rational Hodge block is transported from a $(1,1)$-substructure of a product of Fermat curves, hence algebraic; (2) algebraicity follows for characters that, after adjoining two vanishing pairs, decompose into an Aoki standard sextuple and a grade-$2$ Hodge quadruple; (3) the exceptional class at $m=33$ has a quasi-decomposable lift to level $66$, whose algebraicity descends along $X_{66}^4\to X_{33}^4$. The census covers the $89$ levels $21\le m\le 199$, $m\ne 23$, classifies all $78,299$ Galois-orbit representatives, and isolates thirteen orbits beyond decomposability, quasi-decomposability and Aoki's standard cycles: six close by the $\ast$-split criterion, seven by the two-pair and level-lifted closures, leaving none. Every orbit carries a machine-checked witness (negative screenings for the terminal ones), and the census is reproduced by an algorithmically independent implementation and brute force through $m=143$. Seven of the thirteen are gap classes outside Aoki's lattice calculus, new to the author's knowledge; for the other six, algebraicity is also derivable from that calculus, the explicit presentations being the new content. An exact Jacobi-sum computation at $p=67$ shows no cycle defined over $\mathbb{Q}({\zeta }_{33})$ projects nontrivially onto the exceptional $m=33$ block; more generally, over any finite extension of $\mathbb{Q}({\zeta }_{33})$ carrying a certifying cycle, every residue degree above $67$ is divisible by $6$.

19 pages; ancillary verification package (code, data, machine-checked certificates; ~55 s smoke tier). Companion paper on the even-degree census submitted separately. Code and Lean formalisation: https://github.com/rifmj/fermat-fourfolds-boundary