Kodaira Prefilters and No-Go Results

Provides three precise, directly applicable computation-free shortcuts, followed by a formal ban to conclude. Shortcut 1: Exact no-go (potentially multiplicative): a potentially multiplicative curve twisted by a quadratic character $\psi$ becomes a Tate curve, its residual semisimplification is $1\oplus\omega$, and twisting back yields $\psi\oplus\psi\omega$. Since $\psi^2=1$, this falls exactly into the prohibited form of FW Theorem 1.7—therefore ADDITIVE + POTENTIALLY_MULTIPLICATIVE ⟹ FW17_H2_FAIL, requiring absolutely no local backend computation. Shortcut 2 ($p=3$): Since $\mathbf F_3^\times=\{\pm1\}$, any 1-dimensional local constituent is automatically quadratic or trivial, so p=3+LOCAL_REDUCIBLE⟹FAIL and p=3+LOCAL_IRREDUCIBLE⟹PASS, again requiring no general algorithm. Shortcut 3 (rational local $p$-torsion): If $E(\mathbf Q_p)[p]\ne0$, a trivial line exists, automatically yielding FAIL, which can be cheaply detected using Pannekoek's method—but the document specifically emphasizes that the converse does not hold: NO rational p-torsion ≠ H2 PASS, because the kernel character could still be nontrivial quadratic. Finally, the document formally rejects an entire class of "Kodaira-only" inference tables (such as unproven shortcuts like "potentially supersingular⟹PASS"), explicitly stating they must not be used unless supported by another residual-character theorem; Kodaira/potential-reduction information can only be used for prioritization, and the final determination must fall back on local residual irreducibility or the $p$-isogeny character/kernel certificates established in documents 35 and 36.