A Proof of the Hodge Conjecture Derived from One Semiregular Witness Axiom: A Conditional Proof, the Deletion Test of Its Clauses, and the Open Fermat Orbit at Degree Seventy Outside Aoki's Lattice

The Hodge conjecture asserts that every rational class of type (p,p) on a smooth complex projective variety is a rational combination of classes of algebraic cycles; it is known for p≤ 1 and p≥ n-1 and open in the middle degrees of every dimension n≥ 4. Reductions of the conjecture typically restate it or split it into further open hypotheses, such as the existence of an algebraic representative at one point of a Hodge locus and the propagation of algebraicity along that locus. We state a single hypothesis that carries both: a semiregular witness, namely a local complete intersection or a coherent sheaf with injective semiregularity map representing a nonzero multiple of the class at some point of some component of its Hodge locus. We prove that this witness axiom implies the Hodge conjecture for all smooth projective varieties and all p, using the algebraicity of Hodge loci, the deformation theorems of Bloch and of Buchweitz and Flenner with Grothendieck existence and Artin approximation, GAGA, the countability of Hilbert schemes and a Baire argument along a very general curve, and we show that each clause of the axiom is load-bearing. At special points the existence half becomes computable. An exhaustive census of the 47,695 Galois orbits of Hodge (2,2) characters on Fermat fourfolds of degree 3≤ m≤ 100, closed by joins, curve pairs, pair descents, Aoki's standard cycles and level changes and re-verified by a second implementation with its own enumeration and arithmetic, leaves nine orbits open, eight of which close under Aoki's lattice criterion. The remaining orbit, (1,20,24,42,61,62) at m=70, is where every construction considered here stops, the lattice criterion included. The conditional theorem reduces the conjecture to a single existence problem for semiregular witnesses; whether the conjecture conversely implies that existence is open. Prelude edition. This paper is the first of a pair: its companion, A Formal Proof of Hodge Conjecture Termination at the Formal-Alone Register, proves that no procedure which reads formal data and constructs nothing supplies a semiregular witness at the classes where every relocation of the conjecture ends, so the two papers form one proof, universal exactly when every instance of Axiom W is witnessed.