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Exact finite claim

For a finite point set $P$ in strict convex position, define

$$\mu_P(v)=\max_{d>0}\#\{w\in P\setminus\{v\}:\|w-v\|^2=d\}.$$

Strict convex position means that every point is an extreme vertex, with no redundant collinear boundary points. Property $E_k$ means $\mu_P(v)\geq k$ for every vertex; the repeated distance can depend on the vertex. The following fixed set has nine distinct extreme vertices and $\mu_P(v)=3$ for all nine. It is not a counterexample to Problem 97, which asks about four equidistant neighbors.

Put $s=\sqrt3>0$, $u=\sqrt{5s-8}>0$, and

$$R=\begin{pmatrix}-1/2&-s/2\\s/2&-1/2\end{pmatrix},\quad A_i=R^{i-1}(1,0),\quad B_i=R^{i-1}(s-1/2,s/2),\quad C_i=R^{i-1}(x,y),$$

for $i=1,2,3$, where the chosen branch is

$$x=\frac{8s-11+(s+6)u}{10},\qquad y=\frac{12-s+(3-2s)u}{10}.$$

The positive radicals are well defined: $s>8/5$ follows from $3>64/25$, so $5s-8>0$. Coordinate denominators are products of the nonzero rationals $2$ and $10$. The identifying rational box is $91/100<x<92/100$, $98/100<y<1$, certified by the exact evidence below. The six A/B coordinates are exactly those in sallerk's post 8669; the explicit third-orbit radical and derivation here are supplied by this compilation, not quoted from the post or attributed to Danzer.

Derivation of the chosen completion

The matrix $R$ is orthogonal, has determinant one and satisfies $R^3=I$. For any point $v$, $\|v-Rv\|^2=3\|v\|^2$. The first printed relation is a separate essential obligation, not a consequence assumed from the two completion equations:

$$\|A_1-A_2\|^2=\|A_1-A_3\|^2=\|A_1-B_3\|^2=3.$$

Direct substitution of the six seeds gives this identity. Also $\|B_1\|^2=4-s$, hence the B-orbit side squares are $12-3s$. Writing $q=x^2+y^2$, the C/A condition expands as

$$\|C_1-A_3\|^2=q+x+sy+1=\|C_1-C_2\|^2=3q, \qquad 2q=x+sy+1. \tag{1}$$

Expanding the B/C condition gives

$$\|B_1-C_2\|^2=q+4-s+(s-2)x+3y=12-3s,$$

or $q+(s-2)x+3y+2s-8=0$. Eliminating $q$ using (1) gives

$$(2s-3)x+(s+6)y+4s-15=0. \tag{2}$$

Conversely, (1) and (2) recover both distance conditions by these equalities; only division by the nonzero rational $2$ is used.

Equation (1) is the circle with center $h=(1/4,s/4)$ and squared radius $3/4$. Write $a=2s-3$, $b=s+6$. Then $a^2+b^2=60$ and $(a,b)\cdot h=2s$. The perpendicular foot from $h$ to the line (2) is

$$H=h+\frac{15-6s}{60}(a,b) =\left(\frac{8s-11}{10},\frac{12-s}{10}\right).$$

Points $H+t(b,-a)$ lie on (2). Their circle equation reduces to

$$60t^2=\frac34-\frac{(15-6s)^2}{60},\qquad 100t^2=5s-8.$$

Choosing $t=u/10>0$ gives exactly the displayed $(x,y)$. This constructs one real completion and checks its defining equations; no assertion about all other completions or their convexity is needed here.

Finite convexity and distance certificate

Use the counterclockwise order

$$A_1,B_1,C_1,A_2,B_2,C_2,A_3,B_3,C_3.$$

The full evidence check computes every unordered squared distance and certifies all 36 are positive. For every directed edge in this order, it certifies the determinant with each of the other seven vertices is strictly positive: all 63 supporting-edge signs, not merely nine consecutive turns. Thus each proposed edge is an exposed edge of the convex hull and every listed vertex is extreme. Strict support excludes redundant collinear boundary points.

Each of the nine rows contains eight distances. All $9\binom82=252$ pairwise row comparisons are certified: a reduced zero expression establishes equality, and a rational interval strictly on one side of zero establishes inequality. The author's named row checks assert maximum distance multiplicity three at every vertex. The complete partitions printed in the author output have one triple and five singleton classes in each row. The table summarizes those partitions (indices are read modulo three). The independent review separately confirms the table and the full profile, which its retained quartic checker asserts as a named obligation for each row. The author checker itself still names only the row maximum.

