The side-rank inequality and a failed matching reduction

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Let a strictly convex $v$-gon have $q$ distinct distances, and suppose one of its sides has descending distance rank $k$. The proposed inequality

$$v\le q+k+1 \tag{1}$$

is unproved for arbitrary convex polygons in this work. No convex counterexample to (1) was found. It was considered because applying it to a chord as a side of its longer subpolygon would yield bounded upper rank errors when $q=v/2+O(1)$. An upper rank bound alone would still not establish the monotone row-error hypotheses of quantitative_rank_inverse.

Concyclic sets satisfy the inequality

Let $\alpha$ be the empty angular gap of the chosen side $AB$. If $\alpha\ge\pi$, all vertices occupy a closed arc of length at most $\pi$ starting at $A$, so the $v-1$ distances from $A$ are distinct and $q\ge v-1$.

Otherwise unwrap the occupied arc from $A$ as $[0,2\pi-\alpha]$. A chord shorter than $AB$ has minor angle $\beta<\alpha$. Its two possible angular positions from $A$ are $\beta$ and $2\pi-\beta$, but the latter lies in the empty gap. Thus each distance shorter than $AB$ occurs at most once from $A$; every distance at least $AB$ occurs at most twice. There are $q-k$ values of the former kind and $k$ of the latter. Hence

$$v-1\le(q-k)+2k=q+k,$$

proving (1) in this restricted case.

A matching bound does not prove (1)

A proposed shortcut would bound the matching number of the graph of edges longer than that side by $k+O(1)$. This is false, even for concyclic sets satisfying (1) with equality.

For integers $k\ge2$ and $q\ge3k$, take consecutive vertices $0,1,\ldots,q+k$ of a regular $(2q+1)$-gon. They determine all $q$ chord lengths. The closing side has minor chord order $q+1-k$, hence descending rank $k$. However the edges

$$\{i,q+2-k+i\},\qquad 0\le i\le2k-2,$$

form a matching of size $2k-1$, all at chord order $q+2-k$, strictly longer than the side. Both endpoint intervals are disjoint and lie in $[0,q+k]$ by $q\ge3k$. As $k$ grows, this rules out every bound $k+C$ with absolute $C$. It does not disprove (1) itself or all possible linear matching bounds.

What the maximal-side source does provide

Altman's chain argument in Some Theorems on Convex Polygons, §III controls the minimum length in a chain of ordered internal diagonals opposite a diameter side. It does not bound individual entries' rank errors. Replacing that statement by a row-wise rank estimate was not justified here.

The circular proof and exact matching family were derived by a separate author and checked by the author; they are author-recorded. (1) and the arbitrary-convex inverse remain unresolved. Investigating (1) further is only a provisional direction, not an established reduction completing either Erdős-132 assertion.