Radius of comparison equals half the mean dimension

1 · The essentials

The central claim connects a measurement of a moving system with a measurement of an algebra built from that system. The setting is an infinite space with a reversible continuous motion, where every point’s path comes arbitrarily close to every part of the space. The space must also be compact and admit a metric describing its topology. For every such system, the manuscript shows that the algebra’s measurement is exactly half the system’s measurement, even when both are infinite. It also identifies the case where the system’s measurement is zero with several other conditions on the system and its algebra. A companion manuscript supplies a compression result that keeps existing boundary positions unchanged, under a strict size restriction. These results are presented in manuscripts; the main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two gauges attached to the same machine: one measures its motion, while the other measures a structure assembled from that motion. The claimed equality says that, under the specified conditions, the second gauge always reads exactly half the first—not merely something close to it.

Why it matters

The equality gives an exact relationship between mean dimension on the dynamical side and radius of comparison on the operator-algebra side. Manuscript 2 describes this as resolving the integer-action case of the Phillips–Toms conjecture. Its zero-mean-dimension statement also brings several different conditions into one equivalence, with nuclear dimension at most one in that case.

2 · Going further

Let $h$ be a minimal homeomorphism of an infinite compact metrizable space $X$ . Manuscript 2 shows the identity $rc (C (X) ⋊_{h} Z) = \frac{1}{2} mdim (X, h)$ , including equality at infinity. Here the crossed product is the operator algebra associated with the space and its integer-time action. The abstract identifies this statement as the integer-action case of the Phillips–Toms conjecture.

For the same class of systems, Manuscript 2 also shows that zero mean dimension, the small boundary property, Jiang–Su stability of the crossed product, and finite nuclear dimension of the crossed product are equivalent. When these conditions hold, the nuclear dimension is at most one. Thus the zero case carries additional structural conclusions beyond the numerical equality.

Manuscript 1, designated the main manuscript in the supplied list, concerns a topological compression mechanism. Its abstract states that an $M U$ -null restriction to a finite pointed subcomplex becomes stably null, in sufficiently large smash powers, on the entire union of products having a fixed positive proportion of restricted factors. This coherent vanishing yields boundary-preserving compression of cube-valued maps on compact metrizable spaces. If the torus slot bundle embeds continuously into a trivial bundle of rank strictly less than twice its source rank, the compression puts a positive proportion of slots on their boundaries in arbitrarily large powers while fixing every original boundary slot exactly. An example at equality shows that the strict rank condition cannot be removed. This is the compression result accompanying the operator-algebra statement, but the supplied abstracts do not spell out the steps connecting it to the radius-of-comparison identity.

Useful notions

Compact metrizable spaces and homeomorphisms
Minimal dynamical systems and mean dimension
Crossed-product operator algebras
Radius of comparison and nuclear dimension
Bundles, smash products, and complex cobordism

What was known before / what is new

Before

The source identifies the equality as the integer-action case of the Phillips–Toms conjecture. It does not describe earlier partial results or specify which parts of the zero-mean-dimension equivalence were previously known.

New

Manuscript 2 claims the exact equality for every minimal homeomorphism of an infinite compact metrizable space, including infinite values. It also states the four-way equivalence in the zero case and the bound of one on nuclear dimension. Manuscript 1 presents coherent vanishing and boundary-preserving compression, together with an example showing the necessity of its strict rank inequality.

Limits and assumptions

The operator-algebra statement assumes a minimal homeomorphism of an infinite compact metrizable space and concerns integer actions; the supplied text does not state an extension beyond that setting. The bound on nuclear dimension applies in the zero-mean-dimension case. The compression conclusion involving a positive proportion of boundary slots requires a continuous bundle embedding with target rank strictly less than twice the source rank, and the equality example rules out simply dropping strictness. The abstracts do not provide the full argument linking the two manuscripts or specify further open problems. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Minimal homeomorphism: A reversible continuous map with continuous inverse for which every point’s orbit is dense in the space.
Mean dimension: The dynamical dimension quantity denoted by $mdim (X, h)$ in the stated equality.
Radius of comparison: The numerical invariant of the crossed-product algebra that Manuscript 2 identifies with half the mean dimension.
Crossed product: The operator algebra $C (X) ⋊_{h} Z$ associated with the space $X$ and the integer action generated by $h$ .
Boundary-preserving compression: The compression described in Manuscript 1 that moves a positive proportion of slots to their boundaries while fixing every original boundary slot exactly.

Keywords : Mean dimension · Radius of comparison · Boundary-preserving compression

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of $C (X) ⋊_{h} Z$ equals $\frac{1}{2} mdim (X, h)$ , including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

Filtered products and boundary-preserving compression in complex cobordism

25/09/2026

To be verified

We prove that, in sufficiently large smash powers, an $M U$ -null restriction to a finite pointed subcomplex becomes stably null on the entire union of products with a fixed positive proportion of restricted factors. This coherent vanishing theorem yields boundary-preserving compression of cube-valued maps on every compact metrizable input space. When the torus slot bundle embeds continuously into a trivial bundle of rank less than twice the source rank, the compression places a positive proportion of slots on their boundaries in arbitrarily large powers and fixes every original boundary slot exactly. An example at equality shows that the strict rank inequality cannot be removed.

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