Research proof index

The geometric sum ∑_{j<d/2k} (X^{4ki})^j = 0 when d/2k ∤ i. The ratio r = X^{4ki} satisfies r^{d/2k} = X^{2di} = 1, and r - 1 is a unit (Xpow_sub_one_isUnit, since X^{4ki·2^t} = -1 for a suitable t extracted from the 2-adic valuation of i).

Claim 1 reindex: ∑_{a<n} X^{i·((4k+1)^a mod 2d)} = X^i·∑_{j<n} (X^{4ki})^j. The subgroup ⟨4k+1⟩ = {(4k+1)^a mod 2d} equals the arithmetic progression {4k·j+1 : j<n} (both have n distinct elements, and (4k+1)^a ≡ 1 mod 4k), which linearizes the exponent and exposes the geometric series.

(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.