Candidate Sieve: Why did 696.e1 emerge?

Opening of v0.3, and the first time the titular Phase 2 curve 696.e1 truly appears. The document first alters the filtering order: it adopts a cheap-to-expensive sequence of "rank/optimal/Manin → 2-part $L^{alg}$ valuation → base $\operatorname{BSD}(E,2)$ → odd multiplicative reservoirs → fixed additive primes → residual images → finally local Iwasawa/FW", rather than performing expensive local Galois analysis on every curve upfront. The base data for $E=696.\mathrm{e}1=[0,1,0,8,-16]$ is extremely clean: rank $0$, torsion trivial, optimal, Manin constant $1$, conductor $696<5000$, $\Sha_{\rm an}=1$, Tamagawa product $1$, yielding exactly $L^{alg}(E,1)=1,\ v_2(L^{alg})=0$, which falls perfectly into the "irrational 2-torsion, negative discriminant" branch of Theorem 2.14. Local structure at odd primes: $2$ is additive $\mathrm{II}^*$ ($v_\Delta=11$), $3$ is split multiplicative $\mathrm I_1$, $29$ is nonsplit multiplicative $\mathrm I_1$, hence $W_{\rm mult}^{\rm odd}=\{3,29\}$, $W_-=\{29\}$, and all relevant gcds are exactly $1$. LMFDB records maximal residual images for all primes; support prime irreducibility, fixed multiplicative irreducibility, no rational isogeny, and twist irreducibility preservation are all extremely clean. The document then provides a truly valuable control group: $116.\mathrm b1$ also has rank $0$, a beautiful 2-part anchor, and a nonsplit multiplicative $29$, but its odd bad structure has only $29$ as a single multiplicative reservoir. The fixed multiplicative check fails to find another $q\ne29,\ q\parallel N$ at $p=29$ to serve as a residual-ramification witness, thus it is eliminated by FAIL_FIXED_MULTIPLICATIVE_WITNESS. This comparison precisely points out that the real key to 696.e1 is not simply "having a nonsplit prime", but rather "at least two odd multiplicative reservoirs, at least one of which is nonsplit".