Unresolved steps and failed unrestricted approaches

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Neither assertion of Problem 132 has been proved or disproved. No counterexample with $n\ge5$ and $R\le1$, and no arbitrarily large bounded-$R$ family, has been obtained.

Latest continuation: finite orders, long gaps, and precise obstructions

The full primary six-distance classification of Szöllősi--Östergård (2020) was checked. Its three thirteen-point models all have at least two rare distances. Together with the unique-diameter theorem this extends the finite-order proof to orders 13 and 14. The established range is 5 through 14 except 11. The rotational thirteen-point report is unnecessary for this conclusion and remains unreviewed.

An angular matching inequality proves that, under $R(A)\le K$, every rich fixed largest rank $k$ satisfies $r_{k-1}-r_k=O_{K,k}(\Delta/n^2)$. In an unbounded $R=1$ sequence, all fixed top ranks therefore lie within $O_k(\Delta/n^2)$ of the diameter. For general bounded rarity, gaps with a rare lower endpoint are not controlled. The same note extends the odd-cycle contraction to cycles whose edges are merely at least the disk radius, and proves odd girth $\Omega_{K,k}(n^{1/8})$ for every fixed top-rank union. A rich bipartite second-distance example explains why even cycles do not give the same contraction.

The immediate-shell surplus lemma localizes the cycle surplus of an $r$-graph core to pairs in $(r,2r)$. It counts pairs rather than distinct lengths, so supplies no iteration forcing additional rare classes. The deletion-criticality examples show that one spanning bicyclic minimum graph can become rare after every deletion, even with nonstar diameter and a sublinear global palette. Joint richness of the other classes is still the missing input.

The highest invariant-valuation level always contains a rare class; this does not iterate or cover split valuations. The new exact six-point shell example further excludes a cactus-graph shortcut. The monic-quadratic cluster model and local patch field-alignment lemma retain concrete subproblems, not global reductions. General cluster sizes and near-linear many-anchor expansion remain unproved.

Targeted checks established the scopes above, but neither unrestricted question was resolved.

Previous continuation: stronger estimates, still no unrestricted resolution

The three-off-circle theorem proves $R(A)\ge2$ whenever $n-3$ points are concyclic. Its proof is finite: small differences give a regular polygon with at most three holes, two new palette-extension lemmas handle the extra hole patterns, and first and second moments handle an added center. This does not by itself settle arbitrary eleven- or thirteen-point configurations; the latter order is excluded by the separate classification above.

The refined short-incidence argument counts each induced motion with its actual anchor multiplicity. It gives

$$R(A)+1\gg\frac{n^{1/64}}{L^{7/8}(\log(2n))^{1/8}}$$

for $L$ convex covering subsets, with an explicit exception-point variant. In particular, divergence holds for $L=o(n^{1/56}/(\log n)^{1/7})$ and $o(n)$ uncovered points.

The arithmetic-palette obstruction uses a uniform lattice-point bound for ellipses. Under bounded rarity, covering the squared-distance palette by progressions of polynomial length requires at least $n^{1/10-o(1)}$ progressions. This is a restriction on the global palette, not a progression-cover theorem for arbitrary configurations.

The weighted-circle argument proves $R(A)\ge m/4$ when a circle contains $m\ge C(q(A)+1)^{7/8}$ points. It weights an anchor length by its number of anchor pairs, rather than discarding it after any outside coincidence. This removes the angular covering losses. The remaining $7/8$ threshold comes from the earlier pinned-spectrum occupied-radius step, not from the new weighted count. The resulting diameter-inactivity tradeoff is $R(A)+t(A)+1\ge c n^{1/24}$.

The moving-frame reduction now applies to every bounded-rarity sequence, not just $R=1$. Deleting a positive-sized convex layer from a rich fixed-short-rank graph leaves linearly many roots of noncollinear triangles. The bounded local palette supplies a single triangle congruence type at linearly many roots. Convex difference energy, the checked affine-group energy theorem, and the rich-motion second moment then give

$$\left|\bigcup_g g^{-1}C\right| \gg_K n^{17/16}/\sqrt{\log n}.$$

Every point of this union has its three anchor distances in the global palette, but the updated, checked Solymosi--Zahl three-anchor bound gives only $O_K(n^{12/7})$. No contradiction follows. The many two-dimensional anchor strengthening needed to close this exponent gap was not established. Long collinear anchor sets cannot suffice, as an exact quadratic-size example shows.

