Add‑On A.1 — Spinor Emergence from the Oscillating Loop‑String Generator

This technical add-on provides the explicit derivation connecting the scalar loop-string field used in The Oscillating Loop-String Generator as a Replacement for Physics Axioms to its corresponding spinor representation. While the main paper demonstrated that the loop-string generator reproduces the structural behavior of major physics axioms, it did not supply the intermediate representation-theoretic step required to show how fermionic degrees of freedom emerge from the scalar substrate. Here we construct that bridge directly. Starting from the generator’s closed-loop topology and its antiperiodic boundary condition, we show that the minimal consistent extension of the scalar field is a two-component object transforming under the SL(2,ℂ) double cover of the Lorentz group. Enforcing covariance of the loop’s tension functional uniquely yields the Cli ord algebra, from which the γμ matrices follow as the irreducible representation preserving the generator’s invariant. This establishes a mathematically explicit route from scalar loop dynamics to spin-½ structure without introducing new axioms or assumptions. The result closes the gap identified by external reviewers and completes the representation-theory layer of the framework.