Characterize the space of ultrametric absolute values on the integers

Characterize the point-free space of Dedekind-valued ultrametric absolute values on the ring of integers of the rational numbers, including both the trivial and non-Archimedean absolute values.

Background

The paper constructs a point-free topos of places by applying lax descent to the geometric space of absolute values on the rationals. It successfully identifies each non-Archimedean place, after fixing a prime, with a singleton and identifies the Archimedean place with the space of upper reals bounded between 0 and 1.

However, reconciling the trivial ultrametric absolute value with the non-trivial non-Archimedean absolute values remains unresolved. The discussion explains that an earlier treatment could characterize multiplicative seminorms valued in the upper reals, but could not characterize the corresponding Dedekind-valued ultrametric absolute-value space. Resolving this issue is a prerequisite for a complete global description of the places.

References

Curiously, when we considered the space of Dedekind-valued absolute values, we were unable to even characterise the space of ultrametric absolute values.

— The Archimedean place is a blurred interval at infinity  (2609.09117 - Ng, 8 Sep 2026) in Section 4, Discussion “The trivial place”

One would still need to understand how the resulting Archimedean and non-Archimedean pieces assemble into a single point-free space of places.

— The Archimedean place is a blurred interval at infinity  (2609.09117 - Ng, 8 Sep 2026) in Section 4, Discussion following the problem on general lax descent