The Dynamic Zeros Under Closure: Irreducible Core of a Discrete Physical 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 howthis naturally aligns with the Cayley–Dickson hierarchy through octonions. Modern quantum field theory already rejects the notion of a trivial vacuum: zero-point energy, vacuum polarization, and renormalization reveal that “empty space” is a structured ground state rather than an absence of structure. However, the algebraic status of this structured vacuum is typically introduced through subtraction schemes and regularization procedures that are formally consistent but conceptually layered atop the theory. The dynamic zero principle provides a minimal algebraic model of a non-absorbing ground state in which compression, cancellation, and holonomy arise from closure itself rather than from external adjustment. In this sense, the present work offers a foundational template for thinking about vacuum structure without ad hoc null-state assumptions. 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.