Research proof index

σ_{-1}(X^e) = -X^{d-e} for 0 < e < d: the conjugate of a sub-d/2 monomial is minus the complementary monomial. The only nonzero coefficient of σ_{-1}(X^e) sits at d - e, with sign -1: from e·conjExp ≡ 2^{α+1} - e (mod 2^{α+1}) we get σ_{-1}(X^e) = X^{d + (d-e)} = -X^{d-e}.

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.

X^e + σ_{-1}(X^e) ∈ R_q^H when d/2k ∣ e: X^e is then σ_{4k+1}-fixed (its exponent is a multiple of d/2k), and the sum is symmetric under σ_{-1} (which has order 2).

Full coefficient of a (possibly high-degree) scaled monomial. For p < d, the p-th coefficient of the reduced c·X^m is (-1)^{m/d}·c at the folded position p = m mod d, and 0 elsewhere , the sign records how many times the exponent wrapped past X^d = -1.

The j-th coefficient of the reduced c·X^i (for i < d) is c·[j = i].

The scaled monomial c·X^j ∈ Rq Φ factors as (constant c)·X^j.