696.e1 All-Prime Router

Apply the abstract routing table from document 12 specifically to $E_q$ ($q\in\mathcal P$, the twist family of 696.e1) and verify class by class. Taking $d=q$, Theorem 2.14 already gives $L(E_q,1)\ne0$ and $\operatorname{BSD}(E_q,2)$, so now we only need to handle odd primes. Case A ($p=q$, additive): witness $\ell=29$ is valid; since $v_{29}(\Delta_E)=1$, we have $q\nmid v_{29}(\Delta_E)$, residual ramified, and $29$ splits in $\mathbb Q(\sqrt q)$ making the local twist trivial at $29$. Case B (good ordinary, $p\nmid q$): quadratic twist preserves residual irreducibility; except for $p=3,29$ (which are already bad primes and will not occur), all other $p$ can uniformly use $29$ as a witness, since $p\nmid1=v_{29}(\Delta_E)$. Case C (fixed multiplicative, only $p=3,29$): for $p=3$ take $q_0=29$, for $p=29$ take $q_0=3$; both valuations are $1$, and residual ramification holds. Case D (good supersingular) is the branch specifically highlighted by the document—"this is the branch where semistability originally truly got stuck": FW-H1 (good supersingular local representation irreducible, hence global residual absolutely irreducible), FW-H2 (local residual irreducible, cannot be semisimplified into a forbidden type), FW-H3 (taking $\ell=29$, nonsplit multiplicative and $v_{29}(\Delta)=1$, any good supersingular odd prime $p\ne29$ has $p\nmid1$, residual Steinberg extension ramified) are all successfully verified. The document also honestly addresses the period/Manin issue: a good supersingular $p$ is a good-reduction prime ($p\nmid N_{E_q}$), hence there is no $p$-adic Manin contribution (citing existing Manin-constant support results); all mod-$\ell$ images of the base curve are maximal, and the twist preserves odd-$\ell$ irreducibility and the 2-torsion field, thus $E_q$ has no rational prime-degree isogeny. The final Exhaustion paragraph concludes: odd primes for $E_q$ can only fall into one of these four classes—additive, good ordinary, multiplicative $3/29$, good supersingular—all four classes are covered, and the prime router has no missing branches.