At a fixed smallest-distance rank $\ell$, the quantitative short-incidence estimate is $O_{K,\ell}(n^{55/56}(\log n)^{1/7})$. In the $R=1$ reduction this permits complete triangular patches of radius $c n^{1/112}(\log n)^{-1/14}$ at linearly many centers. Their orientations need not agree; growing patches alone do not provide the missing many-anchor estimate.

The same improved three-anchor input gives $O(n^{5/7})$ centers per minimum-lattice orientation, at least $\Omega(n^{2/7})$ orientations, and $\Omega(n^{9/14})$ exceptional interior vertices in a one-rare sequence. Any common finer triangular lattice must have superpolynomial refinement. These bounds remain compatible with the short-incidence upper bound.

The four-point collision theorem extends the balanced analytic rarity increment to $b=4$, without generic-scale exceptions. A three-value strict Monge matrix is excluded by nine positive integer defects telescoping to at most six; a reflection-pair analysis excludes nonflat cluster pairs. This does not prove the corresponding theorem for arbitrary $b$, nor reduce arbitrary point sets to collision families.

Finally, the mixed-valuation certificate rules out choosing maximal triadic valuation followed by maximal length inside a dyadic shell. Its unit graph is two equally oriented equilateral triangles joined by a bridge, with seven edges on six vertices. It obeys the proposed simultaneous three-part residue restriction and has no four-cycle. The other five lengths in the selected shell are rare, so the per-shell existence question remains open.

These arguments were checked at the scopes stated in their proof notes. No unrestricted proof or counterexample was produced for either target. Stronger noncollinear-anchor bounds and larger analytic clusters remain provisional research directions; no missing implication is being assumed.

The new finite theorem, quantitative incidence proof, and ellipse/palette argument were checked by the author and are author-recorded. The new three-anchor theorem was checked in its primary paper, including uniformity. The unsuccessful attempt to improve the $7/8$ circle threshold by ordinary spectrum second moments is recorded as an abstract set-system obstruction in weighted circle energy, not a Euclidean counterexample. No unrestricted proof results.

Preceding continuation: further restrictions, not a resolution

The diameter-triangle theorem settles the first assertion whenever the diameter graph contains a triangle. A Reuleaux triangle contraction controls inactive points, and an explicit lens calculation handles its boundary paths; the sole golden-ratio exception is resolved using the inactive set's diameter. The same contraction for odd diameter cycles, combined with the long-core surplus, proves rarity divergence for cycle length $o(n^{1/32})$. A cyclic $R=1$ family would need a cycle of at least $(1/5-o(1))n$ vertices.

The quantitative circle criterion gives $R(A)\ge m/4$ from $m\ge C(q(A)+1)^{31/32}$ concyclic points. Consequently $R(A)+t(A)+1\ge c n^{1/96}$, with $t(A)$ the number of diameter-inactive points. This still permits linearly many inactive points.

The translated-triangle bound combines convex difference energy with the checked Sharir--Solymosi three-anchor theorem. If $q(A)=O(n)$ and there is a linear-sized convex subset, any translated family of a fixed noncollinear triangle has only $O(n^{5/6})$ members. Thus a hypothetical $R=1$ sequence must have $\Omega(n^{1/6})$ minimum-lattice orientations, no macroscopic minimum-distance component, and a first nonlattice length. At least $\Omega(n^{7/12})$ minimum pairs touch its outer two layers. This does not contradict their $o(n)$ upper bound.

For arbitrary triangle orientations the same anchor theorem gives $|\bigcup_g g^{-1}C|=O((q(A)+1)^{11/6})$, but the needed lower bound for this union is missing. Convex difference multiplicity controls translations, not the rotational overlaps. No common orientation is inferred.

The new moment identity detects the earlier three-dimensional rank countermodels, but abstract one-rare shell data can satisfy it and the gap test simultaneously. A rich connected unit graph can have no nonzero equilibrium stress, as the explicit example in variational_reduction shows. The exact dyadic examples exclude both choosing either shell extremum and the proposed three-residue maximum lemma. They do not exclude a rare class somewhere in each level.

The separate thirteen-point rotational report is retained as unreviewed input, not included among the proved cases. No counterexample to either target was found, and neither unrestricted proof was completed.

Previous investigation: profiles and nonlinear collisions

The rich-degree profile lemma proves that, under $R\le K$, a positive fraction of points having the same degrees in every rich distance graph forces a positive circular fraction. Consequently every such profile fiber is $o(n)$ in a hypothetical bounded-rarity sequence. No large fiber or bounded equitable partition was derived from the global multiplicity assumptions; the attempted partial-isometry extension does not supply one.

