Rule (3.7) gives $1/\gamma_{k+1}^2 = (1+2\gamma_k(\mu_B-\gamma_kL_C^2/2))/\gamma_k^2$

The statement concerns a real-valued sequence $(\gamma_k)_{k \in \mathbb{N}}$, where $\gamma_k$ is written $\gamma_k := \mathrm{stepsPart2}(\mu_B, L_C, \gamma_0)(k)$. This function comes from the imported module Definitions.Def_ThreeOpSplitting_Accel_Stepsizes, and its definition is not shown here. Nothing in this statement says how $\gamma_k$ is computed from $\mu_B$, $L_C$, $\gamma_0$ and $k$. In particular, it does not say whether $\gamma_0$ as the sequence's value at index $0$ equals the input parameter $\gamma_0$. The statement also does not say whether the $\gamma_k$ are nonzero or positive. The attached documentation comment describes the result as a stepsize identity from a "Theorem 3.3, Part 2" and a "rule (3.7)". That description is commentary only; the code does not enforce it.

The theorem takes three real parameters $\mu_B$, $L_C$, $\gamma_0$ and a natural number $k \ge 0$, under these hypotheses:

• $\mu_B > 0$;
• $L_C > 0$;
• $0 < \gamma_0 < \dfrac{2\mu_B}{L_C^2}$.

For every such choice of parameters and every $k \in \mathbb{N}$, it asserts the exact equality

The hypotheses can all be satisfied, for example with $\mu_B = L_C = \gamma_0 = 1$, so the theorem is not vacuously true. The hypotheses appear only as assumptions; the equation itself does not mention them. Their only effect on the claim is to restrict it to the parameter choices above.

Degenerate cases:

• Small $k$: $k = 0$ is included, and there the claim links $\gamma_1$ to the sequence's value $\gamma_0$ at index $0$. No other special cases arise from $k$.
• Division by zero: the formal statement uses total real division, where $x/0 = 0$. If some $\gamma_k = 0$, the right-hand side is $0$ whatever its numerator is. If $\gamma_{k+1} = 0$, the left-hand side is $1/0 = 0$. So if $\gamma_k = 0$, the statement asserts only $1/\gamma_{k+1}^2 = 0$, which holds exactly when $\gamma_{k+1} = 0$. If $\gamma_k \neq 0$ but $\gamma_{k+1} = 0$, it asserts that the right-hand numerator $1 + 2\gamma_k(\mu_B - \gamma_k L_C^2/2)$ is $0$.
• Signs: the statement fixes $\gamma_k$ only through its square and the displayed numerator. Whether the $\gamma_k$ stay positive, or stay below $2\mu_B/L_C^2$, depends entirely on the unshown definition of the sequence. The hypothesis $\gamma_0 < 2\mu_B/L_C^2$ applies only to the input parameter $\gamma_0$.