Cone domains separate FS-domains from RB-domains

Source: arXiv:2607.02251 · Published 2026-07-02 · By Yuxu Chen

TL;DR

This paper addresses a long-standing question in domain theory posed by Keimel regarding the classification of cone domains as FS-domains and RB-domains. Specifically, given a proper closed convex cone C in a finite-dimensional real vector space V, the associated cone domain DC = (-C) ∪ {⊥} ordered by the cone order is known to be an FS-domain. The open problem was whether DC is always an RB-domain (a retract of a bifinite domain). The author provides a definitive characterization: DC is an RB-domain if and only if the cone C is simplicial (generated by a basis of V). Thus, non-simplicial cones yield FS-domains that are not RB-domains, answering Keimel's question negatively for important examples like the Lorentz cone.

The proof combines domain-theoretic analysis and finite-dimensional convex geometry. The domain-theoretic part identifies the way-below relation for DC and shows the FS property via explicit Scott-continuous approximations finitely separated from identity. Assuming the RB property yields finite-valued C-monotone maps approximating the identity on compact sets. The core analytic obstruction shows that the identity operator must lie in the cone generated by positive rank-one operators v⊗ℓ induced by C and its dual cone C*. This is equivalent to C being simplicial. Non-simplicial cones fail this, thereby separating FS-domains from RB-domains via basic convex geometry.

This provides the first natural infinite family of FS-domains not arising as RB-domains and clarifies the finite approximation landscape in domain theory with a geometric criterion. The paper develops detailed analytic arguments using Rademacher theorem, Lipschitz epigraphs, and integration by parts to establish the rank-one cone obstruction, along with explicit construction of finite-range deflations in the simplicial case.

Key findings

• The domain DC = (-C) ∪ {⊥} is always an FS-domain for any proper cone C (Proposition 3.5).
• DC is an RB-domain if and only if C is simplicial; equivalently, the identity operator lies in the rank-one cone generated by {v⊗ℓ : v ∈ C, ℓ ∈ C*} (Theorem 1.1, Lemma 4.2).
• For non-simplicial proper cones such as the Lorentz cone L3, DC provides an FS-domain that is not an RB-domain, resolving Keimel's question negatively.
• Finite-valued C-monotone maps approximating the identity uniformly on compact sets do not exist for non-simplicial cones (Proposition 4.7, Corollary 4.8).
• The rank-one cone RK generated by positive rank-one operators is a closed convex cone in End(V) (Lemma 4.1).
• Lipschitz epigraph representations of cone-upper sets enable an integration formula that encodes the cone structure analytically (Lemmas 4.4 and 4.5).
• Explicit finite-range deflations approximating the identity are constructed for simplicial cones via coordinatewise grid rounding, proving DC is RB in this case (Lemma 5.1, Proposition 5.2).
• The way-below relation on DC is characterized by y - x lying in the interior of C, linking order theory to cone geometry (Lemma 3.3).

Threat model

The paper does not consider an adversarial threat model in the security sense. Instead, it studies an abstract theoretical adversary who attempts to approximate the identity map on a cone domain DC by finite-valued C-monotone maps. The capabilities include choosing monotone approximations and exploiting geometric structure of the cone. The impossibility results prove that for non-simplicial cones, no sequence of such finite-valued monotone approximations can converge uniformly to identity, thus ruling out RB approximation. There are no computational or probabilistic attacks considered.

Methodology — deep read

The paper integrates domain theory, convex analysis, and finite-dimensional geometric functional analysis to solve the classification problem.

1. Threat model & assumptions: The adversary is abstract—the problem is purely theoretical, concerning domain approximation properties of cone domains DC associated with a proper closed convex cone C in finite-dimensional V. The assumption is that DC is an FS-domain, and the question is whether it is RB.

2. Data: No empirical data is used. The objects of study are the cone C, its dual cone C*, the domain DC = (-C) ∪ {⊥}, and continuous monotone maps on these.

