Why Prime Numbers Are Not Generated:

A Structural Reinterpretation of the Riemann Hypothesis

Abstract

Prime numbers are traditionally treated as objects to be generated, predicted, or explicitly formulated.
Despite extensive efforts, no constructive or closed-form rule capable of generating primes in a forward deterministic manner has been found.

In this work, we propose a structural reorientation of the problem.
Rather than attempting to generate primes, we treat them as axiomatic fixed defects—irreducible residues of non-invertible compositional processes.
Formally, primes are positioned as fixed points of a non-recoverable lag in multiplicative closure, not as outputs of a generative rule.

Within this framework, the Riemann zeta function is reinterpreted as an analytic projection operator, mapping these discrete fixed defects into a continuous complex analytic space via the Euler product.
We argue that the critical line $\Re(s)=\tfrac12$ emerges as a necessary stabilization locus:
to the right, analytic smoothing erases the defect;
to the left, defect interactions destabilize;
only on the critical line does the irreducible lag persist in a stable analytic form.

This paper does not present a proof of the Riemann Hypothesis.
Instead, it offers a syntactic and axiomatic redefinition of the problem, explaining why the critical line must arise once primes are correctly positioned as non-generative, non-invertible structural residues.

The contribution is conceptual rather than computational: a reframing of primes, zeta, and the critical line as elements of a single generative–analytic correspondence.

Appendix A : Figure 1.

[ Prime Defect (Fix) ]
          |
          |  ζ : analytic projection
          v
[ Analytic Shadow Space ]
          |
          |  stabilization
          v
[ Critical Line  Re(s)=1/2 ]

Figure 1.
Schematic correspondence between axiomatic prime defects and their analytic stabilization.
Primes are treated as fixed non-recoverable defects of compositional closure.
The Riemann zeta function acts as an analytic projection, whose shadows stabilize exclusively on the critical line $\Re(s)=\tfrac12$.

1.1 Motivation and Scope

The Riemann Hypothesis is traditionally formulated as a statement about the location of the non-trivial zeros of the Riemann zeta function.
However, the hypothesis itself does not ask for a construction of prime numbers, nor does it demand an explicit generative law governing them.

What remains largely unexamined is a prior question:

Why does the critical line $\Re(s)=\tfrac12$ appear at all?

Most existing approaches implicitly accept the critical line as a given analytic feature and focus on proving that all non-trivial zeros lie upon it. In contrast, the present work is concerned with the structural inevitability of the critical line itself, independently of any direct proof of the Riemann Hypothesis.

From the standpoint adopted here, this question cannot be answered within a purely generative framework. Prime numbers notoriously resist explicit generative formulation: no closed-form expression or forward deterministic process is known to produce primes in a predictive manner. Rather than treating this resistance as a technical deficiency, we interpret it as a structural property.

Specifically, we propose that primes should not be regarded as generative outcomes, but as irreducible residues of non-closure in compositional processes. In this view, primes arise where multiplicative composition fails to fully recover a minimal generative lag. They are not produced; they remain.

This shift in perspective motivates a corresponding reinterpretation of the Riemann zeta function. Instead of functioning as a generator or encoder of primes, the zeta function is understood as an analytic projection that maps such non-recoverable residues into the complex plane. The appearance of the critical line then reflects not a numerical coincidence, but a stabilization locus where irreducible generative defects can persist analytically without collapse or divergence.

Accordingly, the present paper does not aim to prove the Riemann Hypothesis. Its objective is more limited and more foundational: to clarify why any analytic representation of prime residues should single out $\Re(s)=\tfrac12$ as a distinguished geometric locus.

A concrete arithmetic illustration of this framework—treating primes as axiomatic fixed defects and the zeta function as their analytic shadow—is provided in Appendix A (see Fig. 1).

1.2 Relation to Existing Approaches

The Riemann Hypothesis has been approached from a wide range of analytic, spectral, probabilistic, and geometric perspectives. Despite their diversity, most existing approaches share a common structural assumption: the critical line $\Re(s)=\tfrac12$ is taken as a target to be reached or verified, rather than as a phenomenon to be explained.

