Research proof index

Multiplicativity of the computable automorphism, transported from galoisAutₛ (a RingHom) through the soundness bridge.

(S5) The core polynomial identity behind soundness: the monomial-remapped sum (before reduction) equals aeval (X^i) of the underlying polynomial. Both sides are ∑_{k<d} X^{ki}·a_k.

σ_m (X^i) = X^{i·m} (reduced). For i < 2^α the monomial-remap defining galoisAut collapses to the single surviving term. This is the key input to the trace-of-monomial computation in TraceInnerProduct.lean.

Core unit lemma. In S = Z_q[X]/(X^{2^α}+1), every nonzero σ_{-1}-fixed element is a unit. If b = mk g is not a unit then g shares a factor p_i with X^{2^α}+1 = p₁p₂; the reverse identity plus the swap p₁.reverse ~ p₂ force g to be divisible by both p₁ and p₂, hence by X^{2^α}+1, so b = 0.

Linear independence of 1, ζ, …, ζ^{2^{α-1}-1} makes each collected coefficient vanish.

|H| = d/k = 2^α / k (Hachi [NOZ26, §3], from |⟨4k+1⟩| = d/(2k) and the ± factor). The hypotheses match Hachi [NOZ26, §3, Claim 1] / [LS18, Lem 2.4]: k is a power of two (hk2pow) and divides d/2, i.e. 2k ∣ d = 2^α (hk). Both are needed for 4k+1 to have order exactly d/(2k) in (Z/2^{α+1})ˣ; the weaker k ∣ 2^α (= k ∣ d) does not suf...