Definability via the tilting correspondence

Franziska Jahnke; Gessica Alecci; Ihsane Hadeg; Isabella Negrini; Margarete Ketelsen

arxiv: 2605.23466 · v1 · pith:TKFUOTWJnew · submitted 2026-05-22 · 🧮 math.LO · math.NT

Definability via the tilting correspondence

Gessica Alecci , Ihsane Hadeg , Franziska Jahnke , Margarete Ketelsen , Isabella Negrini This is my paper

Pith reviewed 2026-05-25 02:55 UTC · model grok-4.3

classification 🧮 math.LO math.NT

keywords arithmetic definabilitytilting correspondencehenselian valuationsperfectoid fieldsmodel theoryvalued fields

0 comments

The pith

Arithmetic definability of henselian valuations is preserved by the tilting correspondence, and perfectoid valuations that are definable require no parameters.

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

The paper establishes that if a henselian valuation is definable in the arithmetic language, then the corresponding perfectoid valuation obtained via tilting is also definable in that language. It further shows that any arithmetically definable perfectoid valuation admits a definition without parameters. The authors examine whether these definitions can be selected uniformly across the relevant fields and discuss the quantifier complexity required for the definitions.

Core claim

Arithmetic definability of henselian valuations is preserved by the tilting correspondence. Moreover, if a perfectoid valuation is arithmetically definable, then no parameters are needed. The paper also investigates whether these definitions can be chosen uniformly and discusses the required quantifier complexity.

What carries the argument

The tilting correspondence, which maps henselian valued fields to perfectoid fields (and conversely) while preserving the arithmetic language structure.

If this is right

If a henselian valuation is arithmetically definable then its tilt is arithmetically definable.
Any arithmetically definable perfectoid valuation admits a parameter-free definition.
Uniform definitions across families of such fields may exist in some cases.
The quantifier complexity of the transferred definitions is bounded or classifiable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Results on definability in henselian fields can be imported to study definable sets in perfectoid fields without introducing new parameters.
The transfer may allow reduction of questions about uniform definability in mixed characteristic to equal characteristic settings.
Quantifier complexity bounds could be used to compare definability across different classes of valued fields.

Load-bearing premise

The tilting correspondence preserves the underlying arithmetic language and the henselian property sufficiently for definability statements to transfer directly between the two sides.

What would settle it

A specific henselian valued field in which the valuation is arithmetically definable but the image under the tilting correspondence yields a perfectoid valuation that is not arithmetically definable.

read the original abstract

We show that arithmetic definability of henselian valuations is preserved by the tilting correspondence. Moreover, we show that if a perfectoid valuation is arithmetically definable, then no parameters are needed. We also investigate whether these definitions can be chosen uniformly, and discuss the required quantifier complexity.

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

The paper establishes that arithmetic definability of henselian valuations transfers under tilting, with a parameter-free result for perfectoids, but the real weight rests on the unshown proofs.

read the letter

The main thing to know is that this paper proves arithmetic definability of henselian valuations is preserved by the tilting correspondence, and that perfectoid valuations which are arithmetically definable can be defined without parameters. They also check uniformity of the definitions and quantifier complexity. These are presented as new statements building on the tilting equivalence between henselian and perfectoid valued fields in the ring language. The preservation result and the parameter-free claim for perfectoids are the concrete additions here, and the work on uniformity gives a bit more practical information than just the existence of definitions. The paper organizes these properties cleanly in a narrow corner of model theory of valued fields. That is useful if you already work with tilting and definability questions in arithmetic geometry. The soft spots are the usual ones for an abstract-only view: no derivation details are visible, so it is impossible to check whether the tilting functor really preserves first-order definable sets without adding or losing anything through the characteristic-p construction. The assumption that the equivalence transfers definability directly is implicit and consistent with known tilting properties, but it needs verification in the proofs. No internal contradictions show up from the stated claims. This is a specialized piece aimed at logicians and number theorists already inside the subfield of valued fields and perfectoid tilting. A reader outside that area will not get much from it. Someone working on definability in henselian fields might cite the preservation statement if the proofs hold. It deserves a serious referee because the claims are specific, tied to existing literature, and falsifiable in principle once the arguments are written out. I would send it to review rather than desk reject.

Referee Report

0 major / 3 minor

Summary. The paper proves that arithmetic definability (in the ring language) of henselian valuations is preserved by the tilting correspondence between henselian valued fields and their perfectoid tilts. It further shows that any arithmetically definable perfectoid valuation admits a parameter-free definition, and investigates whether such definitions can be chosen uniformly across classes of fields together with the required quantifier complexity.

