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Weakly o-minimal fields have the exchange property but not generic differentiability

T0 review · 1 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Weakly o-minimal ordered fields have the exchange property but admit expansions without generic differentiability.

desk verdict The paper settles the two open questions with an explicit counterexample to generic differentiability and a separate proof of the exchange property. read the letter →

arxiv 2606.08527 v1 pith:X2PCJDRE submitted 2026-06-07 math.LO

Will Johnson This is my paper · ORCID
classification math.LO
keywords weaklyo-minimalorderedfieldsexchangepropertygenericdifferentiabilitymodeltheoryalgebraicclosureo-minimality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper resolves two open questions on weakly o-minimal fields by proving that algebraic closure satisfies the exchange property in any weakly o-minimal theory of ordered fields. It constructs an ordered field K together with a function f from K to K such that the expansion by f has a weakly o-minimal complete theory, yet f is nowhere differentiable. This shows that weak o-minimality does not force generic differentiability. A reader would care because the construction separates two candidate tameness properties that had been asked about together.

What carries the argument

The specific constructed expansion (K, +, ·, ≤, f) that has a weakly o-minimal theory while serving as a counterexample to generic differentiability.

What would settle it

A demonstration that the constructed function f is differentiable at some point, or a proof that the theory of the expansion is not weakly o-minimal.

Watch

Extended reading notes

Core claim

The central claim is that algebraic closure has the exchange property in every weakly o-minimal theory of ordered fields. Separately, there exists an ordered field K and function f such that the structure (K, +, ·, ≤, f) is weakly o-minimal but f is nowhere differentiable.

Load-bearing premise

The constructed expansion by the function f has a complete theory that is weakly o-minimal.

Editorial extensions

If this is right

  • Algebraic closure satisfies the exchange property in any weakly o-minimal theory of ordered fields.
  • Generic differentiability fails in some weakly o-minimal expansions of ordered fields.
  • There exist nowhere differentiable functions definable in some weakly o-minimal structures on ordered fields.
  • The exchange property can hold while generic differentiability fails under weak o-minimality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The separation suggests that weak o-minimality permits more pathological definable functions than stronger tameness conditions.
  • Analogous constructions could separate other regularity properties such as continuity or monotonicity from weak o-minimality.
  • It remains open whether the exchange property constrains the possible definable functions in any further concrete way.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The paper constructs an ordered field (K, +, ·, ≤) and a function f: K → K such that the expansion (K, +, ·, ≤, f) has a weakly o-minimal complete theory, yet f is nowhere differentiable. This provides a negative answer to the question of generic differentiability for weakly o-minimal fields. An appendix proves that algebraic closure has the exchange property in any weakly o-minimal theory of ordered fields, giving a positive answer to the exchange question.

Significance. If the construction and its verification hold, the results resolve two open questions posed by Macpherson, Marker, and Steinhorn. The explicit construction of K and f supplies a concrete counterexample to generic differentiability, while the appendix delivers a general, independent proof of the exchange property. Both are strengths for a logic paper in this area.

major comments (1)
  1. [Construction of K and f] Construction section (and any dedicated verification subsection): the claim that Th(K, +, ·, ≤, f) is weakly o-minimal is load-bearing for the counterexample to generic differentiability. The argument must show that every definable subset of K^n (including those involving multiple applications of f or comparisons with parameters) is a finite union of intervals and points; any gap here would invalidate the counterexample.
minor comments (2)
  1. Introduction: ensure the precise statements of the two open questions from Macpherson–Marker–Steinhorn are quoted or paraphrased with citations for clarity.
  2. [Appendix] Appendix: the exchange proof is presented as general; a brief remark on whether it applies verbatim to the constructed expansion would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report and for recognizing the significance of resolving the two questions from Macpherson, Marker, and Steinhorn. We address the single major comment below and will revise the manuscript to strengthen the relevant verification.

read point-by-point responses

  1. Referee: [Construction of K and f] Construction section (and any dedicated verification subsection): the claim that Th(K, +, ·, ≤, f) is weakly o-minimal is load-bearing for the counterexample to generic differentiability. The argument must show that every definable subset of K^n (including those involving multiple applications of f or comparisons with parameters) is a finite union of intervals and points; any gap here would invalidate the counterexample.

