ProofAtlas

Model theory · real exponentiation · decidability · transcendence

Tarski’s Exponential-Function Problem

Collaboration beta

The problem asks for an algorithm that always decides whether any first-order statement about the real numbers with exponentiation is true. The packet narrows the missing step to exact comparisons between separately described exponential roots, or to a strong new finiteness theorem; it does not supply either one.

∃M∀ϕ(M(ϕ)↓∧[M(ϕ)=YES⇔ℝexp⊨ϕ])?
Known results and sources
Research cover for Tarski’s exponential-function problem, showing the real exponential field and an unresolved algorithmic yes-or-no question.
Can one algorithm decide every first-order sentence about the ordered real exponential field? The packet reports reductions and special cases, not a full decision procedure.

Research problem

Exact mathematical statement

Construct a Turing machine which, for every first-order sentence ϕ\varphi in the language of ordered exponential fields, halts and returns whether

ℝexp⊨ϕ.\mathbb R_{\exp} ⊨ \varphi.

The submitted source explicitly reports no full unconditional decision procedure.

Problem infographic

Problem at a glance

Three-region deterministic explainer of Tarski’s exponential-function problem: the universal decision target, source-reported decidable footholds, and the open cross-root equality barrier.
The packet reduces the decision problem to exact equality certificates and reports rank-zero, one-generator, and conditional finite-core footholds; exact two-graph comparison remains open.

Current mathematical picture

Where work on Tarski’s Exponential-Function Problem stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureVanishing-derivation ideal stabilization

The source gives D=sd/dsD=s\,d/ds and f=sf=s: every repeated derivative vanishes at zero and the ideals stabilize, but the germ is not identically zero. A route using a local parameter and a derivation nonvanishing at the point may still contribute after exact zeroth-order equality is controlled.

Route status · Narrowed route
Main reductionExact name-equality criterion

The source reports full decidability equivalent to computably enumerating true equalities of rational Khovanskii names.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeImplement a sound exact-rational verifier and nontrivial recursively checkable certificate families for equality of rational Khovanskii names.Task status · Ready to work on

Work mapped so far

Tarski’s Exponential-Function Problem in numbers

845retained lines of mathematical investigation845 in the current working snapshot
Argument development
646 · 76%
Explored or eliminated routes
23 · 3%
Computational analysis
70 · 8%
Open obligations
49 · 6%
Definitions and setup
57 · 7%
7selected mapped statements1routes investigated5open questions5contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byAlfred TarskiStatus checkedAug 29, 2026
Source recordedAug 29, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Tarski’s Exponential-Function ProblemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can every first-order question about the real exponential field be decided algorithmically? — Depends on missing premiseCan every first-orderquestion about the realexponential…Current reduction — Depends on missing premiseCurrent reductionExact name-equality criterion — Depends on missing premiseExact name-equalitycriterionClosing target — Depends on missing premiseClosing targetFinite-core conditional theorem — Depends on missing premiseFinite-core conditionaltheoremOne shared generator — Depends on missing premiseOne shared generatorVanishing-derivation shortcut invalid — Depends on missing premiseVanishing-derivationshortcut invalidVanishing-derivation ideal stabilization — stoppedVanishing-derivation idealstabilizationImplement a sound exact-rational verifier and nontrivial recursively checkable certificate families for equality of rational Khovanskii names. — OpenImplement a soundexact-rational verifier andnontrivial…Classify every pair from two separately isolated one-generator root sets by a common-root certificate or a positive rational cross-separation. — OpenClassify every pair from twoseparately isolatedone-generator…Prove or refute finite-dimensionality of the global exponential-differential anomaly space, the exact SYZ-FIN premise of the conditional finite-core route. — OpenProve or refutefinite-dimensionality of theglobal…Two-Graph Separation — OpenTwo-Graph SeparationFinite anomaly support — OpenFinite anomaly support
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeVanishing-derivation ideal stabilization

The source gives D=sd/dsD=s\,d/ds and f=sf=s: every repeated derivative vanishes at zero and the ideals stabilize, but the germ is not identically zero. A route using a local parameter and a derivation nonvanishing at the point may still contribute after exact zeroth-order equality is controlled.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Implement a sound exact-rational verifier and nontrivial recursively checkable certificate families for equality of rational Khovanskii names.Suggested move: Specify certificate syntax for common roots, ideal identities, graph isomorphisms, rank reduction, and normalized-curve identities, then test the verifier on the current work’s rank-one and rank-two regression names.
Ready to work on
02
Classify every pair from two separately isolated one-generator root sets by a common-root certificate or a positive rational cross-separation.Suggested move: Fix degree and height bounds on normalized curve data and test a complete classifier against arbitrary mixed-degree polynomial relations in log 2 and log 3.
Ready to work on
03
Two-Graph Separation

The first unresolved primitive gate compares roots from two separately specified one-generator systems.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Finite anomaly support

Finite-dimensionality of the global anomaly space remains the exact open premise for the finite-core route.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Prove or refute finite-dimensionality of the global exponential-differential anomaly space, the exact SYZ-FIN premise of the conditional finite-core route.Suggested move: Classify primitive defect-one anomaly extensions and determine whether real order forces them into finitely many effectively enumerable families or permits unbounded towers.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 29, 2026
Current statusOpen problem

It remains unconditionally unknown whether the complete first-order theory of the ordered real exponential field is decidable. Model completeness and o-minimality are known, and the real form of Schanuel’s conjecture implies decidability, but the located 2026 advance remains conditional and supplies no unconditional decision procedure or undecidability proof.

