The Symplectic Bootstrap: Non-Perturbative Topological Closure of the 3D Ising Mode

We propose the Symplectic Bootstrap, a universal macroscopic topological framework that analytically determines the non-perturbative closure of strongly coupled field theories by strictly preserving their quantum phase space measure. At the critical singularity, traditional local kinematics diverge, threatening the macroscopic symplectic volume. To rescue this invariant, we complexify the continuous phase space and map the critical horizon into a Picard-Lefschetz homological intersection problem. We reveal that while the exact continuous measure is intrinsically governed by the global monodromy of Lefschetz thimbles, the phase space boundary must undergo sequential canonical projections to prevent measure degeneration. These holographic cascades inherently twist the geometric boundary, framing the 3D ${\mathbb{Z}}_2$ symmetric critical manifold into a discrete Conway rational tangle. This topological surgery uniquely locks the system into the Lens space $L(157,68)$, yielding an absolute topological multiplier ${\kappa }_{3D}^{exact}=157\mathrm{/}68$. By balancing the continuous homological interference against this discrete geometric anomaly, we exactly lock the topological leading-order anomalous dimension at $\eta \approx 0.036312$. The sub-0.04% quantitative agreement with numerical conformal bootstrap bounds rigorously validates that the 3D Ising universality class is fundamentally dominated by this macroscopic symplectic framing anomaly, providing an absolute analytical separation between global topological invariants and residual local quantum fluctuations.