1---2id: H023title: 'End(E) = Z[ω] and the GLV basis is the Eisenstein ideal (π̄ − 1): closed form of the libsecp256k1 constants'4axis: A5kind: algebraic6statement: 'For secp256k1, t² − 4p = −3f² with f integral; End(E) = Z[ω]; the GLV lattice {(k1,k2): k1 + k2 λ ≡ 0 (mod n)}7 is the principal ideal (π̄−1)Z[ω] and a1 = (t−f)/2 − 1, b1 = −f, a2 = a1 − b1, b2 = a1, with a1² − a1 b1 + b1² = n.'8motivation: 'Research question A1: are there efficiently computable endomorphisms beyond the automorphisms? CM theory says9 no; the closed form explains the ''magic'' constants.'10assumptions: E ordinary, p ≡ 1 (mod 3), j = 0.11expected_contribution: Explanation of constants usually presented as output of the extended Euclidean algorithm; proof that12 no endomorphism outside Z[φ] exists.13related_work:14- gallant2001faster15- sica2002analysis16- galbraith2011endomorphisms17- silverman2009arithmetic18possible_counterexamples: If a1 ≠ (t−f)/2 − 1 exactly, the ideal description would be wrong (it was checked exactly).19verification_strategy: Exact integer check of the closed form; identity search Q1/Q3; extended-Euclid re-derivation.20computational_budget: seconds21status: evaluated22classification: known_reproduced23evidence:24- results/automated_discovery/identity_search.json25- results/baselines/secp256k1_facts.json26verdict: Closed form holds exactly; all 22 identities found by the search are explained by CM theory. Known in principle (ideal27 structure of the GLV lattice), the explicit identification with the published constants is a useful clarification, not new28 mathematics.29commands:30- .venv/bin/python experiments/automated_discovery/identity_search.py31---3233# H02 — End(E) = Z[ω] and the GLV basis is the Eisenstein ideal (π̄ − 1): closed form of the libsecp256k1 constants3435For secp256k1, t² − 4p = −3f² with f integral; End(E) = Z[ω]; the GLV lattice {(k1,k2): k1 + k2 λ ≡ 0 (mod n)} is the principal ideal (π̄−1)Z[ω] and a1 = (t−f)/2 − 1, b1 = −f, a2 = a1 − b1, b2 = a1, with a1² − a1 b1 + b1² = n.36