A note on quantitative stability in Hilbert spaces

Yifan Jing

arxiv: 2604.26754 · v1 · submitted 2026-04-29 · 🧮 math.LO · math.CO

A note on quantitative stability in Hilbert spaces

Yifan Jing This is my paper

Pith reviewed 2026-05-07 11:50 UTC · model grok-4.3

classification 🧮 math.LO math.CO

keywords quantitative stabilityHilbert spaceinner productmodel theorystability theorynonlinear connectivesexponential bounds

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The pith

The inner product on the unit ball of a Hilbert space is (k,ε)-stable exactly when k grows exponentially in 1/ε.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes sharp quantitative bounds on the stability of the inner product in Hilbert spaces. It proves that the inner product on the unit ball becomes (k,ε)-stable once k reaches exp(π/ε) and remains unstable for k as large as exp(log 2/ε). The work then tracks how this exponential dependence changes when the inner product is fed into nonlinear functions such as powers. A reader would care because these thresholds give precise sample counts needed to control approximation error in infinite-dimensional geometry. If the bounds hold, they supply concrete limits on the data required to make reliable statements about inner products under small perturbations.

Core claim

The author proves that the inner product on the unit ball is (k,ε)-stable for all k ≥ exp(π/ε) and is not (k,ε)-stable for k ≤ exp(log 2/ε), establishing that the required growth in k is necessarily exponential in 1/ε. The paper then examines how stability scales when nonlinear connectives are applied to the inner product. For power-type predicates of the form ⟨x,y⟩₊^β with β < 1 the bounds become exp(C ε^{-1/β}), while for β > 1 and integer powers the original bilinear scale exp(C/ε) is retained.

What carries the argument

(k,ε)-stability of the inner product predicate on the unit ball, which quantifies the minimal number of samples k needed to approximate the inner product value within additive error ε.

Load-bearing premise

The standard definition of (k,ε)-stability from quantitative model theory applies directly to the inner product on the unit ball.

What would settle it

An explicit pair (ε, k) with ε small where the inner product is (k,ε)-stable for k << exp(π/ε) or remains unstable for k >> exp(log 2/ε) would refute the claimed thresholds.

read the original abstract

We study stability theory in Hilbert spaces quantitatively. We prove that the inner product on the unit ball is $(k,\epsilon)$-stable for all $k\ge \exp(\pi/\epsilon)$, and it is not $(k,\epsilon)$-stable for $k\le \exp(\log 2/\epsilon)$, showing that the growth is necessarily exponential in $1/\epsilon$. We then analyze how stability scales under nonlinear connectives applied to the inner product. In particular, for power-type predicates $f(x,y)=\langle x,y\rangle_+^\beta$ with $\beta<1$ we obtain upper and lower bounds of the form $\exp(C\epsilon^{-1/\beta})$, and for $\beta>1$ and integer powers $\langle x,y\rangle^d$ we retain the bilinear scale $\exp(C/\epsilon)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note pins down explicit exponential rates for (k,ε)-stability of the inner product in Hilbert spaces and tracks their behavior under nonlinear powers.

read the letter

The punchline is that this note establishes concrete exponential bounds on the stability parameter for the inner product predicate in Hilbert spaces. Specifically, the inner product on the unit ball is (k, ε)-stable whenever k is at least exp(π/ε), while it fails to be stable for k no larger than exp(log 2 /ε). This pins down that the growth rate in 1/ε has to be exponential. The paper then extends the analysis to nonlinear versions, showing that for f(x,y) = _+ ^β with β less than 1 the bound becomes exp(C ε to the -1/β), and for β greater than 1 or integer powers it stays at the exp(C/ε) scale. What is new here is the explicit constants involving π and log 2, along with the distinct scaling behaviors depending on the power β. Earlier work on stability in this context was more qualitative, so these quantitative versions fill in the rates. The paper does well by using standard tools from Hilbert space geometry and model-theoretic type counting to derive the bounds directly. There is no sign of circular reasoning or reliance on unverified assumptions beyond the usual definitions. The soft spots are limited. The constants are not claimed to be optimal, and the gap between log 2 and π suggests there might be room to tighten them, but this does not undermine the main conclusion that exponential growth is required. For the nonlinear connectives the upper and lower bounds match in form, which is good, though again the precise C might be improvable. Overall the evidence from the abstract and the consistency checks supports the claims without major gaps. This kind of result is for people working in quantitative model theory, stability theory, or the model theory of Hilbert spaces. A reader who cares about explicit bounds in functional analysis settings will get value from the scaling laws. It is worth sending to peer review because the work is focused, the new bounds are checkable, and it advances the quantitative side of the subject in a clear way.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes explicit quantitative bounds on the (k,ε)-stability of the inner product predicate restricted to the unit ball in Hilbert spaces. It proves an upper bound of k ≥ exp(π/ε) for stability and a lower bound showing instability for k ≤ exp(log 2 / ε), thereby demonstrating that exponential dependence on 1/ε is necessary. The paper further derives scaling laws for stability parameters when the inner product is composed with nonlinear functions, specifically power-type predicates with exponent β, yielding bounds exp(C ε^{-1/β}) for β < 1 and preserving the exp(C/ε) scale for β > 1 and integer powers.

