1---2id: H093title: Automated search over iteration-function designs cannot beat the random-mapping constant for a given class size4axis: B5kind: algorithmic6statement: Across r ∈ {3, 8, 16, 20, 32, 64}, mixed walks, and DP densities, no configuration achieves expected iterations7 below √(π n/(2·|class|)) by more than statistical noise; small r is measurably worse (Teske).8motivation: 'Research question A4/B4: can automated algorithm search discover a better generic iteration?'9assumptions: Walks are deterministic functions of the current point (generic-group restriction).10expected_contribution: 'Negative result consistent with Shoup''s bound: the design space contains no asymptotic shortcut;11 constant-factor ranking of designs.'12related_work:13- teske2001random14- shoup1997lower15- cheon2012speeding16- bernstein2012two17possible_counterexamples: A configuration with constant < 1.25 at all sizes (would require explanation).18verification_strategy: Rust engine sweep on 32–40-bit curves, 40 instances per configuration.19computational_budget: ≈ 2 node-hours20status: evaluated21classification: rejected22evidence:23- results/synthetic_ecdlp/sweep_r_M3U96a.json24- results/synthetic_ecdlp/analysis.json25verdict: '25 configurations × 80 instances: r = 2 and r = 3 walks cost 3.77·√n and 2.26·√n (negative control), r ≥ 8 cluster26 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 significantly27 below the random-mapping constant. No generic iteration design beats the random-mapping floor.'28---2930# H09 — Automated search over iteration-function designs cannot beat the random-mapping constant for a given class size3132Across 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).33