Consequently, to fill the 4-dimensional gap one needs a family of correspondences Z1,...,ZmZ_1,\dots,Z_mZ1,...,Zm whose induced endomorphisms Zi\Phi_{Z_i}Zi yield images on T(Y)T(Y)T(Y) that span End(T(Y))\mathrm{End}(T(Y))End(T(Y)) sufficiently to produce four independent vectors in TTT\otimes TTT. In the maximal Picard rank case T(Y)T(Y)T(Y) is 2-dimensional, so one needs correspondences whose restrictions to TTT give at least a 4-dimensional span of symmetric tensors (or equivalently enough independent endomorphisms).
6. Why our earlier linear-model test failed (explanation and lesson)
In the linear experiments we produced NSNS generators (400 basis vectors) and appended four "natural" candidate vectors (diagonal-as-NS-diagonal, swap-pattern, involution-pattern, trace-like) placed artificially so that they had support only in the NSNS block. This is why the computed rank remained 400400400: none of the added vectors had any component in the modeled TT coordinates. Two takeaways:
a. Modeling choice matters. A correct geometric modeling of [][\Delta][] or [][\Gamma_\sigma][] must include their projections onto both NS and T blocks. If the numeric model omits the T indices or constrains candidates to NS-only support, it cannot detect genuine TT components.
b. Algebraic correspondences must genuinely act on the transcendental lattice. Candidates built purely from divisor data cannot hit the TTT\otimes TTT summand. One must use correspondences coming from sources known (or likely) to induce nontrivial maps on TTT: graphs of automorphisms with transcendental action, Fourier--Mukai kernels associated with derived equivalences, Shioda--Inose transfers from abelian surfaces, or Kuga--Satake/Andr motivated cycles in arithmetic specializations.
7. Practical next steps to produce viable candidates
To make real progress one should:
a. Pick concrete YYY: choose a K3 with extra structure (e.g. singular K3 with known Shioda--Inose realization; K3 with explicit automorphisms; K3 with CM). Extra structure increases the chance that useful correspondences exist and are explicit.
b. Compute an explicit decomposition H2(Y)=NSTH^2(Y)=\operatorname{NS}\oplus TH2(Y)=NST: fix bases for NS and T (this can be done symbolically from lattice data for singular K3s or numerically via periods).
c. Construct geometric correspondences ZZZ whose induced map Z\Phi_ZZ can be computed (at least numerically) on the chosen basis. Candidate sources:
graphs of known automorphisms,
universal kernels/Fourier--Mukai transforms from moduli of sheaves,
Shioda--Inose/Kummer correspondences via an associated abelian surface,
cycles coming from derived equivalences or moduli of stable objects.
d. Compute the matrix of Z\Phi_ZZ in the NST basis and check the projection ZT\Phi_Z|_{T}ZT. Use these projections to form vectors in TTT\otimes TTT and test linear independence.
e. If linear independence holds heuristically, search literature for algebraicity proofs of the matching constructions (e.g. references by Mukai, Huybrechts, Shioda--Inose, Voisin, Andr). If available, these can upgrade heuristic linear-algebra evidence into rigorous statements in special cases.
ConcludingÂ
The diagonal, swap, involution, and trace candidates are natural first probes; they are conceptually simple and geometrically canonical. However, simplicity is not enough: to capture the T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y) summand one must use correspondences that have nontrivial, independent action on the transcendental lattice. This typically requires stronger geometry (automorphisms with nontrivial action on TTT), derived/Fourier--Mukai constructions, Shioda--Inose transfers from abelian geometry, or arithmetic specializations where motivated-cycle techniques apply. The CAS-6 heuristic precisely predicts this requirement: to attach algebraic weights to the orphan topological nodes, we must introduce richer interaction motifs --- correspondences that genuinely "link" the transcendental degrees of freedom across the two factors.
D. CAS-6 Analysis: Identification of Incomplete Closure in the Topology--Algebra Mapping
In Sections V.A--V.C we isolated a concrete numerical obstruction for the Hodge Conjecture on X=YYX=Y\times YX=YY when YYY is a K3 surface of maximal Picard rank: the rational Hodge subspace
TX:=H2,2(X)H4(X,Q)T_X\;:=\;H^{2,2}(X)\cap H^4(X,\mathbb Q)TX:=H2,2(X)H4(X,Q)
has dimension 404404404, while the nave algebraic subspace generated by products of divisor classes,