The nonlinear collision theorem proves that two analytic clusters of size $b=2$ or $3$, collapsing to distinct points, have at least $b$ cross-distances for all sufficiently small positive parameters. Thus an arbitrary balanced analytic doubling or tripling of an $N$-point limit set $P$ has $R\ge R(P)+1$, even when distance coincidences persist identically throughout the family. This removes the generic-scale qualification for these two cluster sizes.

The general cluster-size statement is not established. Pointwise ordered rows cease to be constant when the cross palette has three or more values, and the proposed general distinct-entry bound for strictly Monge matrices remains unproved here. Exact counterexamples in the collision note and the order-type note exclude the weaker ordinal replacement, the distinct-row shortcut, and a minimum/maximum deletion induction, not the general collision statement. No reduction of arbitrary configurations to the proved collision families is known.

The checked claims, exact limitations, and unfinished $b=4$ status are documented. Neither target assertion is resolved.

Previous continuation

The generic cluster-replacement theorem proves $R(A_t)\ge R(P)+1$ for arbitrary fixed equal-cardinality branch shapes outside finitely many scale parameters. Congruent shapes give $R(A_t)\ge R(P)+R(S)$. This handles nonstationary fine-scale hierarchies, but the exceptional parameters depend on the shapes and base. An arbitrary configuration has not been reduced to a generic hierarchy.

The first-exceptional-length lemma propagates a complete radius-two triangular patch to every point within the first nonlattice length. Its edge lenses avoid the deep lattice centers. In a hypothetical unbounded $R=1$ family, that exceptional length cannot have bounded rank from either end of the spectrum. Independently, the fixed shells $2^j\sqrt7$ and the pair budget show $n/2-q(A)\to\infty$.

Neither empty lenses nor this palette deficit controls the middle-ranked exceptional graph. A proposed universal estimate $m_A(r_2)\le2q(A)+C$ would exclude unbounded $R=1$ families, but no such estimate is established; $C=2$ already fails on an exact ten-point two-ring example. Even that proposed estimate would not by itself answer the finite first assertion or the general bounded-$R$ question.

Finally the even-ring spectrum is computed exactly. Its entire sufficiently small neighborhood, not merely generic deformations, has at least $3n/4-2$ rare distances. Thus the no-five-concyclic second-distance extremizers in that note cannot serve as bounded-rarity examples. Remote portions of their framework variety and arbitrary long-distance supports are not controlled by this local calculation.

Scope of the retained results

The latest checked results are recorded below and in their linked proof notes. The circle-extension theorem is an exact finite result; the triangular-lattice spectrum and component conclusions assume an unbounded sequence with $R=1$. They are eventual assertions at each fixed normalized scale, not estimates uniform over scales growing with $n$, and not conclusions for every bounded $R>1$.

The unresolved step is to control the long distance spectra between the outer layers and the separate lattice components, or the error in the critical multiplicities $n+o(n)$. No argument forces a common lattice, a linear-sized component, or a second rare class from these reductions. Investigating the small global palette at that interface is only a provisional direction. The remote-anchor and component-joining examples refute only the indicated shortcuts, not the unrestricted conjectures.

Further consequences of the checked input

The all-but-two-circle theorem proves $R\ge2$ whenever any $n-2$ points are concyclic, without restricting the other two points. A small-difference lemma puts a hypothetical counterexample's circular subset in a regular polygon with at most two missing vertices; explicit palette tests exclude every extension. Together with the exact convex equality classification, it also handles even-order all-but-two convex sets. The arbitrary odd-order two-exception case is not proved.

The linear-aspect-ratio theorem strengthens the sparse-layer separation consequence: $\Delta/\delta=O(n)$ already forces rarity divergence. Thus a bounded-rarity sequence must satisfy $\Delta/(n\delta)\to\infty$. The proof uses perimeter, local packing, the short-edge inverse, and the circle theorem.

For a hypothetical unbounded $R=1$ family, the triangular-core reduction now determines the exact distance spectrum at every fixed scale relative to the minimum distance. Every entire minimum-distance component eventually lies in a triangular lattice, not merely its good-vertex subcomponents. Fixed lattice shells have the computed multiplicities, but the six-vector shells give only $n+o(n)$ pairs, whose error can still make them rich. Different components need not share a lattice. Their long cross-spectra and their spectra to the outer layers remain uncontrolled.

