Generic balanced cluster replacement increases rarity

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This excludes a class of counterexample constructions; it does not resolve either unrestricted assertion of Problem 132. All multiplicities count unordered pairs.

The replacement theorem

Let $P\subset\mathbb R^2$ have $N\ge2$ points. For each $p\in P$, fix a $b$-point set $S_p\subset\mathbb R^2$, where the same integer $b\ge2$ is used at every point. Put

$$A_t=\bigcup_{p\in P}(p+tS_p).$$

Outside a finite set of positive parameters $t$, the union has $n=Nb$ distinct points and

$$R(A_t)\ge R(P)+1. \tag{1}$$

If every $S_p$ is congruent to one set $S$, then

$$R(A_t)\ge R(P)+R(S). \tag{2}$$

In particular these conclusions hold for all sufficiently small positive $t$, with the threshold depending on the base and the fixed shapes. The shapes may have arbitrary different orientations; (1) does not require them to be congruent, centrally symmetric, or convex.

Proof

For a pair of different base points, write $v=q-p\ne0$ and $d=u-s$, where $u\in S_q$, $s\in S_p$. Its squared distance is the polynomial

$$F_{v,d}(t)=|v|^2+2t\,v\cdot d+t^2|d|^2. \tag{3}$$

Internal squared distances have the form $t^2|s-u|^2$. Exclude all positive roots of nonzero differences between these finitely many polynomials, and all parameters causing a collision of distinct labeled points. This excludes only finitely many parameters. At every remaining parameter, equality of distances is exactly polynomial identity. In particular, the constant term separates different base-distance groups, and separates every cross-cluster class from every internal class.

Fix a rare base distance $r$, with $m_P(r)\le N$. Its entire group contains exactly

$$b^2m_P(r)\le b^2N=bn \tag{4}$$

unordered cross-cluster pairs. For one base pair of this length, $|S_q-S_p|\ge2b-1$. To verify this elementary sumset bound, choose a linear functional injective on both sets, order their images, and use the increasing chain of $2b-1$ sums obtained by first varying one summand and then the other.

A fixed polynomial (3) prescribes both $v\cdot d$ and $|d|^2$. A line intersects a circle in at most two points, so this single base pair already produces at least $\lceil(2b-1)/2\rceil=b$ polynomial classes. By (4), not all classes in its base-distance group can have more than $n$ pairs. Thus each rare base distance supplies a distinct, globally rare distance of $A_t$.

Finally let $D_* =\max_p\operatorname{diam}(S_p)>0$. Only clusters attaining that diameter contribute internal pairs at distance $tD_*$. The planar diameter bound gives at most $b$ pairs in each such cluster, and hence at most $Nb=n$ in total. This class supplies the extra one in (1). If all shapes are congruent, each rare distance $r$ of $S$ instead supplies an internal class with exactly $N m_S(r)\le Nb$ pairs, proving (2). This also permits reflections of the shapes.

Consequence and exact limitation

A finite hierarchy assembled through balanced replacements at sufficiently small scales gains at least one rare distance at each nontrivial stage. This includes branch-dependent, nonstationary rhombus replacements, without a common iterated-similarity system. It extends the mechanisms excluded in full_similarity_orbits, but under a different, parameter-generic hypothesis.

To defeat this argument at a stage, a construction must choose an exceptional parameter: two nonidentical squared-distance polynomials must coincide, or labels must collide. Each such parameter satisfies a polynomial of degree at most two over the coordinate field.

The excluded set depends on the base and on all shapes. Its smallest positive element need not have a uniform lower bound as these vary. Consequently large diameter-to-minimum-distance ratio alone does not allow (1) to be applied to an arbitrary configuration. No reduction of all planar sets to these generic hierarchies has been established.

The later nonlinear collision theorem removes the generic-parameter assumption for analytic balanced doubling and tripling ($b=2,3$), even with varying branch shapes. The analogous statement for arbitrary fixed $b$ remains unproved.

The author checked the unordered-pair budget, the sumset and line--circle counts, and the treatment of exceptional parameters and congruent shapes; the argument is author-recorded. No computation or external structural theorem enters the proof beyond the classical planar diameter bound.