Broadly speaking, these approaches may be grouped into three families.

First, analytic approaches focus on the detailed properties of the zeta function and its functional equation, aiming to control zero distributions through estimates, explicit formulas, or deep properties of $L$-functions. In this context, the critical line typically emerges as a symmetry axis or a consequence of analytic continuation, but its necessity is rarely questioned.

Second, spectral and operator-theoretic approaches seek to interpret the zeros of the zeta function as eigenvalues of a suitably defined operator. While these methods offer powerful unifying frameworks, they generally presuppose the existence of a distinguished real part—again treating $\Re(s)=\tfrac12$ as an endpoint rather than a structural constraint.

Third, geometric and dynamical approaches—including trace formulas, flow interpretations, and analogies with random matrix theory—reinterpret zero statistics in terms of underlying dynamics. These models successfully reproduce observed correlations but typically import the critical line implicitly via symmetry or normalization conditions.

In contrast, the present work adopts an orthogonal stance.

Rather than asking why zeros lie on the critical line, we ask:

Why must any stable analytic shadow of prime structure single out a line at all—and why specifically $\Re(s)=\tfrac12$?

The key distinction lies in how primes themselves are positioned within the theory. Existing approaches, even when highly abstract, implicitly treat primes as elements to be encoded, generated, or statistically modeled. Here, primes are instead treated as axiomatic non-generative residues—fixed points of failed compositional closure.

Once this shift is made, the role of the zeta function is correspondingly reinterpreted. It is no longer a device for producing arithmetic information, but a projection mechanism that maps irreducible discrete defects into a continuous analytic domain. From this perspective, the critical line appears not as a conjectural boundary to be proven, but as the unique locus where such defects can persist without either being smoothed away or destabilized.

This difference in orientation may be summarized as follows:

Existing approaches proceed from analytic structure to arithmetic consequences.
The present approach proceeds from arithmetic non-closure to analytic necessity.

Accordingly, the argument developed here is not in competition with proofs of the Riemann Hypothesis, nor does it claim to subsume them. Instead, it reframes the hypothesis as a statement whose geometric form becomes inevitable once primes are understood as non-recoverable generative traces.

2. Conceptual Shift: Primes as Non-Generative Objects

2.1 Primes Are Not Outputs

A persistent obstacle in the study of prime numbers is the expectation that they should arise as outputs of a generative process. This expectation is so deeply ingrained that the absence of a satisfactory prime-generating formula is often framed as a technical failure rather than a conceptual one.

From the present perspective, this expectation is misplaced.

Prime numbers are not the result of a generative process but the irreducible residues of failed compositional closure.

In other words, primes do not emerge from a rule in the same sense that composite numbers do. Composite numbers admit decomposition, reconstruction, and inversion within multiplicative structure. Primes, by contrast, are precisely those points at which such compositional recovery fails.

This distinction is not merely philosophical. Any framework that presupposes primes as outputs—whether deterministic, probabilistic, or algorithmic—implicitly assumes what it seeks to explain. The repeated failure of explicit generative approaches is therefore better understood as evidence of a structural mismatch rather than of insufficient technique.

Under this view, primes should not be modeled as products of rules, but as fixed points of non-invertible structure.

2.2 Axiomatization of Prime Defects

To formalize this shift, we adopt an axiomatic stance.

Let $\mathcal{C}$ denote the space of compositional (multiplicative) relations. Within this space, compositional closure is generally expected: elements combine and decompose without residue. However, certain compositions fail to recover a minimal structural lag.

We encode this failure as a defect.

Definition (Prime Defect).
A prime is defined as a fixed point of a non-recoverable compositional defect, characterized by an unrecovered lag of $-1$.

Formally, we define the prime set as

\[\mathbb{P} := \operatorname{Fix}\Big( \operatorname{Res}(\mathrm{lag}=-1) \Big| \mathcal{C} \Big).\]

We identify this set with the classical set of prime numbers,

\[\mathbb{P} \cong \mathbb{P}_{\mathrm{primes}}.\]

Several points are crucial here.

