Composing the three consecutive-gap bounds

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Let $\mathcal A$ be all circle sequences and $\mathcal D\subset\mathcal A$ those with distinct terms. Both are nonempty. Use the cyclic labels, real lifts and extrema defined in L165. For $n\ge r$, each cyclic gap occurs in exactly $r$ spans, so

$$0\le nm_n^{(r)}\le r\le nM_n^{(r)}.$$

This also gives $M_n^{(r)}>0$, making the extended ratio convention unambiguous. All following limits ignore the finite prefix $n<r$.

L163 gives an absolute threshold $r_R$ and, for every $r\ge r_R$ and every $X\in\mathcal A$, the ratio lower bound with coefficient $1/200$. L164 gives absolute $c>0,r_O$ and the two separate deviation lower bounds for every $r\ge r_O$ and every $X\in\mathcal A$. Both use circumference one, ordering by position with index breaking ties, the lifts $y_{i+n}=y_i+1$, $r$ consecutive gaps, multiplicity, integer $n\ge r$, and natural logs. The ratio convention is $M/0=+\infty$; the one-sided proof divides by no minimum. Set $r_0=\max(r_R,r_O)$. Then all three assertions hold for every fixed $r\ge r_0$ and every sequence. Their limsups need not be witnessed by the same subsequence.

For any such $r$, put $a=c\sqrt{\log r}$ and $b=\log r/200$. The universal upper-span deviation bound gives $\limsup_n nM_n^{(r)}\ge r+a$ for every sequence. Taking its infimum over $\mathcal A$ gives $\Lambda_r^{\mathcal A}-r\ge a$. Since the scaled minima are in $[0,r]$,

$$\limsup_n(r-nm_n^{(r)})=r-\liminf_n nm_n^{(r)}.$$

The lower-deviation bound therefore gives $\liminf_n nm_n^{(r)}\le r-a$ for each sequence. Taking the supremum proves $r-\lambda_r^{\mathcal A}\ge a$. Taking the infimum of the actual sequence limsup ratios gives $\mu_r^{\mathcal A}\ge1+b/r$, hence $r(\mu_r^{\mathcal A}-1)\ge b$. None of these operations exchanges a limit with an infimum or supremum, requires attainment, or combines separately optimized numerator and denominator.

For the subclass, order alone gives

$$\Lambda_r^{\mathcal D}\ge\Lambda_r^{\mathcal A},\qquad \lambda_r^{\mathcal D}\le\lambda_r^{\mathcal A},\qquad \mu_r^{\mathcal D}\ge\mu_r^{\mathcal A}.$$

Thus the same three lower bounds hold for $\mathcal D$. Equality between these extrema is not needed. As $r$ tends through the positive integers, both $a$ and $b$ tend to infinity, proving each claimed divergence.

In the modified Statement of Problem 1221, the extrema range over $\mathcal A$. The displayed maximum and minimum mean the extrema over the $n$ cyclic starting labels, and its distance is the forward arc through $r$ consecutive gaps, as its Notes specify. These are exactly the definitions used above. Its three requested expressions are exactly the three just bounded. The printed first two expressions $r(\Lambda_r-1)$ and $r(1-\lambda_r)$ are different quantities; no conclusion about them is substituted for the modified Statement. Their historical interpretation is outside this proof.