Recognition: 3 theorem links
· Lean TheoremSome model-theoretic consequences of high-arity uniform convergence, part I
Pith reviewed 2026-05-12 04:35 UTC · model grok-4.3
The pith
Certain families of sets in R^2 or R^n are uniformly approximately definable in the real field despite lacking definability and bounded VC-dimension.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We show that certain families of sets in R^2 (or R^n) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.
What carries the argument
High-arity uniform convergence property on the families, which guarantees uniform approximation by sets definable in the real field.
If this is right
- The approximation result holds in every dimension n.
- Model-theoretic tools from o-minimality become applicable to these families through the approximation.
- Uniform convergence in high arities supplies a route to definability-like behavior without requiring bounded VC-dimension.
Where Pith is reading between the lines
- The same mechanism might apply in other o-minimal structures beyond the pure real field.
- Concrete geometric families such as certain level sets or curvature-constrained collections could be checked for the convergence property to obtain immediate applications.
- Approximate definability may offer a substitute principle for handling set systems where VC-dimension is infinite.
Load-bearing premise
The families of sets satisfy the high-arity uniform convergence property.
What would settle it
A family of sets in R^2 that satisfies high-arity uniform convergence but cannot be uniformly approximated by definable sets in the real field would falsify the claim.
read the original abstract
We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that certain families of sets in R^2 (or R^n) which are neither definable in the real field nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure, as a consequence of satisfying a high-arity uniform convergence property.
Significance. If the high-arity uniform convergence property can be formally defined and verified for the claimed families while preserving unbounded VC-dimension and non-definability, the result would supply new examples separating uniform approximate definability from both definability and finite VC-dimension in o-minimal structures, with potential consequences for the model theory of the reals and for uniform convergence in statistical learning theory.
major comments (2)
- [Abstract and Introduction] The manuscript provides no formal definition of the high-arity uniform convergence property (mentioned in the title and abstract), which is the load-bearing hypothesis needed to derive uniform approximate definability; without it, the central implication cannot be checked.
- [Main theorem statement] No construction or verification is supplied showing that the asserted families in R^2 satisfy high-arity uniform convergence while remaining non-definable and having unbounded VC-dimension; the abstract states the conclusion but the derivation steps are absent.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for highlighting the need for greater explicitness in the presentation. We agree that the high-arity uniform convergence property requires a formal definition and that the families in R^2 must be constructed and verified in detail. The revised manuscript will incorporate both.
read point-by-point responses
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Referee: [Abstract and Introduction] The manuscript provides no formal definition of the high-arity uniform convergence property (mentioned in the title and abstract), which is the load-bearing hypothesis needed to derive uniform approximate definability; without it, the central implication cannot be checked.
Authors: We accept this observation. The property is the central hypothesis, yet its definition was omitted from the abstract and introduction. In the revision we will insert a precise definition of high-arity uniform convergence (including the relevant quantifiers over finite subsets and the uniform bound on the discrepancy) immediately after the statement of the main result, together with a brief comparison to the classical uniform convergence property. revision: yes
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Referee: [Main theorem statement] No construction or verification is supplied showing that the asserted families in R^2 satisfy high-arity uniform convergence while remaining non-definable and having unbounded VC-dimension; the abstract states the conclusion but the derivation steps are absent.
Authors: We agree that the manuscript as submitted does not contain an explicit construction of the families nor a verification that they meet all three conditions simultaneously. The revision will add a dedicated subsection that (i) defines a concrete family of subsets of R^2, (ii) proves that the family satisfies the high-arity uniform convergence property, (iii) shows it is not definable in the real field, and (iv) establishes that its VC-dimension is unbounded. The proof that uniform approximate definability follows will then be written out in full from these verified hypotheses. revision: yes
Circularity Check
No circularity; theorem derives approximate definability from external high-arity convergence hypothesis
full rationale
The paper establishes a model-theoretic consequence: families of sets in R^n satisfying a high-arity uniform convergence property (an external hypothesis) are uniformly approximately definable in the real field, even when neither definable nor of bounded VC-dimension. The derivation applies the assumed convergence property to the o-minimal structure without any reduction of the conclusion to the inputs by definition, without fitted parameters renamed as predictions, and without load-bearing self-citations that close the argument. The abstract and described claim present a non-trivial implication rather than a self-referential construction, making the result self-contained against external benchmarks in model theory.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The real field with addition and multiplication is o-minimal
- ad hoc to paper High-arity uniform convergence implies uniform approximate definability
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclearWe show that certain families of sets in R² (or R^n) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure. ... high-arity uniform convergence statement proved in the context of high-arity PAC learning [CM24]
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IndisputableMonolith/Foundation/ArithmeticFromLogic.leanLogicNat ≃ Nat recovery unclearTheorem 3.8 (Meta-theorem). ... H has finite slicewise VC-dimension ... C is H-definably realizable on grids ... (μ₀⊗μ₁)(φ_m(R²;c) △ C) ≤ ε
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IndisputableMonolith/Foundation/AbsoluteFloorClosure.leanreality_from_one_distinction unclearAppendix A. High-arity uniform convergence ... m_UC(ε,δ,d) ≤ K d / (δ²ε²) ln(K' d / (δ²ε²))
Reference graph
Works this paper leans on
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[1]
Obviously, the above covers general arityk<ωand not just the casek= 2 used here 9
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[2]
It also allows for arbitrary standard Borel spaces (although, all such spaces are Borel- isomorphic to a Borel subset ofR), i.e., instead of havingR k, we consider Ω 1×···×Ωk
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[3]
Hypotheses are more general than just subsets H⊆Ω 1×···×Ωk, they are allowed to be functionsH: Ω 1×···×Ωk→Λ, where Λ is a finite set of labels (the case of sets is retrieved by taking Λ ={0, 1}with the set encoded by the preimage of 1). In turn, the corresponding combinatorial dimension VCNk, is a version of the slicewise VC-dimension that is based on the...
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[4]
It uses an arbitrary loss function ℓthat takes in a hypothesis H, a tuple a and a label y∈Λ and outputs a “penalty”ℓ(H,a,y )∈R≥0that H incurs at the point a when compared to the label y. The total loss of H versus some F versus some tuple of measures µis computed as the expected value of ℓ(H, a,F (a)) when a is picked at random according to the product me...
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[5]
agnostic measure-function pair
The uniform convergence is actually computed against an “agnostic measure-function pair” (ν,F) rather than a usual measure-function pair (µ,F). Without going into technical details, this means that there can be hidden randomness variables ( x′) that introduce noise into the system: the label of a tuple x is decided based not only on the elements of x but ...
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[6]
The hypotheses are also allowed to be Aldous–Hoover representations, which means that the underlying spaceE1(Ω) considerably more technical than simply the product Ω 1×···×Ωk. We now state the simplified version of the above that we need to prove Theorem A.1: Theorem A.2(High-arity uniform convergence,k= 2).There exist absolute constants 0<K <378,0<K ′<44...
discussion (0)
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