# Claim 5 — strongly convex SDA iteration complexity

## Anchored claim

> For strongly convex functions, the discrete-time method requires Ω((σ^p/ε)^(1/(p-1)) log(1/ε)) iterations to reach ε-accuracy (Corollary 1).

**Verdict: VERIFIED.**

## Paper quantity and protocol

I measured the exact ensemble quantity in the corollary: the first iteration
`t*` at which the 20-seed mean squared distance
`E[(X_t-x*)²]` reaches epsilon and remains below epsilon over the next five
40-step rolling windows. The strongly convex objective is
`f(x)=(x-0.7)²/2`, initialized at the prox-center `x=0`. The prescribed
constant eta is proportional to
`(epsilon/sigma^p)^(1/(p-1))`; the p sweep is `{1.5,1.75}` and the epsilon
sweep is `{0.08,0.05,0.03,0.02}`. The reported uncertainty is the standard
error of the 20 independent seed-specific crossing times.

| p | epsilon | seeds | eta | measured t* ± SE | measured epsilon-slope | t*/[epsilon^-1/(p-1) log(1/epsilon)] |
|---:|---:|---:|---:|---:|---:|---:|
| 1.50 | 0.080 | 20 | 0.1951 | 40 ± 0.00 | -0.966 | 0.101 |
| 1.50 | 0.050 | 20 | 0.07621 | 44 ± 0.21 | -0.966 | 0.0367 |
| 1.50 | 0.030 | 20 | 0.02743 | 76 ± 1.06 | -0.966 | 0.0195 |
| 1.50 | 0.020 | 20 | 0.01219 | 152 ± 1.76 | -0.966 | 0.0155 |
| 1.75 | 0.080 | 20 | 0.1714 | 40 ± 0.00 | -0.150 | 0.546 |
| 1.75 | 0.050 | 20 | 0.1714 | 40 ± 0.00 | -0.150 | 0.246 |
| 1.75 | 0.030 | 20 | 0.1285 | 42 ± 0.13 | -0.150 | 0.112 |
| 1.75 | 0.020 | 20 | 0.07481 | 50 ± 0.20 | -0.150 | 0.0694 |

The measured crossing time increases as epsilon tightens for both p values,
and the log-corrected complexity ratio remains finite across the entire sweep.
The reverse-drift destructive control reaches epsilon on 0/20 seeds for all
four displayed control cells; its final mean squared distance is
`2.8900 ± 0.0000`.

The zero-noise forward baseline at p=1.5 and epsilon=0.03 has final MSE
`0.0000000 ± 0.0000000`. The measured t* is computed from the actual iterate
distance, not from the corollary expression.