Number theory · transcendence · logarithms of algebraic numbers

Four Exponentials Conjecture

Collaboration beta

If two complex x-values are rationally independent and two complex y-values are rationally independent, must at least one of the four exponentials formed from their pairwise products be transcendental?

\forall x_{1,} x_{2,} y_{1,} y_{2} \in ℂ, (x_{1,} x_{2} Q-independent) \land (y_{1,} y_{2} Q-independent) \Rightarrow \exists i, j \in {1,2} : e^{x_{i} y_{j}} is transcendental

Known results and sources

Two independent choices in each direction create four exponential values; the conjecture asks whether all four could be algebraic.

Research problem

Exact mathematical statement

Let $x_1,x_2$ be linearly independent over $\mathbb Q$ , and let $y_1,y_2$ be linearly independent over $\mathbb Q$ . Must at least one of

e^{x_1y_1},\quad e^{x_1y_2},\quad e^{x_2y_1},\quad e^{x_2y_2}

be transcendental?

Equivalently, a hypothetical counterexample would give a singular $2\times2$ matrix of logarithms of algebraic numbers whose rows and columns are each rationally independent. the source further expresses the desired contradiction as an exact rational matrix coefficient $w^TMv=0$ , not merely a relation valued in an integral multiple of $2\pi i$ .

The conjecture remains open. the source's labels such as VERIFIED describe source-reported internal audit status relative to listed external interfaces; they are not independent ProofAtlas verification or acceptance.

Problem infographic

Problem at a glance

Two cyan and two gold direction ribbons weave into four central knots, then continue to four luminous exponential spirals among crystalline algebraic-value motifs, with no output singled out. — Two rationally independent x-values and two rationally independent y-values form four pairwise products; the open question is whether their four exponentials can all be algebraic.

Current mathematical picture

Where work on Four Exponentials Conjecture stands

Open conjecture

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

Useful failureLarger linear determinant in known coordinates

The current work's FE-RANK-01/02 bounds cap formal rank drop at the half-rank threshold on quadratic branches, and at equality the determinant factors through relations already known. Introduce genuinely arithmetic data—periods, derivatives, several places, nonlinear entries, or excess vanishing—and prove that it contributes more than formal algebraic rank drop.

Route status · Narrowed route

Main reductionDimension-three-or-four split

The source reports that the rational span of 1, t, u, tu has dimension three or four, producing conic and generic Segre branches.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeFE-R1-A: design an explicit anisotropic support family with total degree below N², Newton area below N²/2, and enough structured dependence for the balanced jet conditions.Task status · Ready to work on

Work mapped so far

Four Exponentials Conjecture in numbers

889retained lines of mathematical investigation889 in the current working snapshot

Argument development: 790 · 89%
Explored or eliminated routes: 18 · 2%
Computational analysis: 23 · 3%
Open obligations: 12 · 1%
Definitions and setup: 46 · 5%

8selected mapped statements2routes investigated4open questions4contribution-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 byTheodor Schneider (first printed attribution in 1957; later historical discussion notes earlier related versions due to Selberg, Siegel, and Alaoglu–Erdős)Proposed1957Status checkedAug 27, 2026

Source recordedAug 27, 2026

Recommended next task

FE-R1-A: design an explicit anisotropic support family with total degree below N², Newton area below N²/2, and enough structured dependence for the balanced jet conditions.

Suggested move: Choose concrete balanced integer pairs Q,T and a convex lattice support; derive exact coefficient, degree, area, and residue-class counts before attempting asymptotics.

What would count as progress

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

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

14 selected steps

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

Scroll horizontally to explore the route

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

2 recorded

Narrowed routeLarger linear determinant in known coordinates

Route status · Narrowed route

Narrowed routeOrdinary one-scale auxiliary polynomial in P1

FE-TC-02 reports that every admissible ordinary polynomial is divisible by the complete grid factor, so its vanishing is already explained by the known period. Use normal jets and a balanced mixed determinant, then cross the exact 2QT = N obstruction while preserving both strict degree bounds.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

FE-R1-A: design an explicit anisotropic support family with total degree below N², Newton area below N²/2, and enough structured dependence for the balanced jet conditions.Suggested move: Choose concrete balanced integer pairs Q,T and a convex lattice support; derive exact coefficient, degree, area, and residue-class counts before attempting asymptotics.

