Interaction Outputs \(O(X)\): \(O(X)_2 = CH^2(X) \otimes \mathbb{Q}\), with realization index \(r(X)_2 = \dim O(X)_2 - \dim S(X)_2 > 0\), indicating potential unrealized cycles.
The gap signals incomplete closure, formalized as a failure of the functorial map \(W(X)_2 \to H^{2,2}(X) \cap H^4(X, \mathbb{Q})\) to be an isomorphism, unlike in \(E \times E\).
C. Candidate Cycles (Diagonals, FM Kernels) with Rank Tests
To address the gap, we explore candidate cycles to augment \(W(X)_2\):
Diagonals: The diagonal \(\Delta = \{(x, x) \in K3 \times K3\}\) lies in \(CH^2(X)\), but its class is in \(H^{2,2}\) and contributes to algebraic cycles, not closing the transcendental gap.
Fourier-Mukai (FM) Kernels: FM transforms, via derived categories, suggest kernels in \(D^b(K3 \times K3)\) that may induce new cycles. For instance, an FM kernel corresponding to a rank-4 bundle may contribute classes not spanned by divisors.
We perform rank tests to assess contributions. The algebraic span is modeled as a matrix of cycle classes (e.g., products of divisors). The transcendental classes (dimension 4) require additional cycles, but FM kernels often lie in the same span, as verified computationally below.
D. Computational Pipeline for Span Verification
We use SymPy to compute dimensions and test spans, focusing on \(H^{2,2}\).
SymPy Implementation:
import sympy as sp
