REVIEW 3 major objections 4 minor 4 references
The set of primes is supernatural: a Lean formalization of the statement of the conjecture
T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The primes-are-supernatural conjecture is now a Lean proposition.
desk verdict A serious Lean formalization of a niche conjecture paper, with real new content in the extended class, but the 'complete no-sorry' claim currently rests on an unpinned repository. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing machinery is the pair of encodings of natural functions: a word-based definition FNatural using an inductive type of operation letters, and an inductive predicate IsNatural that closes the identity and constants under pointwise addition, multiplication, and exponentiation. The equivalence theorem FNatural_eq between the two is what lets every proof carry over from the paper's word formalism to structural induction on derivations. The conjecture is then a named def Conjecture_2_1 : Prop, deliberately left unproved so it contributes no axioms; all conditional results take it as an explicit hypothesis.
What would settle it
Compile the companion source file with the stated theorem prover and library versions, then inspect the named Conjecture_2_1, the correspondence-table theorems, and the #print axioms output; if any listed theorem is missing, contains sorryAx, or uses an undeclared axiom, the completeness and no-sorry claims are false.
Extended reading notes
Core claim
The paper's central claim is that the statement of the primes-are-supernatural conjecture can be, and has been, formalized end to end. The class of natural functions is encoded both as a word construction and as an inductive closure; the two encodings are proved equivalent, and the conjecture -- that no non-constant natural function sends every positive integer to a prime, equivalently that the prime set is supernatural -- is stated as an exact named proposition, Conjecture_2_1, deliberately left unproved. The formalization also covers the extension of Remark 3.4 to Knuth arrows, factorial, and truncated subtraction, and it verifies all 89 experimental table rows, using kernel-checked Lucas certificates for the largest primality claims.
Load-bearing premise
The load-bearing premise is that the companion source file described in the paper actually compiles with the stated toolchain and that the printed declarations correspond one-to-one to that checked file; nothing in the paper itself provides a machine-checked certificate of that correspondence.
Editorial extensions
If this is right
- The conjecture is now a named Lean proposition, so any claimed proof or disproof can be checked by the kernel rather than by human review.
- All proved cases from the source paper -- polynomial functions, linear functions, Fermat's function, and the 89 experimental table rows -- are part of the same verified corpus.
- The kernel-checked Lucas certificates show how large primality claims can be formalized without an hour-long trial division.
- If the conjecture holds, the formalized Corollary 3.2 gives infinitely many composite Fermat numbers as a conditional consequence.
- The extension to Knuth arrows, factorial, and truncated subtraction provides a larger class for which the analogous conjecture can be stated, with a proof that it contains the original class.
Reading between the lines
- A machine-generated coverage check that maps each prose statement to a declaration, and a pinned commit hash, would make the completeness claim independently auditable; until then the correspondence table is only as reliable as the author's transcription.
- Because the formal statement fixes the exact domain and exponent convention, an automated search for a counterexample becomes a bounded, checkable computation: any candidate function can be tested against IsNatural and Conjecture_2_1 mechanically.
- The same pattern of recording a conjecture as a named Prop with explicit dependencies could be applied to other open problems in inductive-closure form, giving automated systems a standardized target.
- The Lucas-certificate approach suggests that future table-heavy formalizations can move certificate data outside the kernel while keeping verification inside, trading hand-rolled primality proofs for short checked arguments.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper reports a Lean 4 formalization, in a single file SPCL.lean, of the author's earlier conjecture-oriented paper "Conjecture: the set of prime numbers is supernatural." It claims exhaustive machine-checked coverage of definitions, examples, remarks, numbered results, and experimental tables, with no `sorry`, and it presents selected declaration listings, discusses encoding choices, and gives a correspondence table. It also formalizes an extended class with Knuth arrows, factorial, and truncated subtraction, states an extended conjecture as an unproved `Prop`, and describes Pratt/Lucas certificates for the large primality conjuncts in the experimental tables.
Significance. If the formalization is complete and the repository compiles as claimed, the paper is a useful artifact: it turns an open conjecture into a named, kernel-checked proposition; it verifies all 89 experimental table rows, including large primality claims; and it makes precise some informal proof steps in the source paper, notably the composition-closure lemma needed for Proposition 3.1. The use of external computation only for certificate search, with all certificate checks performed by the kernel, is a sound and reproducible pattern. The main limitation is that the central claim of exhaustive, sorry-free coverage is not independently auditable from the manuscript alone.
major comments (3)
- [§1, §2] The central claim of a complete, sorry-free formalization cannot be checked from this manuscript. Section 1 gives only a bare URL for SPCL.lean, and Section 2 asserts that `lake build` succeeds and that the file is sorry-free, but no commit hash, archive, build log, or `#print axioms` transcript is supplied. Because the displayed snippets are explicitly partial (§3 omits the seven axiom proofs, §4 omits the `mul` and `elev` cases, and several proofs end with `-- ...`), the reader cannot verify that the printed declarations are exactly the declarations in the checked file. Please pin the exact commit or provide a release archive, and include the build output and the relevant `#print axioms` output for the stated toolchain, so that the completeness and no-sorry claims become reproducible.
- [§11] The correspondence table is author-written rather than machine-generated, and it cannot by itself certify that every claim in the paper has a proof in the repository. The table is valuable as documentation, but the paper should either provide a machine-checkable coverage certificate linking each printed declaration to a name in the file, or state clearly that the table is a human-readable summary. In addition, the table records an explicit gap: the row for [3, §4, question (ii)] says "not a determinate proposition as printed; not encoded." If the abstract's phrase "every ... numbered result" includes that question, the coverage claim needs to be qualified; if it does not, the boundary should be stated in the abstract and in Section 1.
