Making AI Accountability Post-hoc Impossible:Responsibility Fixation as a Mathematical Principle

0. Responsibility Vanishes as Explanations Grow — Fixing it with “Responsibility Fixation”

The ultimate vulnerability of modern AI and large-scale decision-making systems lies in the fact that responsibility can be "explained away" after the fact. This paper defines the phenomenon where increasing explainability leads to the blurring of accountability as "Responsibility Evaporation." As a counter-principle, we introduce Responsibility Fixation.

Responsibility Fixation is not a set of ethical guidelines or normative accountability frameworks; rather, it is a mathematical design principle that embeds Post-hoc Impossibility as a prior constraint within a system. This paper details why Responsibility Fixation can overcome the limitations of traditional Explainable AI (XAI) and governance, and explores its mathematical foundations, such as "Finite Closure" and "Observation Point Fixation."

1. Problem Statement: Why Does Responsibility Vanish?

1.1 Responsibility Evaporation

In AI and Decentralized Autonomous Organizations (DAOs), a recurring phenomenon occurs following critical decisions:

1. Proliferation of Post-hoc Reasons: An infinite number of "plausible reasons" are generated from parameters and logs.

2. Diffusion of Agency: The subject responsible for the explanation is diluted across the entire network.

3. Dynamic Updating of Grounds: The basis for judgment is constantly learned and updated in real-time, making past decisions impossible to reproduce.

The result is a vacuum where "no one is lying, yet no one can be held responsible." Responsibility is not rejected; it is "diluted" by excessive explanation and subsequently evaporates.

1.2 Limitations of Traditional Approaches

Traditional ethical guidelines, XAI, and governance frameworks all treat "being explainable" as a virtue. However, from a structural and mathematical perspective, a paradox emerges: "The more post-hoc explainability is enhanced, the more escape routes (degrees of freedom) responsibility has."

2. Proposal: What is Responsibility Fixation?

2.1 Definition

Responsibility Fixation is a structural principle that, before a decision is made, limits the degrees of freedom for its justification, explanation, and reinterpretation to a finite set, making the post-hoc reallocation of responsibility impossible.

This is neither a norm nor a behavioral requirement. It is a structural constraint that ensures the system cannot physically or mathematically choose any other behavior.

2.2 Comparison: Ethics vs. Responsibility Fixation

Perspective	Traditional Ethics / Guidelines	Responsibility Fixation
Target	Human behavior and consciousness	System structure and mathematical models
Timing	Post-hoc reflection and evaluation	Prior constraint and embedding
Means	Linguistic requirements and interpretation	Mathematical impossibility and fixation
On Failure	Apology and improvement	Forced halt due to inability to deviate

3. Core Principle: Post-hoc Impossibility

3.1 Formal Definition: Explanation and Degrees of Freedom

"Explanation" is an operation that allows for multiple justification mappings for the same decision $D$ through the post-hoc introduction of auxiliary variables, re-weighting, or changes in evaluation perspective.

In this paper, we formalize the explainability of decision $D$ as the following measure of degrees of freedom:

$$\mathcal{E}(D) := \dim \left( \mathcal{J}_D \right)$$

Here, $\mathcal{J}_D$ is a "family" of mappings that can justify decision $D$, having a representation $\{J_\theta\}_{\theta\in\Theta}$ over a pre-defined finite-dimensional parameter space $\Theta$. In this context, $\dim(\mathcal{J}_D)$ is interpreted as the degrees of freedom of $\Theta$.

As $\mathcal{E}(D)$ increases, the degree of freedom for post-hoc interpretation of $D$ grows, and the locus of responsibility disperses. In other words, guaranteeing explainability is equivalent to granting the system the "interpretive freedom" to evade responsibility.

3.2 Impossibility and Non-constructibility

In the context of this paper, "structurally impossible" does not refer to implementation difficulties or operational norms. It refers to the fact that, under a pre-defined set of components and transition rules, no mapping exists in the domain that executes the operation in question.

Similarly, "computationally excluded" does not mean an operation is high-cost; it means it cannot be defined as a finitely describable computational process or is unconstructible within the logic system being verified. Therefore, the impossibility discussed here is not probabilistic or empirical, but rather non-constructibility within the defined model.

Under this framework, the following operations are non-constructible within the model after a decision is made (i.e., they are not defined as transition rules):

• Addition of new variables

• Fine-tuning of evaluation functions

• Retroactive tampering with observation logs

By imposing this non-constructibility as a prior constraint, the locus of responsibility is fixed to the point of decision, and post-hoc reallocation is excluded by definition.

4. Mathematical Background

4.1 Finite Closure

Infinite possibilities for exception handling or interpretive leeway are closed by a finite boundary condition $\mathcal{C}_F$. This mathematically blocks the vectors through which responsibility might escape, confining the decision within a closed set.

4.2 Observation Point Fixation (Beacon / Anchor)

Evaluations must always be conducted from a pre-designated "immobile observation point." The inability to move the observation point (Anchor) means that evaluation criteria cannot be modified after the fact, concentrating the locus of responsibility on a single point.

4.3 Provability and $\Sigma_1$ Structure

The entire process is reduced to a $\Sigma_1$ logical structure that is verifiable in finite steps. This computationally eliminates any room for "adding meaning after the fact."

5. Structural Constraints Characterizing Responsibility Fixation

Responsibility Fixation is not a property achieved through implementation methods or operational rules. It is characterized by structural constraints that the decision-making model must satisfy. The following four constraints are necessary conditions for Responsibility Fixation; if any one of them is violated, the evaporation of responsibility becomes theoretically unavoidable.

1. Prior Fixation of Evaluation Functions: The evaluation function must be fully defined prior to the moment of decision and must not permit expansion operations that introduce new terms, weights, or auxiliary variables post-decision.

2. Immobility of Observation Points: Evaluations are performed solely based on a pre-designated set of observation points; operations to add or relocate evaluation perspectives post-decision are not defined.

3. Finite Closure of Boundary Conditions: The state space involved in decision-making is closed by finitely describable boundary conditions; transitions that refer to undefined regions post-hoc are not permitted.

4. Structural Impossibility of Post-hoc Explanation: For a decision $D$, no justification mappings exist other than the pre-defined set $\mathcal{J}_D$, and additional narrative explanations are non-constructible within the model.

6. Why Responsibility Fixation is Essential Now

In environments where AI acts as an autonomous agent and organizations transition toward decentralized models, strengthening ethics only leads to the evaporation of responsibility through "automatically generated explanations." We require a design that does not increase explanation. A simple, fixed structure from which responsibility cannot escape is the true safety device for an advanced AI society.

7. Positioning: "Responsibility Fixation" in the Conceptual Map

• Responsibility Evaporation: An entropic phenomenon that occurs inevitably if left unaddressed.

• Responsibility Fixation: The design principle to prevent evaporation.

• Ghost Drift: The specific framework/implementation system for achieving structural safety through Responsibility Fixation.

Responsibility Fixation is not a superior version of existing ethics or governance; it is a "Mathematical Safety Principle" situated on an orthogonal axis.

8. Conclusion

Responsibility Fixation is not a "punishment mechanism" designed to force someone to take the blame. It is the very structure that eliminates the space for responsibility to escape in the first place.

In a digital society of ever-increasing complexity, what we must protect is not the "fluency of explanation," but the "irreversible weight of decision." This principle will serve as the mathematical North Star for the next generation of social, AI, and decision-making design.