The proximal-anchor inverse also treats short edges near a convex subset. A grid-and-parabola example rules out extending its annular-occupancy step to arbitrary remote anchors using geometry alone; the example has a quadratic palette, so it does not rule out a stronger argument using few distances. The long-core surplus bounds give linear branching and larger-distance matchings, and force linearly many diameter pairs in an $R=1$ family. These restrictions still do not connect the outer long-distance support to the interior short-distance support.

Comparison with the second independent input

The review of the second independent record verified its $(2k-1)|B_k|$ large-distance bound, which improves the previous $2k|B_k|$ coefficient in the layer note. It does not remove the large-layer/interior-support gap. Its warning that rare distances need not survive restoration of omitted points remains valid; exact top-rank support, not monotonicity of rarity, is what licenses the layer argument.

Two gaps in that record are already closed by established arguments. The unique-diameter theorem excludes its remaining twelve-point spectrum and, more generally, every even $R=1$ counting-boundary spectrum. The short-edge inverse proves convex rarity divergence and its fixed-cover extension. This latter result does not assume the general near-minimum convex classification whose absence the input correctly notes. It uses bounded rarity to supply a rich fixed-short-rank class. On a convex subset of an arbitrary set, that class may instead be supported wholly in the complementary interior, so neither unrestricted assertion follows.

Comparison with the third independent input

The third independent report does not change the unrestricted status. Its checked elementary small-difference argument and explicit circular-majority estimate are recorded in regular cores. The existing positive-fraction circle theorem is asymptotically stronger; the new $6/7$ error exponent is not a circle-concentration threshold and does not improve the weighted theorem's $q^{7/8}$ hypothesis.

Its convex-asymptotic gap is internal to that report, not a current gap: the short-edge inverse already proves divergence for fixed convex covers with $o(n)$ exceptions. Likewise the unique-diameter theorem settles its twelve-point deletion obstruction, and the all-but-two-circle theorem settles a broader geometric family than its cocircular-hull argument. The actual remaining interior-support and global-palette gaps described above are unchanged.

The warning about restoring omitted points is valid; no arbitrary convex-subset rarity transfer is available. Its perturbation warning also remains valid for class-splitting deformations, but must not be read as excluding the proved equality-preserving affine criterion or the equality-preserving monotonicity in variational reduction. The report's unavailable companion files and unused Ramsey bound were not adopted as evidence merely because the archived summary calls them proved.

Short-edge incidence advance

The short-edge incidence inverse proves the asymptotic assertion for all strictly convex sets, and for a fixed convex cover with $o(n)$ exceptions. It requires a fixed smallest-distance rank with linearly many edges touching a convex subset. Narrow convex chains have bounded pinned occupancy in intervals of that length; deleting the overloaded bins leaves quadratically many anchored triangles of only linearly many types. Their congruences supply the rich-motion hypothesis that the older unlabeled energy argument lacked.

For arbitrary bounded-rarity sequences this says that every fixed short rank has $o(n)$ edges touching any convex subset, uniformly over that subset. It does not say that a large convex layer must meet a rich short-distance graph. Such graphs can still be supported in the interior. Deleting the layer preserves their low-rank richness, not bounded total rarity in the remaining set.

For $R=1$, this gives the stronger two-layer/triangular-core reduction: the first two layers have total size $(2/3+o(1))n$, later fixed layers have $o(n)$ points, and almost every point in the remaining $(1/3+o(1))n$-point set has six minimum-distance neighbors. Its regular-hexagon neighborhoods propagate a triangular lattice within each good-vertex component, but the components need not share a lattice. No incompatibility with the outer layers has been proved.

An attempted stability shortcut is false: the even-order analytic deformation attains exact $3N/2$ second-distance multiplicity while no five points are concyclic. It also preserves both longest cyclic chord families of the outer even polygon. The missing inverse must use the small global distance palette, not just extremal second-distance counting.

The two-cover diameter bounds also exclude some additional finite cases. They do not resolve two-cover graphs with few diameter edges and inactive points, much less arbitrary diameter graphs. The $n+4$ matching-two second-distance extremum is excluded, as is full second-distance activity at even order.

The new inverse and asymptotic reduction were checked by the author and are author-recorded. Neither unrestricted target is complete.

Finite orders and star results

The star-diameter theorem proves $R(A)\ge2$ whenever all diameter pairs share one endpoint. The generalized arc inequality shows that three diameter neighbors make the second distance rare; two diameter neighbors give an exact-minimum convex core after deleting the three endpoints. Thus an unrestricted counterexample needs diameter vertex-cover number at least two, not merely multiple diameter pairs. The refined bound for exactly two disjoint diameter pairs is $m_A(r_2)\le n+2+\min_v k_v\le n+4$, with notation and proof in that note; it does not settle that case.

