SFIT Part V: The Diffusion Limit of the Over-Strand Carrier and Analytic Record Laws

This manuscript constitutes Part V of the SFIT (Symbolic Field Interaction Theory) Stochastic-Braid Substrate series. It investigates the small-jump (continuum) limit of an over-strand carrier process on a probability simplex, driven by a discrete, planar three-defect stochastic braid substrate. By analyzing the continuous-time limit of this discrete Euler update, we derive exact analytic record laws and resolve the previously observed discrepancies between the continuum theory and discrete carrier simulations. Key Analytical Results Commuting Fields & Green-Kubo Reduction: We prove that the channel vector fields governing the carrier commute. Under a conditional stationary mixing hypothesis, the carrier converges to a diffusion whose generator coefficients are determined by Green-Kubo sums of over-strand correlations. Single-Parameter Limit: Assuming exchangeability of the defects, the limiting generator reduces to a single substrate-derived parameter, $\theta = (1 + 2\gamma_{GK})^{-1}$, where $\gamma_{GK}$ is the integrated over-strand persistence (distinct from the microscopic substrate reversal rate). Analytic Record Laws: For ideal vertex selection, Born weights hold for all initial populations if and only if $\gamma_{GK} = 0$. On a two-channel simplex edge, the ideal record probability is exactly the regularized incomplete beta function $I_y(\theta, \theta)$. For three channels, the law is a harmonic measure on a truncated simplex. Connection to Wong-Zakai: The limiting generator interpolates between the Itô form (martingale/Born, $\theta=1$) and the Stratonovich form (non-martingale/biased, $\theta \to 0$), demonstrating how persistent topological noise generates a Stratonovich-type drift in the discrete carrier. The Finite-Threshold Resolution A major contribution of this revised manuscript is the resolution of the 6–23% excess bias reported in the preliminary working note. We demonstrate that this discrepancy arose from comparing an ideal-vertex analytic law against a finite-threshold computational measurement ($H = 0.999$). By deriving and solving the exact finite-threshold boundary-value problem, the theory now matches the carrier measurements within $2\sigma$ across all tested two- and three-channel points, without fitting any new parameters. Status and Open Problems The manuscript provides retrospective, parameter-free checks that validate the corrected record laws. It also formulates a prospective Stage 4 test, including a novel threshold scan ($H \in \{0.99, 0.999, 0.9999\}$) to further validate the $(1-H)^\theta$ dependence. We explicitly note that the strong mixing hypothesis (H2) required for the functional central limit theorem of the planar substrate remains unproved, and we outline the mathematical steps required to address it.