Formal Conjectures in Mathematics

• Formal conjectures are precise, unproved statements expressed in formal languages like Lean 4 or RDF, serving as interfaces between mathematical intuition and automated verification.
• They are used to reformulate classical problems across areas such as periods, representation theory, and graph theory, enabling exact comparisons and structural insights.
• Recent advances integrate automated conjecture generation within proof assistants, combining syntactic validation with novel formulations and benchmark-driven curation.

Searching arXiv for papers on “formal conjectures” and related formalization/conjecturing work. Formal conjectures are precise unproved statements encoded in a formal language, a formal semantic framework, or a theorem-prover environment. In current research usage, the term does not denote a single doctrine: it ranges from abelian-category formulations of period relations to Lean 4 declarations with omitted proofs, RDF-level conjectural assertions, machine-generated exact identities, and research benchmarks whose objects are kernel-checked open problems (Hörmann, 2021, Onda et al., 27 Jun 2025, Rolfini, 2021, Firsching et al., 13 May 2026). A common feature is that the conjectural status is separated from syntactic well-formedness: a statement may be exact, type-correct, and semantically specified while remaining unresolved.

1. Definitions and conceptual scope

In proof-assistant practice, a formal conjecture is a well-typed statement whose proof is absent. In Lean 4, there is no separate conjecture keyword; one declares a theorem or lemma and leaves the proof body as a placeholder such as sorry, or omits the proof during generation and inserts it later for syntax checking and verification (Onda et al., 27 Jun 2025). This makes “formal conjecture” primarily a status distinction rather than a syntactic category.

Recent work on autoformalisation sharpens that point by separating conjecturing from translation. For many problems, the missing step is not merely expressing a known conclusion in Lean syntax, but first generating the conclusion itself: an explicit numerical value, a bound, or a proposition. In that sense, conjecturing is treated as an independent capability, prior to formalisation and proof (Sivakumar et al., 13 Oct 2025). This suggests that formal conjectures are often best understood as interface objects between informal mathematical intent and later verification.

In model-theoretic knowledge representation, the notion is different but structurally related. The RDF semantics of conjectures treats a conjecture as a statement whose truth is not asserted as fact but is recorded as a potentially true claim whose truth value is currently unknown. Conjectural predicates live in a distinct semantic layer and can coexist even when their asserted counterparts would be contradictory; “collapse to reality” is a later semantic operation, not part of the original assertion (Rolfini, 2021).

Experimental mathematics uses the term in yet another precise sense: a formal conjecture is an exact mathematical identity, often involving a constant or graph invariant, stated without proof after numerical or combinatorial validation. The Ramanujan Machine’s continued-fraction identities and TxGraffiti’s graph-theoretic inequalities are exemplary instances of this usage (Raayoni et al., 2019, Davila et al., 23 Jul 2025).

2. Exact reformulations inside mathematical theories

A major strand of the literature treats formal conjectures not as mere containers for open questions, but as mathematically improved formulations. In the theory of periods, the space of formal periods of a mixed motive is defined from a $\mathbb{Q}$-linear abelian category $\mathcal{A}$ with Betti and de Rham fiber functors and comparison isomorphism. The associated evaluation map

$\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$

is the formal incarnation of integration, and the paper reformulates the period conjectures through Property (A), injectivity of $\mathrm{ev}_\int$, and Property (B), a short-exact-sequence criterion for every vanishing period. The main result is that (A) and (B) are equivalent, yielding a simplified formal version of the Grothendieck and Kontsevich–Zagier period conjectures (Hörmann, 2021).

In rational homotopy theory, Halperin’s conjecture is recast algebraically through Meier’s criterion that a positively elliptic algebra has no negative-degree derivations,

$\mathrm{Der}^{<0}(H^*(F;\mathbb{Q})) = 0.$

Kennard and Wu prove this up to formal dimension $20$, thereby establishing the TNCZ conclusion and the cohomological tensor decomposition for fibrations with positively elliptic fiber in that range (Kennard et al., 2021).

The weight part of Serre’s conjecture is formulated through several formal weight sets. Gee, Herzig, and Savitt define the Breuil–Mézard-predicted set $W_{\mathrm{BM}}(\bar\rho)$ via Hilbert–Samuel multiplicities, the crystalline-lifts sets $W^{\exists}_{\mathrm{cris}}(\bar\rho)$ and $W^{\forall}_{\mathrm{cris}}(\bar\rho)$, and the explicit set $W_{\mathrm{expl}}(\bar\rho)$ built from obvious weights, shadow closure, and Levi compatibility. For semisimple sufficiently generic inertia, the paper proves generic agreement between the explicit prediction and Herzig’s earlier set $\mathcal{A}$0 (Gee et al., 2015).

Representation-theoretic conjectures are formalized in a similarly exact manner. For unipotent square-integrable representations, the Hiraga–Ichino–Ikeda formal degree conjecture is proved in the form

$\mathcal{A}$1

and the analogous Plancherel-density formula is established for tempered unipotent representations (Feng et al., 2019). For unitary groups, an explicit Plancherel formula for $\mathcal{A}$2 yields the HII formal degree conjecture with canonical normalization, and also completes the local comparison needed for the Ichino–Ikeda period formula (Beuzart-Plessis, 2018).