The activity bounds provide a convex subset of size at least $n-4m_A(D)+1$ when the second distance is rich. This is not a constant-width or bounded-rank-error reduction. The attempt to prove the needed arbitrary-convex inverse through a side-rank inequality did not succeed. That inequality is proved for concyclic sets, while a proposed matching bound $k+O(1)$ is false even there.

The full fixed-similarity orbit theorem removes the formal-density assumption for full word image sets. Distinct-map entropy and a convex-sumset theorem handle generators without a common fixed point; angular-group counts and nested convex hulls handle the common-fixed-point case, including reflections. Arbitrary exact overlaps therefore do not supply a bounded-rarity family through full orbits of fixed similarities. Sparse selections and nonstationary maps are outside this conclusion.

The star theorem and the other documented arguments were checked by the author for their stated scopes and are author-recorded. Neither unrestricted question has been answered.

Results from the preceding attempt

The unique-diameter theorem proves $R(A)\ge2$ whenever the diameter is attained once. In the extremal second-distance case, every point is active for that distance; deleting both diameter endpoints leaves a strictly convex set, because each remaining point has a supporting disk containing the remainder. Under $R=1$, its distance count attains the convex minimum. Altman--Fishburn classification and a two-point extension argument give a contradiction. This also eliminates every even counting-boundary spectrum and settles order twelve. A counterexample must now have at least two diameter pairs, and at even order $q\le n/2-1$.

The quantitative inverse lemma gives a circle of size $\Omega(m/(L+1)^3)$ in the existing constant-width setting. It uses repeated adjacent-side patterns, two-anchor uniqueness on a specified half-plane, and an adjusted index shift. It does not produce the bounded rank omissions needed for an arbitrary layer.

The similarity-word theorem excludes fixed contractive-similarity constructions whose selected distinct images have positive density in formal word space. Its measure argument forces every fixed convex layer to be negligible. Vanishing-density word selections and nonstationary maps are outside its assumptions.

The unique-diameter proof, the quantitative inverse, and the similarity-measure proofs were checked by the author against their full derivations and are author-recorded. These are proved restricted results; neither unrestricted target is complete.

The latest new approaches

The spectral-gap lemma is an unrestricted positive result: any occurring $r$ with no distance in $(r,2r)$ is rare, even when $2r$ occurs. Thus each consecutive ratio at least two supplies a rare class distinct from the diameter. The factor is sharp, and no argument forces such gaps in a bounded-rarity sequence.

Angular interpolation gives exact vanishing moments and one nonzero terminal moment for complex barycentric edge charges. An explicit rich unit class on six points makes both its holomorphic real-part charge and its Hermitian charge take either sign with unbounded magnitude. This rules out the proposed richness-implies-fixed-sign step, not every possible use of the identities under additional joint hypotheses.

The weighted-diameter argument proves the sharp nonnegative quadratic vertex-weight bound, but that bound already holds for all pseudoforests. More strongly, every linear forest is individually a planar diameter graph. Since every induced highest-color graph in the existing abstract obstruction is a linear forest, no collection of tests of those graphs in isolation can exclude it. The issue is their simultaneous realization by one planar point set.

The order/sweep note gives an all-rare affine image with the same sweep sequence, a finite-alphabet strict Monge block violating planar squared-distance rank, and a necessary linear surplus of isosceles configurations for $R=1$. These establish precise limits of the scalar order relaxations. The surplus was not shown to concentrate on a line or circle.

Both target questions remain unresolved.

Subsequent checked additions

The monotone-chain inverse theorem proves that a strictly convex distance-monotone $m$-point chain with $q\le m-1+L$, for fixed $L$, has a positive circular fraction. Thus a chain of at least $n/2-O(1)$ points forces $R(A)\to\infty$. A general convex layer need not supply a chain that large; a maximal diameter side does not even imply distance monotonicity, as the exact quadrilateral in that note shows. The three-cap decomposition alone leaves linear rank excess.

For the first assertion, all but one point in convex position is sufficient. The unique-diameter equality case is settled by deleting both endpoints, without bounding either circular fan. This closes that specific gap, but not the multiple-diameter or asymptotic gaps.

