BCR Lemma 15: regrouping tame subchunks into the next level

In Bubeck-Coester-Rabani's randomized $\\Omega(\\log^2 k)$ lower bound the induction over levels alternates two steps. Claim 13 refines a level-$w$ chunk system into small subchunks on level $w+1$, of maximal size $\\tfrac32\\cdot3^{w}$, escape price $2\\beta3^{w}$ and expected total $\\alpha\\beta(w+1)^2 3^{w+1}+3\\cdot3^{w}$, the extra $3\\cdot3^{w}$ being reserved to pay for the next step. Lemma 15 then regroups these subchunks into chunks of the size the induction hypothesis wants on level $w+1$: sizes in $[\\tfrac12 3^{w+1},\\tfrac32 3^{w+1}]$, expected total $\\alpha\\beta(w+1)^2 3^{w+1}$, escape price $2\\beta3^{w+1}$, and at least $\\lceil\\alpha\\beta(w+1)^2\\rceil$ chunks.

Theorem. For $\\alpha>0$, an integer $\\beta\\ge2$ and a level $w$ with $\\alpha(w+1)^2>1$: if a tame subchunk system on level $w+1$ exists - one whose conditional expected remaining size is branchwise slowly-decreasing and surprise-bounded with parameter $\\tfrac32\\cdot3^{w}$ - then the level-$(w+1)$ induction hypothesis holds.

The regrouping cuts the subchunk sequence at the stopping times at which the conditional expected remaining size first drops below $F_0-i\\Theta$ with $\\Theta=3^{w+1}$, so the resulting window $[\\Theta-\\tfrac32 3^{w},\\Theta+\\tfrac32 3^{w}]$ is exactly the required $[\\tfrac12 3^{w+1},\\tfrac32 3^{w+1}]$; the reserved surplus $3\\cdot3^{w}=\\Theta$ is exactly what makes $\\lceil\\alpha\\beta(w+1)^2\\rceil$ super-chunks affordable; and the price increase $\\Theta+3\\cdot\\tfrac32 3^{w}=\\tfrac{15}{2}3^{w}$ fits inside $2\\beta3^{w+1}-2\\beta3^{w}=4\\beta3^{w}$ precisely because $\\beta\\ge2$.

Formalization note. Tameness is a hypothesis here rather than a consequence of Claim 13 as currently formalized: the abstract statement that a subchunk system with the stated parameters exists does not by itself record how its conditional remaining mass behaves along individual branches, whereas the regrouping is a branchwise stopping-time argument. Closing the gap means strengthening the output of Claim 13 to a tame system.