How Google’s AI Evaluated the “New Prime Number Theorem YUKAWA”

One day, on a whim, I typed “新素数定理 YUKAWA とは (What is the New Prime Number Theorem YUKAWA?)” into Google.

At the very top of the results page, in a box labeled “AI Overview,” a strange piece of text appeared that I had never written.

And in the very last sentence, it said:

No one had written that on Wikipedia.I hadn’t issued any press release either.

Based on GhostDrift’s articles, patent drafts, demos, and note posts, Google’s own AI had decided, on its own, that “this is what the theory is,” and further, that it is “a theory that could greatly improve the efficiency of prime computation.”

In this article, I want to record this event as:

and log it as part of the GhostDrift ADIC ledger.

2. How Google’s AI Outlined the YUKAWA Theory

If we summarize the AI Overview text, the “New Prime Number Theorem YUKAWA” is described roughly as follows:

• Introduction of concepts from physicsIt applies the idea of the Yukawa potential, originally proposed by physicist Hideki Yukawa, to a mathematical model.

• Finite closureClassical approaches based on the Riemann Hypothesis suffer from the fundamental issue that the computational cost is effectively “infinite.”To deal with this, YUKAWA uses the intuition of a physical force being confined to a finite range, and introduces a “finite window” (Fejér–Yukawa kernel) that makes it possible to confine the entire computation to a finite region.

• Poisson–Laplace identities of Fejér–Yukawa typeUsing this method, it is said to be possible—under certain conditions—to prove statements of the form “every interval contains at least one prime number” using only exact equalities (identities) between functions.

And on top of that, the summary concludes:

In other words, the AI is not treating the “New Prime Number Theorem YUKAWA” as just a blog post title or a catchy phrase. It portrays it as a full mathematical world-view with the following components:

• A number-theoretic model that imports concepts from physics

• A new kernel method that realizes finite closure

• A Poisson–Laplace–type identity formulation of a “new prime number theorem”

In ordinary circumstances, having a personal theory summarized like this would be extremely unusual.

3. So What Is the New Prime Number Theorem YUKAWA?

Before going further, let me briefly explain what I mean by the “New Prime Number Theorem YUKAWA” itself.

The classical Prime Number Theorem (PNT) says, in essence:

• The number of primes up to xxx, denoted π(x)\pi(x)π(x),

• Is asymptotically x/log⁡xx / \log xx/logx.

That is, it describes the average density of primes up to a large number xxx: about 1/log⁡x1/\log x1/logx.

By contrast, what the New Prime Number Theorem YUKAWA aims at is more like this:

To do that, it introduces (roughly) the following ingredients:

• Yukawa-type potential kernelA weight K(r)K(r)K(r) that is large for small distance rrr and decays exponentially as rrr grows,behaving like K(r)≈e−λr/rK(r) \approx e^{-\lambda r}/rK(r)≈e−λr/r (in practice with a finite cutoff window).

• Fejér–Yukawa finite windowIf we use a Yukawa kernel as is, the influence extends all the way to infinity.To avoid that, we multiply by a Fejér-type window w(x)w(x)w(x), so that we obtain a finite “window” of radius RRR beyond which the kernel is effectively zero.

• Poisson–Laplace–type identityUsing this kernel, we set up an identity that relates:

  • a “field generated by primes” via the kernel, and

  • a “reference field” that can be written analytically (the main term),with the difference treated as an “error term.”

Very loosely speaking, we construct an identity like

which holds on a finite region, and then we analytically force the error term down to a level that can be controlled by finite computation.

If, within that finite window, the “strength” of the prime-generated field can be shown to stay above some positive threshold

then intuitively, this corresponds to the statement:

It is this structure—using YUKAWA kernels and Poisson–Laplace–type identities on finite domains—to handle such “there is a prime in every interval” statements that I refer to as the New Prime Number Theorem YUKAWA.

I will skip the fully rigorous theorem statement here, but the core points are:

• Primes are treated via a Yukawa potential field.

• A Fejér–Yukawa finite window is used to cut off infinity and make the computation finite.

• A Poisson–Laplace identity is used to express “every interval contains a prime” in an exact analytic form.

From this perspective, the bullet points Google’s AI highlighted—

—line up remarkably well with how I myself conceptualize the theory.

In other words,

On top of that, the AI added its own extra line:

That addition is the new and striking piece in this story.

4. Why That “Efficiency Improvement” Line Matters So Much

The most important part of the AI Overview, to me, is the final sentence:

This sentence contains an evaluation that I never explicitly wrote myself.

