Research proof index

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).

(C-3 helper) aeval (X^n) and aeval (X^{n mod 2^{α+1}}) agree in the quotient.

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.

The reverse identity. In S = Z_q[X]/(X^{2^α}+1), the conjugation σ_{-1} (here the semantic automorphism galoisAutₛ of exponent conjExp = 2^{α+1}-1) and the polynomial reversal are related by mk(reverse p) = σ_{-1}(mk p) · (mk X)^{deg p}. This is the concrete handle behind "σ_{-1} swaps the two factors": since (mk X)^{deg p} is a unit, `σ...