or more canonically using the Mukai vector v(E)=ch(E)td(Y)v(\mathcal E)=\operatorname{ch}(\mathcal E)\sqrt{\mathrm{td}(Y)}v(E)=ch(E)td(Y) and the Mukai pairing. For K3 surfaces many simplifications occur: td(Y)=1+2\mathrm{td}(Y)=1+2\omegatd(Y)=1+2 etc., and one works naturally with the Mukai lattice H~(Y,Z)\widetilde H(Y,\mathbb Z)H(Y,Z).
Crucially, composing a FM kernel with its transpose/adjoint gives an algebraic cycle on the self-product. If PDb(YM)\mathcal P\in D^b(Y\times M)PDb(YM) and QDb(MY)\mathcal Q\in D^b(M\times Y)QDb(MY) is its quasi-inverse kernel, then the kernel
K=QPDb(YY)\mathcal K \;=\; \mathcal Q \star \mathcal P \in D^b(Y\times Y)K=QPDb(YY)
defines a correspondence whose cycle class cl(K)H(YY)\operatorname{cl}(\mathcal K)\in H^*(Y\times Y)cl(K)H(YY) has an explicit expression in terms of ch(P),ch(Q)\operatorname{ch}(\mathcal P),\operatorname{ch}(\mathcal Q)ch(P),ch(Q) and push--pull operations. The induced cohomological endomorphism K:H(Y)H(Y)\Phi_{\mathcal K}:H^\ast(Y)\to H^\ast(Y)K:H(Y)H(Y) is precisely the composition QHPH\Phi_{\mathcal Q}^H\circ\Phi_{\mathcal P}^HQHPH, and its restriction to H2(Y)H^2(Y)H2(Y) (or to the transcendental lattice T(Y)T(Y)T(Y)) is the quantity of interest for filling TTT\otimes TTT.
When P\Phi_{\mathcal P}P is a derived equivalence between YYY and itself (an autoequivalence), K\mathcal KK is simply the class of the autoequivalence's kernel sitting in Db(YY)D^b(Y\times Y)Db(YY) and its cohomological class is an honest algebraic cycle (under suitable conditions), providing direct candidate correspondences.
2. Mapping FM/Derived constructions to CAS-6 components
Interpret FM/data in CAS-6 terms:
Interaction Level LLL: FM transforms are naturally higher-order interactions because they are not simply products of divisors; they encode how entire sheaves (or families thereof) interact across factors. For p=2p=2p=2 on YYY\times YYY they produce codimension-2 correspondences.
Interaction Configuration CCC: the kernel P\mathcal PP encodes a specific configuration of how points/sheaves on one factor pair with subschemes on the other; the moduli space MMM parameterizes these configurations.
Interaction Weights WWW: the cohomological transform PH\Phi_{\mathcal P}^HPH yields rational linear combinations (weights) on cohomology: the entries of the induced matrices on a chosen basis are the algebraic weights.
Interaction Probabilities PPP: the dimension and image of PH\Phi_{\mathcal P}^HPH (and of compositions) determine the likelihood that FM-built correspondences reach the target Hodge summands; numerically this is rank / required dim.
Interaction Stability SSS: FM correspondences coming from universal families or derived equivalences are often deformation-stable (e.g. they vary in families or persist across derived equivalent varieties), giving high SSS.
Interaction Output OOO: the geometric output is the algebraic cycle in YYY\times YYY represented by the composed FM kernel --- the candidate that may occupy a component of TTT\otimes TTT.
Thus FM/kernels are prototypical interaction motifs in CAS-6 that can bridge topology and geometry via algebraic weights computable from characteristic classes.
3. Concrete computational recipe (how to test FM candidates numerically/algebraically)
Below is an explicit, implementable pipeline for integrating FM correspondences into the CAS-6 diagnostic tests (suitable for symbolic/numeric computation in Sage/Python/Magma).
Inputs: a K3 surface YYY; a moduli space MMM of stable sheaves on YYY for which a universal family PDb(YM)\mathcal P\in D^b(Y\times M)PDb(YM) exists (or an explicit autoequivalence kernel on YYY\times YYY); explicit bases for H2(Y)H^2(Y)H2(Y) with an NS/T decomposition.
