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Transcendence theory · exponential algebra · algebraic independence

Schanuel’s Conjecture

Collaboration beta

If z1,…,zn are linearly independent over the rationals, must the numbers z1,…,zn and their exponentials together contain at least n algebraically independent quantities? The source develops conditional reductions and a fixed test case, but explicitly leaves both the test case and the full conjecture open.

trdegℚℚ(z1,…,zn,ez1,…,ezn)≥n
Known results and sources
Landscape card pairing three complex inputs with their exponential images above an algebraic-independence motif, marked open.
Schanuel’s Conjecture links rational linear independence of complex inputs to algebraic independence among the inputs and their exponentials.

Research problem

Exact mathematical statement

For complex numbers z_1,…,z_n that are linearly independent over ℚ\mathbb Q, Schanuel’s Conjecture asserts

trdegℚℚ(z1,…,zn,ez1,…,ezn)≥n.\operatorname{trdeg}_{\mathbb Q}\mathbb Q(z_1,…,z_n,e^{z_1},…,e^{z_n})\ge n.

Revision v6 explicitly reports no proof. Its fixed e^{π²} atom is only a necessary test of one proposed arithmetic mechanism and would not by itself prove the universal conjecture.

Problem infographic

Problem at a glance

Three-panel landscape explainer showing rationally independent complex inputs mapping by one-way arrows to their exponentials, followed by the open transcendence-degree inequality.
The conjecture asks for a universal lower bound on transcendence degree; classical and differential analogues do not settle the complex-number statement.

Current mathematical picture

Where work on Schanuel’s Conjecture stands

Open conjecture

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

Useful failureSource-reported limitation

A fixed-atom proof of the transcendence of e^{π²} would not itself prove Schanuel’s Conjecture, and the all-order three-chain theorem cannot be treated as unconditional. First prove or replace the delta-domain continuation lemma; then prove a growing-family common-unit endpoint estimate and fraction-free normalized-Plücker assembly. Full Schanuel still needs the relative moving-Morse arithmetic-monodromy theorem.

Route status · Narrowed route
Main reductionCurrent reduction

The general route derives differential and Gauss-field structure from a minimal defect-one counterexample, while the fixed-atom route assumes e^{-π²} algebraic, constructs a multiscale contact matrix, and seeks a primitive row-volume bound strong enough to force a boundary polynomial with too many double roots.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeClose the delta-domain continuation lemmaTask status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Schanuel’s Conjecture in numbers

2.6kretained lines of mathematical investigation2,630 in the current working snapshot
Argument development
2,223 · 85%
Explored or eliminated routes
104 · 4%
Computational analysis
33 · 1%
Open obligations
126 · 5%
Definitions and setup
144 · 5%
6selected mapped statements1routes investigated3open questions3contribution-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 byStephen SchanuelProposed1966Status checkedAug 14, 2026
Source recordedAug 14, 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

10 selected steps

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

10 selected steps

Scroll horizontally to explore the route

Working route overview for Schanuel’s ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Do n Q-linearly independent complex inputs force at least n algebraically independent coordinates among them and their exponentials? — Depends on missing premiseDo n Q-linearly independentcomplex inputs force atleast…Current reduction — Depends on missing premiseCurrent reductionFixed e-to-pi-squared atom — Depends on missing premiseFixed e-to-pi-squared atomMinimal-counterexample structure — Depends on missing premiseMinimal-counterexamplestructureClosing target — Depends on missing premiseClosing targetTranscendence-degree target — Depends on missing premiseTranscendence-degree targetSource-reported limitation — stoppedSource-reported limitationClose the delta-domain continuation lemma — OpenClose the delta-domaincontinuation lemmaProve common-unit global assembly — OpenProve common-unit globalassemblyLift from the fixed atom to full Schanuel — OpenLift from the fixed atom tofull Schanuel
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 routeSource-reported limitation

A fixed-atom proof of the transcendence of e^{π²} would not itself prove Schanuel’s Conjecture, and the all-order three-chain theorem cannot be treated as unconditional. First prove or replace the delta-domain continuation lemma; then prove a growing-family common-unit endpoint estimate and fraction-free normalized-Plücker assembly. Full Schanuel still needs the relative moving-Morse arithmetic-monodromy theorem.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Close the delta-domain continuation lemmaSuggested move: Classify inverse singularities, exclude finite asymptotic values, track monodromy through the ramified point, and prove the selected cut-jump sign on a uniform delta-domain.
Ready to work on
02
Prove common-unit global assemblySuggested move: Establish a quantitative growing-family Chebyshev or Andreief determinant estimate and perform fraction-free normalization of the primitive Plücker vector with total scale O(T²).
Ready to work on
03
Lift from the fixed atom to full SchanuelSuggested move: Over the positive-dimensional Gauss field, construct the relative trace–norm module and prove the missing arithmetic-monodromy primitive-volume theorem at the moving Morse point.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The universal conjecture for ordinary complex exponentiation remains open. Differential-algebraic analogues, equivalent reformulations, and almost-everywhere results provide substantial context but do not prove the exact every-tuple statement.