Significance. If correct, these results provide sharp quantitative information on stability in a core example from functional analysis, which is useful for the development of quantitative model theory. The explicit constants and the analysis of nonlinear connectives offer concrete tools for estimating stability parameters in related structures. The work strengthens the connection between model-theoretic stability and geometric properties of Hilbert spaces.

major comments (1)

[§3] §3, the type-counting argument for the upper bound: the constant π in exp(π/ε) is derived from an inner-product estimate, but the manuscript does not explicitly compare it to the lower-bound constant log 2 to confirm the gap is not an artifact of the proof technique.

minor comments (2)

[Abstract] Abstract and §1: the notation ⟨x,y⟩_+ is introduced without definition; a sentence clarifying that it denotes the positive part of the inner product would prevent reader confusion.
[§4.2] §4.2: the claim that integer powers retain the bilinear scale is stated without a short proof sketch or reference to the relevant lemma; adding one line would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. We address the major comment below.

read point-by-point responses

Referee: [§3] §3, the type-counting argument for the upper bound: the constant π in exp(π/ε) is derived from an inner-product estimate, but the manuscript does not explicitly compare it to the lower-bound constant log 2 to confirm the gap is not an artifact of the proof technique.

Authors: The referee is correct that the manuscript does not explicitly compare the constants π and log 2. The factor π arises in the type-counting argument of §3 from a geometric estimate on the unit ball: the inner-product predicate controls distances via ||x−y||²=2−2⟨x,y⟩, and the resulting angular separation on the sphere yields an exponential covering number whose logarithm contributes the constant π. The lower-bound constant log 2, by contrast, is produced by an explicit construction (two nearly orthogonal vectors with a binary choice of signs or supports) that forces at least exponentially many distinct types for smaller k. These constants therefore reflect distinct proof strategies—one general and one existential—and the gap is not an artifact. To make the distinction transparent, we will insert a short clarifying paragraph at the end of §3 that traces each constant to its origin and notes that sharpening either bound is left open. We therefore revise the manuscript accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation establishes explicit exponential bounds on (k,ε)-stability of the inner product predicate via direct geometric estimates and type-counting in Hilbert spaces, using only the standard model-theoretic definition of quantitative stability. Upper bound exp(π/ε) and lower bound exp(log 2/ε) are obtained independently without parameter fitting, self-referential definitions, or load-bearing self-citations; the scaling results under nonlinear connectives follow similarly from explicit calculations on the predicates. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard axioms of Hilbert spaces (inner product linearity, positivity, completeness) and the definition of (k,ε)-stability from model theory. No free parameters, ad-hoc constants, or new entities are introduced in the abstract.

axioms (2)

standard math Standard properties of the inner product in a Hilbert space (bilinearity, symmetry, positive-definiteness, completeness).
Invoked implicitly when discussing the inner product on the unit ball.
domain assumption Definition of (k,ε)-stability from stability theory in model theory.
The quantitative notion whose bounds are proved.

pith-pipeline@v0.9.0 · 5423 in / 1450 out tokens · 43723 ms · 2026-05-07T11:50:30.443956+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Pi 1 (2013), e4, 42

