From stable norms to stable theories

Definitions and mathematical statement

Let $M$ be a continuous metric structure, and work on the bounded sorts of its language. A formula $\varphi(\bar x,\bar y)$ has the order property in $M$ if there are real $r<s$ such that for every $n$ there are tuples $\bar a_i,\bar b_i$ in $M$, $i<n$, with \[ \varphi(\bar a_i,\bar b_j)\leq r,\qquad \varphi(\bar a_j,\bar b_i)\geq s\quad(i<j<n). \] Call it stable when it has no such property; the analogous theory condition requires the absence of these configurations in models of the complete theory. Quantifier-free stability restricts this condition to quantifier-free formulas. A stable complete continuous theory has all its formulas stable. The phrase "stability of structures" in the source is not defined: stability of a chosen formula on one structure, stability of all formulas there, and Banach-space stability must be distinguished.

Primary Question 3. Determine the implications among quantifier-free stability, the intended stability of metric structures, and stability of their complete theories. For a Banach space $B$ in the pure Banach language, the relevant Banach-space condition is the Krivine--Maurey double-limit property \[ \lim_i\lim_j\|x_i+y_j\|=\lim_j\lim_i\|x_i+y_j\| \] for bounded sequences whenever both iterated limits exist. The research aid reports its equivalence with stability on $B$ of every quantifier-free formula. It also reports that, for complete continuous theories, quantifier-free stability together with quantifier elimination implies full stability; quantifier elimination here means uniform approximation by quantifier-free formulas on the specified bounded sorts. Language, bounded-sort conventions and the structure/theory distinction must be retained when applying these statements.

The aid's reported counterexample outside this Banach setting is the following. Start with a random bipartite graph with vertex sorts $A,B$, and introduce a discrete sort $W$ with exactly one witness $w$ for each edge. Take the language consisting of equality, discrete metrics, and $\{0,1\}$-valued incidences $P(a,w),Q(w,b)$, whose value is zero exactly when $a,b$ are the corresponding endpoints of $w$. The aid reports that the complete theory of this three-sorted subdivision has stable quantifier-free formulas, while \[ \psi(a,b)=\inf_{w\in W}\max\{P(a,w),Q(w,b)\} \] is zero exactly on the original edges and has the order property. This is a reported counterexample to the unrestricted implication from quantifier-free stability to theory stability, not a Banach-space counterexample.

The workshop background attributes to Krivine the statement that stability of the norm formula gives an isometric copy of $\ell_p$, for some real $1\leq p<\infty$, in an ultrapower of the normed space. The supplied wording says "$p\geq1$" and does not specify all conventions or auxiliary hypotheses; no assertion about an actual subspace of the original space is substituted for this ultrapower statement. The group "Definable norms and stable Banach spaces" is mentioned.

Appended independent Questions 4--11. The local extraction includes the following questions, which its verification field identifies as subsequent problems rather than part of Question 3. They are preserved here without merging their status with that of the primary question. For a Banach space $B$, write $T=\operatorname{Th}(B)$. The density character is the least cardinality of a dense subset; $\kappa$-categoricity in the usual metric sense means one model of density character $\kappa$ up to isometric isomorphism.

1. 4. Determine the consequences of (a) stability of $T$, and (b) $\aleph_1$-categoricity of $T$. The source reports the ultrapower $\ell_p$ consequence in (a), attributes to Ben-Yaacov the implication from $\aleph_1$-categoricity to categoricity in all uncountable densities in its setting, and records Henson's conjecture that such theories are "essentially" Hilbert-space theories. The intended meaning of "essentially" and the precise language hypotheses are not given.
2. 5. Determine what follows if models of a Banach-space theory are classified up to linear homeomorphism, and in particular if there is a unique model of density $\aleph_1$ up to that equivalence. Here a linear homeomorphism is a bounded linear bijection with bounded inverse. The source compares this with a Morley-type theorem for isometric categoricity; it does not state a corresponding theorem for linear homeomorphism.
3. 6. If any two separable models of a fixed theory are linearly homeomorphic, is there a uniform bound on their deformation coefficients? With the usual Banach--Mazur distortion convention, a precise possible target is a finite $C_T$ such that every pair $M,N$ of separable models admits such a map $U$ with $\|U\|\|U^{-1}\|\leq C_T$. The source does not specify its coefficient convention or whether an infimum or an attained bound is intended.
4. 7. Understand Tsirelson space and determine whether its theory is categorical up to small perturbations. The version of Tsirelson space, the permitted perturbations and the density of categoricity are unspecified. A talk by William Johnson is mentioned.
5. 8. Find $\aleph_1$-categorical structures essentially different from Hilbert space, in the sense intended in Question 4.
6. 9. Determine whether Keisler randomization produces new categorical or stable structures and whether it preserves $\lambda$-stability. The source attributes preservation of countable categoricity as separable categoricity, and preservation of $\omega$-stability, to Keisler. A general preservation question requires specifying the base and randomized languages, randomization construction, cardinal $\lambda$, and the metric type-density convention for stability. The group "Weak First Order and Probability Structures" is mentioned.
7. 10. Find theories not essentially separably categorical. In particular, are the cited non-separably-categorical theories of Nakano spaces separably categorical up to perturbations of the norm? The source points to Pedro Poitevin's UIUC 2006 thesis and a talk by C. Ward Henson.
8. 11. Find stable theories not essentially $\omega$-stable. The source reports Nakano-space theories that are $2^{\aleph_0}$-stable but not $\omega$-stable, and asks whether they become $\omega$-stable up to perturbations of the norm. No precise perturbation system or family of Nakano spaces is supplied.

The meanings of "essentially", "small perturbations", and the exact stability conventions in these appended questions remain formulation gaps. All reported progress and background attributions require verification against their original sources.

Short English statement

Compare formula, quantifier-free and theory stability, distinguishing Banach-space double-limit stability from general metric structures. The aid reports a quantifier-free stable subdivision whose existential edge formula is unstable. Eight subsequently extracted questions about Banach categoricity, distortion, perturbations, randomization and Nakano spaces are retained separately.

Sources

• Local UnsolvedMath registry, permanent ID $20002327$, source code catalogue entry, aim-logic-notes.json, notes[102], workshop Model theory of metric structures, source item 3. Supplied source: https://aimath.org/WWN/continuouslogic/continuouslogic.pdf.
• Supplied background attributions: Krivine on stable norms and ultrapower copies of $\ell_p$; the group "Definable norms and stable Banach spaces"; Ben-Yaacov on uncountable categoricity; Henson's Hilbert-space conjecture; William Johnson's talk on Tsirelson space; Keisler's randomization results and the group "Weak First Order and Probability Structures"; Pedro Poitevin's thesis (UIUC, 2006) on Nakano spaces; and C. Ward Henson's talk. No full bibliographic references or theorem hypotheses are supplied.
• Supplied research summary: Krivine--Maurey stability, the quantifier-elimination implication, and the subdivided random bipartite graph example. Attempt 1 is labelled partial_result, partially_solved, rigor "proved", low novelty confidence. It is a machine-generated research aid, not peer reviewed; independently verify before citation. The local statement-verification field reports contamination by Questions 4--11 and page markers; this draft has not independently visited the PDF.