Upper-branch frequency $\omega(k)$ on a $q$-dimensional lattice, and reversal along axis 0

Definition 1: $\omega$ (omega). The inputs are:

• a natural number $q$, with the standing assumption $q \neq 0$, so $q \ge 1$;
• three arbitrary real numbers $K, c, \beta$, with no sign or other constraints (any of them may be negative or zero);
• a function $k : \{0, 1, \dots, q-1\} \to \mathbb{R}$, written $k = (k_0, k_1, \dots, k_{q-1})$, which is an arbitrary real vector of length $q$.

The index $0$ is the first element of $\{0, \dots, q-1\}$. It exists because $q \ge 1$. The value $\omega_q(K, c, \beta; k) \in \mathbb{R}$ is defined as

Here $\sin$ and $\cos$ are the real sine and cosine, with arguments in radians. The sum runs over all $q$ indices, including $a = 0$. For $q = 1$ it is just $1 - \cos(k_0)$. Each summand $1 - \cos(k_a)$ lies in $[0, 2]$, but $R$ can still be negative when $K < 0$ or $c < 0$.

The square root is Mathlib's total real square root:

• if $R \ge 0$, $\sqrt{R}$ is the usual nonnegative square root;
• if $R < 0$, $\sqrt{R}$ is defined to be $0$. No error or complex value is produced.

So $\omega_q(K,c,\beta;k) = \beta c \sin(k_0) + \sqrt{R}$ when $R \ge 0$, and $\omega_q(K,c,\beta;k) = \beta c \sin(k_0)$ when $R < 0$. At the boundary $R = 0$ both formulas give the same value. $\omega$ is defined for every choice of inputs, and nothing in the definition requires $R \ge 0$.

Definition 2: $\mathrm{flip}_0$ (flip0). The inputs are:

• a natural number $q$ with $q \neq 0$;
• a vector $k = (k_0, \dots, k_{q-1}) \in \mathbb{R}^q$, indexed as above.

$\mathrm{flip}_0(k)$ is the vector in $\mathbb{R}^q$ obtained from $k$ by replacing only the entry at index $0$ with its negative and leaving every other entry unchanged:

For $q = 1$ this is the single vector $(-k_0)$. The negated entry is the original value $k_0$. $\mathrm{flip}_0$ does not use $\omega$, and $\omega$ does not use $\mathrm{flip}_0$. These two definitions only introduce the functions and assert nothing about them.