Thus the missing four dimensions in H2,2(X)H^{2,2}(X)H2,2(X) are exactly the contribution of the transcendental lattice squared.
6. Interpretation: where the difficulty for HC resides
The dimensional analysis above isolates the locus of potential failure for the Hodge Conjecture in the case X=YYX=Y\times YX=YY (with =20\rho=20=20). The algebraic cycles given by products of divisors span an explicit Q\mathbb QQ-subspace of dimension 400400400, but the full Hodge subspace has dimension 404404404; hence there are, a priori, four rational Hodge classes that are not accounted for by these obvious algebraic constructions. These four classes arise from purely transcendental data: they live in T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y), and their algebraicity is precisely the subtle question.
From the CAS-6 perspective, the topological skeleton (L,C)(L,C)(L,C) at the level 2p=42p=42p=4 carries four "nodes" for which the algebraic layer (W,P)(W,P)(W,P) (as generated by divisor--product interactions) supplies no canonical weights: the system exhibits an explicit residual incompleteness of dimension four. Restoring CAS-6 closure therefore requires constructing nontrivial algebraic correspondences or cycles that realize these transcendental tensors as genuine algebraic classes in CH2(X)\mathrm{CH}^2(X)CH2(X).
7. Remarks on non-maximal Picard rank and generality
If <20\rho<20<20 the algebraic span from NSNS\operatorname{NS}\otimes\operatorname{NS}NSNS has dimension 2\rho^22 and the transcendental part has dimension (22)2(22-\rho)^2(22)2; the total Hodge dimension satisfies
h2,2(X)=2+2(22)+(22)2+2,h^{2,2}(X) \;=\; \rho^2 \;+\; 2\rho(22-\rho) \;+\; (22-\rho)^2 \;+\; 2,h2,2(X)=2+2(22)+(22)2+2,
when contributions from H0H4H^0\otimes H^4H0H4 and H4H0H^4\otimes H^0H4H0 are made explicit; specialization to =20\rho=20=20 recovers the numbers above. In general, a larger transcendental dimension increases the gap between algebraically constructed classes and the full Hodge subspace, making the search for algebraic representatives correspondingly harder.
8. ConclusionÂ
The case K3K3K3\times K3K3K3 furnishes a crisply quantifiable challenge: the Hodge subspace H2,2(X)H^{2,2}(X)H2,2(X) exceeds the naively algebraic span by exactly four dimensions in the maximal-algebraic scenario, and by a larger number in lesser-algebraic situations. Any successful strategy toward validating the Hodge Conjecture for XXX must therefore identify algebraic cycles (typically non-obvious correspondences, Fourier--Mukai kernels, or images of constructions via Shioda--Inose/Kummer relations) whose cycle-classes realize the T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y) component. This precise identification of the locus of obstruction is the starting point for the constructive and literature-driven program developed in later sections.
B. Transcendental Contribution and Its Interpretation
