id: H09 title: Automated search over iteration-function designs cannot beat the random-mapping constant for a given class size axis: B kind: algorithmic statement: Across r ∈ {3, 8, 16, 20, 32, 64}, mixed walks, and DP densities, no configuration achieves expected iterations below √(π n/(2·|class|)) by more than statistical noise; small r is measurably worse (Teske). motivation: 'Research question A4/B4: can automated algorithm search discover a better generic iteration?' assumptions: Walks are deterministic functions of the current point (generic-group restriction). expected_contribution: 'Negative result consistent with Shoup''s bound: the design space contains no asymptotic shortcut; constant-factor ranking of designs.' related_work:

• teske2001random
• shoup1997lower
• cheon2012speeding
• bernstein2012two possible_counterexamples: A configuration with constant < 1.25 at all sizes (would require explanation). verification_strategy: Rust engine sweep on 32–40-bit curves, 40 instances per configuration. computational_budget: ≈ 2 node-hours status: evaluated classification: rejected evidence:
• results/synthetic_ecdlp/sweep_r_M3U96a.json
• results/synthetic_ecdlp/analysis.json verdict: '25 configurations × 80 instances: r = 2 and r = 3 walks cost 3.77·√n and 2.26·√n (negative control), r ≥ 8 cluster at 1.1–1.5·√n with CIs overlapping √(π/2); mixed walks indistinguishable; best cell r32 at 1.10 [0.92, 1.28] not significantly below the random-mapping constant. No generic iteration design beats the random-mapping floor.'

# H09 — Automated search over iteration-function designs cannot beat the random-mapping constant for a given class size

Across r ∈ {3, 8, 16, 20, 32, 64}, mixed walks, and DP densities, no configuration achieves expected iterations below √(π n/(2·|class|)) by more than statistical noise; small r is measurably worse (Teske).