SFIT Stochastic-Braid Substrate: Part I - Topological Emergence from Classical Stochastic Dynamics

We introduce and numerically validate the **SFIT Stochastic-Braid Substrate**, a pre-continuum stochastic layer in which topological fields emerge as coarse-grained projections of persistent defect histories. The substrate consists of a discrete lattice of binary degrees of freedom (S-bits) from which persistent topological defects emerge, propagate ballistically, and undergo stochastic velocity reversals governed by a Poisson process. Through a sequence of nine targeted computational experiments, we demonstrate that this minimal classical system generates: **(1) Measurable topological entropy** governed by a thermodynamic equation of state, transitioning in functional form from a ballistic maximum to a diffusive decay as the stochastic switching rate increases. **(2) A mathematically lossy projection onto continuum fields**, establishing that scalar densities are topologically blind and that distinct microscopic histories can yield indistinguishable macroscopic observables. **(3) Active scrambling of internal symbolic memory via topological exchange**, revealing a novel mechanism of topological decoherence in which the braid dynamics themselves act as an internal environment. **(4) Genuine Artin braid group structure in 2+1 dimensions**, validated through the Yang-Baxter relation and non-commutativity controls. **(5) A three-level topological hierarchy** in which the substrate generates full braid-group information at the worldline level, the extraction algorithm faithfully recovers it as signed B₃ generator sequences, and the classical binary helicity exchange representation projects that information onto the S₃ permutation shadow. A **pure braid diagnostic** confirms that the 2+1D trajectories retain genuine over/under crossing information, distinguishing the non-trivial pure braid σ₁² from the trivial braid σ₁σ₁⁻¹ despite both yielding the identity permutation. A **B₃ vs S₃ diagnostic** then establishes that the collapse to S₃ is a property of the classical binary exchange representation, not of the substrate or the trajectories, and that escaping this permutation shadow requires an internal representation capable of supporting generators of infinite order (ρ(σᵢ)² ≠ I). Whether such a representation must be specifically complex—rather than merely real and continuous—to reproduce quantum interference and the Born rule is identified as the sharp open question for Part II. These results establish that the apparent smoothness of continuum fields is not a fundamental property but a blurred echo of a deeper, tangled substrate, and they identify the precise mathematical boundary between classical stochastic topology and emergent quantum state spaces.