[1][2]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. PreprintBerarducci and Gallinaro proved, assuming the real form of Schanuel’s conjecture, that the complete theory of the real exponential field is axiomatized by definably complete exponential fields satisfying exp′…[2]
  2. Peer reviewedKrapp proved that, assuming Schanuel’s conjecture, the prime model of real exponentiation embeds into every o-minimal EXP-field; the embedding was not shown elementary, so the neighboring Transfer Conjecture…[1]
  3. Authoritative summaryWilkie proved model completeness of the theory of the real exponential field, and Macintyre and Wilkie proved decidability assuming the real form of Schanuel’s conjecture. These results do not settle…[1][2]
  4. Authoritative summaryTarski asked whether his decidability result for the real closed field extends to the complete theory of the real field with its standard total exponential function.[1]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusTarski’s Exponential-Function Problem
Dependency or reductionReal Schanuel conjecture

The real form of Schanuel’s conjecture implies decidability of the real exponential field, but it remains an unproved transcendence conjecture and therefore gives only a conditional resolution.

[1][2]
Logical consequenceTransfer Conjecture for o-minimal EXP-fields

A positive answer to the Transfer Conjecture for o-minimal EXP-fields would imply decidability of the real exponential field; the Transfer Conjecture itself remains open in the cited source.

[1]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA machine-checked encoding of first-order syntax, satisfaction, and decidability for the ordered real exponential field.
  • Formalization targetFormal interfaces for the exact model-completeness and conditional Schanuel-dependent results used to describe the known frontier.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements5 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma2 of 72
  • special case1 of 71
  • negative result1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementCan every first-order question about the real exponential field be decided algorithmically?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact name-equality criterionintermediate
  • retained route statementOne shared generatorintermediate
  • retained route statementFinite-core conditional theoremintermediate
  • retained route statementVanishing-derivation shortcut invalidintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureVanishing-derivation ideal stabilizationreported failure
  • Research targetImplement a sound exact-rational verifier and nontrivial recursively checkable certificate families for equality of rational Khovanskii names.open
  • Research targetClassify every pair from two separately isolated one-generator root sets by a common-root certificate or a positive rational cross-separation.open
  • Research targetProve or refute finite-dimensionality of the global exponential-differential anomaly space, the exact SYZ-FIN premise of the conditional finite-core route.open
  • Research targetTwo-Graph Separationopen
  • Research targetFinite anomaly supportopen
  • Narrowed routeVanishing-derivation ideal stabilizationThe source gives D=sd/dsD=s\,d/ds and f=sf=s: every repeated derivative vanishes at zero and the ideals stabilize, but the germ is not identically zero. A route using a local parameter and a derivation nonvanishing at the point may still contribute after exact zeroth-order equality is controlled.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeImplement a sound exact-rational verifier and nontrivial recursively checkable certificate families for equality of rational Khovanskii names.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointImplement a sound exact-rational verifier and nontrivial recursively checkable certificate families for equality of rational Khovanskii names.

Tarski’s Exponential-Function Problem · ready to start

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Research contextPrepared context for any AI agent

The problem asks for an algorithm that always decides whether any first-order statement about the real numbers with exponentiation is true. The current work narrows the missing step to exact comparisons between separately described exponential roots, or to a strong new finiteness theorem; it does not supply either one.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references2 cited works · next context review by Nov 29, 2026

The mathematical context was checked on Aug 29, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Embedding the prime model of real exponentiation into o-minimal exponential fieldspeer reviewed result · Lothar Sebastian Krapp · Bulletin of the London Mathematical Society · 2024-03 · DOI 10.1112/blms.12972 · accessed Aug 29, 2026
  2. 2
    On the elementary theory of the real exponential fieldpreprint · Alessandro Berarducci, Francesco Gallinaro · arXiv · 2026-06-23 (v2) · ARXIV 2603.08365 · accessed Aug 29, 2026

Important qualifications

  • The dated open status combines an explicit peer-reviewed 2024 status statement with the scope of a June 2026 preprint and a scoped current search; it does not prove that no unindexed result exists.
  • The 2026 Berarducci–Gallinaro result is a preprint and is not presented as peer-reviewed.
  • This minimal record does not claim that no formalization, computation, prize, or maintained-list entry exists; those categories were left empty rather than turning a scoped search into a nonexistence claim.

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