Why this family is promising.
If \sigma acts nontrivially on T(Y)T(Y)T(Y), then \Phi_{\Gamma_\sigma} has a nonzero component on TTT, and the cohomology class [][\Gamma_\sigma][] will have a nonzero TTT\otimes TTT projection.
Distinct automorphisms with independent actions on T(Y)T(Y)T(Y) can produce independent vectors in the TTT\otimes TTT summand. For example, if the representation of the automorphism group on T(Y)T(Y)T(Y) is rich enough, finitely many graphs might generate the full TTT\otimes TTT.
Why they may fail in practice.
Many K3 surfaces have only trivial or very small automorphism groups; in particular, for a generic (even singular) K3 the automorphisms acting nontrivially on T(Y)T(Y)T(Y) may be absent.
When \sigma is symplectic (acts trivially on the holomorphic 2-form), its action on T(Y)T(Y)T(Y) may reduce to identity or a very restricted subgroup, producing too little new direction. Conversely, non-symplectic automorphisms can act nontrivially but are rarer.
4. Trace-like / global sum classes
Definition.
A "trace-like" class in our linear model was represented as the sum of NSNS basis elements (or more generally a global sum over generators). Geometrically, one may consider classes of the form
=iDiDi,\Theta \;=\; \sum_{i} D_i\times D_i,=iDiDi,
where {Di}\{D_i\}{Di} is a basis of NS(Y)\operatorname{NS}(Y)NS(Y). Alternatively, consider pushforwards of universal families or trace correspondences coming from moduli constructions.
Induced action.
Such a class acts on H2(Y)H^2(Y)H2(Y) by pairing with basis elements and tends to act nontrivially on the NS-block but only indirectly on the transcendental block. In a decomposition where we index the cohomology basis so that NSNS coordinates occupy the leading block, a nave trace class supported only on NSNS will have zero coordinates in the TT block.
Why it typically fails.
If \Theta is constructed purely from NS classes (divisors), then it lies in NSNS\operatorname{NS}\otimes\operatorname{NS}NSNS and produces no component in TTT\otimes TTT. This is exactly what happened in our linear model: the trace candidate was a vector supported on the NSNS indices and therefore did not contribute to the 4-dimensional complement.
Even when \Theta mixes NS and T components analytically, linear dependence with NSNS generators can render it ineffective at increasing rank.
5. Unifying observation: criterion for an effective candidate
All the preceding remarks can be condensed into a clean necessary (and practically useful) criterion:
Criterion. Let ZCH2(YY)Z\in\mathrm{CH}^2(Y\times Y)ZCH2(YY) be a correspondence. Let Z:H2(Y)H2(Y)\Phi_Z: H^2(Y)\to H^2(Y)Z:H2(Y)H2(Y) be the induced linear map. The correspondence ZZZ has nontrivial projection to the summand T(Y)T(Y)H4(YY)T(Y)\otimes T(Y)\subset H^4(Y\times Y)T(Y)T(Y)H4(YY) iff the restriction ZT(Y)\Phi_Z|_{T(Y)}ZT(Y) is nonzero (equivalently, Z\Phi_ZZ has a nonzero component mapping the transcendental lattice to itself).