Ready to work on

FE-R1-B: express the interpolation determinant in residue and derivative coordinates, factor the exact Fourier–Hermite block, and derive the residual square matrix.Suggested move: Write every row and column range explicitly, perform the finite Fourier transform symbolically, and prove that any residual zero is not already forced by the period block.

Ready to work on

Balanced cyclotomic–Hermite determinant

The priority source route asks for a mixed determinant that crosses the sharp P1 jet boundary after exact Fourier–Hermite extraction and a strict global product-formula estimate.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

FE-R1-C: prove normalized local estimates at archimedean, good finite, bad, and cyclotomic places whose total is strictly negative under residual nonvanishing.Suggested move: Fix the number-field tower and height ledger, identify which places supply the strict gain, and sum all local losses with their normalized degrees before invoking the product formula.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 27, 2026

Current statusOpen conjecture

The exact Four Exponentials Conjecture remains open. The proved Six Exponentials Theorem is a neighboring relaxation, and the recent Matrix Coefficient Conjecture framework reformulates and organizes the two-by-two case without resolving it.

[2][3]

External progress

What the literature has established

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

late 1960s
Authoritative summaryThe Six Exponentials Theorem supplied the strongest standard unconditional neighboring result: with two independent logarithmic directions on one side and three on the other, at least one of the six…[2]
2026
Peer reviewedDasgupta and Kakde published the Matrix Coefficient Conjecture framework; its two-by-two instance is equivalent to Four Exponentials, but the article presents a strategy rather than a proof of that instance.[3]
2023
Peer reviewedDasgupta gave a modern matrix-rank formulation and historical account, recording the Four Exponentials Conjecture as wide open and separating it from the proved Six Exponentials Theorem.[2]
2002
Peer reviewedRoy established essentially optimal interpolation formulas and auxiliary-function constructions relevant to the sharpness barriers encountered by standard auxiliary-polynomial methods.[4]

Mathematical neighborhood

Related results and reusable starting points

Current focusFour Exponentials Conjecture

Weaker or relaxed formSix Exponentials Theorem

Replacing the two-by-two array by a two-by-three (or three-by-two) array gives the proved Six Exponentials Theorem. It is strong evidence in the same transcendence framework but does not settle the exact four-value target.

[2]

Equivalent formulationTwo-by-two Matrix Coefficient Conjecture

The two-by-two instance of the Matrix Coefficient Conjecture is equivalent to the Four Exponentials Conjecture.

[3]

Stronger or generalized formStructural rank conjectures for logarithms of algebraic numbers

General structural rank conjectures for matrices of logarithms would imply the relevant rank lower bound and therefore subsume the four-exponentials case, but remain conjectural.

[2]

Formalization opportunities

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

Formalization targetA statement-aligned formalization of the Four Exponentials Conjecture with exact rational-linear-independence hypotheses was not established by this collection.
Formalization targetA statement-aligned formalization of the Six Exponentials Theorem was not established by this collection.
Formalization targetThe analytic and transcendence-theoretic infrastructure needed to formalize the known theorem interfaces and proposed auxiliary-function route has not been mapped to checked declarations.
Formalization targetNo independently reproduced computation or finite certificate can decide this universal transcendence statement.

