The Dynamic Zeros Under Closure: Irreducible Core of a Discrete Computational Framework

We present a self-contained treatment of the dynamic zero principle---the assertion that the element $0 \in S = \{-1,0,+1\}$ is not an absorbing terminal state but a compression boundary through which the system transitions without annihilation. Beginning from three pre-numeric modal states and four interaction constraints (closure, totality, boundedness, nontriviality), we derive the unique minimal algebra $S$, prove that cancellation is forced rather than postulated, and show that Euler's identity $e^{i\pi}+1=0$ emerges as the algebraic termination certificate of the forced completion sequence $S \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{R} \to \mathbb{C}$. We prove that the primordial states are symmetric under permutation; triadic completion alone admits a symmetric ($S_3$-equivariant) law, but imposing orientation (order sensitivity) forces symmetry breaking. We discuss how this naturally aligns with the Cayley--Dickson hierarchy through octonions. We then formalize the dynamic zero as a $\mathbb{Z}_2$ holonomy on the spinor cover: the minimal nontrivial loop that returns observables to themselves while inverting the internal state. The paper is organized into three epistemic tiers: Established (results with complete proofs from first principles), Derivable (results contingent on the full Kosmoplex axiom set whose proofs are sketched or referenced), and Open (precisely stated questions whose resolution would strengthen or falsify the framework). No free parameters appear. The dynamic zero is not a number; it is the engine of non-termination.