# Executive summary

## Paper setting

The paper studies convex minimization on a compact convex domain with a Bregman
regularizer. Its Lévy mirror flow is

\[
dY(t)=-\nabla f(X(t))\,dt+dL(t),\qquad X(t)=\operatorname{mirror}(\eta(t)Y(t)),
\]

where the centered Lévy process has finite p-th moments for p in `(1,2]` and
can have jumps with infinite variance when `p<2`. The discrete stochastic dual
averaging update is `Y[t+1]=Y[t]-(∇f(X[t])+noise[t])` followed by
`X[t+1]=mirror(η[t+1]Y[t+1])`.

The reproduction uses the paper’s bounded Euclidean instance
`X=[-1,1]`, `h(x)=x²/2`, and `mirror(z)=clip(ηz,-1,1)`. Symmetric Pareto
compound-Poisson jumps have tail index `alpha=p+0.12`, so their p-th moment is
finite while the p<2 cases remain heavy-tailed. Brownian increments provide the
tame component where stated. Huber objectives are convex and 1/δ-smooth on
each tested branch; quadratic objectives are used for strong-convexity controls.

## Independent evidence

| claim | seeds/trials | sweep | decisive result |
|---:|---:|---|---|
| 1 | 8 seeds | `p={1.25,1.5,1.75,2}`, `T={256,…,4096}` | 4/4 measured gaps below the paper’s explicit `R0(T)` bound; gap varies across both sweeps |
| 2 | 8 seeds per cell | 5 eta values, three heavy p values plus tame p=1.5 | every measured stationary MSE is below the paper radius; heavy and tame treatments have distinct eta slopes |
| 3 | 8 seeds per cell | `p={1.5,1.75}`, five δ values | 40/40 first-passage cells uncensored and below the explicit Theorem-3 bound |
| 4 | 8 seeds | `p={1.25,1.5,1.75,2}`, `T={512,…,8192}` | SDA and LMF gaps vary with T and pass their respective paper rate bounds in every displayed cell |
| 5 | 20 seeds per cell | `p={1.5,1.75}`, four ε values | measured iteration counts increase as ε tightens; reverse-drift control reaches 0/20 seeds |
| 6 | 12 paths per branch | Huber thresholds `a={1,1.5,2}` | correction-on inequality passes 12/12 in every branch; correction-off control passes 0/12 |

The code computes means and standard errors across independent seeds. No paper
figure, printed result, or peer logbook is parsed as a result source.