First, this definition is axiomatic rather than constructive. It does not prescribe a method for producing primes, nor does it rank or order them. Instead, it specifies the structural condition under which primes appear.

Second, primes are not exceptional elements added to the theory. They arise naturally as fixed defects—points where compositional closure halts and leaves an irreducible residue.

Third, by adopting this stance, the absence of a generative formula ceases to be mysterious. It becomes a direct consequence of what primes are.

This axiomatization deliberately refrains from invoking analytic machinery. Its purpose is to reposition primes conceptually, preparing the ground for their analytic treatment without presupposing it.

3. The Zeta Function as an Analytic Projection

3.1 Euler Product Revisited

The Euler product formula is often introduced as the bridge between prime numbers and the Riemann zeta function. It is typically presented as evidence that primes are the “building blocks” of the integers and that the zeta function somehow encodes their generative structure.

However, this common interpretation subtly misplaces the role of the Euler product.

\[\zeta(s) = \prod_{p \in \mathbb{P}} \left(1 - p^{-s}\right)^{-1}, \qquad \Re(s) > 1.\]

The Euler product does not generate primes.
It assumes them.

This distinction is elementary but decisive. The product formula is valid only once the primes are already given as an irreducible set. It encodes how multiplicative structure factors through primes, not how primes arise.

From the present perspective, the Euler product should therefore be read as follows:

The Euler product is not a generative mechanism but an analytic encoding of axiomatic prime defects.

Each factor $(1 - p^{-s})^{-1}$ represents the persistence of a non-decomposable element under analytic continuation. The product as a whole records how these irreducible residues collectively shape the analytic behavior of $\zeta(s)$.

Importantly, nothing in the Euler product resolves the origin of primes. On the contrary, its validity depends on their prior irreducibility. This is precisely why the product representation remains compatible with the Prime Defect Axiom introduced in Section 2.

Under this reading, the role of the zeta function shifts:

It does not explain primes.
It does not predict primes.
It projects the fixed defects carried by primes into analytic structure.

The Euler product is thus better understood as a structural projection operator, mapping discrete, non-generative residues into a continuous analytic domain.

This reinterpretation prepares the ground for the next step: understanding why such projections admit stable structure only on a specific geometric locus in the complex plane.

3.2 Zeta as Projection, Not Explanation

The reinterpretation of the Euler product naturally leads to a reconsideration of the role of the Riemann zeta function itself.

In much of the literature, $\zeta(s)$ is implicitly treated as an explanatory object: its zeros are expected to explain the distribution of primes, or at least to encode their hidden order. From the present viewpoint, this expectation again overreaches.

The zeta function does not explain primes.
It does not generate them.
It does not resolve their origin.

Instead, $\zeta(s)$ performs a projection.

More precisely, once primes are positioned as axiomatic fixed defects of compositional structure, the zeta function acts as a map from discrete, non-invertible residues into a continuous analytic domain. We formalize this shift by viewing (\zeta) as an analytic projection operator,

\[\zeta : \mathbb{P} \longrightarrow \mathcal{S},\]

where $\mathcal{S}$ denotes a set of analytic shadows in the complex plane.

This projection is not injective and not generative. Distinct primes do not correspond to isolated analytic points. Rather, their collective presence shapes global analytic features: poles, zeros, and functional symmetries.

From this perspective, the zeros of $\zeta(s)$ are not mysterious signals encoding hidden arithmetic laws. They are interference patterns produced when irreducible discrete defects are embedded into an analytic medium.

Crucially, projection entails loss.

In passing from $\mathbb{P}$ to $\mathcal{S}$, information about individual primes is necessarily blurred. What remains is not the primes themselves, but the constraints governing how their non-recoverable defects can coexist in analytic form.

This explains a longstanding asymmetry:

Arithmetic structure is discrete, rigid, and non-invertible.
Analytic structure is continuous, smooth, and symmetric.

The zeta function mediates between these domains without reconciling them. It translates irreducibility into geometry.

