# Claim 3 — first-passage time

## Anchored claim

> The first-passage (hitting) time to within δ of the optimum for strongly convex objectives is Õ(δ^(-2p/(p-1))) (Theorem 3).

**Verdict: VERIFIED.**

## Paper quantity and protocol

I measured the exact first-passage time
`tau_delta=inf{t:|X(t)-x*|<=delta}` for the LMF quadratic strongly convex
instance, starting at `x(0)=0.9`. For each p and delta, eta is set to the
paper’s heavy-noise choice
`eta=(delta²/(8*sigma_heavy^p))^(1/(p-1))`. The sweep is
`p={1.5,1.75}` and `delta={0.50,0.35,0.25,0.18,0.13}`; each cell has 8
independent seeds, `dt=0.2`, and a 12000-time-unit cap. The cap censored
zero of 80 runs. The paper’s explicit Theorem-3 upper bound is evaluated after
the first-passage time is measured.

| p | delta | seeds | measured mean tau ± SE | measured/bound | paper bound | censored |
|---:|---:|---:|---:|---:|---:|---:|
| 1.50 | 0.50 | 8 | 1.175 ± 0.025 | 0.0769 | 15.288 | 0 |
| 1.50 | 0.35 | 8 | 7.925 ± 0.256 | 0.0610 | 129.95 | 0 |
| 1.50 | 0.25 | 8 | 39.025 ± 0.373 | 0.0399 | 978.45 | 0 |
| 1.50 | 0.18 | 8 | 186.23 ± 0.761 | 0.0265 | 7023.3 | 0 |
| 1.50 | 0.13 | 8 | 814.60 ± 3.51 | 0.0165 | 49490 | 0 |
| 1.75 | 0.50 | 8 | 0.200 ± 0.000 | 0.0622 | 3.214 | 0 |
| 1.75 | 0.35 | 8 | 1.000 ± 0.038 | 0.0589 | 16.979 | 0 |
| 1.75 | 0.25 | 8 | 3.425 ± 0.116 | 0.0420 | 81.629 | 0 |
| 1.75 | 0.18 | 8 | 10.325 ± 0.426 | 0.0273 | 378.12 | 0 |
| 1.75 | 0.13 | 8 | 27.550 ± 0.385 | 0.0160 | 1726.5 | 0 |

The measured first-passage time increases across every tightening-delta
sweep, and all 10 mean values lie below the paper bound. The deterministic
forward-drift baseline at `(p,delta)=(1.5,0.25)` also hits the target; the
reverse-drift destructive control has 8/8 censored runs, so the hitting event
depends on the paper’s descent direction rather than a time-cap artifact.

The observed slopes are empirical instance slopes of the measured first-passage
quantity; the decisive check is the direct first-passage measurement against
the paper’s explicit upper bound, with no fitted bound formula substituted for
the measured event time.