id: H05 title: The automorphism orbit set cannot be made a group of order n/6 (rejected conjecture) axis: A kind: algebraic statement: 'Conjecture: 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).' motivation: A natural 'beyond √6' idea; must be ruled out explicitly (A5). assumptions: Aut(E) ≅ Z/6 acting by automorphisms fixing O. expected_contribution: Negative result with proof and computed counterexamples. related_work:

• wiener1999faster
• silverman2009arithmetic possible_counterexamples: 'None: the statement is refuted by a single pair.' verification_strategy: 'theory/elliptic_curves/orbit_quotient_counterexample.py: 500 random pairs on 16/20/24-bit curves.' computational_budget: seconds status: evaluated classification: rejected evidence:
• results/theory/orbit_quotient_counterexample.json verdict: 'Compatibility fails for 1500/1500 random pairs; proof: a congruence would make the orbit of O a subgroup H with cosets the other orbits, but orbits have size 6 ≠ |H| = 1.' commands:
• .venv/bin/python theory/elliptic_curves/orbit_quotient_counterexample.py

# H05 — The automorphism orbit set cannot be made a group of order n/6 (rejected conjecture)

Conjecture: 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).