Compactness transfer from exact isometry to uniform almost isometry

Definitions and mathematical statement

Problem 27 asks for a compactness or ultraproduct mechanism turning an isometric theorem into an almost-isometry theorem. The source does not specify a class of spaces, maps, or an exact conclusion. The aid supplies the following conditional transfer scheme, whose hypotheses are part of the statement.

Let $\mathcal K$ be a class of bounded continuous structures in a fixed signature with fixed continuity moduli. Let $D(M),C(M)\in[0,1]$ be hypothesis and conclusion defects, defined consistently on $\mathcal K$ and its metric ultraproducts, and assume \[ D\left(\prod_{\mathcal U}M_i\right)=\lim_{\mathcal U}D(M_i),\qquad C\left(\prod_{\mathcal U}M_i\right)=\lim_{\mathcal U}C(M_i). \] Uniformly definable predicates provide the intended setting for this requirement. Let the ultraproduct hull include all metric ultraproducts of members of $\mathcal K$, including principal ones. The proposed transfer principle is the equivalence of \[ D(N)=0\ \Longrightarrow\ C(N)=0\quad\text{for every }N\text{ in that hull} \] and \[ \forall\varepsilon>0\ \exists\delta>0\ \forall M\in\mathcal K, \qquad D(M)\leq\delta\ \Longrightarrow\ C(M)\leq\varepsilon. \] Equivalently, with $\sup\varnothing=0$, the sharp error envelope \[ \omega(t)=\sup\{C(M):M\in\mathcal K,\ D(M)\leq t\}\quad(0\leq t\leq1) \] tends to $0$ as $t\downarrow0$. The aid also asserts the existence of a continuous nondecreasing majorant $\rho$ with $\rho(0)=0$ and $C(M)\leq\rho(D(M))$. No effective rate is asserted. Exact validity merely on $\mathcal K$, without control of its ultraproduct hull, is not the displayed hypothesis.

For a map $f:X\to Y$ between nonempty bounded metric spaces, the aid uses both distortion and coverage: \[ \operatorname{dis}(f)=\sup_{x,x'\in X}|d_Y(fx,fx')-d_X(x,x')|,\qquad \operatorname{cov}(f)=\sup_{y\in Y}\inf_{x\in X}d_Y(y,fx),\qquad D(f)=\max\{\operatorname{dis}(f),\operatorname{cov}(f)\}. \] These maps must have the continuity needed for their ultraproducts to be well defined, or that fact must be checked directly along a zero-defect sequence. For a sequence with $D(f_n)\to0$, the induced metric-ultraproduct map is a surjective isometry: coverage supplies coordinatewise approximate preimages. Distortion alone only controls isometric embedding and does not provide surjectivity. In the complete metric setting, an individual zero-defect map is also a surjective isometry.

Finally, if the class is axiomatized by zero-valued continuous sentences $T$ and the defects are uniformly definable in that setting, the aid gives a finite approximate-axiom certificate: for every $\varepsilon>0$ there are finite $T_0\subseteq T$ and $\eta,\delta>0$ such that every structure $N$ in the fixed signature satisfying \[ \sigma^N\leq\eta\ (\sigma\in T_0),\qquad D(N)\leq\delta \] satisfies $C(N)\leq\varepsilon$. This is conditional on exact validity for all models of $T$ and on the stated definability assumptions. No particular almost-isometry application is fixed by the original question.

Short English statement

Compactness gives a uniform almost theorem when vanishing hypothesis defect forces vanishing conclusion defect in all relevant ultraproducts, and the defects commute with ultraproducts. For maps, coverage must accompany distance distortion to obtain a surjective isometry. The supplied scheme includes an error envelope and finite approximate-axiom certificates, rather than a numerical stability rate.

Sources

• Local UnsolvedMath registry, permanent ID $20002336$, catalogue entry; aim-logic-notes.json, notes[111].
• American Institute of Mathematics, Model theory of metric structures, Problem 27, https://aimath.org/WWN/continuouslogic/continuouslogic.pdf. Its extracted "Background:" is empty. No proposer or authoritative bibliographic citation for the transfer theorem is supplied.
• Research attempt 1: partial result, problem status solved at run, rigor label "proved," novelty confidence low. The registry says solved and solved in literature, and labels the statement exact; these labels do not supply the missing literature reference.