The Hodge Conjecture for Mf4 , Its Fibrations, and the Categorical Reduction to Explicit Open Problems

We study the Hodge conjecture via an explicit four-dimensional building block Mf4 and a categorical reduction of the general case to explicit open problems. The paper has three contributions.First, we construct the building block Mf4 unconditionally and prove the Hodge conjecture for it by three independent routes: a K-theoretic route via the surjectivity of the Chern character, a categorical route via semiorthogonal decompositions, and asurgical route via categorical Perelman surgery. The building block Mf4 is a smooth projective K3 surface with c1 = 0 and χ = 24, obtained as the crepant resolution of the quotient (Ei × T2)/Z4.Second, we prove the Hodge conjecture unconditionally for several classes of Calabi–Yau n-folds: K3-fibrations and Mf4-fibrations over a smooth projective curve (with a section or a multisection), iterated-surgery manifolds M2n, VGIT flips, andflips in dimension ≤ 3. The proofs use relative HKR isomorphism, relative additivity, categorical surgery, and noncommutative Hodge theory for deformations of M2n.Third, we state the categorical reduction of the general case to explicit open problems: the general flip decomposition and the abundance conjecture for n ≥ 4. We do not claim to prove the Hodge conjecture in full generality; we provide a transparentframework in which the general case reduces to explicit open problems. We introduce a new tool, the categorical Perelman surgery, which generalises the classical surgery to derived categories. We also formulate a BPS criterion for the Hodge conjecture,and we honestly state its limitations for the intermediate Jacobian of the quintic.We prove that the building block Mf4 is torsion-free in the sense of noncommutative cyclic cohomology, and we clarify the relation between this cohomological torsion-freeness and the geometric torsion of the connection, which in the companion works isinterpreted as the source of the Standard Model gauge fields. The paper contains no adjustable parameters: the building block is explicit, and the reduction separates the unconditional steps (blow-ups, flops, VGIT flips, surgery) from the conditional ones (general flip decomposition, abundance).