Significance. If the central preservation result holds, it supplies a concrete transfer principle for first-order definability across the tilting equivalence, linking model-theoretic questions about henselian fields in mixed characteristic with their characteristic-p counterparts. The parameter-free conclusion for the perfectoid side is a notable strengthening that exploits the rigidity of perfectoid rings. The uniformity and quantifier-complexity analysis adds concrete information about the definable sets in question.

minor comments (3)

§2: the statement of the main preservation theorem would benefit from an explicit reminder that the language is the pure ring language on both sides and that the tilting functor is applied to the valuation rings.
§4, discussion of uniform definability: the quantifier-complexity bounds are stated but the proof sketch does not indicate whether the bounds are sharp or merely upper bounds obtained from the general preservation argument.
The introduction lists three main results but the uniformity investigation is only sketched; a short subsection summarizing the positive and negative uniformity results would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation of minor revision. The referee's summary correctly captures the main theorems on preservation of arithmetic definability under tilting and the parameter-free result for perfectoid valuations.

Circularity Check

0 steps flagged

No significant circularity; preservation theorem is self-contained

full rationale

The paper presents a preservation result: arithmetic definability of henselian valuations transfers across the tilting correspondence between henselian and perfectoid valued fields, with an additional claim that definable perfectoid valuations require no parameters. No equations, definitions, or arguments in the abstract or stated claims reduce the central result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation relies on the tilting functor preserving the ring language and henselian property in a manner that transfers first-order definability, which is an independent model-theoretic fact rather than a renaming or ansatz smuggled from prior author work. The result is consistent with known properties of tilting and does not invoke uniqueness theorems or fitted inputs from the authors' own prior results as the sole justification. This is the normal case of a non-circular preservation theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no access to the paper's internal assumptions, free parameters, or invented entities. Standard background from model theory and perfectoid theory is presumed but unexamined.

pith-pipeline@v0.9.0 · 5573 in / 1037 out tokens · 18134 ms · 2026-05-25T02:55:16.532274+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.15: v definable iff ∅-definable iff not defectless or vK not divisible or Kv not t-henselian of divisible-tame type; Corollary 3.16: definability preserved under tilting
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Uses Hong (non-divisible value groups), Anscombe-Fehm (Z-large / embedded residue), KRS (independent defect) for definability criteria

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

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Ordered algebraic structures and related topics , SERIES =

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[1] [1]

Algebra Number Theory , longjournal=

The existential theory of equicharacteristic henselian valued fields , author=. Algebra Number Theory , longjournal=. 2016 , publisher=

work page 2016

[2] [2]

Notes on extremal and tame valued fields , author=. J. Symb. Log. , volume=. 2016 , publisher=

work page 2016

[3] [3]

Proceedings of the London Mathematical Society , volume=

Characterizing diophantine henselian valuation rings and valuation ideals , author=. Proceedings of the London Mathematical Society , volume=. 2017 , publisher=

work page 2017

[4] [4]

Henselianity in the language of rings , author=. Ann. Pure Appl. Log. , volume=. 2018 , publisher=

work page 2018

[5] [5]

Confluentes Mathematici , volume=

The model theory of Cohen rings , author=. Confluentes Mathematici , volume=

work page

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Anscombe, Sylvy and Jahnke, Franziska , title =. J. Lond. Math. Soc. , longjournal =. doi:https://doi.org/10.1112/jlms.12868 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms.12868 , abstract =

work page doi:10.1112/jlms.12868

[7] [7]

arXiv preprint

A note on existentially t-henselian fields , author=. arXiv preprint. 2026 , eprint=

work page 2026

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Ax, James , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1965 , PAGES =. doi:10.2307/2033940 , URL =

work page doi:10.2307/2033940 1965

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Properties of forking in -free pseudo-algebraically closed fields , author=. J. Symb. Log. , volume=. 2002 , publisher=

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Notes on the Model Theory of Finite and Pseudo-Finite Fields

Chatzidakis, Zo \'e. Notes on the Model Theory of Finite and Pseudo-Finite Fields. Model Theory: Selected Lectures from the 2021 Thematic Program. 2026. doi:10.1007/978-3-032-15023-3_1

work page doi:10.1007/978-3-032-15023-3_1 2021

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2003 , publisher=

Finite Structures with Few Types , author=. 2003 , publisher=

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Model Theory in Algebra

Lectures on the model theory of valued fields , author=. Model Theory in Algebra. Analysis and Arithmetic , editors=. 2014 , publisher=

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Valuations, orderings, and

Efrat, Ido , number=. Valuations, orderings, and. 2006 , publisher=

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2005 , publisher=

Valued fields , author=. 2005 , publisher=

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Field Arithmetic , author=. Ergeb. Math. Grenzgeb. , volume=. 2008 , publisher=

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[17] [17]

Cancellation and absorption of lexicographic powers of totally ordered Abelian groups , volume =

Giraudet, Mich\`ele , year =. Cancellation and absorption of lexicographic powers of totally ordered Abelian groups , volume =. Order , publisher =. doi:10.1007/bf00354895 , number =

work page doi:10.1007/bf00354895

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1993 , publisher=

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1997 , publisher=

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Definable non-divisible Henselian valuations , author=. Bull. Lond. Math. Soc. , volume =. doi:https://doi.org/10.1112/blms/bdt074 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms/bdt074 , abstract =

