Functorial Characterization of the Prime Numbers

In the paper's words

This paper presents a complete theoretical characterization of the set of prime numbers, P, by demonstrating that it is the geometric support of a canonical object constructed via a three-stage functorial lift. This process begins in the logical stratum of Homotopy Type Theory (HoTT), where a decidable Prime type is constructed. This logical object is then mapped via an interpretation functor into the computational stratum of the effective realizability topos, where its algorithmic content is made explicit as a modest set whose elements are witnessed by computable programs (specifically, the Baillie–PSW primality test).

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. The framework’s primary utility is not in providing a new algorithm for primality testing, but in offering a complete, foundational model that explains how the logical definition of a prime number gives rise to its computational and geometric manifestations.

   from A

2. A synthetic approach that attempts to unify these aspects by combining constraints from different mathematical domains—for instance, by constructing a ‘synthetic prime sheaf’ as the fiber product of sheaves encoding modular, analytic, and geometric properties—is fundamentally inadequate.

   from 1.0 The

3. Within this system, the Prime type is formally constructed as a dependent sum over the natural numbers, $\sum_{n:\mathbb{N}} \| \mathsf{isPrime}(n) \|_{-1}$, which is then proven to be a 0-type (a set), ensuring it faithfully represents the prime numbers as a discrete collection.

   from 3.0

4. The Baillie–PSW algorithm, with its deterministic $O((\log p)^3)$ complexity (Baillie & Wagstaff, 1980), provides the explicit family of realizers for this relation, thus giving a concrete computational embodiment to the abstract Prime type.

   from 4.0

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. It consists of the carrier set $P$ (the standard set of primes) and a realizability relation $\Vdash$, where $e \Vdash p$ holds if and only if $e$ is the Gödel number of a program that correctly certifies the primality of $p$.

   from 4.0

2. Theorem: Assuming the Baillie–PSW conjecture, the direct image of the prime modest set $\underline{\mathcal{P}}$ under the canonical geometric morphism $\gamma: \mathbf{RT}(\mathcal{K}_1) \to \mathrm{Sh}(\mathrm{Ét}(\mathrm{Spec}(\mathbb{Z})))$ is a constructible étale sheaf whose support is exactly the set of prime numbers.

   from 7.0 The Main Theorem

3. It does not propose a new, faster primality test.

   from 8.2. Computational

Where to read

1. 1.0 The187 words · 1 min
2. 2.0217 words · 1 min
3. 3.0159 words · 1 min
4. 4.0166 words · 1 min
5. 5.0160 words · 1 min
6. 6.0135 words · 1 min
7. 7.0 The Main Theorem61 words · 1 min
8. 8.0358 words · 2 min
9. Appendix263 words · 1 min
10. Appendix C: Construction of the Modest Set $\underline{\mathcal{P}}$ in $\mathbf{RT}(\mathcal{K}_1)$ |212 words · 1 min
11. Appendix D:210 words · 1 min