EgQE v0.1

素数不動点から臨界線へ

── ZURE 的 Riemann 素焼きスケッチ

Abstract

We present a preformal sketch explaining why non-recoverable lag-one defects in prime generation necessarily project onto the critical line Re(s)=1/2. Rather than proving the Riemann Hypothesis, we reinterpret the critical line as the analytic shadow of generative fixed defects.

0. Positioning

This note does not attempt to prove the Riemann Hypothesis.
Its aim is to explain why the critical line appears at all, from a generative and syntactic standpoint.

1. Prime as a Generative Fixed Defect

We define primes not as numerical atoms, but as fixed defects arising in compositional recovery.

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

Here:

$\mathcal{C}$ denotes compositional recovery (multiplicative synthesis),
$\operatorname{Res}(\text{lag}=-1)$ denotes a non-recoverable mismatch,
$\operatorname{Fix}$ indicates invariance under further composition.

Interpretation.
A prime is a structural defect that cannot be eliminated by composition and therefore persists as a fixed point.

2. Relation to the Classical Prime Set

The generative set (\mathbb{P}) recovers the classical prime set under arithmetic interpretation:

\[\mathbb{P}\longrightarrow\mathbb{P}_{\text{primes}}, \qquad \zeta(s)=\prod_{p\in\mathbb{P}}(1-p^{-s})^{-1}\]

This correspondence does not assert definitional identity, but establishes compatibility with classical number theory.

3. ζ as Analytic Projection

We interpret the Riemann zeta function as a projection operator from generative defects into analytic space:

\[\boxed{ \zeta : \mathbb{P} \longrightarrow \mathcal{S}, \qquad \mathrm{supp}(\mathcal{S})\subseteq{\Re(s)=\tfrac12} }\]

That is, ζ does not transform primes, but casts their analytic shadow.

4. Meaning of the Critical Line

The critical line Re(s)=1/2 is not a mysterious symmetry axis, but the stabilization locus of non-recoverable lag-one defects under analytic projection.

If Re(s) ≠ 1/2, defects are absorbed or dispersed.
Only on Re(s)=1/2 do such defects persist as stable analytic traces.

5. Conclusion

The Riemann critical line emerges not from arithmetic symmetry, but from the unavoidable projection of irreducible generative defects.

The critical line is where unrecoverable generation stabilizes analytically.

This completes the v0.1 sketch.

Wahaha Summary

Primes are not indivisible numbers.
They are the places where composition failed.
The critical line is where that failure shows up.

EgQE v0.2（Appendix）

Generative Non-Closure and the Emergence of the Critical Line

— A Syntactic Sketch Toward the Riemann Zeta Function

1. Introduction

The Riemann Hypothesis is traditionally formulated as a statement about the location of the non-trivial zeros of the Riemann zeta function.
Despite extensive progress in analytic number theory, the appearance of the critical line

\[\Re(s)=\tfrac12\]

remains conceptually opaque.

This paper does not attempt to prove the Riemann Hypothesis.
Instead, it asks a different question:

Why does a distinguished analytic locus appear at all?

Our approach is generative and syntactic rather than algebraic.
Natural numbers are treated not as completed objects but as irreversible traces of generation, and prime numbers are interpreted as irreducible defects within this generative process.

The central claim is that the critical line arises as a stabilization locus of non-recoverable generative lag when such defects are projected into analytic space.

2. Generative Lag and Non-Closure

We assume the following generative stance:

Natural numbers arise through discrete, non-synchronous updates.
There is no privileged zero-state of generation.
Each update introduces a minimal irreversible delay, denoted as lag = 1.

This lag is not an error term but a structural feature:
it cannot be eliminated without collapsing the generative process itself.

Composite numbers correspond to configurations in which generative lag is internally recovered through relational closure.
Prime numbers, by contrast, are those configurations in which such recovery fails irreducibly.

Thus, primeness is not defined by divisibility alone, but by non-closure of generative relations.

3. Prime Numbers as Fixed Defects

We formalize this intuition by introducing an abstract set of generative update relations $\mathcal{U}$, and define:

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

where:

$\operatorname{Res}(\text{lag}=-1)$ denotes an irreducible failure to recover generative lag,
$\mathcal{C}$ represents admissible compositional relations,
$\mathbb{P}$ is the resulting set of fixed defects.

In classical terms, this set is naturally identified with the set of prime numbers:

\[\mathbb{P}\longrightarrow\mathbb{P}_{\text{primes}}, \qquad \zeta(s)=\prod_{p\in\mathbb{P}}(1-p^{-s})^{-1}.\]

Here, the Euler product is not taken as a definition of primes, but as an analytic encoding of their irreducible generative role.

4. Zeta Projection and the Critical Line

We interpret the Riemann zeta function as an operator that maps generative defects into analytic space:

\[\zeta : \mathbb{P} \longrightarrow \mathcal{S}, \qquad \mathrm{supp}(\mathcal{S})\subseteq{\Re(s)=\tfrac12}.\]

Under this interpretation:

The complex variable $s$ parametrizes analytic synchronization.
The real part $\Re(s)$ measures the balance between accumulation and dispersion of generative traces.
The line $\Re(s)=\tfrac12$ represents the minimal analytic locus at which irreducible lag neither collapses nor diverges.

