The Szilard Threshold Principle Spectral Onset of Self-Sustaining Reaction and Controlled Cascade Arrest

We establish a rigorous operator-theoretic foundation for the onset of self-sustaining nuclear chain reactions, formalizing Leo Szilard's 1933 heuristic threshold as a spectral phase transition. The reproduction operator governing cascade dynamics exhibits a sharp bifurcation at unit spectral radius, separating subcritical decay, critical balance, and supercritical amplification. We construct a Birman-Schwinger formulation of this instability and derive a regularized determinant criterion (det₂) for cascade onset. By embedding the discrete generation map into the continuous Boltzmann transport equation, we map the phenomenological threshold to the essential and point spectra of the transport operator. To stabilize supercritical regimes, we introduce an Ω-Σ orthogonal projection framework that systematically excludes unstable modes and enforces strict L²-contractivity, achieving global cascade arrest. The manuscript elevates the threshold criterion via zeta-function regularization, demonstrating that controlled reactor dynamics correspond to projection-constrained spectral manifolds, while unconstrained amplification signifies the loss of analyticity of the regularized determinant through the emergence of a zero mode. Finally, we formulate Floquet and spinorial extensions of the cascade dynamics, proving that periodic operator transport and internal orientation-dependent modes admit the exact same projection-constrained arrest mechanism on an enlarged Hilbert space. The theoretical framework includes fully closed trace-ideal operator proofs and exact finite-dimensional JAX numerical validation, generalizing strictly beyond nuclear physics to universal self-amplifying cascades.