3. Architecture/algorithm:

• The domain DC is ordered by the cone order induced by C, with a bottom element ⊥ added.
• FS-domain property: Construct an explicit increasing sequence of Scott-continuous maps fn finitely separated from identity using functionals φ ∈ int C* and interior points a ∈ int C. These maps truncate inputs below a threshold and shift by a/n.
• RB-domain property: Characterized as existence of directed families of finite-range deflations increasing to identity.
• Key step: From RB property, construct finite-valued C-monotone maps Qε on compact subsets of int C approximating identity within εa.
• Analytic proof: Represent cone-upper sets as Lipschitz epigraphs. Use Rademacher's differentiability theorem, Fubini, and integration by parts to express integrals of monotone maps against smooth test functions.
• Show these integrals lie in the rank-one cone RK generated by positive rank-one operators v⊗ℓ.
• If Qε approximate identity, integrate-by-parts forces identity operation to lie in RK, which corresponds exactly to C being simplicial.
• Non-simplicial cases give a contradiction.

4. Training regime: Not applicable.

5. Evaluation protocol:

• Prove equivalences via constructive and contradiction arguments.
• Use geometric characterizations, measure theory, and functional analysis.
• Proposition 4.7 disproves uniform convergence of finite-valued monotone maps to identity on non-simplicial cones.

6. Reproducibility:

• Full proofs and constructive algorithms (explicit deflations) are given.
• No software or datasets.

Concrete example: Consider the three-dimensional Lorentz cone L3 = {(t,x,y): t ≥ sqrt(x^2 + y^2)} which is non-simplicial. DC = (-L3) ⊥ is an FS-domain by construction but not an RB-domain. Attempts to construct finite-valued monotone maps approximating identity fail by the rank-one cone obstruction argument. This fully answers the posed problem with this geometry example.

Technical innovations

• Reduction of the RB approximation property to existence of finite-valued C-monotone maps approximating the identity operator on compact subsets of int C.
• Use of Lipschitz epigraph representation for cone-upper sets combined with Rademacher's differentiability theorem and integration by parts to translate order-theoretic approximation into an analytic rank-one cone inclusion problem.
• Identification of the rank-one operator cone RK = cone{v⊗ℓ : v ∈ C, ℓ ∈ C*} in End(V) and characterization of simplicial cones as precisely those for which the identity operator lies in RK.
• Explicit construction of finite-range deflations approximating identity on (−R^d_+) ⊥ via coordinatewise step functions on dyadic grids to show the RB property for simplicial cones.

Limitations

• The results are restricted to finite-dimensional real vector spaces; infinite-dimensional analogs are not addressed.
• The study applies to proper closed convex cones; non-proper cones or more general partially ordered structures are not considered.
• No adversarial robustness or noisy perturbation analyses, since this is a pure mathematical domain theory and convex geometry paper.
• The non-existence results hold only under exact uniform approximation assumptions; approximate or randomized variants are not studied.
• The techniques rely heavily on Euclidean geometry and linear isomorphisms; extensions to other ordered topological vector spaces are unclear.

Open questions / follow-ons

• Is there a characterization or construction of FS-domains that are not RB-domains outside the family of cone domains associated with non-simplicial cones?
• How do these geometric classification results extend to infinite-dimensional cones or topological vector spaces?
• Can approximate or randomized finite-range maps approximate identity on non-simplicial cones under relaxed assumptions?
• What are the implications of this distinction between FS and RB domains for computational domain theory and semantics of programming languages beyond finite approximations?

Why it matters for bot defense

While this paper is theoretical and mathematical, its results elucidate fundamental limitations on certain types of finite approximation processes in ordered structures defined by cones. From a bot-defense or CAPTCHA perspective, the methods involving approximations by monotone maps and finite range approximants connect conceptually to approaches in designing finite-state or finite-resolution defenses against automated adversaries. The rank-one cone obstruction showcases how geometric constraints fundamentally limit approximation abilities, which might inform the design of challenge-response or approximation-based bot-detection mechanisms that rely on hardness of monotone finite approximations. Moreover, the careful analytic decomposition of approximations via Lipschitz epigraphs and integration by parts suggests mathematical tools that could be adapted to analyze continuous relaxations or probabilistic approximations in machine-learned bot detectors. However, the paper does not directly propose algorithms or datasets for bot-defense; its main value is in providing deep theoretical insight into finite approximation boundaries in ordered structures.

Cite

@article{arxiv2607_02251,
  title={ Cone domains separate FS-domains from RB-domains },
  author={ Yuxu Chen },
  journal={arXiv preprint arXiv:2607.02251},
  year={ 2026 },
  url={https://arxiv.org/abs/2607.02251}
}

Read the full paper

• arXiv:2607.02251 (abstract)
• arXiv:2607.02251 (PDF)