Quantitative bridge from Vershik genericity to existential closure

Definitions and mathematical statement

Problem 18. Find connections between different notions of genericity, especially Vershik genericity and existential closure. The source notes that the Urysohn space is generic in both senses, but does not specify a language, diameter bound, coding topology or probability notion. The terminal "Background:" contains no further data.

Supplied pure bounded-metric setting. Work in the distance-only language with complete metric structures of diameter at most one. Code separable spaces by infinite symmetric distance matrices on $\omega$, with zero diagonal, entries in $[0,1]$ and triangle inequalities, in the product topology; pseudometric codes may be quotiented before completion. Vershik genericity here refers to the Baire-category universal-request construction in this coding space, not automatically to a probabilistic or expanded-language genericity.

A vector $r=(r_i)_{i=1}^n\in[0,1]^n$ over a finite metric space $(a_i,d_{ij})$ is Katětov if \[ |r_i-r_j|\le d_{ij}\le r_i+r_j\quad\text{for all }i,j. \] The approximate one-point extension property says that for every such finite request and every $\varepsilon>0$ there is a point $x$ with $|d(x,a_i)-r_i|<\varepsilon$ for every $i$. In this setting the aid reports Usvyatsov's 2008 theorem identifying the completions of Vershik-generic universal distance matrices with the unique separable existentially closed metric space, the Urysohn sphere. Existential closure means that every quantifier-free approximate existential condition over the space which is achievable in an extension is already arbitrarily well achievable in the space. The approximate extension property is the concrete bridge. The bounded sphere is distinguished from the unbounded Urysohn space named in the broad source question.

Supplied quantitative repair. For a nonempty finite diameter-at-most-one metric space and $q\in[0,1]^n$, define the two-sided defect \[ \eta=\max_{i,j}\max\left(0,\ |q_i-q_j|-d_{ij},\ d_{ij}-q_i-q_j\right). \] The aid's construction is \[ s_i=\min_j(q_j+d_{ij}),\qquad r_i=\min(1,s_i+3\eta/2). \] It asserts that $r$ is an exact Katětov vector and \[ 0\le r_i-q_i\le3\eta/2\quad\text{for every }i. \] The coordinate bound $q_i\in[0,1]$ is essential for the one-sided conclusion after truncation at one; no statement for arbitrary real vectors is inferred. The constant is a sufficient bound, not claimed optimal.

The aid uses this repair to obtain a countable dense-open request scheme equivalent to approximate one-point extension, with antecedent/witness tolerance margin $\delta=\varepsilon/4$. The actual enumeration of requests and open sets is not supplied here. This quantitative local statement supports the stated bounded-metric bridge; it does not establish that existential closure and every notion of genericity coincide for arbitrary metric theories, extra structure, perturbation systems or measures.

Short English statement

The supplied result connects Baire-generic distance matrices with existential closure in the pure diameter-one metric setting, giving the Urysohn sphere. Its extra lemma repairs an approximately admissible finite distance vector to an exact one with an explicit one-sided error. The broader source request is not treated as solved for every structure or notion of genericity.

Sources

• Local UnsolvedMath registry, permanent ID 20002332, source code catalogue entry; canonical aim-logic-notes.json, notes[107].
• AIM Questions on Model theory for metric structures, Problem 18: https://aimath.org/WWN/continuouslogic/continuouslogic.pdf. Vershik genericity and the Urysohn example are named in the original question.
• The aid cites Usvyatsov (2008) for the pure bounded-metric bridge, without a full title, theorem label or publication entry. No separate proving citation for the quantitative repair is supplied.
• Registry research attempt 1, dated 2026-08-14: partial result, problem status at run solved, rigor "proved", novelty confidence low. The explicitly machine-generated, non-peer-reviewed aid requires independent verification before citation.