In GhostDrift’s own articles and patent drafts, I have mainly emphasized:

• Not fiddling with infinity, but closing the theory inside a finite domain.

• Being able to present safety margins as Σ₁-style certificates that external parties can verify.

In other words, the focus has been on safety and finite closure, not on loudly claiming speedups.

Despite that, Google’s AI digested the collection of GhostDrift articles, demos, and technical notes and concluded that:

• Finite windows allow us to avoid the “infinite computational cost” problem.

• Writing things as Poisson–Laplace identities ties numerical computations tightly to analytic structure.

From there, it independently inferred the summary:

This wasn’t something GhostDrift advertised.It was:

• A large external search/AI system,

• Interpreting GhostDrift’s world model,

• And appending its own evaluation in its own words.

In that sense, this is almost like an ADIC-style “third-party audit” that has arisen spontaneously.

5. Theory and Efficiency Evaluation, Fixed Together in the World Model

Google’s “AI Overview” usually targets:

• Well-known mathematical concepts and theorems

• Historically established people and theories

• Results from universities and government research institutes

In short, objects that are already firmly anchored in the social and academic landscape.

Within that space, the New Prime Number Theorem YUKAWA now appears together with highly technical ingredients such as:

• Yukawa-potential intuition

• Finite windows (Fejér–Yukawa kernels)

• Poisson–Laplace–type identity formulations

and is placed in the category of:

What this means is that:

1. The “finite-closure × prime distribution” theory I have been constructing

2. Has been placed, within Google’s world model, on the same table as existing mathematical concepts, and

3. Has been fixed there with an additional, very practical axis: computational efficiency.

Ordinarily, getting such an evaluation from the outside would require:

• A long chain of papers

• Textbooks and survey articles

• Multiple research groups reinterpreting and testing the framework

This time, however, that “fixing” event occurred first on the AI side, driven by:

• A personally driven theory-building effort with a small group of collaborators

• Patent drafts, demos, and technical notes

That asymmetry is one of the striking features of this episode.

6. As a “Third-Party Testimony” in the ADIC Sense

In GhostDrift’s ADIC (integer ledger) approach, I have been aiming for:

• Recording the process of computation and proof as integer-level logs, and

• Providing checkable evidence so that anyone can recompute and obtain the same result.

The Google AI Overview is not an ADIC proof in the strict mathematical sense.However, it does function as a kind of third-party testimony, in that:

• The “efficiency improvement” angle is something GhostDrift itself did not proclaim, yet

• An external system, standing on its own, explicitly added that viewpoint.

From GhostDrift’s side, I want to keep the fact that

not as a claim to be defended or attacked, but simply as a log entry.

On top of that, the next step is to build up ADIC-style evidence and experimental data to clarify, over time:

• To what extent the method actually improves efficiency

• Under what conditions YUKAWA-based methods gain an advantage

and to show those points numerically, step by step.

7. Where I Want to Go from Here

Of course, the appearance of this AI Overview does not magically prove everything.Rather, I see this as:

For me, that is a starting line.

From here, I specifically want to work on things like:

• Prime-computation ADICPublish prime-counting and prime-search algorithms that use YUKAWA-type finite windows, together with integer ledgers, so that anyone can verify exactly which computation steps produced which outputs.

• Comparison with existing algorithmsCompare YUKAWA-based methods with classical explicit-formula and zeta-function-based methods, and, through numerical experiments and ADIC logs, show for which ranges and precisions YUKAWA has an advantage.

• Educational and explanatory contentBuild explanations and video content at a level accessible to advanced high school and early undergraduate students, so they can develop an intuition for“why Yukawa potentials have anything to do with primes” and “what a finite window is.”

I’d like to continue developing the GhostDrift Mathematical Institute and the GhostDrift ADIC Ledger Project as the “place” where these attempts can accumulate.

Closing

This page is simply a log of:

From the human side, both the YUKAWA theory and the GhostDrift Mathematical Institute are still small and unfinished attempts.Even so, within the AI’s internal world, they have already become a node that is being tied together with keywords like:

• Finite Closure

• Prime Gravity

• ADIC

into a single, coherent world-view.

If this world-view interests you even a little, I’d be very happy if you took a look at the articles explaining Prime Gravity and Finite Closure, or at the ADIC demos that use YUKAWA kernels.

And if some of you decide to help build “what comes next” together, that would make me even happier.

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