Planar distance fibers and a unit-distance power saving

Source and verification

This article explains family 167 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Distance fibers and pinned distances

Let $P\subset {\mathbb{R}}^2$ have $n$ points. For distinct $x,y\in P$, define the multiplicity of the distance from $x$ to $y$ by

$$\begin{array}{c}{k}_P(x,y)\\ =|\{z\in P:\Vert z-x\Vert =\Vert y-x\Vert \}|.\end{array}$$

The point $y$ itself is counted. The pinned distance set at $x$ is

$$D_x=\{\Vert y-x\Vert :y\in P\setminus \{x\}\}.$$

The first manuscript controls large multiplicities uniformly over all configurations. For fixed $s>0$, let $F_n(s)$ be the supremum, over all $n$-point sets, of the fraction of ordered distinct pairs $(x,y)$ with $k_P(x,y)\ge n^s$.

First manuscript claim

For every fixed $s>0$, $F_n(s)\to 0$ as $n\to \infty$. Consequently, for every fixed $\epsilon >0$, all but $o(n)$ pins determine at least $n^{1-\epsilon }$ distances, uniformly over planar $n$-point sets. (OpenAI, 2026)

The theorem does not give a rate for the $o(n)$ term. It also permits exceptional pins. For instance, if a configuration consists of a center and many points on a circle around it, the center determines only one distance.

🏷️ Deriving the almost-all pinned conclusion

Suppose a pin $x$ has fewer than $n^{1-\epsilon }$ distinct distances. Its distance classes of size less than $n^{\epsilon /2}$ together contain fewer than

$$n^{1-\epsilon }{n}^{\epsilon /2}=n^{1-\epsilon /2}$$

neighbors. All its remaining $n-1-n^{1-\epsilon /2}$ neighbors belong to classes of multiplicity at least $n^{\epsilon /2}$.

If there were a positive fraction of such bad pins, a positive fraction of all ordered pairs would therefore have multiplicity at least $n^{\epsilon /2}$. That contradicts $F_n(\epsilon /2)\to 0$. This is an elementary deduction from the fiber theorem; the uniform fiber estimate is the difficult part.

🏷️ A product identity and multiscale cancellation

Write a point as $(u,v)$ and introduce complex coordinates $z_1=u+iv$, $z_2=u-iv$. Squared distance factors as

$$\begin{array}{rl}\Vert x-y{\Vert }^2& =(z_1(x)-z_1(y))\\ & \cdot (z_2(x)-z_2(y)).\end{array}$$

The proof first replaces a hypothetical bad configuration by algebraic coordinate data preserving the relevant distance equalities and inequalities. It can then use the product identity at several embeddings and absolute values of the coordinate field. At nonphysical embeddings the two coordinates need not be complex conjugates; treating them as conjugates everywhere would invalidate the comparison.

At each place, nested partitions record which points are close at each scale. Random shifted grids handle complex coordinates. Large cells are treated together with their complements to keep the comparison balanced. Integrated overlap functions for distinct pairs decompose into one-point terms plus the information that measures genuine clustering.

The source compares three pair laws: two samples from a distance fiber, a fiber sample paired with an ambient sample, and two ambient samples. The combination with coefficients $1,-2,1$ has zero one-point marginals. Thus the additive one-point terms cancel, leaving a variance expression with controlled finite-population corrections.

This cancellation requires distinct-pair conventions. Diagonal singleton terms can accumulate over infinitely many scales, so casually replacing sampling without replacement by independent sampling is not harmless.

The proof divides according to scale spread. Unbounded spread produces limiting trees whose transport variances are incompatible with the distance constraints. Bounded spread produces nonatomic coordinate limits, where the transformations induced by a fixed distance fiber force a collapse inconsistent with positive spread. Both branches use the same cancellation identity to connect local fibers to global geometry.

🏷️ A separate unit-distance power saving

Let $u(P)$ be the number of unordered pairs of points of $P$ at distance one.

Companion manuscript claim

There are absolute constants $C>0$ and $1\le \beta <4/3$ such that every planar $n$-point set satisfies $u(P)\le C{n}^{\beta }$. (OpenAI, 2026a)

This is a fixed power improvement over the exponent $4/3$. It does not claim $u(P)=n^{1+o(1)}$, and the statement does not specify a numerical value of $4/3-\beta$.

The proof assumes configurations at scale $n^{4/3-o(1)}$ and studies their point-circle incidence graph. After regularizing degrees and expansion, it samples a center and two conditional neighbors. The relevant marginal is degree-biased: it is not the uniform law on all points. A low-information prediction argument must preserve that marginal and use a fresh independent test after the prediction is chosen.

The complex product identity again supplies a multiscale profile. If its variation remains bounded, normalization at a suitable embedding gives a predicted point approximately at unit distance from three separated neighbors, contradicting a determinant constraint. If variation is unbounded, the proof extracts rare configurations with a large height ratio but a small rectangle-product height. An algebraic limiting argument then rules them out.

The rare branch needs height-weighted error bounds. A set with tiny probability can still dominate an expectation if its height is enormous. The source therefore controls the contribution of discarded events, not just their probability, and separately removes popular-circle concentrations.

The two manuscripts share multiscale product-formula ideas, but neither main statement is an immediate corollary of the other: rarity of large distance fibers gives no fixed power rate by itself.

