1---2id: H053title: The automorphism orbit set cannot be made a group of order n/6 (rejected conjecture)4axis: A5kind: algebraic6statement: 'Conjecture: the quotient E(F_p)/Aut(E) inherits a group law making the DLP a problem of size n/6. Reality: the7 orbit relation is not a congruence (P ~ φP does not imply P + Q ~ φP + Q).'8motivation: A natural 'beyond √6' idea; must be ruled out explicitly (A5).9assumptions: Aut(E) ≅ Z/6 acting by automorphisms fixing O.10expected_contribution: Negative result with proof and computed counterexamples.11related_work:12- wiener1999faster13- silverman2009arithmetic14possible_counterexamples: 'None: the statement is refuted by a single pair.'15verification_strategy: 'theory/elliptic_curves/orbit_quotient_counterexample.py: 500 random pairs on 16/20/24-bit curves.'16computational_budget: seconds17status: evaluated18classification: rejected19evidence:20- results/theory/orbit_quotient_counterexample.json21verdict: 'Compatibility fails for 1500/1500 random pairs; proof: a congruence would make the orbit of O a subgroup H with22 cosets the other orbits, but orbits have size 6 ≠ |H| = 1.'23commands:24- .venv/bin/python theory/elliptic_curves/orbit_quotient_counterexample.py25---2627# H05 — The automorphism orbit set cannot be made a group of order n/6 (rejected conjecture)2829Conjecture: the quotient E(F_p)/Aut(E) inherits a group law making the DLP a problem of size n/6. Reality: the orbit relation is not a congruence (P ~ φP does not imply P + Q ~ φP + Q).30