Sequential composition of chunk systems at a geodesic junction

Sequential composition of chunk systems at a geodesic junction. Let $Y$ be a metric space and $a, b, c \in Y$ with $d(a,b) + d(b,c) \le d(a,c)$ (so $b$ lies on a geodesic from $a$ to $c$). Let $C_1$ be a chunk system with online escapes from $a$ to $b$ and $C_2$ one from $b$ to $c$, at a common escape price $p_e \ge 0$, with trivial initial histories and size-total variances at most $V_1$ and $V_2$. Then on the product sample space there is a chunk system from $a$ to $c$ whose request sequence is the concatenation of the two phases, with the concatenated chunks and sizes, expected total $T_1 + T_2$, chunk count $m_1 + m_2$, any size window containing both input windows, trivial initial history, and variance at most $V_1 + V_2$:

by independence of the two phases. The two-phase filtration reveals the first system's history verbatim, then pairs its complete history with the running second-phase history; the second phase's cost premises transfer to the composite by conditioning each evader on the fixed first-phase request prefix, and the offline bound concatenates because the first sequence ends with the pinned singleton request $\{b\}$. This combinator chains the stage constructions of the BCR lower-bound induction along the junctions of the glued level step.