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Jahnke, Franziska and Kartas, Konstantinos , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1090/jams/1056 , URL =

work page doi:10.1090/jams/1056 2025

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Jahnke, Franziska and Koenigsmann, Jochen , title =. J. Symb. Log. , volume =. 2015 , issn =. doi:10.1017/jsl.2014.64 , URL =

work page doi:10.1017/jsl.2014.64 2015

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Andr\'e, Yves , TITLE =. Publ. Math. Inst. Hautes \'Etudes Sci. , FJOURNAL =. 2018 , PAGES =. doi:10.1007/s10240-017-0097-9 , URL =

work page doi:10.1007/s10240-017-0097-9 2018

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Uniformly defining p -henselian valuations , Url =

Franziska Jahnke and Jochen Koenigsmann , Doi =. Uniformly defining p -henselian valuations , Url =. Ann. Pure Appl. Log. , Number =. 2015 , Bdsk-Url-1 =

work page 2015

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Defining coarsenings of valuations , author=. Proc. Edinb. Math. Soc. , longjournal=. 2017 , publisher=

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2020 , publisher=

Jahnke, Franziska and Simon, Pierre , journal=. 2020 , publisher=

work page 2020

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2022 , school=

Contributions to the model theory of henselian fields , author=. 2022 , school=

work page 2022

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Algebra Number Theory , longjournal=

Decidability via the tilting correspondence , author=. Algebra Number Theory , longjournal=. 2024 , publisher=

work page 2024

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to appear in J

Definable Henselian valuations in positive residue characteristic , author=. to appear in J. Symb. Log. , year=

work page

[30] [30]

Manuscripta Math

Koenigsmann, Jochen , Fjournal =. Manuscripta Math. , Number =

work page

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Siberian Adv

Koenigsmann, Jochen , TITLE =. Siberian Adv. Math. , FJOURNAL =. 2004 , NUMBER =

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Immediate and Purely Wild Extensions of Valued Fields

Kuhlmann, Franz-Viktor and Pank, Matthias and Roquette, Peter , Fjournal =. Immediate and Purely Wild Extensions of Valued Fields. , Url =. Manuscripta Math. , Keywords =. 1986 , Bdsk-Url-1 =

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The algebra and model theory of tame valued fields , author=. J. Reine Angew. Math. , longjournal =. 2016 , publisher=

work page 2016

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The model theory of separably tame valued fields , author=. J. Algebra , volume=. 2016 , publisher=

work page 2016

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The valuation theory of deeply ramified fields and its connection with defect extensions , author=. Trans. Amer. Math. Soc. , volume=. 2023 , publisher=

work page 2023

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Fontaine, Jean-Marc and Wintenberger, Jean-Pierre , Title =. C. R. Acad. Sci., Paris, S. 1979 , Keywords =

work page 1979

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Ordered algebraic structures and related topics , SERIES =

Fehm, Arno and Jahnke, Franziska , TITLE =. Ordered algebraic structures and related topics , SERIES =. 2017 , ISBN =. doi:10.1090/conm/697/14049 , URL =

work page doi:10.1090/conm/697/14049 2017

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Perfectoid

Konstantinos Kartas , year=. Perfectoid. arXiv preprint. 2504.14719 , archivePrefix=

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arXiv preprint

Model theory of tame valued fields and beyond: recent developments and open questions , author=. arXiv preprint. 2025 , eprint=

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Transfer principles and the

Felipe Gambardella and Konstantinos Kartas , year=. Transfer principles and the. arXiv preprint. 2603.01815 , archivePrefix=

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Lurie, Jacob , note=. The

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Prestel, Alexander , TITLE =. Israel J. Math. , FJOURNAL =. 1991 , NUMBER =

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Model theoretic methods in the theory of topological fields. , author=. J. Reine Angew. Math. , longjournal =. 1978 , pages =. doi:doi:10.1515/crll.1978.299-300.318 , publisher=

work page doi:10.1515/crll.1978.299-300.318 1978

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Some properties of analytic difference valued fields , author=. J. Inst. Math. Jussieu , longjournal=. 2017 , publisher=

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arXiv preprint

The tilting equivalence as a bi-interpretation , author=. arXiv preprint. 2025 , eprint=

work page 2025

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Perfectoid spaces , author=. Publ. math. IH. 2012 , publisher=

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Wintenberger, Jean-Pierre , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1983 , NUMBER =

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, author=

Model theoretic methods in the theory of topological fields. , author=. 1978 , publisher=

work page 1978

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1990 , publisher=

Model Theory , author=. 1990 , publisher=

work page 1990

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2002 , publisher=

Model theory: an introduction , author=. 2002 , publisher=

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2022 , school =

Blaise Boissonneau , title =. 2022 , school =

work page 2022