---

The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0091, E0099, E0103, E0217, E0982, E1082.

Exact optimizers and finite geometric obstructions

E0091: one finite nonuniqueness example is not eventual nonuniqueness

A regular hexagon has three lengths. A regular pentagon together with its center also has three: its side, diagonal and circumradius. The inspected few-distance statement $g(2)=5$ excludes a two-distance set of six points. Both examples therefore minimize the global count at six points. Their convex hulls have six and five vertices, so they are not similar. Kovács's inspected six-point theorem concerns sets in which every triangle is isosceles; a regular hexagon has scalene triples. Thus its uniqueness classification does not classify every distance minimizer. A construction extending this pair to all sufficiently large orders while preserving minimality is still missing. The $g(2)$ source statement was inspected in Shinohara's introduction; its classification proof was not rechecked.

E0099: there is no positive gap to triangle-free near-optimizers

Fix $n\ge3$, and let an optimal normalized packing have minimum distance one and diameter $D_n$. In the space of labeled planar coordinates, pair coincidences and equality between two different unordered pair lengths are proper polynomial zero sets. Their finite union has empty interior. Hence configurations with all pair lengths distinct approximate the optimizer arbitrarily closely. Minimum distance and diameter are continuous. Rescale each approximation by its positive minimum distance; the resulting minimum is exactly one, its diameter converges to $D_n$, and exactly one pair has length one. In particular there is no unit equilateral triangle. Therefore

$$\inf\{\operatorname{diam}P:\min|p-q|=1, \ P\text{ has no unit triangle}\}=D_n.$$

This does not produce a triangle-free exact optimizer. It rules out a uniform positive diameter gap or an asymptotic density argument as a sufficient substitute for an exact contact argument.

One possible exact sufficient condition is more than $2n-4$ contacts in every optimizer. The straight-line contact graph of a minimum-one packing is planar: two crossing unit segments would have two endpoints at distance less than one, by the triangle inequality applied through their crossing (equality would force collinearity). A triangle-free planar graph has at most $2n-4$ edges. No lower bound of this strength for every diameter optimizer was obtained. The Bezdek--Fodor publisher abstract was inspected for its exact order-eight result, not a general theorem or its full proof.

E0103: a sufficient flex criterion has two active constraints

Suppose an exact minimum-diameter packing contains a point $p$ with $1<|p-q|<D_n$ for every other point $q$. Its finitely many strict inequalities persist under small motions of $p$. Both the minimum one and maximum $D_n$ are realized by pairs among the fixed other points. Therefore a small open disk of positions of $p$ consists of exact optimizers. There are continuum many congruence classes: an isometry taking one of these finite sets to another has only finitely many choices on two fixed distinct points and on their images, and hence only finitely many choices in each class. The strict-slack hypothesis is not known for an optimizer. A flexible contact graph by itself is insufficient, because moving along the flex may increase an active diameter. The concrete next test is a simultaneous contact-and-diameter constraint calculation on an exact optimizer.

E0217: a local obstruction for one nine-point partition

The exact point list, distance spectrum and Jacobian computation are retained in L30. The determinant $-1102248$ isolates this realization modulo similarities. Separated neighborhoods of its eight lengths then show that every sufficiently close eight-distance configuration preserves the same partition. Its forbidden collinearity and circles cannot be removed by a small perturbation. A different partition or a remote component of the realization space remains possible; no global order-nine exclusion or enumeration is claimed.

E0982 and E1082: global and pinned targets separate at order eight

For E0982, Dumitrescu's explicit bound at eight vertices is $\lceil(13\cdot8-6)/36\rceil=3$, below the requested four. Altman's planar bound forces at least four global distances. If there are exactly four, Shinohara's inspected Lemma 2.2(a)(ii), specialized to $k=4$, says the convex eight-point set is a regular octagon or a regular nonagon with one vertex removed. In the octagon every pin sees four lengths. In the nonagon each length occurs twice about each pin; deleting one point leaves at least one occurrence of every length. Thus a counterexample with all pinned counts at most three would have at least five global lengths. This narrows a finite realization test; it does not carry out that test. The convex classification is an external statement premise, not a newly checked proof.

For E1082, the inspected facts $g(2)=5,g(3)=7,g(4)=9$, together with the elementary maximum of three equidistant planar points, prove $D\ge\lfloor n/2\rfloor$ for every $n\le11$, without even using no-three-collinearity. They supply no all-order conclusion.

The pinned counterexample $H_8$ can be checked directly. Put $h=1+\sqrt3$, take square vertices $(\pm1,\pm1)$ and apexes $(0,\pm h),(\pm h,0)$. Its four global lengths are

$$2,\quad2\sqrt2,\quad\sqrt2h=\sqrt2+\sqrt6,\quad2h.$$

A square vertex sees the first three; an apex sees the first, third and fourth. There are no three collinear: a line through two square vertices is one of $x=\pm1,y=\pm1,y=\pm x$, containing no apex; a line through two apexes is one of $x=0,y=0,x\pm y=\pm h$, containing no square vertex. Each set of four alone has no collinear triple. The square vertices satisfy $|x|+|y|=2<h$, so all lie strictly inside the apex diamond. Hence $H_8$ refutes the pinned version, meets the global lower bound, and is not a convex-polygon example for E0982.

The [[research/leads/concentric_polygon_census/_index|exact concentric and affine polygon census]] uses $H_8$ as a convexity-filter control. It finds no strictly convex counterexample among half-step concentric regular $m$-gons for $3\le m\le32$, over all positive radii, or the specified diagonal stretches of regular $n$-gons for $8\le n\le24$, over all positive stretch parameters. Every critical parameter is enumerated in exact cyclotomic arithmetic, and the generic complement is covered symbolically. The exclusion is author-recorded and bounded to those families and order ranges.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/distance_problems/dumitrescu_2006_distinct_distances_vertex_convex_polygon/_index|dumitrescu 2006 distinct distances vertex convex polygon]]: Theorem 2 statement (account reading).
• [[../library/distance_problems/kovacs_2024_note_erdos_s_mysterious_remark/_index|kovacs 2024 note erdos s mysterious remark]]: Abstract; Sections 1-2 (statement reading).
• [[../library/distance_problems/shinohara_2008_uniqueness_maximum_planar_five_distance_sets/index|Shinohara: Uniqueness of maximum planar five-distance sets]]: Section 1, g(2), g(3), Theorem 1.1(a) (statement reading). Section 1, g(2)=5 and similarity convention (statement reading). Section 2, Lemma 2.2(a)(ii), M(2k)(k) for k>=4 (statement reading).