The sumset note handles selected subsets of bounded-factor digit sums with formal size $N$, giving growth when $|A|/N^{2/3}\to\infty$, and $\Omega(|A|^{1/6})$ for fixed positive density. This closes the earlier deletion gap only in that regime. The dyadic note establishes a rare class at the lowest norm level for split residue field $\mathbb F_2$, and a characteristic-two good-reduction criterion. Neither gives an iteration or forces a second class different from the ordinary diameter.

These are checked restricted results, not answers to either target.

The checked results in this wiki do not cover a large arbitrary convex layer. The layer reduction shows that every hypothetical bounded-$R$ sequence must have such an outer-layer concentration; it does not show that the large layer is circular or arithmetically suitable for the valuation theorem.

What bounded rarity implies by counting

If $q$ is the number of distinct distances and $b=R(A)$, then

$$\binom n2\ge(q-b)(n+1)+b=q(n+1)-bn,$$

and therefore

$$q\le\frac{n(n-1)/2+bn}{n+1} =\frac n2+b-1+\frac{1-b}{n+1}. \tag{1}$$

For a convex set, Altman's lower bound is $q\ge\lfloor n/2\rfloor$. Thus bounded $R$ forces a bounded additive excess over the minimum number of distances. The exact-minimum configurations are classified in the checked supplied sources, but the needed general near-minimum stability theorem is not available there. Fishburn's source explicitly conjectures only unboundedness of its rigidity allowance $f(n)$, not $f(n)\to\infty$.

Also, for a convex configuration with $b\le B$,

$$\sum_{m_A(r)>n}(m_A(r)-n) =\binom n2-nq+\sum_{m_A(r)\le n}(n-m_A(r))\le Bn.$$

Small total edge excess does not force distance graphs to be close to 2-regular: many isolated vertices can be balanced by a denser component. Nor does $m>n$ imply planar bar-framework rigidity, whose generic edge threshold is $2n-3$. No low-rank distance-matrix argument converting these counts into a large circular subset has been proved.

A regular core would be enough

Here is a proved completion lemma, retaining one concrete possible route beyond exact-minimum classifications. Let $V$ be a regular $m$-gon, with center $o$, and put $\ell=\lfloor m/2\rfloor$. If

$$|D(V\cup\{p\})|\le\ell+L,\qquad \ell>5L+2,$$

then $p\in V\cup\{o\}$.

Normalize the circumradius to one and suppose $p$ is neither a vertex nor the center. If $p$ lies off every reflection axis, its $m$ pinned distances differ, contradicting $m>\ell+L$. On an axis, write $p=re^{i\phi}$, $r>0$, and $h=2\pi/m$. The increasing squared chord list is $x_i=2-2\cos(ih)$, $1\le i\le\ell$. The distinct increasing pinned list has the form

$$y_j=r^2+1-2r\cos(\alpha+jh),$$

with consecutive angles in $[0,\pi]$ and length $k\in\{\ell,\ell+1\}$. At most $L$ pinned entries are absent from the chord list. If $s$ is the number of matches, then $s\ge k-L\ge\ell-L$, and the two lists have at most $\ell+k-2s\le2L$ unmatched entries altogether. The ordered matches therefore break into at most $2L+1$ runs consecutive in both lists. Since $s>4L+2$, one run has three matches.

Writing $c=\cos h$, the two lists satisfy

$$x_{i-1}+x_{i+1}=2cx_i+4(1-c),$$ $$y_{j-1}+y_{j+1}=2cy_j+2(r^2+1)(1-c).$$

Three consecutive matches force $r^2=1$. But a new circle point on a reflection axis has half-integer angular steps, whereas the old chord list has integer steps. Their folded angles, hence squared lengths, are disjoint. This contradicts $L<\ell$ and proves the lemma.

More generally, suppose $A$ contains $V\setminus S$ with $s=|S|<m/2$. All chord lengths of $V$ survive: deletion removes at most $2s<m$ edges of a nondiametral class, and at most $s<m/2$ diameters. If $q=|D(A)|$, restoring the missing vertices creates at most $s$ new pinned distances from each $p\in A$. Hence

$$\ell>5(q-\ell+s)+2 \quad\Longrightarrow\quad A\subseteq V\cup\{o\}. \tag{2}$$

In particular an almost-complete regular core would suffice in the regime $q=n/2+o(n)$, $m=n+o(n)$, $s=o(n)$. No argument obtaining such a core from bounded rarity has been found.

Stronger core results

The regular-core theorem removes the need for an almost-complete core: a regular polygon containing a fixed positive fraction of all points already forces linearly many rare distances. The same holds for a fixed-density subset of a regular polygon whose order is comparable to the subset size.

