id: H07 title: An exhaustive small-coefficient identity search finds only CM-explained relations and no cross-modulus or generator-specific relation axis: A kind: algebraic statement: Among 58 064 candidate relations (binary quadratic forms, modular quadratics, GLV linear relations, cross-modulus and generator relations), every identity found is implied by β, λ being cube roots of unity and by the Eisenstein-ideal structure; the negative-control spaces Q4/Q5 return 0 hits. motivation: 'Stage 2–3 automation: can a dumb generator rediscover structure, and does it hallucinate structure where there is none?' assumptions: Coefficient bound 6; constant set {p, n, t, f, a1, b1, a2, b2, β, λ, Gx, Gy}. expected_contribution: Calibration of the discovery engine (true positives explained, false positives zero). related_work:

• gallant2001faster
• sica2002analysis possible_counterexamples: A relation involving Gx or Gy would indicate a non-random generator (none). verification_strategy: experiments/automated_discovery/identity_search.py computational_budget: seconds status: evaluated classification: empirical_observation evidence:
• results/automated_discovery/identity_search.json verdict: 22 identities found, 22 explained, 0 unexplained, 0 false positives (expected chance hits 10^-73). commands:
• .venv/bin/python experiments/automated_discovery/identity_search.py

# H07 — An exhaustive small-coefficient identity search finds only CM-explained relations and no cross-modulus or generator-specific relation

Among 58 064 candidate relations (binary quadratic forms, modular quadratics, GLV linear relations, cross-modulus and generator relations), every identity found is implied by β, λ being cube roots of unity and by the Eisenstein-ideal structure; the negative-control spaces Q4/Q5 return 0 hits.