Why the Zeta Function Is Necessary for AI Safety — The Mathematics That Separates Cause and Effect and Defines Accountability Boundaries —

The current crisis in AI safety lies not merely in failures of alignment or control, but in the structural inability to definitively establish “responsibility” within a finite operational scope. While Reinforcement Learning from Human Feedback (RLHF) and Interpretability research have made significant strides, they largely rely on the premise of statistical convergence — the assumption that with sufficient data and training, errors will asymptotically vanish. However, in high-stakes environments such as finance, healthcare, and autonomous driving, we require a structure that can halt explanation and fix responsibility “on the spot,” under constraints of finite time and data.1

This report demonstrates that the most rigorous archetype for such a stopping structure is found in the Riemann Zeta function ($\zeta(s)$), specifically in its mechanism for separating “results” (integers) from “causes” (primes) via the Euler product, and extracting those causes via the logarithmic derivative. While the Renormalization Group in physics manages infinity through approximation and cutoffs, the Zeta function maintains a strict duality of information between primes and zeros through the Explicit Formula.

We examine how the ADIC (Arithmetic Digital Integrity Certificate) and the “Ledger” approach, as proposed by GhostDrift theory, represent an engineering implementation of this mathematical structure.3 By treating computational steps as “primes” (atomic units of responsibility) and binding infinite regress with Finite Closure, this framework offers a mathematically rigorous “stopping definition” for AI accountability — one that stands apart from the probabilistic or thermodynamic approaches.

Part I:

The Core of AI Safety — Where “Infinite Deferral” Happens

1.1 Where to Stop the “Why”: The Problem of Infinite Regress

In modern AI systems, particularly Large Language Models (LLMs) and autonomous agents, the answer to “Why was this decision reached?” risks falling into an infinite regress.1 Causal factors are diffused across billions of parameters, making it structurally difficult to pinpoint a specific “cause.” When an AI makes a critical error, attributing the cause to the complex interplay of weights and activation functions effectively blurs the locus of responsibility.4

This structure mirrors the philosophical “Agrippa’s Trilemma” (infinite regress).5 In current AI operations, probabilistic metrics like “Confidence Scores” are often used to halt this regress. However, treating edge cases simply as “low probability events” effectively reserves judgment in the finite “here and now,” deferring the issue to a theoretical long run.2

GhostDrift Mathematics Institute identifies this structure as “Structural Postponement”.2 The limit theorem stating that “error converges as sample size $N$ goes to infinity” is mathematically correct, but in field operations requiring immediate decisions with finite resources, reliance on this theorem can amount to delegating solutions to an infinite future. Instead of seeking complete explanations beyond the “Infinite Fog,” we require a mechanism to definitively fix responsibility within a finite scope.7

1.2 Division of Roles: Probability vs. Responsibility Stopping

Probability theory is a powerful tool for prediction, but it serves a different role when the goal is stopping responsibility. By definition, probabilistic statements describe “ensembles” (collections of events), not definitive responsibility for a single, isolated event. Stating a system is “99% safe” leaves a 1% margin where the locus of responsibility remains undefined.8

Furthermore, systems based on continuous variables and floating-point arithmetic can increase precision indefinitely but struggle to create logical “cuts” or boundaries. GhostDrift theory proposes a clear separation between the “Excess” (fluctuations, errors, continuous variability) and the “Skeleton” (discrete, verifiable structure).2 The “black box” nature of AI arises not just from complexity, but from the lack of an explicit discrete structure (skeleton) to halt the regression of responsibility.3

Therefore, AI safety requires a distinct mathematical layer — one that does not negate probabilistic models but complements them by “binding infinity with finiteness.” This is the structure provided by the Zeta function.

1.3 The Applicability Limits of Explainable AI (XAI)

Current Explainable AI (XAI) techniques (e.g., SHAP, LIME) have succeeded in approximating black-box behaviors in human-understandable terms.10 However, these are largely approaches that “trace the chain of causality backward,” and the criterion for where to stop this tracing is often left to external human judgment.

Moreover, if the XAI tool itself uses a “Surrogate Model” for approximation, a meta-problem arises: “Who guarantees the explanation is correct?” This risks triggering a new chain of surveillance.10

What is needed here is not a more refined approximation, but a structural “Cut.” We need an operation that, based on a rigorous definition, fixes the continuous flow of causality into “units that cannot be further decomposed.”