Kneser's addition theorem also gives a concrete geometric consequence. For fixed $\varepsilon>0$, if $m\ge(1/2+\varepsilon)n$ points of $A$ are concyclic, then, for sufficiently large $n$,

$$R(A)\ge(2m-n)/6.$$

The proof uses bounded overlap of distinct cosine-grid spectra to bound the number of occupied dihedral orbits, then counts chord classes whose entire multiplicity stays on one circle. This improves the available completion mechanism but does not prove that an arbitrary bounded-$R$ configuration has a regular core or a circular majority.

Arbitrary circular subsets

The subsequent cosine-incidence theorem removes both those hypotheses. For any fixed $c>0$, an $m$-point circular subset with $m\ge cn$ gives

$$R(A)\ge(m-1)/2-O_c(n^{2/3}).$$

Under $q(A)\le n$, the stronger estimate is $R(A)\ge q(A)-O_c(n^{2/3})$. The proof uses an incidence bound for translated cosine curves and Ruzsa covering, not regular-polygon rigidity.

The later hyperbola-incidence argument proves the same qualitative conclusion for a positive collinear fraction. Together with the Pach--de Zeeuw curve theorem, it implies that every fixed-degree algebraic curve contains only $o(n)$ points in a hypothetical bounded-rarity sequence. The degree is fixed, but the curve can vary. The layer argument still forces a linear number of points into finitely many outer convex layers. Ruling out this combination remains the unrestricted gap.

An attempted connection with distinct-angle expansion did not close it. It would need an independent $O_K(n)$ angle bound on a linear-sized convex layer under $R(A)\le K$. The known pair count only gives $q(A)\le n/2+K-1$; the direct side-length description gives at most $O(q^3)$ possible angles, not the required linear bound. No stronger implication was established. This is a provisional possibility, not a proved reduction.

A restricted constant-width inverse theorem

The constant-width boundary theorem places all diameter-active points on one boundary whose subdiametral distance graphs have maximum degree two. The new inverse theorem proves that a subset $C$ of such a boundary with $q(C)\le |C|/2+L$, for fixed $L$, has a positive fraction on one circle. Its proof uses bounded omissions in ordered pinned-distance lists, finite van der Waerden, and propagation of one rotation through three anchors.

This proves rarity growth when at most a fixed number of points are diameter-inactive. A merely positive-fraction boundary anchor does not meet its hypotheses: if $|C|=cn$, the pair budget permits $q(C)-|C|/2$ to be linear in $n$. Equivalently, with $d_r=|C|-m_C(r)$ and with all global distance values included,

$$\sum_r d_r=|C|q(A)-\binom{|C|}{2}$$

can be $c(1-c)n^2/2+O(n)$. The per-row omission count therefore need not be bounded. No argument has bounded the number of diameter-inactive points in a hypothetical bounded-rarity sequence.

For shortest-distance graphs, the linear convex local-spectrum and cross-matching bounds are proved. They still do not control numerous isolated or widely separated components. A near-spanning boundary-chain conclusion, bounded branching, and an inference to two uniform long-diagonal families remain unproved. The relevant fixed-rank literature was identified, but no unread theorem from it is used to assert these missing conclusions.

A remaining arithmetic question

In an invariant dyadic valuation with nontrivial residue conjugation, it is unproved here that every occurring valuation level contains some rare distance. This would imply growth for a fixed finite residue field. Taking the maximum distance of the level is false, as the exact six-point certificate in the obstruction note shows. Finite checks of one grid domain do not establish the weaker per-level statement.

The invariance assumption matters: a split-prime construction now disproves per-norm-level rarity without it, even for rational squared distances. This does not decide the invariant-prime question.

There is one limited positive observation. If a level contains only one Euclidean distance $D$, that class is rare. Its blocks are complete multipartite graphs with every cross-part distance equal to $D$. Two parts can form only a star or $K_{2,2}$: two circle centers have at most two common neighbors. With three parts, each part has size at most two, and $K_{2,2,1}$ is impossible. Indeed, taking its singleton as center puts the other four points on a radius-$D$ circle, on which two distinct points have at most one common chord-$D$ neighbor. Four parts would give four equidistant planar points. The only remaining graph with more edges than vertices is $K_{2,1,1}$, the two equilateral triangles sharing an edge. Its same-part tips have squared distance $3D^2$, at the same dyadic norm valuation as $D^2$, contradicting their membership in one child. Thus each block has at most its vertex count in edges. This does not handle levels with two or more rich classes.