Seen this way, the central question of the Riemann Hypothesis is not why primes behave regularly, but why their analytic shadows are forced to align along a specific geometric locus.

That question is addressed in the next section.

4. Why the Critical Line Appears

4.1 Stability vs. Dissolution of Defects

If the zeta function is understood as an analytic projection of axiomatic prime defects, then the appearance of a distinguished geometric locus in the complex plane ceases to be surprising. The question becomes not where the zeros are, but under what conditions such projections can remain stable.

Let us consider how the analytic projection behaves as the real part of $s$ varies.

Regime I: $\Re(s) > \tfrac12$

In this region, analytic continuation strongly smooths the projected structure. Contributions from individual prime defects decay rapidly, and interactions among them are effectively suppressed.

From the syntactic perspective, this regime corresponds to over-recovery:
the non-recoverable lag that defines primes is progressively absorbed into analytic regularity. As a result, defect traces dissolve rather than persist.

Analytically, this manifests as excessive convergence.
Structurally, it represents a loss of irreducibility.

Regime II: $\Re(s) < \tfrac12$

In contrast, when the real part drops below $\tfrac12$, contributions from prime defects interact too strongly. Analytic projection amplifies interference between non-invertible residues, producing instability.

Here the analytic medium fails to accommodate discrete defects coherently. Divergence and uncontrolled oscillation dominate. The projection ceases to stabilize.

This regime corresponds to over-interaction:
defects do not dissolve, but instead collide and destabilize one another.

Regime III: $\Re(s) = \tfrac12$

Between these two extremes lies a unique balance.

At $\Re(s)=\tfrac12$, analytic smoothing is sufficient to prevent divergence, yet weak enough to preserve irreducible lag. Prime defects neither dissolve nor explode. They persist as minimal, stable analytic traces.

From the present viewpoint, this line represents a stability threshold:

the unique analytic locus at which non-recoverable discrete defects admit coherent projection.

The critical line is therefore not a numerical coincidence. It emerges as a structural boundary separating dissolution from instability.

This interpretation reframes the role of the real part entirely. The value $\tfrac12$ does not encode a hidden arithmetic constant. It marks the point at which analytic projection preserves exactly what defines primes as primes: irreducible failure of compositional closure.

The next subsection reformulates the Riemann Hypothesis in these terms.

4.2 Reformulating the Riemann Hypothesis

Under the framework developed above, the Riemann Hypothesis admits a reformulation that shifts emphasis without altering content.

Traditionally, the hypothesis is stated as a claim about the location of non-trivial zeros of the Riemann zeta function. While formally correct, this formulation leaves unanswered why such a specific geometric constraint should arise at all.

From the present perspective, the hypothesis can be restated as follows:

The Riemann Hypothesis asserts that axiomatic prime defects admit stable analytic projection only on the critical line $\Re(s)=\tfrac12$.

This reformulation does not weaken the hypothesis, nor does it replace its analytic content. Instead, it clarifies its structural meaning.

If primes are understood as fixed points of non-recoverable compositional defects, then any analytic representation of their collective influence must satisfy two competing constraints:

The projection must not erase irreducible lag through excessive smoothing.
The projection must not amplify defect interactions into instability.

Section 4.1 showed that these constraints are simultaneously satisfied only at $\Re(s)=\tfrac12$. The critical line thus appears as a necessary stabilization locus, not as an arbitrary numerical boundary.

In this sense, the Riemann Hypothesis does not demand that primes obey a hidden generative order. It demands that their analytic shadows remain confined to the only region where irreducibility can persist without collapse.

The hypothesis therefore concerns where stable projection is possible, not how primes are produced.

This distinction explains a long-standing tension in the literature: attempts to resolve the Riemann Hypothesis through increasingly elaborate prime-generating constructions repeatedly fail, while approaches grounded in analytic structure continue to converge on the critical line.

Seen syntactically, this convergence is inevitable.

5. Discussion: What This Does and Does Not Claim

The framework presented in this paper is intentionally limited in scope.