Part II: The Riemann Architecture — Binding Infinity with Finiteness

2.1 The Euler Product: Separation of Result and Cause

The Riemann Zeta function bridges the world of analysis (continuum, waves) and number theory (discrete, primes). This dual nature offers a crucial paradigm for “separating cause and effect” in AI safety. This relationship is encapsulated in the Euler product formula:

The “Sum” on the left represents the world of observed “Results” (integers $n$). The “Product” on the right represents the world of constitutive “Causes” (primes $p$).12

The significance of this equation lies in its nature as an identity, not an approximation. Every integer (result) is uniquely composed of a combination of primes (causes) — the Fundamental Theorem of Arithmetic. In the context of AI safety, this provides a model for “Atomic Responsibility.” The question “What is $p$ composed of?” stops here, by the very definition of a prime. Thus, the prime acts as a mathematical “Stopping Point” for responsibility inquiry.14

2.2 Logarithmic Derivative and the von Mangoldt Function: Signal Extraction

If the Euler product defines the “relationship,” its Logarithmic Derivative provides the mechanism for “audit” and extraction. Taking the logarithm of the product converts it into a sum; differentiating it reveals the von Mangoldt function $\Lambda(n)$:

Here, $\Lambda(n)$ is defined as a filter that isolates prime powers 16:

$$\Lambda(n) = \begin{cases} \ln p & (n = p^k, \text{ for prime } p \text{ and integer } k \ge 1) \\ 0 & (\text{otherwise}) \end{cases}$$

This operation corresponds to an “Audit” in an AI system. By applying the logarithmic derivative ($-\zeta’/\zeta$) to the “Result” (the complex state $\zeta$), one dissolves the noise (interactions of composite numbers) and extracts only the “Atomic Responsibility” ($\Lambda(n)$). This mathematical structure models how to assign specific weight ($\ln p$) to distinct causes without ambiguity.19

2.3 The Explicit Formula: Information Duality and Finite Closure

Furthermore, Riemann’s “Explicit Formula,” which connects primes (discrete causes) with the Zeta function’s zeros (systemic waves/vibrations), offers a vital perspective on information preservation:

This formula demonstrates a “Dual” relationship: microscopic causes (primes) and macroscopic behavior (zeros) hold the same information in different representations.21

In AI safety, this implies “Information Integrity.” Interpretations like “Prime Gravity” view primes as “sources generating a field” and zeros as the “field’s resonance,” suggesting that information within the system is not randomly dissipated but follows a conservation law.2

Crucially, by introducing weight functions (like the Gaussian or Yukawa potential) to handle this infinite sum, we enable “Finite Closure” — the ability to dampen distant influences and close the calculation within a finite “window”.7 This opens the path to fixing responsibility and safety within a finite operational scope, even while dealing with infinite potential.

2.4 Consistency of “Skeleton” and “Excess”

GhostDrift theory formalizes this structure as “Structure = Skeleton + Excess”.2

• Skeleton: $\log rad(n)$. The prime factors (without multiplicity) that support the system’s core structure.

• Excess: $\delta(n)$. The “surplus mass” or fluctuation arising from higher powers of primes.

In standard models, this “Excess” risks unbounded growth (error explosion). However, the Consistency Principle, based on Zeta structure, posits conditions where the expected value of Excess remains finite and the Skeleton dominates the system.2 This serves as a mathematical requirement for an AI system to avoid being buried in noise and to maintain an explainable structure.

Part III: Engineering Implementation — ADIC and the Ledger System

3.1 The “Ledger” as Prime Factorization

The ADIC (Arithmetic Digital Integrity Certificate) developed by GhostDrift Mathematics Institute is an engineering implementation of the Zeta function’s structure described above.3

The correspondence is as follows:

Zeta Function World (Number Theory)	Ledger System World (Engineering/AI Safety)
Integer $N$	System State $S$ (Final Output)
Prime $p$	Ledger Row $L_i$ (Atomic Operation)
Unique Prime Factorization	Reproducibility & Verifiability
von Mangoldt Function $\Lambda(n)$	Audit Function Verify($L$)
Stopping Definition (Prime)	Stopping Point of Responsibility (Ledger Row)

3.2 Ledger Rows as “Isomorphic to Primes”

In conventional AI, computation is a fluid sea of floating-point numbers where identifying a definitive “turning point” is difficult. The ADIC Ledger system transforms this process into a “record of integer addition and multiplication”.3

Here, each row in the ledger (a single step of operation) plays the same role as a “Prime” in number theory: it is the “minimum unit of responsibility that requires no further decomposition.”

When asked, “Why is the system in this state?”, the ledger answers, “Because operations $L_1, L_2, \dots$ occurred,” and the explanation Stops there. Just as a prime has no internal generation rule that needs explaining, the ledger row cuts off infinite regress through a consensus (definition) that “we do not decompose further.”

3.3 Finite Closure: Engineering Prevention of Infinite Regress

To prevent the log itself from growing infinitely, the concept of weight functions from the Explicit Formula is applied as “Finite Closure.”

• Mathematical Process: Infinite sums of zeros are made computable by converging them within a specific window using kernels like the Yukawa potential ($e^{-\lambda |x|}$).7

• Engineering Process: The infinite causal chain of the AI is segmented by a “Safety Shell” (or Window $\Delta$). A complete integer log (ledger) is kept only inside this window. Influences from outside are mathematically dampened and bounded by strict inequalities.2

This establishes a “Rigorous Stopping Definition.” The statement, “Within this window $\Delta$, these are the causes, and the error from outside is strictly less than $\epsilon$,” becomes a provable boundary (ADIC) rather than a probabilistic estimate.3

3.4 $\Sigma_1$ Proof and Asymmetry of Verification

This approach relies on $\Sigma_1$ completeness (proving the existence of a valid computation trace). While generating the proof (the ledger) may require significant computational resources, Verifying it can be done instantly on a smartphone.3 This “Verification Asymmetry” is key to social implementation. It allows any user to confirm, with number-theoretic rigor, that a massive AI’s behavior is contained within a “finite box.”

