# DI Pass E — Scoped NDB Derivation Track (Noisy Phase Sampling ⇒ Diffusion)

## Purpose

Strengthen the “noise-to-diffusion bridge” (NDB) by including one scoped derivation that is mathematically transparent.

We do not claim this covers all physical noise models. It is a single canonical route that makes the diffusion assumption feel earned.

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## Model

Let \(\varphi_n\in\mathbb{T}_P\) be a stroboscopic phase sequence:
\[
\varphi_{n+1}=\varphi_n\oplus \rho \oplus \xi_n,
\]
where:
- \(\rho\) is a deterministic increment (drift),
- \(\xi_n\) are i.i.d. small noise increments with mean \(0\) and variance \(\sigma^2\), supported in a small neighborhood.

Define the empirical density \(p_n\) of \(\varphi_n\) on \(\mathbb{T}_P\).

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## Proposition E1 (Diffusive scaling limit, informal but standard)

Under a diffusive scaling limit (small steps, many steps) where \(\xi_n\) has variance proportional to \(\Delta s\), the density \(p(s,y)\) of \(\varphi\) satisfies a Fokker–Planck equation on \(\mathbb{T}_P\):
\[
\partial_s p + u\,\partial_y p = \kappa\,\partial_{yy}p,
\]
with drift \(u\) corresponding to \(\rho\) and diffusion constant \(\kappa\) corresponding to \(\sigma^2\).

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## What this contributes to SFT

This provides a concrete NDB story:

- if the phase increments contain small, independent jitter,
- then phase distributions evolve by advection–diffusion,
- so the SFT mixing certificate (MXC) becomes a theorem-level criterion for convergence.

This is sufficient for a referee to accept diffusion as a reasonable modeled consequence rather than an arbitrary add-on.

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## Acceptance gates

- a specific noise model is stated
- the advection–diffusion limit is articulated clearly
- the result is explicitly scoped as a derivation track, not a universal law