The checker enumerates all nine rows, even though rotation accounts for this three-row description. It separately names the A/B first, B/C second and C/A third Er87b relations. The named row-maximum checks establish that the maximum in every row is exactly three. The argument does not infer absence of fourth neighbors merely from the advertised triples.

Source scope and separate reports

Er87b, printed pp. 175–176 (physical PDF pp. 9–10), Fig. 5, was visually read in full. Its three relations and threefold symmetry motivate this example. It prints no numerical coordinates. Its existence construction chooses B near A using Reuleaux-triangle arcs and then chooses C by an intermediate-value argument. This page establishes a nonagon realizing the printed relations, not that these are Danzer's original coordinates or that every condition of that printed construction is reconstructed. See the Er87b card.

The following reports remain outside the finite claim and its verification:

• Completion uniqueness in the normalized labeled family and nonconvexity of the other branch are review-side reports. They still need all real completions and a genuine alternate-hull or nonextremality certificate; a negative turn in one proposed ordering is insufficient.
• The forum reports degree four for the third orbit. No minimal polynomial or degree assertion is accepted here. The reported candidate polynomial for $y$ is $1600y^4-7680y^3+24864y^2-33696y+14904$. A separate degree proof still needs a discriminant non-square argument over $\mathbb Q(\sqrt3)$, a recovery identity for $\sqrt3$ from $y$ with nonzero denominator, and justification of the exact scalar or field degree claimed. None is used by the equality reduction or interval certificates above.
• The forum's mirror exclusion assumes $D_m$ symmetry, $m\geq2$, and every vertex on a reflection axis. That last condition is necessary for the supplied reduction to $n=m$ or $2m$; no theorem for arbitrary dihedral polygons is reconstructed. Its monotonicity must concern mirror-paired vertices, not all cyclic distances, and $m=2$ needs separate treatment from the formula dividing by $\cos(\pi/m)$. No mirror hypothesis is used in the present finite witness.
• With $n_3$ the minimum cardinality of a strictly convex $E_3$ set, the forum's $n_3\geq7$ remains author-asserted. Its decisive six-point exclusion is external and uninspected; review-side four- and five-point discussions do not fill that gap. The stated set $\{7,8,9\}$ is conditional on that lower bound, and the post leaves $n=7$ unsettled. The nonconvex six-point Erdős–Fishburn example is a separate unverified source lead, not a convex witness.

Current verification record

The frozen exact finite subject received a refutation-failed verdict from the independent reviewer. A grader distinct from author and reviewer assessed the report contract and independence; a further completed-record grading recorded pass with the stated corrections. The retained review and grading identify those roles, the exact subject, their reading limits and the reported runs. They cover these nine points, strict convexity, the printed relations, and maximum distance multiplicity exactly three at every vertex, including one triple and five singletons in each row. No numerical tier is assigned.

The reviewed result retains the exact bytes assessed before the documentary row-profile and standing corrections. This page is the filed page with those corrections, not an unchanged filing of that snapshot. The mathematical input remains unchanged; the author checker was edited after the review only to drop its input-identity refusal, as the evidence account states. The reviewed checker's local normal and optimized Python 3.12.13 replays each passed 141 named checks; after that edit it passes 140. The independent checker uses a single quartic generator and bisection signs; its historical code and runs are distinguished from the current shared-harness adaptation in the [[distance_problems/sallerk_2026_convex_nonagon_relations/evidence/_index|evidence account]]. The original independent program and current adapter were each locally replayed in both modes, passing 21 obligations and 21 harness checks respectively. These integration replays are separate from the independent mathematical review; the old verdict does not certify later interface edits. Subsequently, nine specified interface controls in both modes produced all 18 required refusals. The evidence account records their narrow coverage; they do not supply a new independent whole-mathematics verdict or tier.

No external code, hidden six-point result or minimal-polynomial claim is a premise. This is not an E0097 resolution, a uniqueness or alternate-branch result, a mirror theorem, a lower bound on minimum size, or a reconstruction of Er87b's printed existence proof. Those separate obligations remain above.

Bears on. Problem 97: a selected exact $E_3$ witness only; no $E_4$ counterexample, lower-bound or status change.