In our earlier numerical linear-model test we represented the NSNS block as the first 400 coordinates and created a "diagonal" candidate that placed nonzero entries only within that NSNS block (i.e. we did not include projections onto the transcendental indices). That modeling choice artificially forced the diagonal to lie in the algebraic block, so the computed rank did not increase. In true geometry, the diagonal does have a TTT\otimes TTT component --- but whether that component alone spans the full 4-dimensional TTT\otimes TTT depends on its independence relative to other algebraic classes.
More fundamentally: even though cl()\operatorname{cl}(\Delta)cl() has a TTT\otimes TTT part, it is a single vector in the 4-dimensional TTT\otimes TTT space. One algebraic correspondence (the diagonal) at best supplies a one-dimensional subspace of the 444-dimensional target; additional independent correspondences are needed to span the whole.
2. The swap / transpose class (symmetrization)
Definition. Consider the involution :YYYY\tau: Y\times Y\to Y\times Y:YYYY by (y1,y2)=(y2,y1)\tau(y_1,y_2)=(y_2,y_1)(y1,y2)=(y2,y1). The graph of \tau equals the diagonal \Delta pushed by the swap, so the "swap class" can be taken as either the class of the graph of \tau (which is the diagonal again under identification) or one may consider the symmetrizer/antisymmetrizer correspondences built from \Delta. One useful algebraic class is the symmetric projector
sym=12(cl()+cl()),\Pi_{\mathrm{sym}} \;=\; \tfrac12\big(\operatorname{cl}(\Delta) + \operatorname{cl}(\Delta\circ\tau)\big),sym=21(cl()+cl()),
or alternately the antisymmetric projector.
Induced action.
 Projectors constructed from swap/transpose act on H2(Y)H2(Y)H^2(Y)\otimes H^2(Y)H2(Y)H2(Y) by symmetrizing or antisymmetrizing tensors. They thus isolate symmetric or alternating parts of H2H2H^2\otimes H^2H2H2. The TTT\otimes TTT summand decomposes into symmetric and antisymmetric subspaces; the swap projector therefore can produce components inside TTT\otimes TTT.
Why it may fail.
As with the diagonal, the symmetric/antisymmetric projections are natural but are still low-dimensional: they reduce some redundancy but do not automatically produce a basis of the entire TTT\otimes TTT.
If all algebraic projectors and simple symmetrizations have images lying in the NSNS\operatorname{NS}\otimes\operatorname{NS}\oplusNSNS (low-dim) subspace, they will not reach the full TTT\otimes TTT.
3. Graphs of involutions / automorphisms (Nikulin-type graphs)
Definition.
If :YY\sigma:Y\to Y:YY is an algebraic automorphism (e.g. a Nikulin involution, a symplectic or non-symplectic involution), consider its graph
={(y,(y))}YY,\Gamma_\sigma \;=\; \{(y,\sigma(y))\}\subset Y\times Y,={(y,(y))}YY,
a codimension--2 algebraic cycle whose class [][\Gamma_\sigma][] defines a correspondence.
Induced action.
The induced endomorphism \Phi_{\Gamma_\sigma} of H2(Y)H^2(Y)H2(Y) equals the pull--push along the graph; it is precisely :H2(Y)H2(Y)\sigma_*:H^2(Y)\to H^2(Y):H2(Y)H2(Y). Thus the action on the transcendental lattice is the linear action of \sigma restricted to T(Y)T(Y)T(Y).
