# Mixing Bridge Ndb

## Purpose

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

This is not a claim about all noise models. It is a single canonical route that makes the diffusion assumption explicit and auditable.

## Model

Let φ_n ∈ T_P be a stroboscopic phase sequence:

  φ_{n+1} = φ_n ⊕ ρ ⊕ ξ_n,

where

- ρ is a deterministic increment (drift)
- ξ_n are i.i.d. small noise increments with mean 0 and variance σ^2

Define the empirical density p_n of φ_n on T_P.

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

Under a diffusive scaling limit (small steps, many steps) where ξ_n has variance proportional to Δs, the density p(s,y) of φ satisfies a Fokker–Planck equation on T_P:

  ∂_s p + u ∂_y p = κ ∂_{yy} p,

with drift u corresponding to ρ and diffusion constant κ corresponding to σ^2.

## Mixing certificate inequality

The advection–diffusion evolution on a circle has an explicit spectral picture in Fourier modes.

For mean-zero modes on T_P, diffusion alone produces exponential decay with rate comparable to κ(2π/P)^2 on the first nontrivial mode. When u has shear, the decay can be slowed, and a conservative sufficient inequality is:

  λ := 2κ(2π/P)^2 − ||∂_y u||_∞  > 0.

This inequality is not presented as a sharp theorem for every PDE model. It is used as an engine-level criterion that forces every mixing claim to declare a quantitative mechanism.

## MXC

MXC (mixing certificate) means:

- an explicit NDB model is declared (what generates κ)
- a rate lower bound λ is declared via a documented inequality
- the artifact records the target and baseline the rate applies to

Operationally, MXC is represented by a convergence certificate of kind NDB_MXC, storing either a numeric rate_lambda or a ledger_ref.

## What this contributes to SFT

This provides a concrete NDB story:

- if phase increments contain small, independent jitter
- then phase distributions evolve by advection–diffusion
- so an MXC inequality can be used as an admissible criterion for convergence toward the baseline

This is sufficient for diffusion to be a modeled consequence rather than an untracked add-on.

## Empirical diagnostics (evidence)

SFT supports optional empirical diagnostics that are useful in benchmarks:

- random-walk circle simulations can measure decay of low Fourier modes
- the resulting rates can be compared to declared thresholds as an evidence signal

These diagnostics are not proofs. They exist to keep modeling assumptions checkable and to prevent unearned claims.

## Acceptance gates

- a specific noise model is stated
- the advection–diffusion limit is articulated clearly
- an explicit MXC inequality is stated with a declared rate symbol
- empirical diagnostics are labeled as evidence, not proofs
