# Claim 1 — convex LMF rate

## Anchored claim

> Under the Lévy Mirror Flow model with centered Lévy noise having finite p-th moments (1<p≤2), time-averaged orbits achieve ε-optimality within Õ(ε^(-p/(p-1))) time for convex objectives (Theorem 1).

**Verdict: VERIFIED.**

## Paper quantity and protocol

The paper’s exact quantity is the time-averaged objective gap
`f(X̄(T))-min f`. I simulated the LMF dual equation with
`eta(t)=(1+t)^(-1/p)` and measured the Huber objective centered at `x*=0.6`
on `[-1,1]`. Each cell uses 8 independent seeds, `dt=0.5`, and the horizon
sweep `T={256,512,1024,2048,4096}`. The table reports the slope fitted to
the measured gap across that horizon sweep, not to a theorem expression.

The paper’s explicit `R0(T)` is evaluated only as a direct upper-bound check.
The centered treatment combines Brownian tame noise and centered compound-
Poisson Pareto jumps. The deterministic run is a sane baseline. The
reverse-drift run is a destructive control: it uses the same domain and
objective but moves away from the minimizer.

| p | seeds | measured gap at T=4096 ± SE | measured T-slope ± SE | paper exponent | explicit R0(4096) | gap/R0 | bound |
|---:|---:|---:|---:|---:|---:|---:|---|
| 1.25 | 8 | 0.0027976 ± 0.000075 | -0.3778 ± 0.0192 | -0.2000 | 0.122119 | 0.0229 | PASS |
| 1.50 | 8 | 0.0003501 ± 0.000020 | -0.6440 ± 0.0257 | -0.3333 | 0.035356 | 0.00990 | PASS |
| 1.75 | 8 | 0.0000786 ± 0.0000079 | -0.8160 ± 0.0429 | -0.4286 | 0.015032 | 0.00523 | PASS |
| 2.00 | 8 | 0.0000267 ± 0.0000041 | -0.9304 ± 0.0687 | -0.5000 | 0.008144 | 0.00328 | PASS |

The measured gaps vary by more than two orders of magnitude across the
horizon and p sweeps. All four p branches satisfy the paper’s explicit bound
on the measured quantity. The deterministic baseline at p=1.5 is
`0.0003378 ± 0.0000000`; the reverse-drift destructive control is
`1.3497 ± 0.0000`, demonstrating the sign-sensitive objective decrease.

The result is an instance-level numerical verification of the theorem’s exact
time-averaged objective gap, with the full p range represented and a control
that removes the descent direction.