They concentrate the most difficult instances of HC, where transcendental phenomena are expected to play decisive roles.
They connect HC to other deep conjectures and tools---mirror symmetry, derived categories, motives---providing multiple channels for CAS-6 to integrate with broader mathematical theories.
They allow systematic stress-testing of CAS-6: if the framework can illuminate closure vs. incompleteness in these settings, it demonstrates real explanatory power beyond the toy-model level.
4. Summary
Extending CAS-6 to Calabi--Yau varieties and higher K3K3K3 products will provide:
a. A stress test of scalability (can CAS-6 handle exponential growth of nodes and structures?).
b. A deeper probe into transcendental obstructions (do incomplete systems persist systematically?).
c. A fertile ground for discovering new emergent geometries, guided by the systemic balance of topology, algebra, and geometry.
These extensions will push the heuristic program towards the "hard frontier" of HC, where the conjecture is most uncertain and where systemic insights may prove most valuable.
B. Integration of Fourier--Mukai and Derived Categories into CAS-6 Modeling
One of the most powerful geometric sources of non-obvious correspondences on K3 surfaces (and more generally on varieties with rich derived categories) comes from Fourier--Mukai (FM) transforms and derived equivalences. In CAS-6 language these constructions supply new interaction motifs (correspondences ZZZ) that can assign algebraic weights to otherwise orphan topological nodes (notably the TTT\otimes TTT block). In this subsection I (i) summarize the relevant mathematics in precise form, (ii) explain how FM kernels map to CAS-6 components, (iii) outline concrete computational recipes to test their effectiveness, and (iv) flag the principal obstacles and how to address them.
1. Quick mathematical prcis (FM kernels and induced cohomological maps)
Let YYY and MMM be smooth projective varieties (in our focus: YYY a K3 surface and MMM a moduli space of stable sheaves on YYY, which often itself is a hyperkhler/K3-type variety). A Fourier--Mukai transform is an exact functor between derived categories
PYM=RpM(PLpY()):Db(Y)Db(M),\Phi_{\mathcal P}^{\,Y\to M} \;=\; R p_{M*}\big( \mathcal P \overset{L}{\otimes} p_Y^*( - ) \big) : D^b(Y) \longrightarrow D^b(M),PYM=RpM(PLpY()):Db(Y)Db(M),
determined by a kernel (universal family) PDb(YM)\mathcal P \in D^b(Y\times M)PDb(YM). When P\Phi_{\mathcal P}P is an equivalence (a derived equivalence), it induces isometries on cohomological invariants.
The cohomological (or Mukai) action of a kernel P\mathcal PP is given by the usual push--pull with characteristic classes. For H(Y,Q)\alpha\in H^\ast(Y,\mathbb Q)H(Y,Q) one defines
PH()=pM(pY()ch(P)pYM(td(YM))),\Phi_{\mathcal P}^H(\alpha) \;=\; p_{M*}\Big( p_Y^*(\alpha)\cup \operatorname{ch}(\mathcal P)\cup p_{Y\times M}^*\big(\sqrt{\mathrm{td}(Y\times M)}\big) \Big),PH()=pM(pY()ch(P)pYM(td(YM))),
