FW-H2 Compiler and Ordinary Obstruction

Checking FW-H2 reduction type by reduction type marks the most important strategic turning point of the entire document. Good supersingular: the residual local representation is controlled by niveau-2 fundamental characters, and the local residual type is automatically irreducible, so $\mathrm{FW\text{-}H1=PASS,\ FW\text{-}H2=PASS}$ holds directly—this is exactly why documents 09 and 11 could focus on H3. Good ordinary: the standard ordinary semisimplification is $\bar\rho^{ss}\simeq\bar\alpha\oplus\bar\chi_{\rm cyc}\bar\alpha^{-1}$, and deducing H2 failure is exactly equivalent to $\bar\alpha^2=1$, which translates into curve language as a cheap exact criterion: $a_p(E)^2\equiv1\pmod p$. However, the document immediately points out that this path is a dead end: to cover all ordinary primes using FW, one must exclude all primes satisfying this congruence, and generally there is no clean finite-exception theorem available. Thus, the document makes a crucial strategic decision: ordinary primes will continue to use the existing ordinary theorem (echoing the BCS supporting route in the route matrix of document 00), leaving FW exclusively for additive and supersingular—the two reduction types not covered by existing Banwait–Huang theory. Potentially multiplicative is also excluded: its local semisimplification inherently takes the quadratic Tate twist form $\psi\oplus\psi\bar\chi_{\rm cyc}$, falling directly into the prohibited form of FW-H2, so it also does not use FW. This document formally narrows the scope of the Fouquet–Wan Compiler from "attempting to cover all odd primes" to "only handling additive + supersingular", which is an honest scope reduction rather than a forced expansion of coverage.