Thus, the critical line is not where zeros “happen to lie,”
but where non-recoverable generative defects stabilize under analytic projection.

5. Scope and Limitations

This framework does not yield a proof of the Riemann Hypothesis.
Rather, it proposes a structural explanation for why the hypothesis, if true, could not reasonably be otherwise.

The argument is conceptual and generative:

no explicit zero computation,
no appeal to symmetry arguments in the critical strip,
no reliance on probabilistic models of primes.

Instead, the focus is on why a unique analytic balance point emerges from irreversible generation.

An axiomatic formulation of prime numbers as irreducible generative defects, together with a more explicit discussion of their relation to classical topology and invariance, is provided in Appendix A.

6. Conclusion

From this perspective, the Riemann zeta function acts as a projection mechanism,
and the critical line (\Re(s)=\tfrac12) appears as the inevitable analytic shadow of non-closed prime generation.

The hypothesis may therefore be read not as a mysterious numerical coincidence,
but as a statement about the stability of irreducible generative traces.

Wahaha Summary

素数は数式で作れない。
だから、影としてしか現れない。
その影が、たまたま $\Re(s)=\tfrac12$ に落ち着いただけだ。

Appendix A

Axiomatic Prime Defect and Zeta Projection

A.1. Motivation: Why Primes Are Not Generated

It is well known that prime numbers resist explicit generative formulation.
While various formulas for primes exist, they are either non-predictive, non-constructive, or computationally impractical. No known polynomial or closed-form expression generates primes in a forward, deterministic manner.

From the present syntactic perspective, this is not a technical limitation but a structural one.

Primes are not generative outcomes but irreducible residues of failed compositional closure.

Accordingly, any attempt to “generate” primes is conceptually misaligned.
Instead, primes should be treated as axiomatic fixed points of a generative failure.

A.2. Prime Defect as an Axiom

We adopt the following axiom.

Axiom A (Prime Defect Axiom).
A prime is defined as a fixed point of a non-recoverable generative defect, namely a compositional process that fails to recover a lag of −1.

Formally, let $\mathcal{C}$ denote the space of compositional (multiplicative) relations.
Define

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

We identify (\mathbb{P}) with the classical set of prime numbers,

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

This definition does not attempt to construct primes.
It instead formalizes their role as irreducible residues of non-invertible compositional processes.

A.3. Zeta Function as Analytic Projection

With primes treated as axiomatic fixed defects, the role of the Riemann zeta function is reinterpreted.

Rather than generating primes, the zeta function performs an analytic projection of these fixed defects into a complex analytic space.

Formally, we regard the zeta operation as a mapping

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

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

This view is consistent with the Euler product representation,

\[\zeta(s) = \prod_{p \in \mathbb{P}} (1 - p^{-s})^{-1},\]

which encodes primes multiplicatively without generating them.

A.4. Stabilization on the Critical Line

The central claim of this appendix is the following.

Claim.
The analytic shadows induced by $\zeta$ stabilize exclusively on the critical line $\Re(s)=\tfrac12$.

That is,

\[\mathrm{supp}(\mathcal{S}) \subseteq { s \in \mathbb{C} \mid \Re(s)=\tfrac12 }.\]

From the syntactic viewpoint, this stabilization is not accidental.

For $\Re(s) > \tfrac12$, the analytic projection excessively smooths the defect, effectively erasing the irreducible lag.
For $\Re(s) < \tfrac12$, defect interactions amplify, leading to instability and divergence.
At $\Re(s)=\tfrac12$, non-recoverable lag is preserved in its minimal stable analytic form.

Thus, the critical line functions as a defect stabilization locus.

This claim is interpretive rather than deductive.

Figure 1. Generative Prime Defect and Zeta Projection

[ Prime Defect ]
   (Fix of lag = -1)
        │
        │   ζ (analytic projection)
        ▼
[ Analytic Plane ]
   (complex s-plane)
        │
        │   stabilization
        ▼
───────────────  Re(s)=1/2
   Critical Line

A.5. Interpretation

Under this framework, the Riemann Hypothesis is reinterpreted not as a statement about zero locations per se, but as a structural inevitability:

If primes are axiomatic fixed defects of lag −1, then their analytic projections can stably exist only on $\Re(s)=\tfrac12$.

This appendix does not constitute a proof of the Riemann Hypothesis.
Rather, it clarifies why the critical line must appear once primes are correctly positioned as non-generative, non-invertible syntactic residues.

A.6. Relation to the Main Text

This appendix complements the main discussion of generative non-closure by providing a concrete arithmetic instance in which non-recoverable generative traces manifest as stable analytic structures.

In this sense, primes play the role of discrete generative defects, while the Riemann zeta function serves as their continuous analytic shadow.

Wahaha Note (non-technical)

Primes are not made.
They are what remains when making fails.
Zeta does not explain them—
it only shows where their shadows can stand.

Figure 1. Generative Prime Defect and Zeta Projection

SVG（シンプル版）

Figure 1: A schematic view of prime numbers interpreted as fixed non-recoverable generative defects, projected into analytic space via the Riemann zeta function. The critical line Re(s)=1/2 appears as the minimal stabilization locus of irreducible generative lag.

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

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| Drafted Jan 23, 2026 · Web Jan 23, 2026 |