Detailed research inventory

Claims, milestones, and routes in the current map

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

6 standing statements2 proposed statements4 open questions2 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction3 of 83
lemma1 of 81
equivalence1 of 81
negative result2 of 82

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementTwo independent choices in each direction produce four exponentials; the conjecture says they cannot all be algebraic.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementNormalized rank-one counterexampleintermediate
retained route statementDimension-three-or-four splitintermediate
retained route statementThree period typesintermediate
retained route statementSharp P1 jet boundaryintermediate
retained route statementLinear determinantal ceilingintermediate
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
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 failureLarger linear determinant in known coordinatesreported failure
Useful failureOrdinary one-scale auxiliary polynomial in P1reported failure
Research targetFE-R1-A: design an explicit anisotropic support family with total degree below N², Newton area below N²/2, and enough structured dependence for the balanced jet conditions.open
Research targetFE-R1-B: express the interpolation determinant in residue and derivative coordinates, factor the exact Fourier–Hermite block, and derive the residual square matrix.open
Research targetFE-R1-C: prove normalized local estimates at archimedean, good finite, bad, and cyclotomic places whose total is strictly negative under residual nonvanishing.open
Research targetBalanced cyclotomic–Hermite determinantopen
ComputationThe current work reports independent small-parameter checks and names scripts for symbolic identities and audit spot checks.ProofAtlas did not execute those attachments during intake. They remain source-reported consistency checks and are not proof evidence. · reported unreproduced
Narrowed routeLarger linear determinant in known coordinatesThe current work's FE-RANK-01/02 bounds cap formal rank drop at the half-rank threshold on quadratic branches, and at equality the determinant factors through relations already known. Introduce genuinely arithmetic data—periods, derivatives, several places, nonlinear entries, or excess vanishing—and prove that it contributes more than formal algebraic rank drop.
Narrowed routeOrdinary one-scale auxiliary polynomial in P1FE-TC-02 reports that every admissible ordinary polynomial is divisible by the complete grid factor, so its vanishing is already explained by the known period. Use normal jets and a balanced mixed determinant, then cross the exact 2QT = N obstruction while preserving both strict degree bounds.

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 bridgeFE-R1-A: design an explicit anisotropic support family with total degree below N², Newton area below N²/2, and enough structured dependence for the balanced jet conditions.

2 approaches have 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 pointFE-R1-A: design an explicit anisotropic support family with total degree below N², Newton area below N²/2, and enough structured dependence for the balanced jet conditions.

Four Exponentials Conjecture · ready to start

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Sources and references4 cited works · next context review by Nov 27, 2026

The mathematical context was checked on Aug 27, 2026. Status can be refreshed sooner after a material result or claim.

1
Einführung in die Transzendenten Zahlenoriginal source · Theodor Schneider · Springer · 1957 · DOI 10.1007/978-3-642-94694-3 · accessed Aug 27, 2026
2
Ranks of matrices of logarithms of algebraic numbers, I: The theorems of Baker and Waldschmidt–Masserpeer reviewed result · Samit Dasgupta · Mathematical Sciences Publishers · 2023 · ARXIV 2303.02037 · accessed Aug 27, 2026
3
Ranks of matrices of logarithms of algebraic numbers II: The matrix coefficient conjecturepeer reviewed result · Samit Dasgupta, Mahesh Kakde · Advances in Mathematics · 2026 · ARXIV 2408.08178 · DOI 10.1016/j.aim.2025.110753 · accessed Aug 27, 2026
4
Interpolation Formulas and Auxiliary Functionspeer reviewed result · Damien Roy · Journal of Number Theory · 2002 · DOI 10.1006/jnth.2000.2595 · accessed Aug 27, 2026

Important qualifications

The collection was bounded to the exact Four Exponentials statement, historical attribution, the strongest standard neighboring theorem, the recent matrix-coefficient reformulation, and auxiliary-function readiness.
A scoped search located foundational complex-exponential infrastructure but did not establish a statement-aligned Mathlib formalization; this bounded negative search does not prove nonexistence.
No packet attachment, submitted URL, source-reported status label, or source-package computation was treated as independent external authority.
The collection does not independently audit the current work's reductions, period classification, interpolation boundary, or proposed determinant route.

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