Leading asymptotics for polygonal-linkage chamber orbits

Definitions and mathematical statement

The question is attributed to Farber: find the asymptotic behavior of $C_n$, the number of chamber orbits for polygonal linkages. The source does not define the arrangement or acting group. The aid uses positive length vectors $\ell\in{\mathbb{R}}_{>0}^n$, $n\geq3$, with walls \[ \sum_{i\in I}\ell_i=\sum_{i\notin I}\ell_i \quad(\varnothing\ne I\subsetneq\{1,\ldots,n\}), \] and chamber orbits under the permutation action of $S_n$. A chamber is a connected component of the complement of all these walls. Positive scaling does not change chamber membership, so one may instead normalize $\sum_i\ell_i=1$.

Let $L(m)$ count strict-sign Boolean linear threshold functions on $\{\pm1\}^m$: each is $x\mapsto\operatorname{sign}(a_0+\sum_{i=1}^m a_ix_i)$, with the affine form nonzero at every cube vertex, and constants included. The aid reports the orbit squeeze \[ \frac{L(n-1)}{2^n n!}\leq C_n\leq L(n-1). \] The finite factors account for coordinate-sign and permutation choices in relating the full threshold arrangement to positive-length chamber orbits; they are not an equality of orbit counts with threshold functions.

Combining this with the supplied Zuev asymptotic yields \[ \lim_{n\to\infty}\frac{\log_2C_n}{n^2}=1, \qquad C_n=2^{n^2+o(n^2)}. \] The aid also quotes an error bound transferring to \[ \log_2C_n=n^2+O\left(\frac{n^2}{\ln n}\right). \] This is a leading logarithmic asymptotic, not a multiplicative estimate $C_n\sim2^{n^2}$.

If one excludes infeasible generic polygon length vectors, feasibility means $\max_i\ell_i<\sum_{j\ne i}\ell_j$. There are $n$ chambers in which one length strictly exceeds all others combined; they form one permutation orbit. The aid therefore gives the exact convention correction \[ C_n^{\mathrm{feas}}=C_n-1. \] It does not affect the displayed leading logarithmic bounds. The relevant object is the full threshold arrangement restricted to positive lengths, not its trace-zero resonance-arrangement slice. A sharper asymptotic or a different orbit convention would require additional data beyond this reduction.

Short English statement

Under the aid's permutation-orbit convention, polygon-linkage chambers have $C_n=2^{n^2+o(n^2)}$, with a quoted logarithmic error bound from Zuev's threshold-function theorem. Excluding infeasible lengths removes exactly one orbit. This gives leading-logarithmic growth, not a full multiplicative asymptotic.

Sources

• Local UnsolvedMath registry, permanent ID $20001855$, code catalogue entry; canonical aim-geometry-notes.json, notes[192].
• AIM, "Configuration spaces of linkages," Problem 6, attributed in the source to Farber, https://aimath.org/pastworkshops/linkagesproblems.pdf. The extracted "cham-bers" is a line-break artifact. No PDF inspection is claimed here.
• Zuev's threshold-function theorem is named by the aid, without title, date, theorem number or complete authoritative reference for the asymptotic and quoted error bound.
• Research attempt 1: "partial_result," "partially_solved," rigor label "proved," novelty confidence "low." The machine-generated aid is not peer reviewed; independently verify before citation.