[3][4][2]
External progress

What the literature has established

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

  1. PreprintWaldschmidt proved strong algebraic-independence properties for almost all tuples in elliptic and quasi-elliptic settings; this metric result does not settle the universal exponential conjecture.[4]
  2. Authoritative summaryWaldschmidt surveyed the Four Exponentials Problem and Schanuel’s Conjecture, including equivalent algebraic-independence formulations and partial strategies.[3]
  3. Peer reviewedJames Ax proved Schanuel-type statements for exponentiation in a differential-algebraic setting, an influential analogue rather than the complex-number conjecture.[2]
  4. Historical sourceSerge Lang published the conjecture attributed to Stephen Schanuel in Introduction to Transcendental Numbers.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusSchanuel’s Conjecture
Weaker or relaxed formAx–Schanuel differential analogue

Ax’s differential-field theorem proves a Schanuel-type transcendence-degree bound under differential hypotheses, not for arbitrary complex numbers with ordinary exponentiation.

[2]
Dependency or reductionFour Exponentials Problem

The Four Exponentials Problem is a central weaker consequence and testing ground in transcendence theory.

[3]
Weaker or relaxed formalmost-everywhere Schanuel property

Almost-all tuple results establish metric genericity but do not imply a statement for every rationally independent tuple.

[4]

Formalization opportunities

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

  • Formalization targetA checked statement must encode rational linear independence of arbitrary complex tuples and the transcendence degree of the field generated by inputs and exponentials.
  • Formalization targetA formal account must distinguish ordinary complex exponentiation from Ax’s differential-field analogue and from model-theoretic or elliptic variants.
  • Formalization targetThe private packet’s named standard imports, moving-Morse reduction, analytic continuation, determinant normalization, and relative primitive-volume theorem need exact statement alignment.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

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

4 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction3 of 63
  • lemma2 of 62
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 statementDo n Q-linearly independent complex inputs force at least n algebraically independent coordinates among them and their exponentials?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementTranscendence-degree targetintermediate
  • retained route statementMinimal-counterexample structureintermediate
  • retained route statementFixed e-to-pi-squared atomintermediate
  • 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
  • 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 failureSource-reported limitationreported failure
  • Research targetClose the delta-domain continuation lemmaopen
  • Research targetProve common-unit global assemblyopen
  • Research targetLift from the fixed atom to full Schanuelopen
  • Research targetUniversal conjecture remains opensuperseded
  • Research targetAll-order shifted-minor gapsuperseded
  • Research targetNormalized-Plücker assembly gapsuperseded
  • ComputationThe source lists exact rational symbolic expansions and determinant values, and warns never to infer a theorem from a finite sign pattern; no packet computation was executed in this staging lane.Reported finite identities support the local block algebra, but they do not establish all-order shifted minors, primitive row-volume bounds, e^{π²} transcendence, or Schanuel’s Conjecture. · reported unreproduced
  • Narrowed routeSource-reported limitationA fixed-atom proof of the transcendence of e^{π²} would not itself prove Schanuel’s Conjecture, and the all-order three-chain theorem cannot be treated as unconditional. First prove or replace the delta-domain continuation lemma; then prove a growing-family common-unit endpoint estimate and fraction-free normalized-Plücker assembly. Full Schanuel still needs the relative moving-Morse arithmetic-monodromy theorem.
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 bridgeClose the delta-domain continuation lemma

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.

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Prepared starting pointClose the delta-domain continuation lemma

Schanuel’s Conjecture · ready to start

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Research contextPrepared context for any AI agent

If z1,…,zn are linearly independent over the rationals, must the numbers z1,…,zn and their exponentials together contain at least n algebraically independent quantities? The source develops conditional reductions and a fixed test case, but explicitly leaves both the test case and the full conjecture open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Introduction to Transcendental Numbersoriginal source · Serge Lang · Addison-Wesley · 1966 · accessed Aug 14, 2026
  2. 2
    On Schanuel’s conjecturespeer reviewed result · James Ax · Annals of Mathematics 93(2), 252–268 · 1971 · DOI 10.2307/1970774 · MR 0277482 · accessed Aug 14, 2026
  3. 3
    The Four Exponentials Problem and the Schanuel Conjecturesurvey or monograph · Michel Waldschmidt · Springer Lecture Notes in Mathematics 2313 · 2023 · accessed Aug 14, 2026
  4. 4
    Schanuel Property for Elliptic and Quasi-Elliptic Functionspreprint · Michel Waldschmidt · arXiv · 2025-04-18 · ARXIV 2504.14041 · accessed Aug 14, 2026

Important qualifications

  • This source-separated metadata does not verify any mathematical claim in the private packet or grant review, credit, publication, or deployment authority.
  • Ax’s theorem is a differential-algebraic analogue and does not prove the complex-number conjecture.
  • Almost-everywhere algebraic-independence results do not prove the universal statement for every rationally independent tuple.
  • The current work’s moving-Morse reduction, delta-domain lemma, normalized-Plücker program, and e-to-pi-squared atom require separate exact review.
  • No checked formalization of the exact universal complex-number conjecture was identified in this bounded collection; this is not evidence of nonexistence.

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