3. Candidate mechanisms to realize T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y) algebraically
Although the transcendental summand is subtle, the literature and known constructions suggest several promising mechanisms (i.e. families of algebraic correspondences) that one should examine carefully when attempting to realize the 444-dimensional residue:
a. Diagonal-type and multi-diagonal cycles.
The small diagonal (and its variants) or diagonals in higher self-products sometimes project nontrivially onto transcendental factors. One may consider cycles supported on loci of the form {(y1,y2)f(y1)=g(y2)}\{(y_1,y_2)\mid f(y_1)=g(y_2)\}{(y1,y2)f(y1)=g(y2)} for suitable morphisms f,gf,gf,g between K3s, or combinations of diagonals and divisorial corrections. These are natural first candidates because they are canonical correspondences between the two factors.
b. Correspondences arising from automorphisms or involutions (Nikulin-type).
If YYY admits nontrivial automorphisms (Nikulin involutions, symplectic involutions, etc.), the graph of such an automorphism produces a correspondence whose action on cohomology can have nontrivial components on the transcendental lattice. In special geometric situations these graphs can generate classes in T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y).
c. Shioda--Inose / Kummer transfers.
For singular K3 surfaces (those with maximal Picard rank), there often exists a Shioda--Inose structure relating YYY to a Kummer surface associated with an abelian surface AAA. Since the theory of cycles on abelian varieties is comparatively richer and more tractable, pushforward/pullback along such birational or rational correspondences can transfer algebraic cycles from AAA\times AAA to YYY\times YYY, potentially producing classes in T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y).
d. Fourier--Mukai transforms and moduli-of-sheaves correspondences.
Moduli spaces MMM of stable sheaves (or complexes) on YYY are often hyperkhler varieties and admit universal families (or kernels) inducing Fourier--Mukai type correspondences between YYY and MMM. Composing the universal correspondence with its adjoint can produce algebraic cycles on YYY\times YYY whose cohomology classes have transcendental projections. Mukai's and subsequent work shows this is a fertile source of nontrivial algebraic correspondences for K3 surfaces.
e. Kuga--Satake and comparison with abelian varieties.
The Kuga--Satake construction attaches to the Hodge structure H2(Y)H^2(Y)H2(Y) an abelian variety AKSA_{KS}AKS whose first cohomology controls the original Hodge structure (up to certain tensor operations). If one can realize algebraic cycles on AKSAKSA_{KS}\times A_{KS}AKSAKS that correspond under the Kuga--Satake correspondence to elements of T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y), then pushing these cycles back (through explicit correspondences, when available) may yield algebraic realizations on YYY\times YYY. Note, however, that implementing this program concretely is subtle: the Kuga--Satake map is Hodge-theoretic and not known to be algebraic in general.
f. Motivic / Andr's "motivated cycles" approach.
Andr's framework of motivated cycles provides a conceptual route by which Hodge classes that are motivated by "natural" algebraic correspondences can be shown to be algebraic under additional hypotheses. If the classes in T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y) can be exhibited as images of motivated correspondences (for example via moduli constructions or reduction arguments), then one may be able to upgrade motivation to algebraicity.
Each of these mechanisms has been used successfully in various contexts to produce algebraic cycles that capture parts of transcendental cohomology; the current problem is to identify which of them (alone or in combination) can produce a basis for the full T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y) in the concrete geometry of the chosen YYY.
4. Heuristic CAS-6 interpretation: "orphan nodes" and restoration of closure
In the CAS-6 language the four-dimensional T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y) is a cluster of orphan nodes: topological configurations for which the obvious algebraic interactions (products of divisors) provide no weights. To restore closure one must introduce new interaction motifs --- i.e., nontrivial correspondences or kernels --- that assign rational weights to these nodes and produce stable geometric outputs.
The candidate mechanisms above correspond precisely to introducing such motifs: graphs of automorphisms add pairwise links; Fourier--Mukai kernels produce higher-order correspondences that act nontrivially on transcendental Hodge summands; Shioda--Inose correspondences import algebraic structure from abelian worlds. The CAS-6 heuristic predicts that if such motifs exist and are sufficiently independent, they will raise the algebraic span to full dimension and thereby realize the missing Hodge classes.
5. Practical implications for a constructive program
Given the delicacy of the transcendental contribution, a practical research program aimed at resolving the four-dimensional residue should proceed along these lines:
a. Select concrete geometric models of YYY where added structure is present (e.g. singular K3s, K3s with known automorphisms, or K3s admitting Shioda--Inose descriptions). Extra structure increases the chance that one of the mechanisms above produces algebraic realizations.
b. Compute the transcendental basis explicitly (numerically or symbolically) for the chosen YYY, and compute its tensor-square basis in H2,2(X)H^{2,2}(X)H2,2(X). This yields explicit target vectors that candidate correspondences must hit.
c. Construct and test correspondences (diagonal variations, graphs, Fourier--Mukai kernels, Kummer/Shioda--Inose induced cycles) and compute their cohomological projections onto the transcendental subspace. This step can be done heuristically by linear algebra in a model of the cohomology (as in our earlier experiments) to assess whether the chosen correspondences produce independent vectors spanning the 444-dimensional target.
d. Investigate arithmetic specializations (CM points, reductions mod ppp, Tate-type results) where additional structure can be used to prove algebraicity more directly. For example, CM-type phenomena and reductions to finite fields sometimes allow invocation of Tate or motivated-cycle arguments.
6. Concluding
The 4-dimensional transcendental residue in H2,2(YY)H^{2,2}(Y\times Y)H2,2(YY) epitomizes the heart of the Hodge Conjecture's difficulty in higher codimension: although small in numerical size, these purely transcendental tensors are protected by Hodge-theoretic symmetries and are not produced by the elementary algebraic operations that generate the bulk of algebraic cycles. The CAS-6 heuristic precisely characterizes this as a failure of the algebraic layer (W,P)(W,P)(W,P) to assign weights to certain topological nodes (L,C)(L,C)(L,C); remedying this requires introducing sophisticated algebraic correspondences---objects like Fourier--Mukai kernels, Shioda--Inose transfers, or Kuga--Satake mediated cycles---that are capable of targeting the transcendental subspace. The subsequent sections will explore, constructively and by reference to the literature, candidate correspondences and computational checks aimed at filling this residue.
C. Attempted candidates (diagonal, swap, involution, trace)
