Functorial Invariance Across Presentations of the Prime Numbers, with Context-Relative Quantitative Illustrations

In the paper's words

The prime numbers admit many simultaneous presentations: as the atomic subset of the natural numbers, as the support of the von Mangoldt function, as the zero data of the Riemann zeta function, as the output of sieve procedures, as primes inside Beatty-type sequences, and as the output of concrete algorithms. Working number theorists move between these presentations freely, but the literature rarely states which theorems survive a change of presentation and which are artifacts of the encoding. We formalize the question by defining a category RepPrimes whose objects are presentations of the primes and whose morphisms are structure-preserving translations.

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

Numbers it reports

1. The Legendre sieve presentation counts a prime $p \le x$ by sieving the interval $[2, x]$ by the primes $q \le \sqrt{x}$: an integer survives if it is divisible by no $q \le \sqrt{x}$, and the survivors are $1$ together with the primes in $(\sqrt{x}, x]$.

   from 4.4 Non-invariance of the error term (continued)

2. We prove that the Euler product, Möbius inversion, and the leading asymptotic $\pi(x) \sim x/\log x$ are invariants of this category, while the error term $E(x) = \pi(x) - \operatorname{Li}(x)$ is not: the Legendre-sieve presentation carries a main term inflated by the factor $2e^{-\gamma} \approx 1.1229$ relative to the analytic one.

   from Abstract

3. Numerical verification at $x = 10, \ldots, 10^6$ confirms the invariant asymptotics (ratio $\pi(x)\log x/x$ drifting from $0.9210$ toward $1.0845$) and the non-invariant error (growing to $-129.55$ at $10^6$).

   from Abstract

4. Worked illustrations with the twin-prime constant $\alpha_2 = 0.6601618605\ldots$ and a Germain-prime lower bound $2.72 \times 10^6$ at $x = 10^6$ anchor the framework in computable data, and a syntactic functoriality test gives a decidable transfer criterion.

   from Abstract

5. Two further worked quantitative illustrations drawn from the prime-literature record: the twin-prime correlation constant $\alpha_2 = 0.6601618605\ldots$ derived from the spectral analyses of [1] and [2], and a weighted Germain-prime lower bound exceeding $2.72 \times 10^6$ at $x = 10^6$ derived from the bound of [6] (Section 4.7).

   from 1.3 Contributions

6. This asymptotic lower bound is what we evaluate numerically at $x = 10^6$ in Section 4.7.

   from 2.2 Specific results from the literature

How it is done

1. We prove that the Euler product, Möbius inversion, and the leading asymptotic $\pi(x) \sim x/\log x$ are invariants of this category, while the error term $E(x) = \pi(x) - \operatorname{Li}(x)$ is not: the Legendre-sieve presentation carries a main term inflated by the factor $2e^{-\gamma} \approx 1.1229$ relative to the analytic one.

   from Abstract

2. The statement $\sum_{n \le x} \Lambda(n) \sim x$ and the statement $\pi(x) \sim x/\log x$ are treated as the same theorem, because the first implies the second by partial summation and the second implies the first by a standard Abelian argument.

   from 1.1 The problem

3. A presentation of the primes is a structure $(X, \phi, \iota)$ where $X$ is a mathematical object (a set, a measure space, a function, a multiset of complex numbers, an algorithm), $\phi$ is a distinguished substructure or function on $X$ intended to encode the primes, and $\iota$ is an interpretation map connecting $\phi$ to the canonical presentation $\mathcal{P} \subset \mathbb{N}$.

   from 1.2 The proposal

4. Definition of the category $\mathbf{RepPrimes}$ and proof that it satisfies the category axioms (Section 3).

   from 1.3 Contributions

What would falsify it

1. The explicit density $1/\alpha$ is a presentation-level quantity: it belongs to the encoding, not to the primes themselves, and our non-invariance mechanism predicts exactly this kind of presentation-bound constant.

   from 2.2 Specific results from the literature

2. Under a morphism to the arithmetic presentation, the encoding $f$ has no image — only the primes it encodes do — so the constant $\alpha_2$ is flagged non-transferable by test condition T3, exactly as the framework predicts for a presentation-level constant.

   from 4.7 Worked quantitative illustrations

3. The central non-invariance claim (Proposition 4.5) would be falsified by a demonstration that the Legendre sieve's natural comparison function, properly transported along a morphism of $\mathbf{RepPrimes}$, has main term $\operatorname{Li}(x)(1 + o(1))$ rather than $2e^{-\gamma}\operatorname{Li}(x)(1+o(1))$ — equivalently, that Mertens' product $\prod_{q \le \sqrt{x}}(1 - 1/q)$ evaluated at $y = \sqrt{x}$ does not produce the factor $2$ relative to $y = x$.

   from 6. Discussion

Where the claim stops

1. What the results do and do not claim.

   from 6. Discussion

2. The main result rests on the following assumptions, each with its sensitivity: Density-one image (Definition 3.1).

   from 6. Discussion

3. Theorem 4.4 fails for presentations with image density $\delta \lt 1$; the transferred asymptotic would then carry the factor $\delta$.

   from 6. Discussion

Where to read

1. Abstract216 words · 1 min
2. 1. Introduction656 words · 3 min
3. 2. Background and Related Work969 words · 4 min
4. 3. Methods: The Category of Prime Presentations506 words · 2 min
5. 4. Analysis2,397 words · 10 min
6. 5. Results421 words · 2 min
7. 6. Discussion530 words · 2 min
8. 7. Practical Implications and Engineering Pathways606 words · 3 min
9. 8. Conclusion155 words · 1 min
10. Provenance and AI Involvement220 words · 1 min