BCR Lemma 15 in situ: regrouping Claim 13's subchunks into the level-$(w{+}1)$ chunk system

Fix a constant $\alpha>0$ and an integer $\beta\ge2$, and let $\mathcal M_w$ be the level-$w$ member of the recursive Bubeck--Coester--Rabani space family, with marked points $s_w,t_w$ at distance $\beta\cdot 3^w$.

Write $\mathrm{Chunks}(\alpha,\beta,w)$ for the inductive invariant of Lemma 12 at level $w$: on $\mathcal M_w$ there is a filtered chunk system with online escapes, with constant initial history, chunk sizes in $[3^w/2,\ 3\cdot 3^w/2]$, escape price $2\beta 3^w$, at least $\lceil\alpha\beta w^2\rceil$ chunks, and expected total size at least $\alpha\beta w^2 3^w$.

Write $\mathrm{Subchunks}(\alpha,\beta,w)$ for the conclusion of Claim 13 at level $w$: on the next-level space $\mathcal M_{w+1}$ there is a chunk system with constant initial history, sizes in $[0,\ 3\cdot 3^w/2]$, escape price $2\beta 3^w$, no lower bound on the number of chunks, and expected total size at least

Statement. For every $\alpha>0$, every integer $\beta\ge2$ and every level $w$ with $\alpha(w+1)^2>1$,

This is the regrouping step that completes the induction of Lemma 12: the subchunks of Claim 13, whose sizes are too small and carry no count guarantee, are grouped into super-chunks of average size $c_{\mathrm{avg}}=3\cdot 3^w=3^{w+1}$. With maximal subchunk size $\tilde c_{\max}=3\cdot 3^w/2$ the regrouped sizes fall in

the surplus $3\cdot 3^w$ of Claim 13 pays for the total lost in the last, incomplete group, the count bound $\lceil\alpha\beta(w+1)^2\rceil$ comes from dividing the expected total by $c_{\mathrm{avg}}$, and the escape price rises from $2\beta 3^w$ to $2\beta 3^{w+1}$, which for $\beta\ge2$ is at least $2\beta 3^w+c_{\mathrm{avg}}+\tilde c_{\max}$.

Formalization note. The metric-space-independent form of this regrouping, KServer.bcr_lemma15_regroup, is false as formalized and has been disproved on the platform: it allows the underlying space to be a two-point space, on which no chunk system with a positive pointwise size floor and at least two chunks exists, and its hypothesis supplies the surplus only in expectation while its conclusion demands the size floor in every branch and at every chunk index. The present statement fixes the spaces and all parameters to those of the induction of Lemma 12, where the space is the level space $\mathcal M_{w+1}$ and the input is the system supplied by Claim 13; the small-space obstruction does not apply to it. The predicates $\mathrm{Chunks}$ and $\mathrm{Subchunks}$ are the platform definitions KServer.BCRInductiveChunks and KServer.BCRInductiveSubchunks; all costs are scaled by $3^w$ relative to the paper's normalization $d_w(s_w,t_w)=\beta$.