Scalar radial Fourier positivity cannot suffice

For an actual configuration, angular averaging of the squared Fourier sum gives

$$1+\frac2n\sum_r m_A(r)J_0(rt)\ge0\qquad(t\in\mathbb R).$$

Even this condition together with the exact integer pair budget and the diameter bound does not force a second rare class. For every sufficiently large odd $n$, set $k=(n-3)/2$ and prescribe

$$m_1=\cdots=m_k=n+1,\qquad m_{k+1}=(n+3)/2.$$

Their sum is $\binom n2$, and only the last is at most $n$. There are increasing positive radii $r_1<\cdots<r_{k+1}$ making the corresponding Bessel sum strictly positive everywhere.

To prove this, first note that $1+2J_0(t)>0$ for every $t$. It is one third of the angular average of the squared exponential sum on a unit equilateral triangle. In the direction perpendicular to one edge, two phases coincide, so the modulus of that sum is at least one; continuity makes its average positive. Since $J_0(t)\to0$ at infinity, there is a positive global lower bound $\delta$. The limit follows, for example, by writing $J_0$ as the Fourier transform of the integrable density $1/(\pi\sqrt{1-u^2})$ on $(-1,1)$. Thus $1+(2+2/n)J_0(t)>0$ uniformly when $2/n<\delta$.

If $f(t)=1+\sum a_iJ_0(r_it)$ has positive infimum $d$, and $0<b<f(0)$, add $bJ_0(Rt)$ with $R$ sufficiently large. On a fixed small interval about zero, $f(t)>b$ and $J_0\ge-1$ give a positive bound. Outside that interval, choose $R$ so large that $b|J_0(Rt)|<d/2$. The sum remains uniformly positive. Start with $f(t)=1+(2+2/n)J_0(t)$, whose positive infimum was just proved, and then add the remaining prescribed coefficients, finally $1+3/n$, while increasing the radii.

These shell data are not planar realizations and may violate even metric triangle constraints. They only disprove a positivity-only relaxation as a proof route.

Status of the work

The subsequent unrestricted attempt

The renewed attempt produced neither a complete proof nor a counterexample. The following are the precise outcomes, not additional assumptions that can be used to close the argument.

The motion-incidence note checks an applicable inverse theorem: linearly many direct motions with linear overlap on a strictly convex set force a positive circular fraction. Its additive alternative is excluded by $E_+(C)\le3|C|^2-2|C|$. But the available bound $q(C)=O(|C|)$ supplies only a cubic second moment of motion overlaps, not those linearly rich motions or a quartic triangle moment. No implication filling that gap was proved. Investigating such a promotion is a provisional direction, not an established final lemma.

The local inheritance estimate after deleting a convex subset of size $cn$ has error $O_c(\lambda s)$ below cutoff $\lambda r_s$. It does not bound the residual set's total rarity, so no induction through convex layers follows. Likewise, the attempted turning-word argument established neither bounded word complexity from short chord lengths nor the necessary long boundary chain.

The matrix certificates show that uncentered squared-distance rank four, conditional negative definiteness, and numerical class restrictions still allow three-dimensional $R=1$ sets. One satisfies the exact even-boundary spectrum; another satisfies all hereditary diameter graph restrictions. They are not planar examples. The centered-rank-two condition remains essential and has yielded no new multiplicity inequality.

The five-point dyadic obstruction rules out a universal arithmetic shortcut: its equality pattern has no injective realization admitting a conjugation-invariant dyadic valuation. Thus a change of algebraic realization cannot bypass the split-prime case. The nontrivial-residue per-level question remains unanswered, and even an affirmative answer to it would not complete the general reduction.

The construction search found no viable unrestricted family. The two-square extension certificate rules out adding at most three points to that eight-point $R=2$ model: its smaller diameter stays rare and remains below the full diameter. A second finite extension test yielded no counterexample and is not used as an exhaustive classification. A proposed general stability statement for tiny clusters around regular polygons was not checked to completion and is not incorporated as a theorem. There is no claimed perturbation transfer or construction beyond the proved scopes in the existing notes. No new exhaustive grid search was run.

The rational and ramified-characteristic-two bounds, line-cover bound, sparse-layer theorem, one-off-circle theorem, and fixed-ring construction obstructions are proved partial results. They do not combine into a proof for arbitrary planar sets. The missing structural or arithmetic step is substantive, not a matter of checking a finite configuration.