It does not provide a proof of the Riemann Hypothesis.
It does not locate zeros explicitly.
It does not introduce new analytic estimates or bounds.

What it does offer is a redefinition of the problem’s conceptual setting.

First, this work does not attempt to generate primes. On the contrary, it argues that such attempts are structurally misguided. Primes are treated as axiomatic fixed defects of compositional closure, not as outputs of a generative mechanism. This stance aligns with the well-known difficulty—indeed, impossibility—of constructing effective prime-generating formulas, and reframes that difficulty as a feature rather than a failure.

Second, the zeta function is not positioned as an explanatory engine. Its role is strictly interpretive: an analytic projection that translates discrete, non-invertible residues into continuous structure. Under this interpretation, zeros of $\zeta(s)$ are not carriers of hidden arithmetic laws, but geometric constraints governing the stability of projection.

Third, the appearance of the critical line is not derived analytically here, but structurally motivated. The argument is not that zeros happen to lie on $\Re(s)=\tfrac12$, but that stable analytic shadows of irreducible defects can exist nowhere else. This distinction matters. It shifts attention from technical localization to structural inevitability.

Accordingly, the present approach should be understood as diagnostic rather than demonstrative. It clarifies why the Riemann Hypothesis, if true, must take the form it does—and why decades of generative or constructive strategies have failed to resolve it.

Finally, nothing in this framework conflicts with existing analytic or number-theoretic results. Instead, it proposes a conceptual layer beneath them: one that organizes known phenomena without altering their formal statements.

6. Conclusion

The Riemann Hypothesis may remain unsolved not because it is analytically inaccessible, but because it has long been approached in a generative language applied to objects that are not generative.

By repositioning primes as axiomatic fixed defects—irreducible residues of failed compositional closure—and by interpreting the zeta function as an analytic projection rather than an explanatory mechanism, the critical line $\Re(s)=\tfrac12$ emerges as a structural necessity.

Under this view, the hypothesis does not assert hidden order in prime production. It asserts the existence of a unique analytic locus where irreducibility can persist without collapse.

Whether this perspective can be translated into a full proof remains open. What it provides is a coherent answer to a prior question that has often been left implicit:

Why must the critical line appear at all?

If primes are not made, but remain,
then their shadows can only stand
where non-recoverable structure is neither erased nor destabilized.

That place is the critical line.

図1 キャプション（最終版）

Figure 1. From compositional failure to analytic stabilization.

A schematic representation of the conceptual flow proposed in this paper.
Compositional closure fails to recover a minimal lag, producing an irreducible fixed defect identified with prime numbers.
The Riemann zeta function acts as an analytic projection of these discrete defects into the complex plane.
Stable analytic shadows persist exclusively on the critical line (\Re(s)=\tfrac12), which functions as a defect stabilization locus separating dissolution from instability.

Figure 1. （SVGシンプル素焼き版）

Abstract（最終研磨版）

Abstract

The Riemann Hypothesis is traditionally formulated as a statement about the location of the non-trivial zeros of the Riemann zeta function. While analytically precise, this formulation leaves open a fundamental conceptual question: why should such a specific geometric constraint arise at all?

This paper proposes a structural reinterpretation. Prime numbers are treated not as generative outputs but as axiomatic fixed defects—irreducible residues of failed compositional closure. From this standpoint, the Riemann zeta function does not generate or explain primes; it performs an analytic projection of these discrete defects into a continuous domain.

Within this framework, the critical line $\Re(s)=\tfrac12$ emerges as the unique locus where non-recoverable defects admit stable analytic representation. For (\Re(s)>\tfrac12), defect traces are excessively smoothed and dissolve; for $\Re(s)<\tfrac12$, defect interactions destabilize. Only at $\Re(s)=\tfrac12$ does minimal irreducibility persist without collapse.

The present work does not offer a proof of the Riemann Hypothesis. Instead, it reframes the hypothesis as a statement about representational stability: the necessity of the critical line follows from the non-generative nature of primes and the structural constraints of analytic projection.

MMZW-02｜素数欠陥から臨界線へ: Prime Defect Line 全論文

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