Arithmetic transfer conjectures exhibit the same pattern. In the ramified quadratic case for unitary groups, regular formal moduli spaces of $\mathcal{A}$3-divisible groups with parahoric level are used to formulate identities of the schematic form

$\mathcal{A}$4

with several variants depending on the ambient product, correspondence, and parahoric level; the lowest-dimensional cases are proved (Luo et al., 2 Jul 2025). For Bessel subgroups, filtered Rapoport–Zink spaces with non-reductive structure group lead to a Bessel AFL conjecture

$\mathcal{A}$5

where the geometric side is a weighted intersection number of Bessel cycles (Zhang, 2021).

Formal conjectures also appear as proved statements about dynamical systems. In one-dimensional Hegselmann–Krause opinion dynamics, three conjectures about $\mathcal{A}$6-switches are established: finiteness of the switch sequence, identical evolution between consecutive switches up to the switch time, and positive-affine invariance of the dynamics together with rescaling of switch values (Molignini, 21 Aug 2025).

3. Proof assistants, theorem provers, and formal conjecturing pipelines

The theorem-proving literature has turned formal conjectures into explicit computational objects.

System	Formal conjecture object	Reported scale or result
LeanConjecturer	Lean 4 theorem statement with omitted proof	12,289 generated; 10,950 syntactically valid; 4,130 novel; 3,776 non-trivial
ConjectureBench / Lean-FIRe	Lean conjecture embedded into autoformalisation pipeline	457 paired informal–formal statements; 13 end-to-end PutnamBench successes with GPT-4.1; 7 with DeepSeek-V3.1
Minimo	Well-typed term $\mathcal{A}$7 with $\mathcal{A}$8	Valid conjectures sampled by construction from axioms
Formal Conjectures benchmark	Lean 4 research statement with open or solved status	2615 statements; 1029 research open; 836 research solved

LeanConjecturer generates theorem statements only, beginning with theorem and ending with := by, then inserts sorry for syntax checking. Its pipeline combines rule-based context extraction, LLM-based theorem generation, compile checking with sorry, novelty filtering through exact?, and non-triviality testing through aesop. The reported corpus contains 12,289 generated conjectures from 40 Mathlib seed files, of which 10,950 are syntactically valid, 4,130 are novel relative to Mathlib and prior generated context, and 3,776 are non-trivial in the paper’s sense that aesop cannot prove them (Onda et al., 27 Jun 2025).

ConjectureBench treats conjecturing as distinct from autoformalisation. It contains 457 paired informal–formal statements derived from PutnamBench and CombiBench, and introduces two targeted notions of correctness: ConJudge, which checks whether a generated formal statement embeds the gold conjecture semantically, and equiv_rfl, which requires definitional equality of the standalone conjecture object in Lean. Lean-FIRe interleaves natural-language Chain-of-Thought with Lean-of-Thought scaffolding, and the paper reports the first successful end-to-end autoformalisation of 13 PutnamBench “no-answer” problems with GPT-4.1 and 7 with DeepSeek-V3.1 (Sivakumar et al., 13 Oct 2025).

Minimo works in a dependent type theory aligned with the Calculus of Constructions and treats a formal conjecture as any term $\mathcal{A}$9 with $\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$0. Constrained semantic decoding plus a type-indexed completion engine guarantees well-formed conjectures by construction, even from a randomly initialized model, while the same model supplies the policy and value used in proof search. The system therefore couples conjecture generation and theorem proving inside a single intrinsic-motivation loop (Poesia et al., 2024).

The benchmark named “Formal Conjectures” takes the repository itself as the object of evaluation. In the bench-v1-lean4.27.0 snapshot it contains 2615 formalized Lean 4 statements, including 1029 research open conjectures and 836 research solved problems. Open problems are intended as a zero-contamination frontier for proof discovery, while solved problems benchmark proof auto-formalization. The benchmark fixes acceptance by Lean kernel validation without forbidden axioms such as sorry (Firsching et al., 13 May 2026).

4. Automated conjecture generation outside proof assistants

Formal conjectures also arise from systems that search mathematical structure directly rather than first targeting proof assistants. The Ramanujan Machine searches generalized continued fractions with polynomial partial numerators and denominators using a meet-in-the-middle hash search and a tailored “Descent–Repel” gradient method. Its outputs are exact identities for constants such as $\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$1, $\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$2, Catalan’s constant, and $\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$3, validated numerically to high precision but initially unproved; several of the resulting formulas were later proved by the community (Raayoni et al., 2019).

TxGraffiti occupies a similar position in graph theory. It searches precomputed invariant tables for algebraic inequalities among graph parameters and ranks candidates by sharpness, structural simplicity, and non-redundancy. The paper highlights four open graph-theoretic conjectures, including

$\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$4

for connected subcubic graphs other than $\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$5, and reports validation across hundreds of graphs; for the harmonic-index conjecture, the displayed scatter plot uses 335 connected graphs (Davila et al., 23 Jul 2025).

