FW Theorem 1.7:Weight-2 Exact Translation

The opening of v0.2 marks the moment when the "H2 exact specialisation" requested all along by Documents 04 and 06 is truly completed—not guessing an equivalent condition, but formally deriving it starting from $\det E[p]=\bar\chi_{\rm cyc}$. H1 remains unchanged: $\bar\rho_{E,p}$ is absolutely irreducible. The core derivation of H2: if the local semisimplification forbidden form $\bar\rho_{E,p}|_{G_{\mathbf Q_p}}^{ss}\simeq\psi\oplus\psi\bar\chi_{\rm cyc}$ exists, it can be deduced from $\det E[p]=\bar\chi_{\rm cyc}$ that the forbidden form must satisfy $\psi^2=1$; written more generally as $V^{ss}=\alpha\oplus\beta$, this yields a directly testable representation-level predicate: $\text{H2 FAIL}\iff\alpha\beta^{-1}\in\{\bar\chi_{\rm cyc},\bar\chi_{\rm cyc}^{-1}\}$—the document explicitly marks that this is exactly the criterion the production compiler should use, no longer an abstract "local degeneracy". H3 is similarly precisely specialised: for weight 2 elliptic curves, FW's auxiliary Steinberg prime condition is equivalent to $\exists\ell\parallel N$ such that $E$ has nonsplit multiplicative reduction at $\ell$, $\ell\ne p$, and $p\nmid v_\ell(\Delta_{\min})$—this locks the FW-H3, which Document 02 warned is "finer than Banwait's $p\nmid\operatorname{ord}_\ell(\Delta_E)$ criterion", into a precise formula that can be directly compared with Banwait's existing criteria. This document formally submits the "first true algebraic task of Phase 2" defined by the entire v0.1 series.