Part IV: Comparative Analysis — Why Not Other Models?

Here we clarify the distinction between the Zeta approach and other mathematical or physical models (Renormalization Group, Fourier Transform, Probability). These fields are not inferior; rather, they have different design philosophies that make the Zeta approach uniquely engaging for the specific problem of “stopping” responsibility.

4.1 Renormalization Group (RG) and Information Handling

The Renormalization Group (RG) in physics is a powerful tool for handling infinity and shares similarities with the Zeta function. RG averages out high-energy (microscopic) details to derive a low-energy (macroscopic) Effective Field Theory (EFT).23

Key Differences:

• Irreversibility of Information: RG inherently involves “Coarse-graining,” where microscopic details are discarded (they become “irrelevant operators”).25 While sufficient for describing physical phenomena, in AI accident analysis, a single “microscopic bug” can be critical. The Zeta Explicit Formula emphasizes Information Preservation, maintaining the link between microscopic (primes) and macroscopic (zeros) data via duality.

• Nature of the Cutoff: Effective theories are valid only below a specific cutoff scale $\Lambda$.27 In contrast, the Zeta function possesses “Primes” as internally defined stopping points, forming a complete algebraic structure rather than an approximation.

4.2 Fourier Transform

Fourier analysis decomposes signals but differs in its compatibility with “Atomic Responsibility.”

• Uncertainty Principle: In Fourier analysis, one cannot make both time and frequency finite simultaneously.28 This makes it difficult to rigorously confine both cause and result into a “finite box.”

• Atomicity: Sine waves are useful bases, but they lack the property of being “undecomposable and unique constituents” in the same way primes are (Fundamental Theorem of Arithmetic).

The Zeta Explicit Formula can be interpreted as a “Multiplicative Fourier Transform” 29, but because it is grounded in discrete primes, it uniquely guarantees decomposition uniqueness and stoppability.

4.3 Graph Zeta Functions as an “Extension”

The Ihara Zeta Function is a graph-theoretic analogue of the Riemann Zeta, mapping “prime cycles” on a graph to prime numbers.31 This is not a competitor but an “Expansion of the Application.” Since neural networks have graph structures, the Ihara Zeta is useful for analyzing topological robustness 34 and complements the GhostDrift theoretical direction.

Part V: Conclusion — The Zeta Function as a Reference Principle

5.1 Approaches to “Conservation of Responsibility”

This analysis reveals that the fundamental challenge in AI safety is not just improving prediction accuracy, but “how to conserve and define the total amount of responsibility.”

Standard AI models tend to diffuse responsibility into probability distributions. In contrast, approaches based on Zeta structures (such as Legal ADIC) aim to clarify the locus of responsibility by decomposing contracts and computational processes into “Atomic Responsibilities” (isomorphic to primes) and recording them in Finite Closure Ledgers.2

5.2 Conclusion: The Role of the Zeta Function

The Riemann Zeta function serves as a unique “Reference Principle” for AI safety:

1. Providing a Stopping Point: Through the Euler product and von Mangoldt function, it defines “Primes” as the stopping point of decomposition. This offers a mathematically justified “Hard Stop” to the infinite “Why?” regression.

2. Enabling Finite Closure: It provides the mathematical basis for evaluating infinite interactions within a finite “window” and rigorously bounding external influences.

3. Unique Position: While physics and other mathematical models are powerful, the Zeta function’s property of “maintaining consistency through a definition of stopping, even without full causal explanation,” holds unique value for high-stakes AI operations.

Thus, ADIC and Ledger systems can be viewed as engineering translations of the Zeta function’s structure — a realistic solution to implement the principle that “Responsibility is finite, atomic, and conserved.”

Appendix: Definitions and Roles

Term	Definition and Role in AI Safety
Finite Closure	A technology using window functions (e.g., Yukawa kernel) to cut causal chains and guarantee verifiable results within a finite range.
Infinite Postponement	The structural tendency to push errors or responsibilities into the infinite "tail" of a distribution or asymptotic limit, deferring immediate responsibility.
$\Sigma_1$ Proof	A proof verifiable in finite time by a third party, establishing the fact that "a valid computation trace exists."
Atomic Responsibility	The minimum unit of causality. Isomorphic to a "Prime" in the Zeta function and corresponds to a "Single Log Entry" in a ledger system.
ADIC	Arithmetic Digital Integrity Certificate. A ledger-based certificate using integer arithmetic to prove computational integrity, eliminating floating-point ambiguity.
von Mangoldt Function ($\Lambda(n)$)	A filter function extracting only "primes (and their powers)" from integers. A model for an audit mechanism extracting "true causes" from system logs.