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…y checks Adds gen_pc_loop.py, which uses Sage to compute a_p and reduction type for elliptic curves over ℤ and renders a Lean file that recomputes each local Euler factor from a point count over 𝔽ₚ and checks it against the LMFDB value by `decide`. The generated MinimalAt predicate applies Kraus' criterion (KrausAt) at p = 2, 3 rather than testing valuations alone: at these primes v_p(c₄) ≥ 4, v_p(c₆) ≥ 6, v_p(Δ) ≥ 12 is not sufficient to conclude non-minimality, and without the extra check on the descended pair (c₄/p⁴, c₆/p⁶) the certificate mis-fires on genuinely minimal models (e.g. 112.c5, minimal with v₂(Δ) = 14, v₂(c₄) = 4, v₂(c₆) = 6); Kraus' criterion is what makes MinimalAt correct at the two primes where naive valuation bounds are unconditionally reliable for p ≥ 5 but not below it. Includes five generated certificates under generated/ for the curves in CURVES.
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Add Euler-factor certificate generator with Kraus-criterion minimality checks
Compiles against Lean v4.34.0 with mathlib and cslib on live.lean-lang.
Collaborators: Zachary Feng, Håvard Damm-Johnsen (the work started during the LMFDB-Lean workshop at UEA in June 2026.)