🏷️ Key steps and proof outlines

An exact finite-population identity supplies the variance

Let $\alpha$ be the uniform law on the $n$ ambient points, and let $\nu$ be uniform on a fiber subset $B$ of size $k>1$. For a cell $C$, write $a=|C|/n$ and $p=|B\cap C|/k$. The “Variance” section compares three laws on distinct ordered pairs: $P_B$ samples two distinct points from $B$, $\pi$ samples two distinct ambient points, and $R_B$ samples the first point from $B$ and gives the second exactly marginal $\alpha$. (OpenAI, 2026b)

To construct $R_B$, after choosing $X\in B$, choose $Y\in B\setminus \{X\}$ with probability $k/n$ and otherwise choose $Y$ outside $B$. For a fixed $y\in B$, the resulting mass is $(k-1)(1/k)(k/n)(1/(k-1))=1/n$; the outside points also have mass $1/n$. When $B$ is the whole set, use $R_B=P_B=\pi$ directly.

The cell probabilities are exactly

$$\begin{array}{rl}{P}_B(C^2)& =p^2-\frac{p(1-p)}{k-1},\\ R_B(C^2)& =ap-\frac{k}{n}\frac{p(1-p)}{k-1},\\ \pi (C^2)& =a^2-\frac{a(1-a)}{n-1}.\end{array}$$

Consequently their signed combination is

$$(p-a{)}^2+\left(\frac{2k}{n}-1\right)\frac{p(1-p)}{k-1}-\frac{a(1-a)}{n-1}.$$

The two marginals of $P_B-2R_B+\pi$ are $\alpha -\nu$ and $\nu -\alpha$; their sum vanishes. The source’s additive overlap identity therefore makes its integrated contribution zero. It is this exact cancellation that leaves a nonnegative variance controlled by finite-population errors.

Separate large cells, small cells, and singleton tails

For fibers of size $k\ge n^{\gamma }$, average with fixed nonnegative family weights, independent of the place and grid, of total mass at most one, whose mean fiber law is dominated by $M\alpha$. On a large cell $|C|>n^{1-c}$, with $c<\gamma$, the correction terms are controlled by $O_M(n^{c-\gamma })\pi (C^2)$. This follows from $a>n^{-c}$ and $\pi (C^2)\ge a^2/2$ for cells of size at least two.

On smaller cells, the useful term is $p^2$ only when $|B\cap C|\ge 2$. Indeed, then $P_B(C^2)\ge p^2/2$. A singleton contributes zero to every distinct-pair probability, exactly. The proof never integrates the separate correction terms over the infinite range of singleton scales. Combining the regions bounds the integrated discrepancy by

$$O_M(n^{c-\gamma }{W}_2+W_{2,\mathrm{sm}})=o(W+1),$$

where $W_2$ is the ambient distinct-pair overlap integral, $W_{2,\mathrm{sm}}$ its small-cell part, and $W$ the source’s total scale spread. Earlier small-cell bounds supply the final little-oh estimate. This comparison applies to outgoing fibers and to incoming conditional source laws; controlling both is needed in the subsequent bounded-spread and unbounded-spread contradictions.

The unit-distance companion predicts against a fresh test

Its prediction lemma starts from a point-center law $\omega (p,q)$. Draw a center and then two conditionally independent neighboring points, giving point-pair law $\pi$. Let $\mu$ be their common, generally degree-biased marginal and let

$$D=D_{\mathrm{KL}}(\pi \Vert \mu \otimes \mu )$$

measure their dependence. Tests encode points into $M$ symbols. For $K$ independent tests, the entropy chain rule bounds the sum of the conditional next-code relative entropies by $D$. Averaging over a uniformly selected step and applying Pinsker gives average conditional total variation at most $\sqrt{D/(2K)}$. (OpenAI, 2026a)

For a prefix history $h$, predict a feature by its mean under the actual posterior ${\mu }_h$. Subtract each endpoint’s own posterior mean. Under the reference conditional product law the two residuals are centered and independent, so their inner product has mean zero. Under the actual wedge law, that same expected inner product is

$${\mathbb{E}}_q{\Vert \mathbb{E}[U(p)\mid q]\Vert }^2\ge 0.$$

The total-variation estimate bounds this square, and another Cauchy–Schwarz yields discrepancy at most $4B{B}^{\prime }(D/K{)}^{1/4}$ for features and center coefficients bounded by $B,B^{\prime }$. Only the strict test prefix is retained; the next test can consequently be replaced by a fresh independent test after fixing the history.

For acceptance sets of codes, the error becomes $4\sqrt{M}(D/K{)}^{1/4}$. One posterior draw can be tested against several neighbors, with a union bound multiplying the error by their number. Its marginal remains $\mu$, and its collision probability with the original point is at most $M^{K-1}{\max }_p\mu (p)$. These facts make the prediction usable in the later geometric contradiction without replacing a degree-biased law by a uniform one or choosing the test after seeing the prediction.

See Also

• Higher-dimensional distinct distances proves a different claimed bound, at the sharp lattice exponent for $d\ge 3$.
• Plane coloring studies the chromatic number of the full unit-distance graph, rather than the number of its edges in a finite set.

🏷️ Manuscripts and verification scope

• The weak pinned planar distance theorem — pinned manuscript and companion source.
• A power saving for planar unit distances — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page has separate comparator entries for the uniform large-fiber limit and the unit-distance power saving. The sentence excluding the latter from the first statement does not exclude its separate second entry. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A power saving for planar unit distances.

🐻  OpenAI 2026b. The weak pinned planar distance theorem.