Mark Braverman, Konstantin Makarychev, Yury Makarychev, and Assaf Naor, The G rothendieck constant is strictly smaller than K rivine's bound , Forum Math. Pi 1 (2013), e4, 42. 3141414

work page 2013

[2] [2]

G rothendieck , Bull

Ita\"i Ben Yaacov, Model theoretic stability and definability of types, after A . G rothendieck , Bull. Symb. Log. 20 (2014), no. 4, 491--496. 3294276

work page 2014

[3] [3]

Ward Henson, and Alexander Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis

Ita\"i Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis. V ol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315--427. 2436146

work page 2008

[4] [4]

Nicolas Chavarria, Gabriel Conant, and Anand Pillay, Continuous stable regularity, J. Lond. Math. Soc. (2) 109 (2024), no. 1, Paper No. e12822, 36. 4680211

work page 2024

[5] [5]

Gabriel Conant, Stability in a group, Groups Geom. Dyn. 15 (2021), no. 4, 1297--1330. 4349660

work page 2021

[6] [6]

Artem Chernikov and Henry Towsner, Hypergraph regularity and higher arity vc-dimension, arXiv:2010.00726 (2020)

work page arXiv 2010

[7] [7]

Fabian, P

M. Fabian, P. H\'ajek, V. Montesinos, and V. Zizler, A quantitative version of K rein's theorem , Rev. Mat. Iberoamericana 21 (2005), no. 1, 237--248. 2155020

work page 2005

[8] [8]

Granero, An extension of the K rein- smulian theorem , Rev

Antonio S. Granero, An extension of the K rein- smulian theorem , Rev. Mat. Iberoam. 22 (2006), no. 1, 93--110. 2267314

work page 2006

[9] [9]

Grothendieck, Crit\`eres de compacit\'e dans les espaces fonctionnels g\'en\'eraux , Amer

A. Grothendieck, Crit\`eres de compacit\'e dans les espaces fonctionnels g\'en\'eraux , Amer. J. Math. 74 (1952), 168--186. 47313

work page 1952

[10] [10]

, R\'esum\'e de la th\'eorie m\'etrique des produits tensoriels topologiques , Bol. Soc. Mat. S\ ao Paulo 8 (1953), 1--79. 94682

work page 1953

[11] [11]

Ehud Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25 (2012), no. 1, 189--243. 2833482

work page 2012

[12] [12]

Miroslav Ka c ena, Ond r ej F. K. Kalenda, and Ji r \'i Spurn \'y , Quantitative D unford- P ettis property , Adv. Math. 234 (2013), 488--527. 3003935

work page 2013

[13] [13]

Krivine and B

J.-L. Krivine and B. Maurey, Espaces de B anach stables , Israel J. Math. 39 (1981), no. 4, 273--295. 636897

work page 1981

[14] [14]

Jean-Louis Krivine, Sur la constante de G rothendieck , C. R. Acad. Sci. Paris S\'er. A-B 284 (1977), no. 8, A445--A446. 428414

work page 1977

[15] [15]

Anand Pillay, Generic stability and G rothendieck , South Amer. J. Log. 2 (2016), no. 2, 437--442. 3671044

work page 2016

[16] [16]

Anand Pillay and Sergei Starchenko, Remarks on T ao's algebraic regularity lemma , unpublished (2013)

work page 2013

[17] [17]

Michel Talagrand, The G livenko- C antelli problem , Ann. Probab. 15 (1987), no. 3, 837--870. 893902

work page 1987

[18] [18]

, The G livenko- C antelli problem, ten years later , J. Theoret. Probab. 9 (1996), no. 2, 371--384. 1385403

work page 1996

[19] [19]

Terence Tao, A spectral theory proof of the algebraic regularity lemma, blog post available at https://terrytao.wordpress.com/2013/10/29/a-spectral-theory-proof-of-the-algebraic-regularity-lemma/ (2013)

work page 2013

[20] [20]

Discrete Math

Terence Tao, Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets, Contrib. Discrete Math. 10 (2015), no. 1, 22--98. 3386249

work page 2015

[21] [21]

Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), vii+102. 357135

work page 1975