Birational invariance conjecture for the K3 atom's BPS spectrum

Let $X \subset \mathbb{P}^5$ be a smooth cubic fourfold with Kuznetsov component $\mathcal{A}_X$, and let $\widetilde{X} \to X$ be a birational modification obtained by a sequence of blow-ups along smooth centers. Let $\{E_j\}$ be the exceptional objects appended to the semiorthogonal decomposition of $D^b\operatorname{Coh}(\widetilde{X})$ by Orlov's formula. Let $\mathcal{M}$ be the parameter space whose fundamental group acts by monodromy, and let $S$ be the global Stokes matrix with K3 block $S_A$. Birational invariance conjecture. (i) For every exceptional object $E_j$ and every state $A \in \mathcal{A}_X$, the full BPS tunneling amplitude vanishes, $T_{E_j \to A}=0$, as a consequence of the acyclicity of $\operatorname{RHom}(E_j,A)$. (ii) The monodromy representation of $\pi_1(\mathcal{M})$ on $D^b\operatorname{Coh}(\widetilde{X})$ preserves the block-diagonal structure of $S$, so $S_A$ is invariant under analytic continuation around the irregular singularity. (iii) The physical spectrum associated to the K3 atom, encoded in the Mukai lattice $\widetilde{H}(\mathcal{A}_X,\mathbb{Z})$, the JLO cyclic cocycle, and the monodromy weight filtration, is invariant under all birational modifications of $X$. Thus the K3 Hodge atom represents a dynamically protected quantum phase whose spectral data is not altered by non-perturbative tunneling associated with birational surgery. The preceding discussion establishes acyclicity in one tunneling direction but explicitly says that proving obstruction in both directions requires a full analysis of BPS stability conditions; consequently, the broader three-part assertion remains open.