Finite Exceptional Prime Problem

Directly dismantling the mother problem left by document 00: how does $\forall p>2,\ \mathrm{FW}(E,p)$ become a finite certificate? The document examines the "finiteness" of each of the three hypotheses point by point. H1 is good news: absolute reducibility (i.e., $\bar\rho_{E,p}$ is reducible) is equivalent to the existence of a residual stable line of rational $p$-isogeny type, which for a given $E/\mathbb Q$ only occurs at finitely many primes, and can be directly generated as a finite set $P_{\rm red}(E)$ from LMFDB isogeny/Galois-image metadata, Sage isogeny classes, and known rational isogeny theorems. H3 has a finite-exception heuristic: if a fixed multiplicative $\ell\parallel N$ can serve as a witness, then residual ramification is often related to $p\nmid v_\ell(\Delta_E)$, and multiple witness candidates can intersect to form $P_{\rm ram}(E)$—but the document explicitly warns that Fouquet–Wan's precise H3 condition is finer than pure ramification; this formula can only serve as a compiler heuristic and cannot be directly treated as a theorem. H2 is directly singled out by the document as the most unclear part at present; whether it can be reduced to $a_p$ congruences, local reduction types, or a finite exceptional set requires formal derivation—this is exactly the first real algebraic task of Phase 2. The document gives a precise success criterion: find a theorem such that $p\notin P_{\rm red}(E)\cup P_{\rm loc}(E)\cup P_{\rm ram}(E)\Rightarrow\mathrm{FW}(E,p)$, and the three sets on the right are all finite, effectively computable, and certificate-producing. Equally precise is the failure criterion: if the precise conditions for H2/H3 require incompressible local Galois computations for infinitely many $p$, and there is no generic-large-$p$ theorem, the route can only be downgraded to a "per-prime theorem, not a full-BSD family closure"—the document explicitly forbids using "tested up to $B$" as a substitute for the universal quantifier.