Algorithm cost equals $(1 + 1/c)$ times its dual objective

Binders. The statement quantifies over exactly two variables, both explicit: natural numbers $B$ and $k$. It carries exactly one hypothesis, $0 < B$. There are no implicit arguments, no typeclass assumptions, no other side conditions — nothing constrains $k$ relative to $B$, and nothing constrains the free parameter $c$, because the statement instantiates it at one specific value.

The constant. $c^*(B) = \left(1 + \frac{1}{B}\right)^{B} - 1$, a real base raised to a natural-number power. The scalar on the right-hand side is

For $B \ge 1$ this is $2$ when $B = 1$ and lies in $\left(\tfrac{e}{e-1},\,2\right]$ in general; the statement does not name, bound, or characterize it beyond writing it as that expression.

The sequences. With $c = c^*(B)$ fixed throughout:

and, each looking at the same index $j$ (not $j+1$), $z_j = 1 - x_j$ if $x_j < 1$ else $0$, and $y_j = 1$ if $x_j < 1$ else $0$.

The two sides, expanded. The left-hand side is the "primal cost" of the pair $(x_k, (z_j)_{j<k})$: $\text{LHS} = B \cdot x_k + \sum_{j=0}^{k-1} z_j$. The right-hand side is the scalar times a plain sum: $\text{RHS} = \left(1 + \frac{1}{c^*(B)}\right)\sum_{j=0}^{k-1} y_j$. Both sums run over the $k$ indices $0,\dots,k-1$ and are empty when $k = 0$.

What each side evaluates to combinatorially. Since $y_j$ indicates $x_j < 1$, the dual sum counts indices: $\sum_{j<k} y_j = \#\{j < k : x_j < 1\}$. Unwinding the recursion with $c = c^*(B)$, as long as the guard holds $x_j = \frac{(1+1/B)^j - 1}{c^*(B)}$, so $x_j < 1$ exactly for $j < B$, $x_B = 1$ exactly, and $x_j = 1$ for $j \ge B$; hence the count is $\min(k,B)$. The primal side collapses correspondingly, and both sides equal

Exactness. The claim is an equality of real numbers, not an inequality in either direction and not an asymptotic or approximate relation: no error term, no limit, no "for sufficiently large $k$". The factor is asserted to be attained on the nose, for every $k$ and every $B \ge 1$.

Index asymmetry. The two sides do not range over the same indices. The left side evaluates $x$ at index $k$ — one past the last index appearing in either sum — and adds the $k$ terms $z_0, \dots, z_{k-1}$; the right side involves only $y_0, \dots, y_{k-1}$ and has no term at index $k$. Moreover $z_j$ and $y_j$ are both computed from $x_j$, the value before the $j$-th update, whereas $B \cdot x_k$ uses the value after $k$ updates.

Degenerate cases. $k = 0$: both sums empty, $x_0 = 0$, so the claim reads $0 = 0$. $k = B$: $x_B = 1$ exactly and both sides equal $B\left(1 + 1/c^*(B)\right)$; $y_B = 0$ but index $B$ is outside the summation range anyway. $k > B$: the sequence is frozen at $x_k = 1$ and all summands with index $\ge B$ vanish, so both sides are constant in $k$; the equality for large $k$ carries no more information than at $k = B$. $B = 1$: $c^*(1) = 1$, factor exactly $2$, and for $k \ge 1$ the claim reads $1 \cdot 1 + (1 - 0) = 2 \cdot 1$. $B = 0$ is excluded, though all expressions remain well-formed under $1/0 = 0$.

What is NOT asserted. No optimality, minimality, or competitive-ratio claim: no offline optimum, adversary, or comparison quantity appears, so nothing says the ratio compares the cost to anything other than $\sum_{j<k} y_j$. Nothing asserts $c^*(B)$ is the best, unique, or only constant with this property, and nothing for any other $c$. No feasibility claim: not that $x_k + z_j \ge 1$, not that $y_j \le 1$, not that $(y_j)$ is feasible for any dual program; the words "primal" and "dual" in the definition names carry no asserted linear-programming content, and no weak- or strong-duality statement is present. No structural facts about the sequence are asserted, even though they follow from the definitions: not $x_j \le 1$, not monotonicity, not $z_j \ge 0$, not the closed form. Neither side is asserted to equal $\min(k,B)$ or the scaled version — that identification is a consequence of unfolding, not part of the statement. Nothing about limits as $B \to \infty$ or about $e/(e-1)$, nothing about any interpretation as days, prices, or decisions, and nothing about any algorithm's behaviour beyond the literal arithmetic identity.