Finite thresholds rule out an r-independent VC covering bound

The real-power definition gives ${16}^1=\exp (\log 16)=16$. Its exponent law gives ${16}^{j+1}={16}^{j}16$ for every natural $j$, so induction identifies the real power ${16}^m$ with the natural product defining $N$. By [F3], $N^{-1\mathrm{/}m}=({16}^m{)}^{-1\mathrm{/}m}={16}^{-1}=1\mathrm{/}16.$

The subgraphs $C_{f_k}=\{(i,t):t<f_k(i)\}$ are nested in increasing $k$. A point $(1,1\mathrm{/}2)$ belongs to $C_{f_N}$ and not to $C_{f_0}$, so a singleton is shattered. No two-point set is shattered by a nested family, since shattering would require both opposite singleton traces, which nested sets cannot realize. Therefore the VC-subgraph dimension is exactly one and its index is $V=2$.

For $k\mathrm{\ne }\mathrm{\ell }$, the functions differ on exactly $\mathrm{\mid }k-\mathrm{\ell }\mathrm{\mid }$ atoms. By [F2] and [F4], $\mathrm{\parallel }{f}_k-f_{\mathrm{\ell }}{\mathrm{\parallel }}_{Q_m,m}={\left(\frac{\mathrm{\mid }k-\mathrm{\ell }\mathrm{\mid }}{N}\right)}^{1\mathrm{/}m}\ge N^{-1\mathrm{/}m}=1\mathrm{/}16.$

Since $1\mathrm{/}16>2ϵ=1\mathrm{/}32$, a closed $L^m(Q_m)$ ball of radius $ϵ$ centered at a class member cannot contain two distinct class members: the triangle inequality would make their distance at most $2ϵ$. Thus every cover needs at least $N+1$ centers; taking each member as its own center gives a cover with $N+1$. This proves the exact covering number.

Fix any proposed finite right side that depends only on $ϵ$, the VC index, and constants independent of $r$. At $ϵ=1\mathrm{/}64$ and $V=2$ it is a fixed finite number. By [F5], choose $m$ so large that ${16}^m+1$ exceeds that number. Step 3.1 then contradicts the proposed bound, so this family rules out a VC covering bound whose right-hand side is independent of $r$.