Constraint Migration: A Formal Theory of Throughput in AI Cybersecurity Pipelines

1. Problem Statement

The paper addresses the lack of formal mathematical grounding in informal arguments regarding the deployment of AI in cybersecurity operations. Common slogans—such as "AI speeds up both attackers and defenders," "AI cannot replace human judgment," or "more alerts lead to more noise"—are widely circulated but lack rigorous definitions. Specifically, there is no formal theory to determine:

• Under what exact conditions does AI-induced speedup actually increase system throughput?
• How do human oversight constraints limit these gains?
• How does AI acceleration affect the relative advantage between attackers and defenders?
• Does increasing detection sensitivity inevitably lead to a decline in "useful" throughput due to false positives, or is this a modeling artifact?

The author aims to replace these informal assertions with a formal axiomatic theory based on the Theory of Constraints (TOC) applied to finite serial pipeline systems.

2. Methodology

The paper constructs a deterministic, bufferless, serial pipeline model using formal mathematical logic.

• Pipeline Definition: A pipeline $\Pi$ is defined as a tuple $(V, \prec, c)$, where $V$ is a finite set of stages, $\prec$ is a strict total order (serial structure), and $c: V \to \mathbb{R}_{>0}$ is a capacity function.
• Throughput Metric: System throughput $T(\Pi)$ is defined strictly as the minimum capacity among all stages: $T(\Pi) = \min_{v \in V} c(v)$. This abstracts the "bottleneck" concept.
• AI Modeling: AI intervention is modeled as a multiplicative perturbation. An admissible multiplier function $\alpha: V \to \mathbb{R}_{\ge 1}$ assigns an improvement factor to each stage. The perturbed capacity is $c^\alpha(v) = \alpha(v) \cdot c(v)$.
• Extensions:
  • Human Authority: A subset of stages $H$ is constrained such that $\alpha(h) = 1$ (no acceleration).
  • Adversarial Model: A pair of independent pipelines (Attacker $A$, Defender $D$) is analyzed to compare relative throughput gains.
  • False Positive Model: A scalar function is introduced to model "useful throughput" considering alert rates and investigation capacity, testing both fixed-fraction and rate-dependent precision models.

3. Key Contributions

The paper provides six primary contributions, moving from foundational definitions to specific theorems:

1. Formal Axiomatization: Establishes a rigorous framework for pipeline throughput, bottleneck sets, and multiplicative perturbations.
2. Exact Invariance Conditions: Proves necessary and sufficient conditions for when throughput remains unchanged versus when it strictly increases.
3. Human Ceiling Theorem: Quantifies the hard upper bound on throughput imposed by non-acceleratable human stages.
4. Adversarial Ratio Characterization: Derives an algebraic condition determining when AI favors the attacker over the defender.
5. False Positive Analysis: Proves that the "paradoxical decline" in useful throughput is impossible under a fixed false-positive fraction model and identifies the specific precision function required to model such a decline.
6. Scope Analysis: Clearly delineates what the model proves (deterministic bottlenecks) and what it excludes (stochasticity, queueing, game theory).

4. Key Results and Theorems

Throughput Invariance and Improvement (Theorems 1 & 2)

• Theorem 1 (Invariance): Throughput remains unchanged ($T(\Pi^\alpha) = T(\Pi)$) if and only if at least one original bottleneck stage retains a multiplier of 1 (i.e., is not improved).
  • Implication: Improving non-bottleneck stages yields zero throughput gain. If there are multiple tied bottlenecks, improving only one of them is insufficient; the throughput is pinned by the unimproved one.
• Theorem 2 (Strict Improvement): Throughput strictly increases ($T(\Pi^\alpha) > T(\Pi)$) if and only if every original bottleneck stage has a multiplier strictly greater than 1.
  • Implication: To increase system capacity, all current bottlenecks must be accelerated simultaneously.

Human Authority Ceiling (Theorem 3)

• Result: If a non-empty set of stages $H$ (human stages) cannot be accelerated ($\alpha(h)=1$), the system throughput is bounded above by the minimum capacity of those human stages: $T(\Pi^\alpha) \le \min_{h \in H} c(h)$.
• Tightness: This bound is tight. If machine stages are accelerated sufficiently (unbounded), the throughput converges exactly to the capacity of the slowest human stage.
• Significance: Formalizes the intuition that "AI cannot replace human judgment" as a hard throughput ceiling.

Adversarial Dynamics (Theorem 4)

• Result: The attacker-defender throughput ratio worsens for the defender if and only if the attacker's relative throughput gain exceeds the defender's relative gain.
  • Mathematically: $\frac{T(\Pi^\alpha_A)}{T(\Pi_A)} > \frac{T(\Pi^\alpha_D)}{T(\Pi_D)}$.
• Corollary: If the defender invests in non-bottleneck stages (gain = 1) while the attacker improves its bottleneck (gain > 1), the defender's relative position deteriorates, even if the defender's absolute throughput is unchanged.

False Positive Modeling (Theorem 5 & Proposition 4)

• Theorem 5 (Impossibility of Decline): Under a model where the false positive fraction $f$ is fixed, useful throughput $U(\lambda) = (1-f)\min(\lambda, c_{inv})$ plateaus once the alert rate $\lambda$ exceeds investigation capacity $c_{inv}$. It does not decline.
  • Significance: The common argument that "more alerts = less useful work" is mathematically false under a fixed-fraction assumption.
• Proposition 4 (Repair): To recover the intended "decline" behavior, the model must assume a rate-dependent precision function $p(\lambda)$ that is strictly decreasing. If precision degrades as alert volume increases, useful throughput will genuinely decline beyond saturation.

5. Significance and Limitations

Significance:

• Precision over Intuition: The paper transforms vague cybersecurity slogans into precise mathematical theorems. For example, it clarifies that "improving AI" only helps if it targets all bottlenecks, not just the most obvious one.
• Policy Implications: It suggests that security operations centers (SOCs) should focus investment on human bottleneck stages or ensure that AI improvements cover all tied bottlenecks simultaneously.
• Modeling Rigor: It exposes flaws in common operational models, specifically showing that fixed false-positive rates cannot explain throughput collapse, necessitating more complex precision models.

Limitations:

• Deterministic & Bufferless: The model assumes a deterministic, serial pipeline without buffers or stochastic variability. Real-world SOCs involve queues, parallel branches, and random workloads.
• No Optimization: While it characterizes when throughput changes, it does not solve for the optimal allocation of AI resources under budget constraints (though it implies investing in bottlenecks is optimal).
• No Game Theory: The adversarial model treats attacker and defender as independent; it does not model strategic interactions or arms races.
• Idealized Human Constraint: The assumption that human stages have a multiplier of exactly 1 is an idealization; in reality, AI might provide partial assistance ($\alpha > 1$ but $< \infty$).

Conclusion:
The paper provides a foundational formal theory for AI in cybersecurity pipelines. It proves that throughput is governed strictly by the minimum capacity of the system and that AI's impact is binary (either it pins the bottleneck or it doesn't). The work serves as a critical baseline for future research into stochastic extensions and game-theoretic applications in cybersecurity operations.