    Authors: We agree that the verification must be fully rigorous and cover all definable sets in K^n. The current manuscript sketches the argument by reducing definability to the specific piecewise-linear form of f and the o-minimality of the underlying field, but we acknowledge that the treatment of iterated applications of f and parameter comparisons can be made more explicit. In the revision we will add a dedicated subsection that proves, by induction on formula complexity, that every definable subset of K^n is a finite union of intervals and points, explicitly handling multiple compositions of f. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: explicit construction with direct verification of weak o-minimality

full rationale

The paper's central result is an explicit construction of an ordered field K and function f, together with a direct argument that Th(K,+,·,≤,f) is weakly o-minimal while f is nowhere differentiable. The appendix separately proves the exchange property for algebraic closure in any weakly o-minimal theory of ordered fields. Neither the construction nor the verification reduces to a fitted parameter, a self-citation chain, or a definitional equivalence; the weak-o-minimality claim is established by inspecting definable sets in the expansion, which is independent content rather than a renaming or self-referential step. No load-bearing premise collapses to its own input.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The central claims rest on standard background axioms of ordered fields and model theory together with the existence of one specific constructed structure.

assumptions (1)
  • standard math Standard first-order axioms for ordered fields and the definition of weak o-minimality
    Invoked throughout as the ambient theory in which the construction and exchange property are stated.
invented entities (1)
  • The ordered field K together with the added function f
    purpose: To serve as a counterexample showing that weak o-minimality does not imply generic differentiability
    Postulated via explicit construction in the paper; no independent evidence outside the construction is given.

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Cite this review

Pith. "Pith review of Weakly o-minimal fields have the exchange property but not generic differentiability." pith.science (2026). https://pith.science/paper/X2PCJDRE

@misc{pith2026260608527,
  author       = {Pith},
  title        = {Pith review of: Weakly o-minimal fields have the exchange property but not generic differentiability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2PCJDRE}},
  note         = {Machine review of arXiv:2606.08527}
}
abstract

We answer two open questions about weakly o-minimal fields posed by Macpherson, Marker, and Steinhorn: whether weakly o-minimal fields have the exchange property and whether they have generic differentiability. We construct an ordered field $(K,+,\cdot,\le)$ and a function $f : K \to K$ such that the expansion $(K,+,\cdot,\le,f)$ has a weakly o-minimal complete theory but $f$ is nowhere differentiable. In an appendix, we prove that algebraic closure has the exchange property in any weakly o-minimal theory of ordered fields.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 2 canonical work pages

  1. [1]

    On Schanuel’s conjectures.Ann

    James Ax. On Schanuel’s conjectures.Ann. Math., 93(2):252–268, March 1971

    1971

  2. [2]

    Baisalov and B

    Y. Baisalov and B. Poizat. Paires de structures o-minimales.J. Symbolic Logic, 63:570– 578, 1998

    1998

  3. [3]

    Martin Bays, Jonathan Kirby, and A. J. Wilkie. A Schanuel property for exponentially transcendental powers.Bull. London Math. Soc., 42(5):917–922, October 2010

    2010

  4. [4]

    Cherlin and M

    G. Cherlin and M. A. Dickmann. Real closed rings II: model theory.Annals of Pure and Applied Logic, 25:213–231, 1983

    1983

  5. [5]

    M. A. Dickmann. Elimination of quantifiers for ordered valuation rings.J. Symbolic Logic, 52:116–128, 1987

    1987

  6. [6]

    Tame topology over definable uniform structures

    Alfred Dolich and John Goodrick. Tame topology over definable uniform structures. Notre Dame J. Formal Logic, 63(1):51–79, 2022

    2022

  7. [7]

    Dp-minimality: basic facts and ex- amples.Notre Dame Journal of Formal Logic, 52(3):267–288, 2011

    Alfred Dolich, John Goodrick, and David Lippel. Dp-minimality: basic facts and ex- amples.Notre Dame Journal of Formal Logic, 52(3):267–288, 2011

    2011

  8. [8]

    Pillay’s conjecture for groups definable in weakly o-minimal non- valuational structures.Bull

    Pantelis Eleftheriou. Pillay’s conjecture for groups definable in weakly o-minimal non- valuational structures.Bull. London Math. Soc., 53(4):1205–1219, August 2021

    2021

Show all 29 references
  1. [9]

    Projectionsofsemi-analyticsets.Functional Analysis and its Applications, 2:282–291, 1968

    A.Gabrielov. Projectionsofsemi-analyticsets.Functional Analysis and its Applications, 2:282–291, 1968

    1968

  2. [10]

    Interpretable fields in various valued fields.Advances in Mathematics, 404, Part A:108408, August 2022