- [§2, §10] The paper's description of the proof of the large table rows is plausible but not fully demonstrated: Section 10 says the heavy conjuncts are proved by Lucas certificates, with `lucasCert` converting an explicit factorization into Mathlib's `lucas_primality`, but the only fully displayed certificate is for the single prime `18446744073709551709`. The paper should state precisely how many certificate theorems are used, which table rows depend on them, and how the external certificate data is represented and checked, so that the claim that all certificate data is untrusted and kernel-verified can be audited.
minor comments (4)
- [§8] The prose statement of Proposition 2.2(ii) writes "f(n)=a n + b", while the formal statement `prop_2_2_ii` and the surrounding proof use `a ^ (n :N) + b`; please correct the displayed formula in the prose.
- [§4] The theorem `isNatural_of_mem_word` is presented in two slightly different forms: the earlier declaration has `{σ : Word} {f : N+ → N+}` before the colon, while the proof block writes `: ∀ {σ : Word} {f : N+ → N+}, ...`. Please align the two displays.
- [§11] The correspondence table lists `Question_i` as "stated (open)", but no declaration text or section of the paper displays this Lean statement; please include the declaration or a clear pointer to it.
- [§3] The category instance `ElevCat` proves the category laws by `rfl`, so it would be helpful to state explicitly whether the bundled `ElevationHom` structure uses proof irrelevance or definitional equality in a way that makes the three laws hold definitionally; without that explanation, a reader may be surprised that `id_comp`, `comp_id`, and `assoc` close by `rfl`.
Circularity Check
No circularity: the paper formalizes an open conjecture as an unproved Lean proposition, and every use of it is conditional or definitional.
full rationale
The paper does not derive the primes-are-supernatural conjecture from itself: Conjecture_2_1 is deliberately introduced as `def Conjecture_2_1 : Prop`, with no proof attached, and every theorem that uses it takes it as an explicit hypothesis. The equivalence `conjecture_2_1_iff_supernatural` is presented as an unfolding of the definitions of `IsNaturalSet`, `IsSupernatural`, and `Conjecture_2_1`, which the paper itself states plainly: "an unfolding of quantifiers once IsNaturalSet is unfolded... with no further mathematical content beyond [3, Conjecture 2.1] itself." That is a theorem about formulations, not a concealed input. The extended conjecture `Conjecture_KnuthFactorialSub` is likewise stated as an open problem and neither proved nor assumed. The experimental table theorems are verified by `norm_num` and kernel-checked Lucas certificates; the certificates' external data is explicitly "not trusted," with all congruence and divisibility conditions checked in Lean. The formalization's strongest claim depends on the public repository being exactly as described, but that is an auditability or reproducibility concern, not circularity: no fitted parameter is renamed as a prediction, and no load-bearing step reduces to a self-citation. The self-citations to [3] and [4] identify the object of formalization and the author's broader program; they do not replace independent mathematical justification. Hence no significant circularity is present.
Assumptions & free parameters
assumptions (4)
- standard math Mathlib's classical axioms (propositional extensionality, quotient soundness, choice) are accepted.
- standard math Mathlib's arithmetic results, including power laws, monotonicity of exponentiation, Fermat's little theorem, and the Lucas primality test, are sound.
- domain assumption Knuth's up-arrows are identified with Mathlib's hyperoperation index j+2 for j at least 1.
- domain assumption The author-provided correspondence table in Section 11 correctly maps every item of [3] to the named Lean declaration.
invented entities (1)
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IsNaturalKnuthFactorialSub and Conjecture_KnuthFactorialSub
Cite this review
Pith. "Pith review of The set of primes is supernatural: a Lean formalization of the statement of the conjecture." pith.science (2026). https://pith.science/paper/CL3K6PJO
@misc{pith2026260808643,
author = {Pith},
title = {Pith review of: The set of primes is supernatural: a Lean formalization of the statement of the conjecture},
year = {2026},
howpublished = {\url{https://pith.science/paper/CL3K6PJO}},
note = {Machine review of arXiv:2608.08643}
}
read the original abstract
The paper \emph{Conjecture: the set of prime numbers is supernatural} conjectures that no non-constant function built from the identity and constants by finitely many pointwise additions, multiplications, and exponentiations maps every positive integer to a prime. We give a complete Lean~4 formalization of that paper over Mathlib: every definition, example, remark, numbered result, and experimental table row has a machine-checked counterpart, with no \lcode{sorry}. The conjecture and similar generalizations are stated exactly, as named open problems. So stated, the conjecture becomes a precise target: an automated reasoning system can now attempt a kernel-checked proof.
Reference graph
Works this paper leans on
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[1]
L. de Moura and S. Ullrich,The Lean 4 theorem prover and programming language, in: Automated Deduction – CADE 28, Lecture Notes in Computer Science, vol. 12699, Springer, 2021, pp. 625–635
work page 2021
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[2]
The mathlib Community,The Lean mathematical library, in: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), ACM, 2020, pp. 367–381
work page 2020
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[3]
A. Mayeux,Conjecture: the set of prime numbers is supernatural, Proceed- ings of the Bulgarian Academy of Sciences78(2025), no. 11, 1585–1592. doi:10.7546/CRABS.2025.11.01
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[4]
A. Mayeux,Formalizing all indexed mathematics as a benchmark for general reasoning, in: Intelligent Systems and Applications: Proceedings of the 2026 Intelligent Systems Conference (IntelliSys), Lecture Notes in Networks and Systems, Springer, to appear. arXiv:2606.03835. University of Wisconsin–Madison, Madison, WI, USA Email address:mayeux@wisc.edu
arXiv 2026
Reviewed August 14, 2026 · model on record in the stance chip above.
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