Research proof index

Hachi [NOZ26, §3, Theorem 2]: Tr_H(ψ(a)·σ_{-1}(ψ(b))) = (d/k)·⟨a,b⟩, where ⟨a,b⟩ = Σ_i a_i b_i is the inner product over R_q^H. Expand the product as a double sum, pull the R_q^H-coefficients out of Tr_H (traceH_smul_fixed), evaluate the monomial kernel (traceH_kernel, which is (d/k)·[p=q]), and collapse the diagonal.

Tr_H distributes over finite sums (it is additive).

(Claim 2) Tr_H(X^i) = 0 whenever d/2k ∤ i. Splitting H = ⟨σ_{-1}, σ_{4k+1}⟩ into the ⟨4k+1⟩-orbit {p_a} and its conjugate {q_a = -p_a}, the orbit sum is X^i·∑_{j}(X^{4ki})^j (four_pow_i_reindex), which vanishes (four_pow_i_geom_zero); the conjugate sum is its image under σ_{-1}, hence also 0.

Generalized Claim 3: Tr_H(X^j) = 0 for any j with X^{2j} = -1 (equivalently 2j ≡ d mod 2d, i.e. j an odd multiple of d/2). Proven by a fixed-point-free involution on H: conjugation m ↦ -m sends X^{j·m} to its negation, since (X^{j·m})² = (X^{2j})^m = (-1)^m = -1 (m odd) and conjugation inverts a square-root of -1.

Divisibility characterization: 2^{α+1} ∣ gⁿ − 1 ↔ 2^{α-κ-1} ∣ n for g = 4·2^κ+1, when κ + 1 ≤ α (so α - κ - 1 is the genuine order d/(2k)).

Bridge: u^m = 1 in the unit group iff 2^(α+1) ∣ q^m - 1.