1---2id: H043title: Automorphism orbits give exactly √6 on Pollard rho, with measurable fruitless-cycle overhead4axis: A5kind: algebraic6statement: On synthetic j = 0 prime-order curves, rho on classes of size 6 needs √6 fewer iterations than plain rho (and √27 for the negation map alone), up to fruitless-cycle handling costs.8motivation: Research question A2 (equivalence classes reduce redundant computation).9assumptions: Random-mapping heuristic; Teske r-adding walk r = 20.10expected_contribution: Precise measurement of the achieved versus theoretical factor and of the cycle overhead.11related_work:12- wiener1999faster13- duursma1999speeding14- gallant2000improving15- bos2010negation16- bernstein2011correct17possible_counterexamples: Achieved factor materially below √6 because of fruitless cycles or canonicalisation cost.18verification_strategy: 'Rust engine: rho_dp1 vs rho_dp1_neg vs rho_dp1_aut3 vs rho_dp1_neg_aut3 on 20 j = 0 curves per size.'19computational_budget: ≈ 1 node-hour20status: evaluated21classification: known_reproduced22evidence:23- results/synthetic_ecdlp/scaling_j0_M2U64.json24- results/synthetic_ecdlp/analysis.json25verdict: '180 paired instances at 44–52 bits: iteration speed-up 2.21 [2.00, 2.45] (r = 128) and 2.42 [2.19, 2.70] (r = 20)26 vs √6 = 2.449; constant 0.499·√n vs √(π/12) = 0.512; negation alone 1.29 vs √2, φ-orbits alone 1.52 vs √3; fruitless cycles27 ≤ 0.11 per 1000 iterations. Wall-clock speed-up only 0.95 (order-6) and 0.73 (negation) because canonicalisation costs 56–8328 ns vs 38 ns per plain iteration in this engine.'29---3031# H04 — Automorphism orbits give exactly √6 on Pollard rho, with measurable fruitless-cycle overhead3233On synthetic j = 0 prime-order curves, rho on classes of size 6 needs √6 fewer iterations than plain rho (and √2 for the negation map alone), up to fruitless-cycle handling costs.34