A different style of formal conjecture appears in additive number theory through gcd-driven recurrences. The proposed sequences reformulate or strengthen statements related to twin primes, Polignac gaps, Goldbach, and Schinzel’s Hypothesis H. The central pattern is that a recursively defined integer sequence hits zero at special indices, and explicit expressions in those indices are then conjectured to be prime or simultaneously prime (Cloitre, 2011). Likewise, the explicit conjectures

$\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$6

are studied by modular analysis; complementary residue classes are proved composite via divisibility by $\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$7 or $\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$8, and the remaining residue classes are posed as open compositeness conjectures (Asimi, 2023).

These examples show that a formal conjecture need not be prover-native. It may instead be an exact symbolic statement discovered by structured search and stabilized by modular, combinatorial, or high-precision numerical evidence.

5. Semantics, verification, and correctness criteria

The semantics of formal conjectures depends strongly on the host formalism. In RDF, the extended Simple Interpretation introduces a structure

$\mathrm{ev}_\int : \mathcal{P}(\mathcal{A}) \to \mathbb{C}$9

where IPC is a set of conjectural properties disjoint from asserted properties IP, IEXTC maps each conjectural property injectively to a unique ordered pair, and CONJFORM connects a real predicate with its conjectural form. This semantics remains two-valued: conjectures are not third-valued propositions, but true statements in a separate semantic layer. The framework also defines collapse to reality, nested conjectures, and cascading collapses (Rolfini, 2021).

In theorem proving, correctness is typically stratified. LeanConjecturer uses syntactic validity, novelty relative to Mathlib and prior generated context, and non-triviality measured by failure of aesop; the paper explicitly notes that aesop failure does not imply truth, only that the statement was not discharged automatically (Onda et al., 27 Jun 2025). ConjectureBench adds a finer hierarchy: Typecheck, BEq+, LLM Grader, ConJudge, and equiv_rfl, reflecting the difference between well-typedness, conservative equivalence, semantic equivalence, and exact definitional identity (Sivakumar et al., 13 Oct 2025).

Benchmark curation makes verification itself part of the topic. The “Formal Conjectures” repository accepts a proof or disproof if and only if the Lean 4 kernel validates it on the tagged toolchain without forbidden axioms. At the same time, the benchmark documents misformalization risk: 291 issues were fixed, with a taxonomy distinguishing translation errors, underspecification, and source errors. AI-generated proofs and disproofs are treated as an auditing mechanism for improving statement fidelity (Firsching et al., 13 May 2026).

A recurring misconception is that formal conjectures are automatically more trustworthy than informal ones merely because they are formal. The literature does not support that simplification. Formal syntax guarantees well-formedness, and kernel checking guarantees proof correctness once a proof exists, but the conjectural status can still hide false statements, incomplete answer construction, or misformalization of the intended mathematics (Sivakumar et al., 13 Oct 2025, Firsching et al., 13 May 2026).

6. Significance and open directions

The significance of formal conjectures lies in the separation they impose between statement, semantics, and proof. In motives and periods, that separation enables conjectures to be reformulated in more computable or more structural language, such as the equivalence between injectivity of $\mathrm{ev}_\int$0 and the single-period exact-sequence criterion (Hörmann, 2021). In arithmetic geometry and representation theory, it yields exact comparison formulas whose unresolved status is concentrated in transfer identities or multiplicity predictions rather than in informal narrative (Luo et al., 2 Jul 2025, Zhang, 2021, Gee et al., 2015).

In automated reasoning, formal conjectures have become a data modality in their own right. LeanConjecturer shows that domain-constrained statement generation can create training data at scale; Minimo shows that conjecturing and proving can be learned jointly from axioms; ConjectureBench shows that conjecturing cannot be collapsed into autoformalisation; and the “Formal Conjectures” benchmark shows that open formalized problems can supply a reproducible zero-contamination frontier (Onda et al., 27 Jun 2025, Poesia et al., 2024, Sivakumar et al., 13 Oct 2025, Firsching et al., 13 May 2026).

Several open directions recur across these literatures. One is broader semantic filtering: both LeanConjecturer and ConjectureBench identify the need for stronger ways to separate false, vacuous, and genuinely difficult statements before proof search (Onda et al., 27 Jun 2025, Sivakumar et al., 13 Oct 2025). Another is structural computability: in formal periods, the question remains how far principality and endomorphism-coinvariant descriptions extend beyond the currently controlled classes (Hörmann, 2021). A third is benchmark evolution: the research frontier itself changes as open statements are solved and reclassified, so a living corpus requires immutable snapshots together with continuous curation (Firsching et al., 13 May 2026).

A plausible synthesis is that formal conjectures are becoming a common currency across pure mathematics, formal verification, and machine-assisted discovery. They are not proofs, and they are not merely informal guesses encoded in stricter notation. Rather, they are rigorously specified unresolved claims whose exact representation makes semantic analysis, computational search, and eventual verification possible.