A table from even numbers under the map x ↦ → 2x − 3. Part II. The negative branch and its mirror symmetry

We continue the study of the iteration table T_{n,k} = f^k(2n), where f(x) = 2x - 3, introduced in Part I. Part I considered only the rows n ≥ 1 and identified the first row T_{1,k} = 3 - 2^k as the unique row going to negative values. In the present part we extend the table to all integer n and show that the negative branch is a mirror image of the positive one: the table is symmetric under the involution σ(x) = 6 - x combined with the reflection of rows n ↦ 3 - n, yielding the identity T_{3-n,k} = 6 - T_{n,k}. The symmetry follows from a single fact: σ commutes with the generating map f, so σ ∘ f^k = f^k ∘ σ for every k ≥ 0. Consequences: every Mersenne number M_p = 2^p - 1 has a mirror image 7 - 2^p at the reflected row n' = 2 - 2^{p-3}, and every Fermat number F_m = 2^{2^m} + 1 (for m ≥ 2) has a mirror image 5 - 2^{2^m}; the first Fermat number F_1 = 5 reflects into the first Mersenne number M_1 = 1, since 5 + 1 = 6. The pair of rows n = 1 and n = 2 forms an axis of the table: their entries sum to 6 in every column, at equal distance 2^k from the fixed point x = 3. The correction K_j introduced in Part I acquires a natural meaning there: K_{k-1} = 2^k - 3 is the radius 2^k minus the shift 3. The negative branch has a clean binary form T_{1,k} = -1^{k-2} 0 1 (k ≥ 2), and its mirror is T_{2,k} = 1 0^{k-2} 1 1. Finally, the inverse map g(x) = (x + 3)/2 implements 2-adic halving: after the shift y = x - 3 it becomes a pure division by two, dual to the pure doubling of f. The odd-orbit length of an odd x ≠ 3 is exactly v_2(x - 3), and one application of g recovers either the left Mersenne neighbour G_{p-1} = 2^{p-1} + 1 of M_p, or the seed T_{n_m,0} of the Fermat row. All statements are verified numerically by an accompanying Python script: the mirror symmetry is checked on 4816 pairs (n, k) with n ∈ [-100, 200] and k ∈ [0, 15], together with monotonicity of each row, the 2-adic identity v_2(T_{n,k} - 3) = k on all these pairs, and the inverse-step identity g(T_{n,k}) = T_{n,k-1} on every odd entry.