Holographic Cascade Conjecture and Symplectic Bounds for the 3D Ising Model

Abstract

We propose a heuristic non-perturbative framework to investigate the 3D Ising model at criticality by mapping the continuous 3D $ϕ^4$ field theory into an operator-valued Stroh matrix governed by the infinite-dimensional symplectic Lie algebra $\mathfrak{sp}(\infty)$. By enforcing the symplectic topological constraint $\mathbfΓ^2 = -\mathbf{I}$ on the boundary operators, we derive a Symplectic Bootstrap equation. This framework establishes two foundational bounds. First, a pure Euclidean classical geometric projection yields the universal factor $κ_{3D}^{\text{classical}} = 1.4$, locking the anomalous dimension to $η\approx 0.0185$, perfectly recovering the microscopic 2-loop perturbative scattering limit. Second, to reach the non-perturbative horizon, we propose a Holographic Cascade Conjecture: the Stroh spatial foliation induces a topological framing anomaly. The conformal dimensional reduction ($5 \to 4 \to 3 \to 2$) acts as framing charges in a rational tangle, yielding the topological invariant $[2; 3, 4, 5] = 157/68$. This absolute multiplier locks the strong-coupling root to $η\approx 0.036312$. Its astonishing $\sim 0.04\%$ proximity to rigorous numerical bootstrap bounds uncovers this pure-geometric dimensional cascade as the dominant topological backbone of the 3D Ising universality, with the residual gap rigorously quantifying non-topological local quantum fluctuations.