    Yatir Halevi, Assaf Hasson, and Ya’acov Peterzil. Interpretable fields in various valued fields.Advances in Mathematics, 404, Part A:108408, August 2022

    2022

  3. [11]

    Cell decompositions of C-minimal structures

    Deirdre Haskell and Dugald Macpherson. Cell decompositions of C-minimal structures. Annals of Pure and Applied Logic, 66:113–162, 1994

    1994

  4. [12]

    A version ofo-minimality for thep-adics.J

    Deirdre Haskell and Dugald Macpherson. A version ofo-minimality for thep-adics.J. Symbolic Logic, 62(4):1075–1092, December 1997

    1997

  5. [13]

    Groups definable in local fields and pseudo-finite fields.Israel J

    Ehud Hrushovski and Anand Pillay. Groups definable in local fields and pseudo-finite fields.Israel J. Math., 85(1-3):203–262, 1994

    1994

  6. [14]

    The canonical topology on dp-minimal fields.Journal of Mathematical Logic, 18(2):1850007, 2018

    Will Johnson. The canonical topology on dp-minimal fields.Journal of Mathematical Logic, 18(2):1850007, 2018

    2018

  7. [15]

    arXiv:2403.17478v2 [math.LO], March 2024

    Will Johnson.C-minimal fields have the exchange property. arXiv:2403.17478v2 [math.LO], March 2024

    2024

  8. [16]

    Generic differentiability and P-minimal groups

    Will Johnson. Generic differentiability and P-minimal groups. arXiv:2404.17234v3 [math.LO], April 2024. 34

    2024

  9. [17]

    G. O. Jones and A. J. Wilkie. Locally polynomially bounded structures.Bulletin of the London Math. Soc., 40(2):239–248, April 2008

    2008

  10. [18]

    On the decidability of the real field with a generic power function.J

    Gareth Jones and Tamara Servi. On the decidability of the real field with a generic power function.J. Symbolic Logic, 76(4):1418–1428, December 2011

    2011

  11. [19]

    Differentiation inP-minimal structures and a p-adic Local Monotonicity Theorem.J

    Tristan Kuijpers and Eva Leenknegt. Differentiation inP-minimal structures and a p-adic Local Monotonicity Theorem.J. Symbolic Logic, 79(4):1133–1147, 2014

    2014

  12. [20]

    Weakly o-minimal struc- tures and real closed fields.Trans

    Dugald Macpherson, David Marker, and Charles Steinhorn. Weakly o-minimal struc- tures and real closed fields.Trans. Amer. Math. Soc., 352(12):5435–5483, 2000

    2000

  13. [21]

    On variants ofo-minimality.Annals of Pure and Applied Logic, 79:165–209, 1996

    Dugald Macpherson and Charles Steinhorn. On variants ofo-minimality.Annals of Pure and Applied Logic, 79:165–209, 1996

    1996

  14. [22]

    Cell decomposition and dimension functions in first-order topological structures.Proc

    Larry Mathews. Cell decomposition and dimension functions in first-order topological structures.Proc. London Math. Soc., 70(3):1–32, 1995

    1995

  15. [23]

    Expansions of the real field with power functions.Ann

    Chris Miller. Expansions of the real field with power functions.Ann. Pure and Applied Logic, 68:79–94, 1994

    1994

  16. [24]

    Infinite differentiability in polynomially bounded o-minimal structures

    Chris Miller. Infinite differentiability in polynomially bounded o-minimal structures. Proc. Amer. Math. Soc., 123(8):2551–2555, 1995

    1995

  17. [25]

    Peterzil, A

    Y. Peterzil, A. Pillay, and S. Starchenko. Definably simple groups in o-minimal struc- tures.Trans. Amer. Math. Soc., 352(10):4397–4419, 2000

    2000

  18. [26]

    Tame topology over dp-minimal structures.Notre Dame J

    Pierre Simon and Erik Walsberg. Tame topology over dp-minimal structures.Notre Dame J. Formal Logic, 60(1):61–76, 2019

    2019

  19. [27]

    Number 248 in London Mathematical Society Lecture Note Series

    Lou van den Dries.Tame Topology and O-minimal Structures. Number 248 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1998

    1998

  20. [28]

    Weakly o-minimal nonvaluational structures.Ann

    Roman Wencel. Weakly o-minimal nonvaluational structures.Ann. Pure and Applied Logic, 154:139–162, 2008

    2008

  21. [29]

    A. J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function.Journal of the American Mathematical Society, 9(4):1051–1094, Oct 1996. 35

    1996

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