Quadratic-Twist Invariance Bridge

The Fouquet–Wan theorem speaks to a single $(E_d,p)$, but what Banwait–Huang needs is "one base $E$ ⟹ infinitely many $d$," so the FW hypotheses must be brought down from the twist level back to the base level—this is exactly half of the Level 2 "quantifier compression" goal from Document 02. Starting from $\bar\rho_{E_d,p}\cong\bar\rho_{E,p}\otimes\chi_d$, the document proposes three candidate lemmas: Lemma A proves that absolute irreducibility is completely invariant under twist (tensor by 1-dimensional character is a category auto-equivalence, this item can be completely base-curve-ified); Lemma B handles how the local semisimplification degeneracy type is preserved under twist, and the document honestly marks its status as "standard representation theory derivation candidate, formal documents need to check against theorem versions one by one"; Lemma C proves that if the conductor prime $\ell\mid N$ splits in $K_d=\mathbb Q(\sqrt d)$, the local representation after twist remains completely unchanged over $G_{\mathbb Q_\ell}$, thus any FW-H3 local certificate using that $\ell$ as a witness can be preserved along the entire admissible twist family. If A/B/C are all formalized, then for a fixed $p$, $\mathrm{FW}(E,p)\Rightarrow\mathrm{FW}(E_d,p)$ holds for all admissible twists satisfying the splitting conditions—this is highly significant: there is no need to rerun the residual representation theorem for infinitely many $d$; one only needs to establish the FW certificate once for the base curve. However, the document explicitly draws a boundary at the end: this bridge only solves the $\forall d$ half; $\forall p>2$ is completely untouched.